**7400 **

**Settlement **

Andrew F. Daughety *Department of Economics and Business Administration, VanderbiltUniversity *

© Copyright 1997 Andrew F. Daughety

**Contents **

* A. Basic Issues, Notions and Notations *

9. An Example of a Model of Settlement Negotiation

* B. Basic Models of Settlement Bargaining *

10. Perfect and Imperfect Information Models: Axiomatic Models forthe Cooperative Case

11. Perfect and Imperfect Information Models: Strategic Models forthe Non-Cooperative Case

12. Analyses Allowing for Differences in Player Assessments Due toPrivate Information

13. Comparing the Two-Type Models: Imperfect and AsymmetricInformation

* C. Variations on the Basic Models *

Bibliography on Settlement (7400)

*Figures *

Figure 1: Settlement Under Perfect Information

Figure 2: Screening with a Continuum of Types

Figure 3: Signaling with a Continuum of Types

*Tables *

Table 1: Hands of Cards for A and B

Table 2: Hands of Cards for A and B

Table 3: Ultimatum Game Results Under Imperfect and AsymmetricInformation

This survey of modeling of pretrial settlement bargaining organizes currentmain themes and recent developments. The basic concepts used are outlinedas core models and then a number of variations on these core models arediscussed. The focus is on papers that emphasize formal models of settlementnegotiation and the presentation in the survey is organized in game-theoreticterms, this now being the principal tool employed by analyses in this area,but the discussion is aimed at the not-terribly-technical non-specialist. Thesurvey also illustrates some of the basic notions and assumptions ofinformation economics and of (cooperative and noncooperative) game theory.

*JEL classification: * K41, C70

*Keywords: * Settlement Bargaining

This survey of the modeling of pretrial settlement bargaining organizescurrent main themes and recent developments. The basic concepts used areoutlined as core models and then a number of variations on these coremodels are discussed. The emphasis in the survey on main themes andmodeling of basic concepts is for two reasons. First, Robert Cooter andDaniel Rubinfeld (1989) provide an excellent survey of this literature (in thecontext of a broader consideration of the economics of dispute resolution andthe law) up to that date, while Geoffrey Miller (1996) provides a recent non-technical review addressing policies that encourage settlement. Second, aswith much of law and economics, a catalog of even relatively recent researchwould rapidly be out of date. The focus here is on papers that emphasizeformal models of settlement negotiation and the presentation is organized ingame-theoretic terms, this now being the principal tool employed by analysesin this area. The discussion is aimed at the not-terribly-technical non-specialist. In this survey some of the basic notions and assumptions of gametheory are presented and applied, but some of the more recent models ofsettlement negotiation rely on relatively advanced techniques; in those cases,technical presentation will be minimal and intuition will be emphasized. Forthe interested reader, a recent source on game theory applications in law andeconomics is Douglas Baird, Robert Gertner and Randal Picker (1994) ;extensive reviews of that book appear in Peter Huang (1995) and StephenSalant and Theodore Sims (1996) . Two quite readable books on game theory,modeling and a number of related philosophical issues are Ken Binmore(1992) and David Kreps (1990) . Finally, chapters seven through nine of Andreu Mas-Colell, Michael Whinston and Jerry Green (1995) provide thetechnically sophisticated reader with a convenient, efficient and carefulpresentation of the basic techniques of modern (non-cooperative) gametheory, while chapters thirteen and fourteen provide a careful review of thebasics of information economics.

What is the basic image that emerges from the settlement bargainingliterature? It is that settlement processes act as a type of screen, sortingamongst the cases, presumably causing the less severe (e.g., those with lowertrue damages) to bargain to a resolution (or to do this very frequently), whilethe more severe (e.g., those with higher damages) may proceed to be resolvedin court. The fact that some cases go to trial is often viewed as aninefficiency. While this survey adopts this language (as does much of theliterature reviewed) one might also view the real possibility of trial asnecessary to the development of case law and as a useful demonstration ofthe potential costs associated with decisions made earlier about levels ofcare. In other words, the possibility of trials may lead to greater care and tomore efficient choices overall. Moreover, the bargaining and settlementliteratures have evolved in trying to explain the sources of negotiationbreakdown: the literature has moved from explanations fully based onintransigence to explanations focused around information. This is not toassert that trials don't occur because of motives outside economic analysis,just that economic attributes contribute to explaining an increasing share ofobserved behavior.

In the next few sections (comprising Part A) significant features ofsettlement models are discussed and some necessary notation is introduced;this part ends with a simplified example indicating how the pieces cometogether. Part B examines the basic models in use, varying the level ofinformation that litigants have and the type of underlying bargaining storiesthat are being told. Part C considers a range of "variations" on the Part Bmodels, again using the game-theoretic organization introduced in Part A.

* * **A. Basic Issues, Notions and Notation ** * *

In this part the important features of the various approaches are discussedand notation that is used throughout is introduced. Paralleling thepresentation of the models to come, the current discussion is organized toaddress: 1) players; 2) actions and strategies; 3) outcomes and payoffs; 4)timing; 5) information and 6) prediction. A last section provides a briefexample. Words or phrases in italics are terms of special interest.

The primary participants (usually called litigants or __players __) are the plaintiff(P) and the defendant (D); a few models have allowed for multiple P's ormultiple D's, but for now assume one of each. Secondary participants includeattorneys for the two litigants (A _{P } and A _{D }, respectively), experts for the twoparticipants (X _{P } and X _{D }, respectively) and the court (should the case go totrial), which is usually taken to be a judge or a jury (J). Almost all modelsrestrict attention to P and D, either ignoring the others or relegating them tothe background. As an example, a standard assumption when there is someuncertainty in the model (possibly about damages or liability, or both) is that,at court, J will learn the truth and make an award at the true value (the awardwill be the actual damage, liability will be correctly established, etc.).Moreover, J is usually assumed to have no strategic interests at heart, unlikeP, D, the A's and the X's. Section 15 considers some efforts to incorporate J'sdecision process in a substantive way.

Finally, uncertainty enters the analysis whenever something relevant isnot known by at least one player. Uncertainty also arises if one player knowssomething that another does not know, or if the players move simultaneously(for example, they simultaneously make proposals to each other). Theseissues will be dealt with in the sections on timing (6) and information (7), butsometimes such possibilities are incorporated by adding another "player" tothe analysis, namely nature (N), a disinterested player whose actionsinfluence the other players in the game via some probability rule.

An __action __ is something a player can choose to do when it is their turn tomake a choice. For example, the most commonly modeled action for P or Dinvolves making a __proposal __. This generally takes the form of a demand from Pof D or an offer from D to P. This then leads to an opportunity for anotheraction which is a __response __ to the proposal, which usually takes the form of anacceptance or a rejection of a proposal, possibly followed by yet anotheraction such as a counterproposal. Some models allow for multiple periods ofproposal/response sequences of actions.

When a player has an opportunity to take an action, the rules of the gamespecify the allowable actions at each decision opportunity. Thus, in theprevious example, the specification of allowed response actions did notinclude delay (delay will be discussed in Section 18). Actions chosen at onepoint may also limit future actions: if "good faith" bargaining is modeled asrequiring that demands never increase over time, then the set of actionspossible when P makes a counterproposal to D's counterproposal may belimited by P's original proposal.

Other possible actions include choosing to employ attorneys or experts,initially choosing to file a suit or finally choosing to take the case to courtshould negotiations fail. Most analyses ignore these either by not allowingsuch choices or by assuming values for parameters that would makeparticular choices "obvious". For example, many analyses assume that the netexpected value of pursuing a case to trial is positive, thereby making crediblesuch a threat by P during the negotiation with D; this topic will be exploredmore fully in Section 16.1.

In general, a * strategy *for a player provides a complete listing of actions tobe taken at each of the player's decision opportunities and is contingent on:1) the observable actions taken by the other player(s)in the past; 2) actionstaken by the player himself in the past; 3) the information the player currentlypossesses; 4) any exogenous relevant actions that have occurred that theplayer is aware of (for example, speculations about the award a specific Jmight choose). Thus, as an example, consider an analysis with no uncertaintyabout damages, liability, what J will do, etc., where P proposes, D respondswith acceptance or a counterproposal, followed by P accepting thecounterproposal or choosing to break off negotiations and go to court. Astrategy for P would be of the form "propose an amount x; if D accepts, makethe transfer and end while if D counterproposes y, choose to accept this if yis at least z, otherwise proceed to court." P would then have a strategy foreach possible x, y and z combination.

An analogy may be helpful here. One might think of a strategy as a book,with pages of the book corresponding to opportunities in the game for thebook's owner to make a choice. Thus, a typical page says "if you are at thispoint in the book, take this action." This is not a book to be read from cover-to-cover, one page after the previous one; rather, actions taken by playerslead other players to go to the appropriate page in their book to see whatthey do next. All the possible books (strategies) that a player might use formthe player's personal library (the player's strategy set).

There are times when being predictable as to which book you will use isuseful, but there can also be times when unpredictability is useful. A sportsanalogy would be to imagine yourself to be a goalie on the A team, and amember of the B team has been awarded the chance to make a shot on yourgoal. Assume that there is insufficient time for you to fully react to the kick,so you are going to have to generally move to the left or the right as thekicker takes his shot. If it is known that, in such circumstances, you alwaysgo to your left, the kicker can take advantage of this predictability andimprove his chance of making a successful shot. This is also sometimes truein settlement negotiations: if P knows the actual damage that D is liable for,but D only knows a possible range of damages, then D following apredictable policy of never going to court encourages P to make high claims.Alternatively, D following a predictable policy of always going to court nomatter what P is willing to settle for may be overly costly to D. __Mixedstrategies __ try to address this problem of incorporating just the right amountof unpredictability and are used in some settlement models. Think of theindividual books in a player's library as __pure __ strategies (pure in the sense ofbeing predictable) and think of choosing a book at random from the library,where by "at random" we mean that you have chosen a particular set ofprobability weights on the books in your library. In this sense your chosenset of weights is now your strategy (choosing one book with probability oneand everything else with probability zero gets us back to pure strategies). Alist of strategies, with one for each player (that is, a selection of books fromall the players' libraries), is called a __strategy profile __.

An __outcome __ for a game is the result of a strategy profile being played. Thus,an outcome may involve a transfer from D to P reflecting a settlement or itmight be a transfer ordered by a court or it might involve no transfer as Pchooses not to pursue a case to trial. If the reputations of the parties are ofinterest, the outcome should also specify the status of that reputation. In pleabargaining models, which will be discussed in Section 17.4, the outcomemight be a sentence to be served. In general, an outcome is a list (or vector)of relevant final attributes for each player in the game.

For each player, each outcome has an associated numerical value calledthe *payoff *, usually a monetary value. For example, a settlement is a transfer ofmoney from D to P; for an A or an X the payoff might be a fee. For modelsthat are concerned with risk preferences, the payoffs would be in terms of theutility of net wealth rather than in monetary terms. Payoffs that are strictlymonetary (for example, the transfer itself) are viewed as reflecting risk-neutralbehavior on the part of the player.

Payoffs need not equal expected awards, since parties to a litigation alsoincur various types of costs. The cost most often considered in settlementanalyses is called a __court cost __ (denoted here as k _{P } and k _{D }, respectively). Anextensive literature has developed surrounding rules for allocating such coststo the litigants and the effect of various rules on the incentives to bring suitand the outcome of the settlement process; this is addressed in Section 17.2.Court costs are expenditures which will be incurred should the case go to trialand are associated with preparing for and conducting a trial; as such they areavoidable costs (in contrast with sunk costs) and therefore influence thedecisions that the players (in particular P and D) make. Generally, costsincurred in negotiating are usually ignored, though some recent papersreviewed in Sections 11.2 and 18 emphasize the effect of negotiation costs onsettlement offers and the length of the bargaining horizon. Unless specificallyindicated, assume that negotiation costs are zero.

The total payoff for a player labeled i (that is i = P, D, ...) is denoted $ _{i }.Note that this payoff can reflect long-term considerations (such as the valueof a reputation or other anticipated future benefits) and multiple periods ofnegotiation. Generally, players in a game maximize their payoffs and thus, forexample, P makes choices so as to maximize $ _{P }. For convenience, D's payoff iswritten as an expenditure (if D countersues, then D takes the role of a plaintiffin the countersuit) and thus D is taken to __minimize __ $ _{D } (rather than maximize -$ _{D }). While an alternative linguistic approach would be to refer to thenumerical evaluation of D's outcome as a "cost" (which is then minimized,and thereby not use the word payoff with respect to D), the use of the termpayoff for D's aggregate expenditure is employed so as to reserve the wordcost for individual expenditures that each party must make.

Finally, since strategy profiles lead to outcomes which yield payoffs, thismeans that payoffs are determined by strategy profiles. Thus, if player i usesstrategy s _{i }, and the strategy profile is denoted s (that is, s is the vector, or list(s _{1 }, s _{2 }, ..., s _{n }), where there are n players), then we could write this dependencefor player i as $ _{i }(s).

The sequence of play and the horizon over which negotiations occur areissues of timing and of time. For example: do P and D make simultaneousproposals or do they take turns? Does who goes first (or who goes when)influence the outcome? Do both make proposals or does only one? Canplayers choose to delay or accelerate negotiations? Are there multiple roundsof proposal/response behavior? Does any of this sort of detail matter in anysubstantive sense?

Early settlement models abstracted from any dynamic detail concerningthe negotiation process. Such models were based on very general theoreticalmodels of bargaining (which ignored bargaining detail and used desirableproperties of any resulting bargain to characterize what it must be) initiallydeveloped by John Nash (1950) . More recent work on settlementnegotiations, which usually provides a detailed specification of howbargaining is assumed to proceed (the strategies employed and thesequencing of bargaining play are specified), can be traced to results in thetheoretical bargaining literature by Nash (1953) , Ingolf Ståhl (1972) and ArielRubinstein (1982) . Nash's 1950 approach is called __axiomatic __ while theStåhl/Rubinstein improvement on Nash's 1953 approach is called __strategic __;the two approaches are intimately related. Both approaches have generatedvast literatures which have considered issues of interest to analyses ofsettlement bargaining; a brief discussion of the two approaches is providedin Sections 10 and 11 so as to place the settlement applications in a unifiedcontext. The discussion below also addresses the institutional features thatmake settlement modeling more than simply a direct application of bargainingtheory.

When considered, time enters settlement analyses in two basic ways.First, do participants move simultaneously or sequentially? This is not limitedto the question of whether or not P and D make choices at different points onthe clock. More significant is whether or not moving second involves havingobserved what the first-mover did. Two players who make choices at differentpoints in time, but who do not directly influence each other's choices(perhaps because the second-mover cannot observe or react to what the first-mover has done) are viewed as moving simultaneously: that my choice andyour choice together influence what each of us receives as a payoff(symbolized in the payoff notation as $ _{i }(s) for player i) does not make movingat different points in time significant in and of itself. The real point here iswhether all relevant decision-makers must conjecture what the others are __likely to do __, or whether some can observe what others actually __did __. This isbecause the second-mover is influenced by the first-mover's choice andbecause they __both __ know this, which affects the first-mover's choice as well.Asymmetry in what choices depend upon (in this sense) is modeled aschoices being made in a sequence; symmetry is modeled as choices beingmade simultaneously. As will be seen in the example to be discussed inSection 9, who moves when can make a very significant difference in what ispredicted. Note that a sequence of simultaneous decisions is possible (forexample, P and D both simultaneously make proposals and then bothsimultaneously respond to the proposals, etc.).

A second way that time enters is in terms of the length of the horizon overwhich decisions are made. The main stream of research in the strategicbargaining literature views the horizon as infinite in length; this is done toeliminate the effect of arbitrary end-of-horizon strategic behavior. Settlementmodels, on the other hand, typically take the negotiation horizon as finite inlength (and often very short, say, two periods). This is done for two reasons.First, while some cases may seem to go on forever, some form of terminationactually occurs (cases are dropped, or resolved through negotiation or meet acourt date). While setting a court date is not an iron-clad commitment, fewwould argue that an infinite number of continuances is realistic. Second, inthe more informationally complex models, this finite horizon restriction helpsprovide more precise predictions to be made than would otherwise bepossible. Thus, in most settlement models there is a last opportunity tonegotiate, after which either the case proceeds to trial or terminates (eitherbecause the last settlement proposal is accepted or the case is dropped). Thisis important because court costs are incurred only if the case actuallyproceeds to trial; that is, after the last possible point of negotiations. Ifnegotiations were to continue during the trial, the ability to use the avoidanceof these costs to achieve a settlement obviously is vitiated: as the trialproceeds the portion of costs that are sunk become larger and the portionthat is avoidable shrinks. This problem has not been addressed generally,though papers by Kathryn Spier (1992) and by Lucian Bebchuk (1996) consider significant parts of this issue; this is discussed in more detail inSection 18.

In Steven Shavell (1982) , the range over which litigants might bargain whenassessments about outcomes may be different is analyzed as a problem indecision theory (a game against nature, N); this raises the issue of whoknows what, when, why and how. Shavell's paper indicated that differencesin assessments by P and D as to the likelihood of success at trial, and thelikely award, can lead to trial as an outcome. While Shavell's paper did notconsider strategic interaction among the players (for the first paper toincorporate strategic behavior, see Ivan P'ng (1983) ), the role of informationhas become a central theme in the literature that has developed since, withspecial emphasis on accounting for informational differences and consistent,rational behavior. Moving momentarily from theory to empirical analysis, Henry Farber and Michelle White (1991) use data from a hospital toinvestigate whether seemingly asymmetrically distributed informationinfluenced settlement rates and the speed with which cases settled; they findthat it did.

Informational considerations involve what players individually know andwhat they must guess about (where such guessing presumably involvessome form of organized approach). Many of the analyses in the literature usedifferent informational structures (who knows what when) and in this surveya variety of such structures will be presented. As a starting example, considerPat, who developed an improved framitz (a tool for making widgets). Pat tookthe tool to the Delta Company (D), with the notion that Delta wouldmanufacture the tool and Pat would become rich from her share of the profits.Delta indicated that the tool was not likely to be financially viable and Patwent back to work on other inventions. Some years later Pat noticed thatmany people who made widgets were using a slightly modified version of hertool made by Delta. Pat (P) decided to sue Delta (D) for misappropriation ofintellectual property, and for convenience assume that while D's liability isclear, the assessment of a level of compensation to P is less clear. D'sfamiliarity with the profits made (and experience with creative accountingprocedures) means that D has a better idea of what level of total profits mightbe proved in court. Both P's attorney and D's attorney have (potentiallysimilar) estimates of what the court is likely to do with any particular evidenceon the level of profits of the tool (how the court, J, might choose to allocatethe costs and revenues of the tool to P and D). Simplifying, there are twosources of uncertainty operating here: uncertainty by P and D about J anduncertainty by P about what D knows.

First, both P and D cannot perfectly predict what J will choose as anaward: here each faces an essentially similar level of uncertainty (there is noobvious reason to assert the presence of an asymmetry in what is knowable).Moreover, we will assume that this assessment of J's likely actions, whileprobabilistic, is __common knowledge __. Common knowledge connotes thenotion that were P and D to honestly compare their assessments of J's likelyactions for __each possible __ set of details about the profits made by D, theirassessments would be exactly the same, and P and D know that the otherknows this, and P and D know that the other knows that the other knows this,and so on (see Robert Aumann (1976) , John Geanakoplos and HeraklisPolemarchakis (1982) and Binmore (1992) for two early technical papers and arecent game theory text with an extensive prose discussion of this topic).

Thus, P and D have the same information with regard to J; we might callthis __imperfect __ or __symmetrically uncertain __ information to contrast it with theclear asymmetry that exists between P and D with respect to the informationabout revenues and costs that D knows. This latter notion of uncertainty isreferred to as __asymmetric __ information, and it is a main attribute of much of therecent work on settlement. Finally, if actual damage was common knowledgeand if P and D truly knew exactly what J would choose as an award and __that __were common knowledge, the resulting information condition is called __perfect __.

A nice story which makes the differences in informational settings clear isdue to John Roberts of Stanford University. Consider a card game played byat least two people, such as poker. If all hands were dealt face up and no morecards were to be dealt, this would be a situation of perfect information. If,instead, hands are dealt with (say) three cards face up and two cards facedown, but no one can look at their "down" cards, this is a setting of imperfectinformation. Finally, asymmetric information (also called __incomplete __information) would involve each player being able to privately look at theirdown cards before taking further actions (asking for alternative cards, betting,etc.). Note that in this last case we see the real essence of asymmetricinformation: it is not that one party is informationally disadvantaged whencompared with the other (as in the case of Pat and Delta) as much as theplayers have __different __ information from each other.

One caution about the foregoing example. The perfect information caseseems to be rather pointless: players without the best hand at the table wouldchoose not to bet at all. Perfect information models are really not as trivial,since they significantly clarify what the essential elements of an analysis areand they provide a comparison point to evaluate how different informationaluncertainties affect the efficiency of the predicted outcome.

Timing in the play of the game is also a potential source of imperfectinformation. If P and D make simultaneous proposals (which might beresolved by, say, averaging), then when they are each considering whatproposal to make they must conjecture what the other might choose topropose: what each will do is not common knowledge. Even if all otherinformation in the analysis were perfect, this timing of moves is a source ofimperfect information.

The incorporation of informational considerations (especially asymmetricinformation) has considerably raised the ante in settlement modeling. Why?Is this simply a fad or an excuse for more technique? The answer is revealedin the discussion of the basic models and their variations. As indicatedearlier, the problem with analyses that assumed perfect or imperfectinformation was that many interesting and significant phenomena were eitherattributed to irrational behavior or not addressed at all. For example, in somecases negotiations fail and a trial ensues, even though both parties mayrecognize that going to court is very costly; sometimes cases fail to settlequickly, or only settle when a deadline approaches. Moreover, agencyproblems between lawyers and clients, discovery, disclosure, various rulesfor allocating court costs or for admitting evidence have all been the subjectof models using asymmetric information.

Models with perfect information specify parameters of the problem with nouncertainty attached: liability is known, damages are known, what J will do isknown, costs are known, etc. Imperfect information models involveprobability distributions associated with one or more elements of theanalysis, but the probability distributions are common knowledge, and thusthe occurrence of uncertainty in the model is symmetric. For example,damages are "unknown" means that all parties to the negotiation itself (all theplayers) use the same probability model to describe the likelihood of damagebeing found to actually have been a particular value. Thus, for example, ifdamage is assumed to take on the value d _{L } (L for low) with probability p andd _{H } (H for high) with probability (1 - p), then the expected damage, E(d) = pd _{L } +(1-p)d _{H } is P's estimate (E _{P }(d)) as well as D's estimate (E _{D }(d)) of the damagesthat will be awarded in court. Usually, the court (J) is assumed to learn the"truth" should the case go to trial, so that the probability assessment by theplayers may be interpreted as a common assessment as to what the court willassert to be the damage level, possibly reflecting the availability andadmissibility of evidence as well as the true level of damage incurred. Notethat such models are not limited to only accounting for two possible events(for example, perhaps the damage could be any number between d _{L } and d _{H });this is simply a straight-forward extension of the probability model. On theother hand, since P and D agree about the returns and costs to trial, there isno rational basis for actually incurring them and the surplus generated by notgoing to trial can be allocated between the players as part of the bargainstruck.

With asymmetric information, players have different information and thushave different probability assessments over relevant uncertain aspects of thegame. Perhaps each player's court costs are unknown to the other player,perhaps damages are known to P but not to D, or the likelihood of beingfound liable is better known by D than by P. Possibly P and D have differentestimates, for a variety of reasons, as to what J will do. All of thesedifferences in information may influence model predictions, but the nature ofthe differences is itself something that must be common knowledge.

Consider yourself as one of the players in the version of the earlier cardgame where you can privately learn your down cards. Say you observe thatyou have an Ace of Spades and a King of Hearts as your two down cards.What can you do with this information? The answer is quite a bit. You knowhow many players there are and you can observe all the up cards. You can'tobserve the cards that have not been dealt, but you know how many of themthere are. You also know the characteristics of the deck: four suits, thirteencards each, no repeats, etc. This means that given your down cards youcould (at least theoretically) construct probability estimates of what the otherplayers have and know what estimates they are constructing about what youhave. This last point is extremely important, since for player A to predict whatplayer B will do (so that A can compute what to do), A needs to think fromB's viewpoint, which includes predicting what B will predict about what Awould do.

To understand this, let's continue with the card game example andconsider a simple, specific example (we will then return to the settlementmodel to indicate the use of asymmetric information in that setting). Assumethat there are two players (A and B), a standard deck with 52 cards (jokers areexcluded) and the game involves two cards dealt face up and one card dealtface down to each of A and B; only one down card is considered to simplifythe presentation. Table 1 shows the cards that have been dealt face up(available for all to see) and face down (available only for the receiver of thecards, and us, to see). For expositional purposes, H, D, C and S stand forhearts, diamonds, clubs and spades while a K represents a King, etc.

Table 1 : Hands of Cards for A and B | ||

UP | DOWN | |

A | 4S, KD | KH |

B | QH, AC | 2C |

(Note: entries provide face value of card and suit) |

By a player's __type __ we mean their down cards (their private information), soA is a KH while B is a 2C, and only they know their own type. A knows that Bis one of 47 possible types (A knows B can't be a KH or any of the up cards)and, for similar reasons, B knows that A is one of 47 possible types.Moreover, A's model of what type B can be (denoted p _{A }(t _{B }|t _{A })) provides Awith the probability of each possibility of B's type (denoted t _{B }) conditional(the vertical line) on A's type (denoted t _{A }). The corresponding model for Bthat gives a probability of each possibility of A's type is denoted p _{B }(t _{A }|t _{B }).Note that all the probability models are formally also conditioned on all theupcards (for example, if we are especially careful we should writep _{A }(t _{B }|t _{A }, 4S, KD, QH, AC _{ }) and p _{B }(t _{A }|t _{B },4S, KD, QH, AC)), which we supress inthe notation employed.

In the particular case at hand t _{A } = KH, so once A knows his type (sees hisdown card) he can use p _{A }(t _{B }|KH) to compute the possibilities about B's downcard (type). But A actually could have worked out his strategy for eachpossible down card he might be dealt (and the possible up cards) __before __ thegame; his strategy would then be a function that would tell him what to do foreach possible down-card/up-card combination he might be dealt in the game.Moreover, since he must also think about what B will do and B will not knowA's down card (and thus must use p _{A }(t _{B }|t _{A }), instead of p _{A }(t _{B }|KH), as hisprobability model for what A will use about B's possible down cards) thenthis seemingly extra effort (that is, A working out a strategy for each possibletype) isn't wasted, since A needs to do it anyway to model B modeling A'schoices. What we just went through is what someone analyzing anasymmetric game must do for every player.

Finally, for later use, observe that these two probability models are __consistent __ in the sense that they come from the __same __ overall model p(t _{A }, t _{B })which reflects common knowledge of the deck that was used. In other words,the foregoing conditional probabilities in the previous paragraph both comefrom the overall joint probability model p(t _{A }, t _{B }) using the usual rules ofprobability for finding conditional probabilities.

So what does this mean for analyzing asymmetric information models ofsettlement? It means, for example, that if there is an element (or there areelements) about which there is incomplete information, then we think of thatelement as taking on different possible values (which are the types) and thatthe players have probability models about which possible value of theelement is the true one. For example, if P knows the true damage level, then ithas a probability model placing all the probability weight on the true value. IfD only knows that it is between d _{L } and d _{H }, then D's model covers all thepossible levels of damage in that interval. The foregoing is an example of a __one-sided __ asymmetric information model, wherein one player is privatelyinformed about some element of the game and the other must use aprobability model about the element's true value; who is informed and theprobability assessment for the uninformed player is common knowledge. *Two-sided * asymmetric information models involve both players havingprivate information about either the same element or about different elements.Thus, P and D may, individually, have private information about what anindependent friend-of -the-court brief (still in preparation for submission attrial) may say, or P may know the level of damage and D may know whetherthe evidence indicates liability or not (this latter example will be discussedfurther in Section 12.4).

** ** *7.2. Consistent versus Inconsistent Priors * ** **

The card game examples above, in common with much of the literature onasymmetric information settlement models, involve games with consistentpriors. A few papers on settlement bargaining appear to use "inconsistent"priors. In this section we discuss what this means.

A game has __consistent priors __ if each player's conditional probabilitydistribution over the other player's type (or other players' types) comes fromthe same overall probability model. In the card game example with A and B,we observed that there was an overall model p(t _{A }, t _{B }) which was commonknowledge, and the individual conditional probability models p _{A }(t _{B }|t _{A }) andp _{B }(t _{A }|t _{B }) could have been found by using p(t _{A }, t _{B }). This was true because themakeup of the deck and the nature of the card-dealing process were commonknowledge. What if, instead, the dealer (a stranger to both A and B) firstlooked at the cards before they were dealt and chose which ones to give toeach player? Now the probability assessments are about the dealer, not thedeck, and it is not obvious that A and B should agree about how to model thedealer. Perhaps if A and B had been brought up together, or if they havetalked about how to model the dealer, we might conclude that the game wouldhave consistent priors (though long-held rivalries or even simpleconversations themselves can be opportunities for strategic behavior). Thus,if there is no underlying p(t _{A }, t _{B }) that would yield p _{A }(t _{B }|t _{A }) and p _{B }(t _{A }|t _{B })through the standard rules of probability, this is a model employing __inconsistent priors __.

Thus, if P and D both honestly believe they will win, they haveinconsistent priors, because the joint probability of both winning is zero.While such beliefs might be held, they present a fundamental difficulty forusing models which assert fully rational behavior: how can both players berational, both be aware of each other's assessment, aware that theassessments fundamentally conflict, and not use this information to reviseand refine their own estimates? The data of the game must be commonknowledge, as is rationality (and more, as will be discussed in the nextsection), but entertaining conflicting assessments themselves is in conflictwith rationality. Alternatively put, for the consistent application of rationalchoice, differences in assessments must reflect differences in privateinformation, not differences in world views. Presented with the sameinformation, conflicts in assessments would disappear.

To understand the problem, consider the following example taken from Binmore (1992, p. 477) . Let A and B hold prior assessments about anuncertain event (an election). A believes that a Republican will win theelection with probability 5/8 and the Democrat will win it with probability 3/8.B believes that the Democrat will win with probability 3/4 and the Republicanwith probability 1/4. Now if player C enters the picture, he can offer A thefollowing bet: A wins $3 if the Republican wins and pays C $5 if the Democratwins. C offers B the following bet: B wins $2 if the Democrat wins and pays C$6 if the Republican wins. Assume that A and B are risk-neutral, are wellaware of each other's assessments, and stick to the foregoing probabilitiesand that C pays each of them a penny if they take the bets. Then both A andB will take the bets and for __any __ probability of the actual outcome, C'sexpected profits are $2.98 ($3 less the two pennies). This is derisively called a"money pump" and works because of the inconsistent priors; that is, neitherA nor B update their assessment in response to the assessments that theother is using and is willing to bet with.

Now inconsistent priors may occur because one or the other or both thinkthat the other player is irrational. Recent laboratory experiments (see LindaBabcock, et. al. (1995) ) have found seemingly inconsistent priors that arisefrom a "self-serving" bias reflecting anticipated opportunities by players in asettlement activity. However, in an analysis employing incentives andrational choice, introducing something inconsistent with rational behaviorcreates a problem in terms of the analysis of the model and the comparison ofany results with those of other analyses.

How important consistent priors are to the analysis has been madeespecially clear in work on analyzing asymmetric information games. Startingfrom basic principles of rational decision-making, anyone making a choiceabout something unknown must make some assumptions about whatcharacterizes the unknown thing (usually in the form of a probabilitydistribution). To have two players playing an asymmetric game means,essentially, that they are playing a family of games, one for each possible pairof types (that is, one game for each pair of possible players). But which oneare they playing? This is solved by superimposing a probabilistic choice byNature (N), where each game is played with the probability specified by theoverall distribution over types (denoted earlier in the card examples as p(t _{A },t _{B })). If this distribution doesn't exist (that is, if priors are inconsistent), wecan't do this and players are left not properly anticipating which game mightbe played. This transformation of something difficult to analyze (anasymmetric information game) into something we know how to analyze (agame with imperfect information) won John Harsanyi a share (along with JohnNash and Reinhard Selten) of the 1994 Nobel Prize (the original papers are Harsanyi (1967, 1968a, 1968b) ).

Thus, while players may hold different assessments over uncertainevents, the notion of consistent priors limits the causes of the disagreementto differences in things like private information, and not to alternative modesof analysis; thus, players cannot paper-over differences by "agreeing todisagree." It is through this door that a literature, initially spawned bydissatisfaction with the perfect (and imperfect) information prediction thatcases always settle, has proceeded to explain a variety of observed behaviorwith asymmetric information models.

One last point before passing on to prediction. Shavell (1993) hasobserved that when parties seek nonmonetary relief and the bargaininginvolves an indivisible item, settlement negotiations may break down, even ifprobability assessments are the same. An example of such a case would bechild custody in a state with sole custody laws. This survey restrictsconsideration to cases involving non-lumpy allocations.

The main purpose of all of the settlement models is to make a predictionabout the outcome of bargaining, and the general rule is the more precise theprediction the better. The main tool used to make predictions in the recentliterature is the notion of __equilibrium. __ This is because most of the recent workhas relied upon the notion of __non-cooperative __ game theory, whereas earlierwork implicitly or explicitly employed notions from __cooperative __ game theory.The difference is that in a cooperative game, players (implicitly or explicitly)bind themselves __ex ante __ to require that the solution to the game be efficient("no money is left on the table"), while the equilibrium of a non-cooperativegame does not assume any exogenously enforced contractual agreement tobe efficient, and may end up not being efficient. We consider these notions inturn (for a review of laboratory-based tests of bargaining models, see AlvinRoth (1995) ).

** ** *8.1. Nash Equilibrium in Non-Cooperative Games * ** **

A strategy profile provides an equilibrium if no individual player canunilaterally change their part of the strategy profile and make themselvesbetter off; this notion of equilibrium is often called __Nash equilibrium __ (after Nash (1951) ), but its antecedents go far back in history. Using the notationintroduced earlier, let s* be an __equilibrium profile __; for convenience, assumethe game has two players, named 1 and 2, so s* = (s _{1 }*, s _{2 }*) and s _{1 } and s _{2 } areall the other strategies that 1 and 2, respectively, could use. Then player 1 isprepared to stay with s _{1 }* if:

$ _{1 }(s _{1 }*, s _{2 }*) > $ _{1 }(s _{1 }, s _{2 }*) |

for every possible s _{1 } player 1 could choose. Player 2 is prepared to stay withs _{2 }* if:

$ _{2 }(s _{1 }*, s _{2 }*) > $ _{2 }(s _{1 }*, s _{2 }) |

for every possible s _{2 } player 2 could choose. As stated earlier, no player canunilaterally improve his payoff by changing his part of the equilibriumstrategy profile.

Without generating more notation, the above conditions for a Nashequilibrium in a perfect information setting can be directly extended to theimperfect information setting. Here, the payoffs shown in the aboveinequalities are replaced by expected payoffs (the expectation reflecting thepresence of uncertainty in one or more elements in the payoff function).Finally, in the case of asymmetric information, strategies and expectations aredependent upon type, and thus the equations must now hold for every playertype and must reflect the individual player's conditional assessment about theother player(s). This last version is sometimes called a Bayesian Nashequilibrium to emphasize the role that the conditional probability distributionshave in influencing the strategies that players use (for more detail, see Mas-Colell, Whinston and Green (1995) , Chapter 8).

Note that in all the variations on the definition of Nash equilibrium, thereis a reference to no individual choosing to "defect" from the strategy profileof interest. What about coalitions of players? Class action suits involveforming coalitions of plaintiffs, joint and several liability impacts coalitions ofdefendants and successful ("real world") bargaining strategy sometimesrequires building or breaking coalitions (see David Lax and James Sebenius(1986) ). Issues of coalitions have been of great interest to game theorists andequilibrium notions have been developed to account for coalition defectionfrom a purported equilibrium strategy profile (see Binmore (1985) , DouglasBernheim, Bezalel Peleg and Whinston (1987) , Binmore (1992) , JosephGreenberg (1994) and Akira Okada (1996) for a small sample of recent work),but this is still an emerging area.

If two people are to divide a dollar between them (and both get nothing ifthey do not come to an agreement), then __any __ allocation of the dollar such thateach player gets more than zero is a Nash equilibrium, meaning that there isno predictive "bite" to our definition of equilibrium in this bargaining context(prediction improvements, called refinements, exist and often haveconsiderable bite; more on this later). In yet another seminal contribution, Nash (1950) provided a remarkable result that still provides context, and areference point, for many analyses (cooperative and non-cooperative) ofbargaining. His approach was to focus on the outcome of the bargaininggame and to ignore the details of the bargaining process entirely, therebyalso skipping the notion of requiring an equilibrium as the predictionmechanism. He posed the question: what desirable properties (called axioms)should a bargaining solution possess in order that a problem have a __unique __prediction? As mentioned earlier, by *solution * we mean that, __ex __ __ante __, the twoplayers would be prepared to bind themselves to the outcome which thesolution provides. This approach is presented in more detail in Section 10.

Nash's axioms can be summarized as follows (see Binmore (1992) ). First,the solution should not depend on how the players' utility scales arecalibrated. This means that standard models of utility from decision theorymay be employed (see, for example, Baird, Gertner and Picker (1994) ). Ifpayoffs are in monetary terms, this also means that players using differentcurrencies could simply use an exchange rate to convert everything to onecurrency. Second, the solution should always be efficient. Third, if theplayers sometimes agree to one outcome when a second one is also feasible,then they never agree to the second one when the first one is feasible.Fourth, in a bargaining game with two identical players, both players get thesame payoffs. The remarkable result is that whether the game is in utilities ormoney terms, the four axioms result in a __unique __ solution (called the __NashBargaining Solution or NBS __) to the bargaining game. We return to this inSection 10.

There is a very important linkage between predictions using refinementsof Nash equilibrium and predictions using a cooperative solution. One of themost remarkable and far-reaching results of game theory which emerged overthe seventies and eighties has been the delineation of conditions underwhich the equilibria for properly structured non-cooperative games would be(in the lingo, would __support __) solutions to properly related cooperative games.In our particular case, there are conditions on the data for the strategicapproach which guarantee that the equilibrium predicted for that model is theNBS of the associated bargaining problem. In other words, under certainconditions, the non-cooperative equilibrium is an efficient outcome.

Since we've not explored the axiomatic or strategic approaches in detailyet, let us consider an example likely to be familiar to most readers: the classicmodel of the conflict between group and individual incentives captured in the"Prisoner's Dilemma" (see, Baird, Gertner and Picker (1994) ). A variety of non-cooperative formulations have been developed wherein individual choices ofstrategies lead to the socially optimal outcome. The same techniques havebeen applied in a variety of settings, including bargaining. Thus, we nowhave a better understanding of how institutions, incentives and behavior mayor may not substitute for artificially imposed binding agreements in achievingan efficient outcome. This also means that sources of inefficiencies ("moneyleft on the table," and thus wasted) brought about by institutionalconstraints, incentives and non-cooperative behavior can be betterunderstood.

** 9. An Example of a Model of Settlement Negotiation **

Before venturing into the section describing the range of settlement models, abrief example will help clarify the concepts raised above. Reconsider Pat andDelta's negotiation with the following further simplifications and somenumerical values. First, assume that the only source of uncertainty is Delta'sliability. Damages are known by all, as are court costs. Moreover, there are noattorneys or experts and J will simply award the actual damage if Delta isfound to be liable. Second, we will consider two simple bargaining stories.

1) P makes a demand of D, followed by D accepting or rejecting thedemand. Acceptance means a transfer from D to P; rejection means that Jorders a transfer from D to P (the two transfers need not be the same) andboth parties pay their court costs. Third, it is also common knowledge that ifD is indifferent between accepting the proposal and rejecting it, D will acceptit.

2) D makes an offer to P, followed by P accepting or rejecting the offer.Acceptance means a transfer from D to P; rejection means that J orders atransfer from D to P (again, the two transfers need not be the same) and bothparties pay their court costs. Third, it is also common knowledge that if P isindifferent between accepting the proposal and rejecting it, P will accept it.

Let d = 100 be the level of damages and L = .5 be the likelihood of D beingfound liable by J for the damage d. Let court costs be the same for bothplayers with k _{P }= k _{D } = 10. Note that the expected compensation Ld exceeds theplaintiff's court cost k _{P }, so that should D reject P's demand in case (1), or offertoo little in (2), it is still worth P's effort to go to trial. Note also that thisignores the possibility of bankruptcy of D. All of the above, that both playersare rational (that is, P maximizes, and D minimizes, their respective payoffs)and the bargaining story being analyzed are common knowledge. One final bitof notation: let s be a settlement proposal.

** ** *9.1. Analyzing the Case Wherein P Makes a Demand * ** **

The first task is to find out if settlement is possible (the *admissible *settlements). We start with D, as P must anticipate D's choice when facedwith P's demand. D and P know that if the case goes to trial, D will expendeither d + k _{D } or k _{D } (110 or 10), the first with probability L and the second withprobability (1 - L); thus D's expected expenditure at trial (payoff from theoutcome go to trial) is Ld + k _{D } (that is, 60). Note that in this circumstance, P'sexpected payoff from the outcome labelled trial is Ld - k _{P } (that is, 40). Thus, Dwill accept any settlement demand not exceeding this expected expenditure attrial:

s < Ld + k _{D }. | (*) |

P wishes to maximize her payoff which depends upon P's demand and thechoice made by D: $ _{P } = s _{P } if D accepts the demand s _{P } or $ _{P } = Ld - k _{P } if Drejects the demand.

More carefully, using our earlier notation that the indicated payoffdepending upon the strategy profile, we would have $ _{P }(s _{P }, accept) = s _{P }; thatis, the payoff to P from her using the strategy "make the settlement demands _{P }" and D using the strategy "accept" is the transfer s _{P }. Similarly, we wouldhave $ _{P }(s _{P }, reject) = Ld - k _{P }. In the rest of this example we will suppress thisnotation when convenient, but understanding it will be of value later.

Observe that the maximum settlement demand that P can make (s _{P } = Ld +k _{D }, as shown in inequality (*)) exceeds P's payoff from court. Thus, Pmaximizes the expected payoff __from the game __ by choosing Ld + k _{D } as hersettlement demand, which D accepts since D cannot do better by rejecting theproposal and facing trial. Thus, to summarize: 1) the players are P and D; 2)the action for P is the demand s _{P } (this is also P's strategy) while the action forD is to accept or reject and D's strategy is to accept if s _{P } satisfies thecondition (*) and to reject otherwise; 3) the outcomes are settlement andtransfer with associated payoffs $ _{P } = s _{P } and $ _{D } = s _{P }, or proceed to trial andtransfer with associated payoffs $ _{P } = Ld - k _{P } and $ _{D } = Ld + k _{D }; 4) the timing isthat P makes a demand and D chooses accept or reject; 5) information isimperfect in a very simple way in that P and D share the same assessmentabout the trial outcome with respect to liability. Note that it is unnecessary tomodel a choice for P about going to court if her demand were rejectedbecause of the assumption that the expected compensation exceeds plaintifftrial costs. Moreover, since nothing in the settlement phase will influence thetrial outcome itself should trial occur, J is not a player in a meaningful sense.The prediction (the equilibrium) of this game is that the case settles, theresulting transfer from D to P is Ld + k _{D } (in the numerical example, 60) and thegame payoffs are $ _{P } = $ _{D } = Ld + k _{D } (60).

** ** *9.2. Analyzing the Case Wherein D Makes an Offer * ** **

We now start with P in order to find the admissible settlements. Given theforegoing, P will accept any settlement offer that yields at least what shewould get in court:

s > Ld - k _{P }. | (**) |

D wishes to minimize its payoff, which depends upon the offer it makes andthe choice made by P: $ _{D } = s _{D } if P accepts the offer s _{D } or $ _{D } = Ld + k _{D } if Prejects the offer. Thus D minimizes its payoff from the game by choosing Ld -k _{P } as its settlement offer, which P accepts since P cannot do better byrejecting the proposal and going to trial. Thus, to summarize: 1) the playersare P and D; 2) the action for D is the offer s _{D } (this is also D's strategy) whilethe action for P is accept or reject and P's strategy is to accept if s _{D } satisfiesthe condition (**) and to reject otherwise; 3) the outcomes are settlement andtransfer with associated payoffs $ _{D } = s _{D } and $ _{P } = s _{D }, or proceed to trial andtransfer with associated payoffs $ _{D } = Ld + k _{D } and $ _{P } = Ld - k _{P } ; 4) the timing isthat D makes an offer and P chooses to accept or reject; 5) information isimperfect in the same way as in the first case. The prediction (the equilibrium)of this game is that the case settles, the resulting transfer from D to P is Ld -k _{P } (40) and the game payoffs are $ _{P } = $ _{D } = Ld - k _{P } (40).

** ** *9.3. Bargaining Range and Bargaining Efficiency *

A clear implication of the foregoing analysis is that who moves last has asignificant impact on the allocation of the surplus generated by not going tocourt. One could think of the process in the following way. D pays Ld to P nomatter what procedure is used. P and D then contribute their court costs to afund (called surplus) which they then split in some fashion. If the bargainingprocess involves P making a demand and D choosing only to accept or reject,then P gets all the surplus. If the roles are reversed, so are the fortunes. Thismight suggest that the two cases studied provide the extremes (the __bargaining range __) and that actual bargaining will yield something inside thisrange. The answer, we shall see, is maybe yes and maybe no. In thepreceding analysis bargaining was efficient (no cases went to trial; again thereader is cautioned to recall the earlier discussion of the use of the word"efficiency") since all information was symmetric and the first mover couldmake a take-it-or-leave-it proposal (cognizant of the second-mover's ability togo to trial). Efficiency will fail to hold when we allow for asymmetricinformation. This will occur not because of mistakes by players, but becauseof the recognition by both players that information which is asymmetricallydistributed will impose a cost on the bargaining process, a cost that oftenfalls on the better informed party.

**B. Basic Models of Settlement Bargaining ** * *

** 10. Perfect and Imperfect Information Models: Axiomatic Models for theCooperative Case **

The perfect information model (and its first cousin, the imperfect informationmodel), versions of which appear in William Landes (1971) , John Gould (1973) and Richard Posner (1973) , is an important starting place as it focusesattention on efficient bargaining outcomes. Many of the earlier modelsemployed risk aversion, which will be addressed in Section 17.1. For now, andso as to make the presentations consistent with much of the more recentliterature, payoffs will be assumed to be in dollar terms with risk-neutralplayers.

We start with the perfect information version. In keeping with the earlierdiscussion, the following model emphasizes final outcomes and suppressesbargaining detail. While the analysis below may seem like analytical overkill,it will allow us to structure the problem for later, more complex, discussions inthis Part and in Part C.

The players are P and D; further, assume that the level of damages, d, iscommonly known and that D is fully liable for these damages. Court costs arek _{P } and k _{D }, and each player is responsible for their own court costs. Eachplayer has an individual action they can take that assures them a particularpayoff. P can stop negotiating and go to court; thus the payoff to P from trialis $ _{P } ^{t } = d - k _{P } (note the superscript t for trial) and the payoff to D from trial is$ _{D } ^{t } = d + k _{D }. Under the assumption that d - k _{P } > 0, P has a __credible threat __ to goto trial if negotiations fail. For the purposes of most of this paper (and most ofthe literature) we assume this condition to hold (the issue of it failing will bediscussed in Section 16.1). By default, D can "assure" himself of the samepayoff by stopping negotiations, since P will then presumably proceed totrial; no other payoff for D is guaranteed via his individual action. The pair($ _{D } ^{t }, $ _{P } ^{t }) is called the __threat or disagreement point __ for the bargaining game(recall that D's payoff is an expenditure).

What might they agree on? One way to capture the essence of thenegotiation is to imagine both players on either side of a table, and that theyactually place money on the table (abusing the card-game story from earlier,this is an "ante") in anticipation of finding a way of allocating it. This meansthat D places d + k _{D } on the table and P places k _{P } on the table. The maximum atstake is the sum of the available resources, d + k _{P } + k _{D }, and therefore anyoutcome (which here is an allocation of the available resources) that does notexceed this amount is a possible settlement.

P's payoff, $ _{P }, is his bargaining outcome allocation (b _{P }) minus his ante(that is, $ _{P } = b _{P }- k _{P }). D's payoff, $ _{D }, is his ante minus his bargaining outcomeallocation (b _{D }); thus D's __cost __ (loss) is $ _{D } = d + k _{D } - b _{D }. Since the bargainingoutcome (b _{P } + b _{D }) cannot exceed the total resources to be allocated (themoney on the table), b _{P } + b _{D } __< __ d + k _{D } + k _{P }, or equivalently, P's gain cannotexceed D's loss: $ _{P } = b _{P } - k _{P } __< __ d + k _{D } - b _{D } = $ _{D }. Thus, in payoff terms, theoutcome of the overall bargaining game must satisfy: 1) $ _{P } __< __ $ _{D }; 2) $ _{P } __> __ $ _{P } ^{t } and3) $ _{D } __< __ $ _{D } ^{t }. In bargaining outcome terms this can be written as: 1') b _{P }+ b _{D } __< __ d +k _{D } + k _{P }; 2') b _{P } __> __ d and 3') b _{D } __> __ 0. For the diagrams to come, we restate (2') as b _{P }- d __> __ 0.

Figure 1a illustrates the settlement possibilities. The horizontal axisindicates the net gain (-$ _{D }) or net loss (that is total expenditure, $ _{D }) to D. Thevertical axis indicates the net gain ($ _{P }) or net loss (-$ _{P }) to P. The sloping linegraphs points satisfying $ _{P } = $ _{D } while the region to the left of it involvesallocations such that $ _{P } < $ _{D }. The best that D could possibly achieve is torecover his ante d + k _{D } and get k _{P } too; this is indicated at the end of the lineat the point (k _{P }, -k _{P }), meaning D has a net gain of k _{P } and P has a net loss of k _{P }.At the other extreme is the outcome wherein P gets all of d + k _{P } + k _{D }, meaningP has a net gain of d + k _{D } which is D's net loss.

Note also that points below the line represent inefficient allocations: thisis what is meant by "money left on the table." The disagreement point (-(d +k _{D }), d - k _{P }) draws attention (observe the thin lines) to a portion of the feasibleset that contains allocations that satisfy inequalities (2) and (3) above. Theplacement of this point reflects the assertion that there is something tobargain over; if the point were above the line $ _{P } = $ _{D } then trial is unavoidable,since there would be no settlements that satisfy (2) and (3) above. Thistriangular region, satisfying inequalities (1), (2) and (3), is the *settlement set *and the end points of the portion of the line $ _{P } = $ _{D } which is in the settlementset are called the __concession limits __; between the concession limits (andincluding them) are all the efficient bargaining outcomes under settlement,called the __settlement frontier __.

Figure 1b illustrates the settlement set as bargaining outcomes, found bysubtracting the disagreement point from everything in the settlement set.Doing this helps adjust the region of interest for asymmetries in the threatsthat P and D can employ. This leaves any remaining asymmetries in power orinformation in the resulting diagram. In the case at hand, the only powerdifference might appear in the difference between the costs of going to trial;other power differences such as differences in risk preferences, patience, etc.will be discussed in Section 17.1, while informational differences will bediscussed in the sections on asymmetric information in Parts B and C.

Notice that, in view of (2') and (3'), the vertical (non-negative) axis islabelled b _{P } - d while the horizontal (non-negative) axis is labelled b _{D }.Moreover, since aggregate trial costs determine the frontier in Figure 1b, thebargaining problem here is symmetric. The Nash Bargaining Solution (NBS)applies in either picture, but its prediction is particularly obvious in Figure 1b:recalling the discussion in Section 8.2, requiring the solution to be efficient(axiom 2) and that, when the problem is symmetric the solution is, too (axiom4), means that splitting the saved court costs is the NBS in Figure 1b. Thus,to find the NBS in Figure 1a, we add the disagreement point __back into thesolution from Figure 1b __. Therefore, P's payoff is d - k _{P } + (k _{P } + k _{D })/2 = d + (k _{D } -k _{P })/2. D's net outflow is -(d + k _{D }) + (k _{P } + k _{D })/2. In other words, D's payoff (hisexpenditure) is d + (k _{D } - k _{P })/2. The result that players should "split thedifference" is always the NBS solution for any bargaining game with payoffsin monetary terms.

Observe that if court costs are the same, then at the NBS P and D simplytransfer the liability d. If D's court costs exceed P's, P receives more from thesettlement than the actual damages, reflecting his somewhat stronger relativebargaining position embodied in his threat with respect to the surplus that Pand D can jointly generate by not going to court. A similar argument holdsfor P in the weaker position, with higher costs of going to court: he settles forless than d.

This is essentially the same model, so only the variations will be remarkedupon. Assume that P and D have the same probability assessments for thetwo court costs and assume that they also have the same probabilityassessments over expected damages. This latter possibility could reflect thatthe level of damages is unknown (for example, as discussed in Section 7.1)but that liability is taken to be assured. Then they both see expected damagesas E(d). Alternatively, perhaps damages are known to be d but liability is lessclear but commonly assessed to be L; that is, L is the common assessmentthat D will be held liable for damages d and (1 - L) is the common assessmentthat D will not be held liable at all. Then E(d) = dL (an admitted but usefulabuse of notation). Finally, if there are common elements influencing thevalues that d might take on and the likelihood of D being held liable, and ifthe assessments of the possible values and their joint likelihood is commonknowledge, then again we will write the expected damages from trial as E(d).Again, this is abusing the notation, but avoids needless technicaldistinctions. The point is that in an imperfect information setting we simplytake the preceding analysis and replace all known parameters with theirsuitably constructed expectations, yielding the same qualitative results: notrials occur in equilibrium, strong plaintiffs settle for somewhat more thantheir expected damages, and so forth.

** 11. Perfect and Imperfect Information Models: Strategic Models for the Non-Cooperative Case **

Again, we start with the perfect information case. Furthermore, since theactual bargaining procedure is now to be specified, the length of thebargaining horizon now enters into the analysis. The generic style of themodels to be considered is that one player makes a proposal followed by theother player choosing to accept or reject the proposal. Concatenating asmany of these simple proposal/response sequences as we choose providesthe basic story.

Some questions, however, remain. First, is the proposer the same playereach time? In general we will assume that if there is more than one round ofproposal/response, then proposers alternate (an important exception is Spier(1992) where the plaintiff always proposes; this will be discussed in Section18). If there is more than one round, the next proposal is often thought of as acounterproposal to the one before it.

Second, how many periods of proposal/response will there be? This turnsout to be a very significant question. Recall that in the cooperative model Pand D were committed to finding an efficient bargaining outcome. Here, nosuch commitment is made; instead, we want to know when non-cooperativebargaining will be efficient. However, certain types of commitments instrategic models still may occur. The reason this is of interest is thatcommitment generally provides some power to the player who can make acommitment. For example, if there is one round of proposal/response, then theproposer has the ability to make an all-or-nothing proposal (more carefullyput, all-or-court proposal). As was seen in the examples in Section 9, this ledto a settlement that was efficient but rather one-sided. In particular, theproposer was able to achieve the point on the settlement frontier that is theresponder's concession limit. This is a reflection of the commitment powerthat the proposer enjoys of __not __ responding to any counterproposals that theresponder might desire to make: these are ruled out by the structure of thegame analyzed. This is why this game is often referred to as an __ultimatumgame __. Ultimatum games form the basis for many of the asymmetric informationsettlement analyses we shall examine.

Almost at the other end of the spectrum of theoretical bargaining analysesis the Rubinstein infinite-horizon model ( Rubinstein (1982) ). In this model aninfinite number of rounds of proposal/response occur wherein the proposer'sidentity alternates. In the settlement setting, each round allows P to choose tobreak off negotiations and go to trial. Here there are two somewhat moresubtle forms of commitment in place of the power to make all-or-nothingproposals. First, if there is a positive interval of time between one round andthe next, and if "time is money," meaning (for example) that costs are accruing(perhaps experts are being kept available, or lawyers are accruing time), thenthe fact that during a round only one proposal is being considered (theproposer's) provides some power to the proposer.

Second, who goes first is still significant. Rubinstein considers a simple"shrinking pie" example wherein each player discounts money received in thefuture relative to money received now. This encourages both players to wantto settle sooner rather than later (all else equal). Thus, delay here yieldsinefficiency. Rubinstein uses a notion of Nash equilibrium that incorporatesthe dynamics of the bargaining process (this extra property of equilibrium iscalled __sequential rationality __, which will be discussed in more detail in Section11.1) which results in a unique prediction for the bargaining game. Inparticular, there is no delay and, if both players are identical, then the playerwho goes first gets more than half of the amount at stake.

Models that shrink the time interval associated with each round find thatboth sources of power go away as the time interval between proposalsbecomes vanishingly small. Note that, even with positive intervals, the effectof commitment (in this case, a short-run commitment to a proposal) is not asstrong as in the ultimatum game, since counterproposals can occur andplayers generally cannot bind themselves to previous proposals they'vemade. In other words, such infinite horizon models can generate efficientsettlement at points on the frontier other than the concession limits. In fact,under certain conditions they predict the NBS as the unique equilibrium ofthe strategic bargaining game. Note that the fact that a strategic modelemploys perfect information does not guarantee that the predicted outcome isefficient. A particularly striking example of this is contained in RaquelFernandez and Jacob Glazer (1991) who consider wage negotiations betweena union and a firm under perfect information and yet get inefficient equilibria.The source of the inefficiency is a pre-existing wage contract. This is anunexplored area for settlement bargaining which might yield some interestingresults.

Finally, a further difference between settlement applications and thegeneral literature on bargaining concerns the incentive to settle as soon aspossible, all else equal. Generally, in the settlement context, P wants to settlesooner but D wants to settle later. While countervailing pressures, such ascosts that increase with time, may encourage D to settle as early as possible,the fact that payment delayed is generally preferred by the payor (due to thetime value of money) means that D's overall incentives to settle soon can bemixed and delay may be optimal. Moreover, as observed in Spier (1992) ,unlike the Rubinstein model, if both P and D have the same discount rate thenthe pie itself is __not __ shrinking (assuming no other costs of bargaining). This isbecause the effect of the opposed interests and the same discount rates is tocancel out. Therefore, in a multiperiod settlement model, delay due to the timevalue of money does not, in and of itself, imply inefficiency. We will return tothis shortly.

Note that in much of the preceding discussion an implicit notion was that aplayer's strategy anticipates future play in the game. A strategy is __sequentially rational __if it is constructed so that the player takes an __optimal __action at each possible decision opportunity that the player has in the future.Earlier, in the discussion of the disagreement point, sequential rationality wasused by P. The threat to go to trial if bargaining failed to satisfy (2) wassequentially rational: it was a credible threat because if P got to that point, hewould choose to fulfill the threat he had made. Applying sequentialrationality to the strategies players use, and to the analysis players make ofwhat strategies __other __ players might use, means that strategies based onthreats that a rational player would not carry out (incredible threats) are ruledout. Many of the improvements in making predictions for asymmetricinformation models that have occurred over the last fifteen years haveinvolved employing sequential rationality, generally in conjunction withfurther amplifications of what rational behavior implies. Rubinstein finds aunique prediction in the infinite-horizon alternating offers game (for short, theRubinstein game) because he predicts Nash equilibria which enforcesequential rationality (called __subgame perfect equilibrium __; for a discussion ofsome applications of subgame perfection in law and economics, see Baird,Gertner and Picker (1994) ). In the Rubinstein game, even though the horizon isinfinite, the (sequentially rational) Nash equilibrium is a unique, specific,efficient bargaining allocation which is proposed and accepted in the firstround. Thus, efficiency (both in terms of fully allocating what is available tobargain over as well as doing it without delay) is a __result __, not an assumption.

** ** *11.2. Settlement Using Strategic Bargaining Models in the PerfectInformation Case *

The discussion in Section 9 provides the details of the ultimatum gameversion of settlement. Since that application technically involved imperfectinformation (the assessment about liability), a careful treatment means that wewould take L = 1, yielding the payoffs for the ultimatum model with P asproposer (the P-proposer ultimatum model) to be $ _{P } = $ _{D } = d + k _{D }, while the D-proposer ultimatum model's payoffs would be $ _{P } = $ _{D } = d - k _{P }. The rest of thissection is therefore devoted to the analysis in the infinite horizon case.

The tradeoff between D's natural interest to delay payment and anyincentives to settle early (in particular, P's credible threat to go to court andany negotiation costs borne by D) is explored in the settlement context in Gyu Ho Wang, Jeong-Yoo Kim and Jong-Goo Yi (1994) ; they also consider anasymmetric information case which will be discussed in Section 12.4. In theperfect information analysis, D proposes in the first round, but is subject ineach round to an additional cost, c, which reflects per period negotiatingcosts but is charged __only __ if the negotiations proceed to the next period. Onecould include a cost of this sort for P, too, but it is the difference between P'sand D's negotiating costs that matters, so letting P's be zero is not ameaningful limitation.

Let ° (a fraction between zero and one) be the common discount rate usedby the players for evaluating and comparing money flows at different pointsin time; that is, a player is indifferent between receiving $1 next period and $°this period. Note that this effect could be undone if, at trial, damages wereawarded with interest from the date of filing the suit; this does not occur intheir model. Wang, Kim and Yi show that if °c/(1 - ° ^{2 }) > d - k _{P }, then the uniquesubgame perfect equilibrium is for D to offer ° ^{2 }c/(1 - ° ^{2 }) to P; if instead °c/(1 -° ^{2 }) < d - k _{P }, the unique subgame perfect equilibrium involves D offering °(d -k _{P }) to P. For a careful proof of this, see Wang, Kim and Yi; for our purposes,let us use some crude intuition (sweeping all sorts of technical details underthe rug) to understand this result.

Consider D's viewpoint. The only reason for D to be indifferent betweenan expenditure in one period and an expenditure in the next period involvesthe delay cost and the discount rate: indifference as to when to spend s inone of two adjacent periods means the discounted value of waiting, incurringc and then spending s (which is °s + °c) should just equal the expenditure ofs now (that is, °s + °c = s). Of course, since D only makes a decision every __other __ period this relationship affects P more directly than D. P will find this tobe advantageous if c is large enough and if °s + °c > d - k _{P }. Assume it is; thismeans that if costs are high enough to get D to not want to delay (or make anoffer that would have P choose to go to trial) the worst that P would getwould be °(°s + °c), which should be the most D would offer; that is, s.Solving °(°s + °c) = s for s yields s = °c/(1 - ° ^{2 }). If this is large enough, D willmake an offer that P will prefer to the payoff from going to trial. Since the offeronly has to make P indifferent between two adjacent time periods (when Doffers and P accepts or rejects, and when P decides about going to trial orcounterproposing), D can offer °(°c/(1 - ° ^{2 })) and P will accept in the period inwhich the offer is made if next period °c/(1 - ° ^{2 }) > d - k _{P } (the equality betweenthe two sides is eliminated as it provides multiple predictions, while the strictinequality yields a unique solution; see Rubinstein (1985) ). In this sense,°c/(1 - ° ^{2 }) > d - k _{P } really is a statement that the cost of delay is high from D'sviewpoint, which is why D must offer something higher than the discountedvalue of going to trial, that is, °(d - k _{P }).

Even if costs are low (that is, °c/(1 - ° ^{2 }) < d - k _{P }), D must still worry aboutP's choice of going to court, but P can no longer exploit D's cost weakness tofurther improve the bargain in his favor. Thus, D can offer the discountedvalue of P's concession limit, namely °(d - k _{P }), since P cannot choose to go tocourt until next period and therefore might as well accept °(d - k _{P }) now.

To summarize, the players are P and D, actions in each period involveproposals followed by accept/reject from the other player, with P able tochoose to go to court when it is his turn to propose. Payoffs are as usual withthe added provisos concerning the discount rate ° reflecting the time value ofmoney, and the per period cost c, for the defendant, which is incurred eachtime negotiators fail to agree. The bargaining horizon is infinite andinformation is perfect. The result is that: 1) P and D settle in the first period; 2)the prediction is unique and efficient; 3) if negotiation costs are sufficientlyhigh then the prediction is on the settlement frontier, between the discountedvalues of the concession limits and 4) otherwise it is at the discounted valueof P's concession limit, reflecting the fact that D moved first.

** ** *11.3. The Imperfect Information Case * ** **

The extension of the ultimatum game results of Section 9 to the imperfectinformation case parallels the discussion in Section 10.2. This is similarly truefor the multiperiod case, which is why this issue has not received muchattention.

** 12. Analyses Allowing for Differences in Player Assessments Due toPrivate Information **

In this section we consider models that account for differences in the player'sassessments about items such as damages and liability based on the privateinformation players possess when they bargain. We focus especially on twomodels: one developed by Bebchuk (1984) and one developed by JenniferReinganum and Louis Wilde (1986) . Most of the analyses in the currentliterature are based on one of these two primary settlement models, both ofwhich analyze ultimatum games and both of which assume one-sidedasymmetric information; that is, that there is an aspect of the game (typically aparameter such as damages or liability) about which P and D have differentinformation, and only one of the players knows the true value of theparameter during bargaining. Since the model structure is so specific (theultimatum game) and the distribution of information is so one-sided, we alsoconsider what models with somewhat greater generality suggest about thereasonableness of the two prime workhorses of the current settlementliterature. For a survey of asymmetric information bargaining theory, see JohnKennan and Robert Wilson (1993) . For a recent discussion of the settlementfrontier under inconsistent priors, see Tai-Yeong Chung (1996) .

** ** *12.1. Yet More Needed Language and Concepts : Screening, Signaling,Revealing and Pooling *

Reaching back to Section 7, a one-sided information model is like a card gamewhere only one player has a down-card and knows the value of that card.That player is privately informed and, because of consistent priors, bothplayers know that the probability model being used by the uninformed playeris common knowledge.

Since the basic bargaining process involves one round of proposal andresponse, the fact that only one of the players possesses private informationabout something that is important to both means that __when __ the informedplayer acts is, itself, important. A __screening __ model (also sometimes called a __sorting __ model) involves the uninformed player making the proposal and theinformed player choosing to accept or reject the proposal. A __signaling __ modelinvolves the reverse: the informed player makes the proposal and theuninformed player chooses to accept or reject it. Note that the word"signaling" only means that the proposal is made by the informed player, notthat the proposal itself is necessarily informative about the privateinformation possessed by the proposer. Bebchuk's model is a screeningmodel and Reinganum and Wilde's model is a signaling model.

These are non-cooperative models of bargaining, so our method ofprediction is finding an equilibrium (rather than a cooperative solution). Inboth cases sequential rationality is also employed. Generally, in the case of ascreening model this yields a unique prediction. In the case of a signalingmodel sequential rationality is generally insufficient to produce a uniqueprediction. The reason is that since the uninformed player is observing theinformed player's action in this case, the action itself may reveal somethingabout the private information of the proposer.

To understand this, consider a modification of the card game story fromSection 7.1. Here are the hands for players A and B and, as before, only Aknows his down card:

Table 2 : Hands of Cards for A and B | ||

UP | DOWN | |

A | AD, QD, JD, 10D | 2H |

B | 10C, 10S, 9S, 5D, 2C | |

(Note: entries provide face value of card and suit) |

To fill out the story, all the above cards have just been dealt, after playersput some money on the table, from a standard 52-card deck. The rules are thatplayer A may now discard one card (if it is the down card, this is donewithout revealing it) and a new card is provided that is drawn from theundealt portion of the deck. A can also choose not to discard a card. If hedoes discard one, the new card is dealt face up if the discard was an up-cardand is dealt face down otherwise. After this, A and B can add money to thatalready on the table or they can surrender their share of the money currentlyon the table ("fold"); for convenience, assume that a player must fold or addmoney (a player can't stay in the game without adding money to that alreadyon the table). The cards are then compared, privately, by an honest dealer,and any winner gets all the money while a tie splits the money evenlyamongst those who have not folded. The comparison process in this casemeans that, for player A, a down card that matches his Ace, Queen or Jackcard with the same face value (that is, A ends up with two Aces or twoQueens or two Jacks) will beat B's hand, as will any Diamond in conjunctionwith A's up-cards currently showing. Other draws mean that A will either tie(a 10H) or lose.

Before A chooses whether and what to discard, B knows that the downcard could be any of 43 cards with equal likelihood. Now if A discards hisdown card, based on sequential rationality, B knows that it was not an Ace, aQueen, a Jack or a Diamond. This information can be used by B __before __ hemust take any action. He may choose to fold, or he may choose to addmoney, but this decision is now influenced by what he believes to be A'sprivate information, A's new down card. These beliefs take the form of animproved probability estimate over A's type (adjusting for what has beenobserved). In fact, these assessments are called __beliefs __, and in an asymmetricinformation model, players form beliefs based upon the prior assessments andeverything that they have observed before each and every decision theymake. The addition of the need to account for what beliefs players canreasonably expect to hold makes the signaling game more complex to analyzethan the screening game.

A few more observations about the card game are in order. First, eventhough this was a signaling game, the signal of discarding did not completelyinform the uninformed player of the content of the private information. Astrategy for the informed player is __revealing __ if, upon the uninformed playerobserving the action(s) of the informed player, the uninformed player cancorrectly infer the informed player's type. In this sense, each type of playerhas an action that distinguishes it from all the other types. For example, in thesettlement context where P is privately informed about the true level ofdamages but D only knows the prior distribution, a revealing strategy wouldinvolve each possible type of P (each possible level of damages) making adifferent settlement demand, such as P demands his true damages plus D'scourt costs. If instead P always asked for the average level of damages,independent of the true level, plus court costs, then P would be using a __pooling __ strategy: different types of P take the same action and therefore areobservationally indistinguishable.

In the card game above, A choosing to discard his down card has someelements of a revealing strategy (not all types would choose this action) andsome elements of a pooling strategy (there are a number of types who wouldtake the same action). This is an example of a __semi-pooling or partial pooling __strategy. Notice that if the deck had originally consisted of __only __ the elevencards AD, KD, QD, JD, 10D, 10C, 10S, 9S, 5D, 2C, 2H, then B could use theaction "discard" to distinguish between the private information "initial down-card = KD" and "initial down-card = 2H" because discarding the down card isonly rational for the player holding a 2H. Thus, discarding or not discardingin this case is a fully revealing action. In this particular example we got this bychanging the size of the deck (thereby changing the number of types), butthis is not always necessary. In many signaling models, extra effort placed onmaking predictions, even in the presence of a continuum of types, leads tofully revealing behavior; we will see this in the signaling analysis below.

Finally, a __revealing equilibrium __ means that the equilibrium involves thecomplete release of all private information. In a revealing equilibrium theprivately informed player is employing a revealing strategy. In a __poolingequilibrium __ the private information is not released. In other words, at the endof the game, no more is necessarily known by the uninformed player than wasknown before play began. More generally, in a __partial pooling equilibrium __,some of the types have been revealed through their actions and some of thetypes took actions which do not allow us to distinguish them from oneanother.

** ** *12.2. Where You Start and Where You End * ** **

As will be seen below, a typical screening model produces partial poolingequilibria as its prediction; in fact, the equilibrium is often composed of twobig pools (a bunch of types do this and the rest do that) and is only fullyrevealing if each pool consists of one type. In other words, if the privateinformation in the model takes on more than two values, some pooling willtypically occur in the equilibrium prediction of a screening model. On theother hand, a typical signaling model has all three types of equilibria(revealing, fully pooling, partial pooling) as predictions, but with some extraeffort concerning rational inference (called __refinements __ of equilibrium) thisoften reduces to a unique prediction of a revealing equilibrium. Either one ofthese types of prediction may be desirable, depending upon the processbeing analyzed.

** ** *12.3. One-Sided Asymmetric Information Settlement Process Models :Examples of Analyses *

In the subsections to follow we will start with the same basic setting and findthe results of applying screening and signaling models. Bebchuk's 1984 paperconsidered an informed defendant (concerning liability) responding to anoffer from an uninformed plaintiff; Reinganum and Wilde's 1986 paperconsidered a plaintiff with private information about damages making ademand of an uninformed defendant. Initially, information will be modeled astaking on two levels (that is, a two-type model is employed) and a basicanalysis using each approach will be presented and solved. The result ofallowing for more than two types (in particular, a continuum of types) willthen be discussed in the context of the alternative approaches.

Since most of the discussion earlier in this survey revolved arounddamages, both approaches will be applied to private information on damages,suggesting a natural setting of an informed P and an uninformed D (note,however, that the earlier example of Pat and Delta was purposely posed withDelta as the informed party to emphasize that the analysis is applicable in avariety of settings). More precisely, the level of damages is assumed to takeon the value d _{L } (L for low) or d _{H } (H for high), meaning that P suffers a lossand it takes on one of these two levels, which is private information for P.Moreover, d _{H } > d _{L } > 0. The levels are common knowledge as is D'sassessment that p is the probability of the low value being the true level ofdamages. If the case were to go to trial then J will find out the true level ofdamages (whether the damages were equal to d _{L } or equal to d _{H }) and award thetrue damages to P. Thus E _{D }(d) = pd _{L } + (1-p)d _{H } is D's prior estimate of theexpected damages that he will pay if he goes to trial; P knows whether thedamages paid will be d _{L } or d _{H }. Should the case go to trial each player pays hisown court costs (k _{P } and k _{D }, respectively) and, for simplicity again, assumethat d _{L } > k _{P }; relaxing this assumption is discussed in Section 16.1. Finally, ineach case the structure of the bargaining process is represented by anultimatum game. In particular, the player who responds will choose to acceptthe proposal if he is indifferent between the payoff resulting from acceptingthe proposal and the payoff resulting from trial. __Without any moreinformation __, D's __a priori __ (that is, before bargaining) expected payoff from trialis E _{D }(d) + k _{D }; P's payoff from trial is d _{L } - k _{P } if true damages are d _{L }, and d _{H } - k _{P }if true damages are d _{H }.

** ** __12.3.1. Screening : A Two-Type Analysis __

In this model D offers a settlement transfer to P of s _{D } and P responds withacceptance or rejection. For the analysis to be sequentially rational (that is,we are looking for a subgame perfect equilibrium) we start by thinking aboutwhat P's strategy should be for __any __ possible offer made by D in order for himto maximize his overall payoff from the game. If s _{D } __> __ d _{H } - k _{P }, then no matterwhether damages are high or low, P should accept and settle at s _{D }. If s _{D } < d _{L } -k _{P }, then no matter whether damages are high or low, P should reject the offerand go to trial. If s _{D } is set so that it lies between these two possibilities, thatis, d _{H } - k _{P } > s _{D } __> __ d _{L } - k _{P }, then a P with high damages should reject the offer,but a P with low damages should accept the offer. This last offer is said to *screen the types *; note that the offer results in the revelation of the privateinformation. Thus P's optimal action is contingent upon the offer made; wehave found P's strategy and it involves rational choice. D can do thiscomputation, too, for each possible type of P that could occur, so from D'sviewpoint he models P as having a strategy that depends both on P's typeand upon D's offer.

As always, D's objective is to minimize expected expenditure and he mustmake an offer before observing any further information about P; thus, Dcannot improve his assessments as occurred in the card game. D knows,however, that some offers are better than others. For example, the lowest offerin the range d _{H } - k _{P } > s _{D } __> __ d _{L } - k _{P }, namely s _{D } = d _{L } - k _{P }, is better than any otheroffer in that range, since it doesn't change the result that H-type P's will rejectand L-type P's will accept, and its cost is least when compared to otherpossible offers in this range. The expected cost of screening the types is p(d _{L }- k _{P }) + (1 - p)(d _{H } + k _{D }) = E _{D }(d) + k _{D } - p(k _{P } + k _{D }), since the offer elicitsacceptance with probability p (the probability of L-types) and generates a trialand attendant costs with probability (1 - p). The payoff from making offersthat both types reject is E _{D }(d) + k _{D }. Finally, the expected cost from making anoffer that both types will accept is simply the cost of the offer that the H-typewill accept, namely d _{H } - k _{P }.

A comparison of the payoffs from the different possible offers that Dcould make indicates that it is always better for D to make an offer of at leastthe L-type's concession limit (that is, the value of s _{D } specified above, d _{L } - k _{P }).It may be optimal to pool the types; that is, to make an offer at the H-type'sconcession limit (d _{H } - k _{P }), which will therefore be accepted by P independentof his actual damages incurred. Finally, comparing the expected cost to D ofthe screening offer, E _{D }(d) + k _{D } - p(k _{P } + k _{D }), with the expected cost to D of thepooling offer, d _{H } - k _{P }, it is optimal to screen the types if E _{D }(d) + k _{D } - p(k _{P } + k _{D })< d _{H } - k _{P }; that is, if:

p > (k _{P } + k _{D })/(d _{H } - d _{L } + k _{P } + k _{D }). | (SSC) |

Inequality (SSC) is the __simple screening condition __ (simple because itconsiders two types only); it indicates that the relevant comparison betweenscreening the types or pooling them involves total court costs (k _{P } + k _{D }), thedifference between potential levels of damages (d _{H } - d _{L }) and the relativelikelihood of H- and L-types. Given court costs and the gap between high andlow damages, the more likely it is that P has suffered low damages rather thanhigh damages, the more likely D should be to screen the types and therebyonly rarely go to trial (and then, always against an H-type). If the likelihood offacing an H-type P is sufficiently high (that is, p is low), then it is better tomake an offer that is high enough to settle with both possible types ofplaintiff. Therefore, with pooling there are no trials, but with screening trialsoccur with probability (1 - p). Condition (SSC) also suggests that, for a givenprobability of low-damage Ps and a given gap between the levels of damages,sizable trial costs auger for pooling (that is, settling with both types of P).

Finally, the model allows us to compute the efficiency loss and torecognize its source. To see this, imagine the above analysis in the imperfectinformation setting; in particular, for this setting both P and D do not knowP's type, and they agree on the estimate of damages, E _{D }(d), and that liabilityof D for the true damages is certain. In the imperfect information version ofthe D-proposer ultimatum game, D's optimal offer is E _{D }(d) - k _{P }, P's concessionlimit under imperfect information. Since in that setting P doesn't know histype, he would settle at E _{D }(d) - k _{P } rather than require d _{H } - k _{P } (which is greaterthan E _{D }(d) - k _{P }) to avoid trial if he is an H-type. Thus, the difference in D'spayoff under imperfect information (E _{D }(d) - k _{P }) and that under the asymmetricinformation analyzed above (E _{D }(d) + k _{D } - p(k _{P } + k _{D })) is (1 - p)(k _{P } + k _{D }). Thisextra cost to D comes from the fact that D recognizes that P knows his owntype and will act accordingly. Note that this is not a transfer to P; it is anefficiency loss. This loss is a share of the surplus that, under perfect orimperfect information, would have been avoided by settling rather than goingto trial, and is a loss that is due to the presence of an asymmetry in theplayers' information.

__ 12.3.2. Screening with Many Types __

The principle used above extends to settings involving finer distinctionsamong levels of private information. In particular, this subsection will outlinethe nature of the model when applied to a continuum of types, such as aplaintiff whose level of damages could take on any value between two givenlevels of damages (that is, d may take on values between, and including, d _{L }and d _{H }; d _{L } __< __ d __< __ d _{H }). This is formally equivalent to Bebchuk's original model( Bebchuk (1984) ), even though his analysis presented a D who was privatelyinformed about liability in a P-proposer setting with known damages. Thus,differences in the presentations between this discussion and Bebchuk's aredue to the shift of the proposer and the source of private information; thereare no substantive differences between the analyses.

As always, D's probability assessment of the likelihood of the differentpossible levels of damages is common knowledge and is denoted p(d), whichprovides the probability that damages are no more than any chosen value ofd. Figure 2 illustrates the intuition behind the analysis. The distribution ofpossible levels of damages as drawn implies equal likelihood, but this is forillustrative purposes only; many (though not all) probability assessmentmodels would yield similar qualitative predictions. Figure 2 illustrates a levelof damages, x, intermediate to the two extremes, d _{L } and d _{H }, and that thefraction of possible damage levels at or below x is given by p(x). Alternativelyput, if D offers s = x - k _{P }, a P who has suffered the level of damages x wouldbe indifferent between the offer and the payoff from going to trial. Moreover,this offer would also be accepted by any P with damages less than x, whileany P with damages greater than x would reject the offer and go to trial. Theexpected expenditure associated with offers that are accepted is sp(x). Notethat the particular value x that made the associated P indifferent betweensettling and going to trial depends upon the offer: x(s) = s + k _{P }. This isaccounted for by explicitly recognizing this dependence: if D makes an offers _{D }, then the expected expenditure associated with accepted offers iss _{D }p(x(s _{D })).

Two observations are in order. First, as s _{D } increases, the "marginal" typex(s _{D }) (also known as the "borderline" type; see Bebchuk (1984) ) moves to theright and this would increase p(x(s _{D })). Thus, this expected expenditure isincreasing in the offer both because the offer itself goes up and as itincreases, so does the likelihood of it being accepted. Second, while thetypes of P that reject the offer and go to trial are "revealed" by the awardmade by J (who learns the true d and awards it), as long as x does not equald _{L } there is residual uncertainty in every possible outcome of the game: theoffer pools those who accept, and their private information is not revealed(other than the implications to be drawn from the fact that they must havedamages that lie to the left of x in Figure 2). The fact that the equilibrium willtherefore involve only partial revelation is the main difference between thetwo-type model (where screening reveals types) and the model with acontinuum of types.

To minimize expected expenditure, D trades off the expected expenditurefrom settling with the expected expenditure for trial, since the fraction (1 -p(x(s _{D })) goes to trial. Under appropriate assumptions on p(x), this latterexpenditure is declining in s _{D }, yielding an optimal offer (s _{D }*) for D that makesthe type of P represented by the level of damages x(s _{D }*) the marginal type.Thus, "P has been screened."

As an example, if all levels of damages are equally likely, as illustrated inFigure 2 above, then as long as the gap between the extreme levels ofdamages exceeds the total court costs (that is, d _{H } - d _{L } __> __ k _{P } + k _{D }), theequilibrium screening offer is s _{D }* = d _{L } + k _{D } and the marginal type is a P withlevel of damages d _{L } + k _{D } + k _{P }; Ps with damages at or below this level acceptthe offer while those with damages in excess of this level reject the offer. Notethat should the gap in levels of damages be less than aggregate trial costs (d _{H }- d _{L } < k _{P } + k _{D }), then D simply pools all the types with the offer d _{H } - k _{P }.

__ 12.3.3. Signaling: A Two-Type Analysis __ ** **

This approach employs a P-proposer model in which P makes a settlementdemand followed by D choosing to accept or reject the proposal; given theassumptions made in the discussion before subsection 12.3.1, a rejectionleads to P going to trial, at which J learns the true level of damages andawards P their value.

As discussed in Section 4, in the circumstances of this ultimatum game, Dshould use a mixed strategy: if demands at or below some level were alwaysaccepted, while those above this level were always rejected, some types of Pwould be compensated more than might be necessary and D would go tocourt more often than necessary. Here a mixed strategy should respond to thedemand made: low demands should be rejected less often than high demands,if only because a high demand is more advantageous to a greater percentageof possible types of Ps, and therefore requires D to be more diligent.

The notion that lower types (those with lesser damages) of P have anincentive to try to be mistaken for higher types (those with greater damages)plays a central role in the analysis. D's use of a mixed strategy, dependentupon the demand made, provides a counter-incentive which can makemimicry unprofitable: a greedy demand at the settlement bargaining stage,triggering a greater chance of rejection, may therefore more readily lead tomuch lower payoffs at trial (where the true level of damages is revealed withcertainty and P then must pay his court costs from the award) than wouldhave occurred at a somewhat lower demand.

P's demand is s _{P }, which D responds to by rejecting it with probabilityr _{D }(s _{P }) or accepting it, which occurs with probability 1 - r _{D }(s _{P }). Clearly, if thedemand is d _{L } + k _{D }, then D should accept this demand as D can do no betterby rejecting it. For convenience, we will define s _{L } to be this lowest-typedemand, and thus, r _{D }(s _{L }) = 0. On the other hand, if P were to make a demandhigher than what would be D's expenditure at trial associated with the highesttype, namely d _{H } + k _{D }, then D should reject any such demand for sure. It issomewhat less clear what D should do with d _{H } + k _{D }, which for conveniencewe denote as s _{H }. As will be shown below (in both the two-type and thecontinuum of types models) D's equilibrium strategy will set r _{D }(s _{H }) to be lessthan one. This will provide an incentive for greedy Ps to demand at most s _{H }(technically, this is for the benefit of specifying an equilibrium, and it turnsout not to hurt D). Since P knows what D knows, P can also construct ther _{D }(s _{P }) function that D will use to respond to any demand s _{P } that P makes. Puses this function to decide what demand will maximize his payoff.

While there are demand/rejection probability combinations that cangenerate all three types of equilibria (revealing, pooling and partial pooling),the focus here is on characterizing a revealing equilibrium. To do this we takeour cue from the appropriate perfect information ultimatum game. In thoseanalyses, if it was common knowledge that P was a high type, he coulddemand and get s _{H }, while if it was common knowledge that P was a low type,he could demand and get s _{L }. Making such demands clearly provides anaction that could allow D to infer that, should he observe s _{L } it must havecome from a low type, while if he observed s _{H }, it must have come from a high-type. While wishing doesn't make this true, incentives in terms of payoffscan, so that a low-damage P's best choice between s _{L } and s _{H } is s _{L } and a high-damage P's best choice between s _{L } and s _{H } is s _{H }. In particular, consider thefollowing two inequalities (since r _{D }(s _{L }) = 0, the following inequalities employthe notation r for the rejection probability; we will then pick a particular valueof r to be the value of D's rejection strategy, r _{D }(s _{H })):

s _{L } > (1 - r)s _{H } + r(d _{L } - k _{P }) | (ICL) |

and

s _{L } < (1 - r)s _{H } + r(d _{H } - k _{P }). | (ICH) |

The first inequality, called ICL for the __incentive compatibility condition forthe low type __ states that D's choice of r is such that when P has the low levelof damages, his payoff is at least as good when he demands s _{L } as what hispayoff would be by mimicking the high-damages P's demand s _{H }, which isaccepted with probability (1 - r), but is rejected with probability r (resulting inthe P of either type going to court). Note that, since J would learn the truetype at court, a low-type P gets d _{L } - k _{P } if his demand is rejected. In otherwords, on the right is the expected payoff to a P with the low level ofdamages from misrepresenting himself as having suffered high damages.Inequality ICH __(the incentive compatibility condition for the high type) __ has asimilar interpretation, but now it is for the high types: they are also no worseoff by making the settlement demand that reflects their true type (the expectedcost on the right side of the inequality) than they would be if theymisrepresented themselves. When r, s _{L } and s _{H } satisfy __both __ (ICL) and (ICH),then these strategies for D and the two types of P yield a revealingequilibrium.

Substituting the values for s _{L } and s _{H } and solving the two inequalitiesyields the following requirement for r:

(d _{H } - d _{L })/[(d _{H } - d _{L }) + (k _{P } + k _{D })] < r < (d _{H } - d _{L })/(k _{P } + k _{D }). |

While the term on the far right is, by an earlier assumption, greater than 1,the term on the far left is strictly less than one. In fact, for __each __ value of r (pickone arbitrarily and call it r' for now) between the value on the left and one,there is a revealing equilibrium involving the low-damages P demanding s _{L },the high-damages P demanding s _{H } and D responding via the rejectionfunction with r _{D }(s _{L }) = 0 and r _{D }(s _{H }) = r'. In the equilibrium just posited a low-damage P reveals himself and always settles with D at d _{L } + k _{D } and a high-damage P always reveals himself, possibly (that is, with probability (1 - r'))settling with D at d _{H } + k _{D } and possibly (with probability r') going to court andachieving the payoff d _{H } - k _{P }. Note that the strategies for the players are verysimple: P's demand is his damages plus D's court costs; D's strategy is toalways accept a low demand and to reject a high demand with a givenpositive, but fractional, probability.

For what follows we will pick a particular value of r in the interval, namely,let r _{D }(s _{H }) = (d _{H } - d _{L })/[(d _{H } - d _{L }) + (k _{P } + k _{D })], the smallest value. There aretechnical reasons (refinements) that have been alluded to earlier, concerningextensions of notions of rationality, to support this choice, but anothermotivation is that the smallest admissible r, r _{D }(s _{H }), is the most efficient of thepossible choices. All the r values that satisfy the incentive conditions (ICL)and (ICH) provide the same expected payoff to D, namely E _{D }(d) + k _{D }. P'sexpected payoff (that is, computed before he knows his type) is E _{D }(d) + k _{D } - (1- p)(k _{P } + k _{D })r, for __any __ r that satisfies both incentive compatibility conditions,so using r _{D }(s _{H }) minimizes the efficiency loss (1 - p)(k _{P } + k _{D })r. Note also thatusing the specified r _{D }(s _{H }) as the rejection probability for a high demandmeans that the likelihood of rejection is inversely related to total court costs,but positively related to the difference between possible levels of damages.This is because while increased court costs minimize the threat of going tocourt, an increased gap between d _{H } and d _{L } increases the incentive for low-damage Ps to claim to be high-damage Ps, thereby requiring more diligenceon the part of D. D accomplishes this by increasing r _{D }(s _{H }).

__ 12.3.4. Signaling with Many Types __ ** **

While the principle used above extends to the case of a continuum of valuesof the private information, the extension itself involves considerably greatertechnical detail. The presentation here will summarize results in much thesame manner as used in Section 12.3.2 to summarize screening with acontinuum of types. This presentation is based on the analysis employed in Reinganum and Wilde (1986) , though that model allows for non-strategicerrors (that is, exogenously specified errors) by J and awards that areproportional to (rather than equal to) damages.

The basic results developed in the two-type model remain: 1) a revealingequilibrium is predicted; 2) P makes a settlement demand equal to damagesplus D's court cost and 3) D uses a mixed strategy to choose acceptance orrejection. The likelihood of rejection is increasing in the demand made, andtherefore in the damages incurred, and is decreasing in court costs. Thismeans that the distribution of levels of damages that go to trial involves,essentially, the entire spectrum of damages, though it consists ofpreponderantly higher rather than lower damages (relative to the initialdistribution).

Figure 3 shows an example which starts with the same assessment overdamages as envisioned in the continuum screening model in Section 12.3.2.As is shown in the graph displayed in the upper left of Figure 3, againassume that each possible level of damages is equally likely. Following thegray arrow, the graph in the upper right shows the equilibrium settlementdemand function for P: it is parallel to the 45¡ line (the dotted line) and shiftedup by the amount k _{D }. Thus, P's settlement demand function s _{P }(d) = d + k _{D },where d is P's type (level of damages actually incurred). Thus, for example,s _{P }(d _{L }) = d _{L } + k _{D } (this is s _{L } on the vertical axis). The graph below thesettlement demand function (follow the fat gray arrow) displays D's rejectionfunction. Demands at or below s _{L } are accepted and demands above s _{L } arerejected with an increasing likelihood up to the demand s _{H } = d _{H } + k _{D }. This isrejected with a positive but fractional likelihood (the dot is to show theendpoint of the curve); anything higher yet is rejected with certainty. Finally,following the gray arrow to the lower left of Figure 3, the posteriorassessment of damages for cases going to trial is shown. The word posterioris used to contrast it with the assessment D used before bargainingcommenced (the prior assessment). The effect of settlement bargaining is tocreate an assessment model at the start of the next stage of the legal processwhich is shifted upwards; that is, which emphasizes the higher-damage cases.This contrasts with the resulting distribution of cases that emerge from ascreening process. The result of the screening model applied to the "box-shaped" prior assessment shown in Figures 2 and 3 would be a box-shapedposterior assessment model over the types that rejected the screening offer.

** ** *12.4. How Robust are One-Sided Asymmetric Information Ultimatum GameAnalyses ? *

As the earlier discussion of the various approaches used in perfectinformation suggests, model structure and assumptions play an importantrole in the predictions of the analysis. Is this a problem of "tune the dial andget another station?" In some sense it seems to be. Such models seem toprovide conflicting predictions which: involve proposing one or the otherplayer's concession limit (not in between, as the Nash Bargaining Solutionprovided); sometimes fully reveal private information, other times do not; andstrongly restrict when and if players can make proposals at all.

However, some consistent threads emerge. Asymmetric information willgenerally result in some degree of inefficiency in the bargaining process dueto some use of trial by the players. The extent of inefficiency is related to thenature of the distribution of the information, the range of the possible valuesthat the private information can take on and the level of court costs. Highercourt costs encourage settlement and influence the transfer between P and D.Asymmetric information means that the relatively less informed player needsto guard against misrepresentation by the more informed player, and must bewilling to employ the threat of court. The signaling model indicated anotheraspect of this: even though P was informed and made the proposals, it was Pwho bore an inefficiency cost (D's expected payoff was what it would havebeen under imperfect information). This is because the private informationthat P possesses cannot be credibly communicated to D without a cost beingincurred by P via the signaling of the information.

Clearly, both models use a highly stylized representation of bargaining.How restrictive is this? While this question is difficult to address verygenerally, some tests of the robustness of the model structure and thepredictions exist. These analyses are of two types. 1) Would changes insequence matter (who moves when, whether moves must be sequential, whatif there were many opportunities to make proposals)? 2) Is the one-sidednature of information important; would each player having information on arelevant attribute of the game affect the outcome in a material way?

Papers by Daughety and Reinganum (1993) , Wang, Kim and Yi (1994) and Spier (1992) address aspects of the first question above. Daughety andReinganum provide a two period model that allows players to movesimultaneously or sequentially. Here, P and D can individually make (orindividually not make) proposals in the first period and then choose to acceptor reject whatever comes out of the first period during a second period; arejection by either individual of the outcome of the first period means goingto court. What comes out of the first period is: 1) no proposal, whichguarantees court; 2) one proposal, provided by whomever made it; or 3) anintermediate version of two proposals if both players make one; theintermediate proposal is a general, commonly known function of the twoindividual proposals. An example of such a "compromise" function would beone that averaged the proposals. Note that this means that if both playersmake proposals then intermediate outcomes are possible candidates asequilibria of the overall game. The model allows one-sided asymmetricinformation, but examines both possible cases in which a player is informed.The general result is that players do not choose to wait: they both makeproposals in the first period. Thus, formally, the ultimatum structure whereinonly one player makes a proposal is rejected as inconsistent withendogenously generated timing. However, the unique equilibrium of the gamehas the same payoffs as either that of the ultimatum signaling game or theultimatum screening game; which one depends only on the compromisefunction used and which player is informed.

The Wang, Kim and Yi (1994) paper discussed earlier in Section 3.2.2 alsocontains a continuum-type, one-sided asymmetric information model basedon Rubinstein (1985) . Wang, Kim and Yi consider the case of an informed Pand an uninformed D, with D as first proposer. In subsequent periodsproposers alternate. They show that the settlement outcome is consistentwith a one-period D-proposer screening ultimatum game as discussed above.Finally, Spier (1992) (discussed in more detail in Section 18 below) alsoemploys a dynamic model (in this case a finite horizon model) withnegotiating and trial costs. In her model if negotiating costs are zero then allbargaining takes place in the last period. Together, the three papers providesome limited theoretical support for using the ultimatum game approach torepresenting one-sided asymmetric information settlement problems.

The second issue, concerning one-sided versus two-sided information, isaddressed in papers by Urs Schweizer (1989) and Daughety and Reinganum(1994) ( Joel Sobel (1989) also considers a two-sided model, but his interest isdiscovery; this paper will be discussed in Section 19). Both Schweizer andDaughety and Reinganum consider ultimatum games where P is privatelyinformed about damages and D is privately informed about liability. Schweizerconsiders a P-proposer model with two types on both sides while Daughetyand Reinganum consider both P- and D-proposer models with a continuum oftypes on both sides. The results are fundamentally the same: the proposersignals and uses the signal to screen the responder. Thus, proposer types arerevealed fully and responders are partially pooled.

In sum, it would appear that the screening and signaling models havereasonably robust qualitative properties that survive relaxation of some of theunderlying structure and that the intuition derived from the separate analysessurvives the integration of both types of models in a more comprehensiveanalysis.

** 13. Comparing the Two-Type Models: Imperfect and AsymmetricInformation **

This section provides two means of comparison. First, employing specificnumerical values, Table 3 below presents computations for the same datafrom imperfect, screening and signaling analyses; it also acts as a convenientsummary of the strategies and payoffs for the different models. While theresults do not purport to indicate magnitudes of differences in the predictionsmade, it will suggest directional differences. The directional differences willbe amplified, based on the two-type analyses provided earlier, as the secondmeans of comparison.

Table 3 considers a two-type model wherein P is informed of the true levelof damages and D is not. D's prior assessment on the two levels of damagesis that they are equally likely (this is to make comparisons easier). Court costsare the same for the two players and liability by D for damages is certain.Specific values of the data are provided at the top of Table 3. Note that (SSC)holds as shown. D's expectation of damages (E _{D }(d)) is the commonexpectation under imperfect information.

Table 3 concentrates on the ultimatum game predictions, but the relevantimperfect information NBS is also provided, as shown near the top. Given theinformation endowments, the only asymmetric information D-proposer modelis a screening model and the only P-proposer asymmetric information modelis a signaling model. The table provides the proposer's proposal, theresponder's strategy in the signaling case and the payoffs to the players.Note that $ _{L } provides the payoff to a P with low damages while $ _{H } providesthe payoff to a P with high damages. Finally, in the asymmetric case $ _{P }provides the expected payoffs to a P before damages are observed so that __exante __ efficiency can be evaluated. The statement "efficient" means that theoutcome is on the settlement frontier, while "inefficient" means that thesolution lies below the frontier, with the efficiency loss calculated as shown.Finally, the source of inefficiency, that some cases go to trial, is indicated byproviding the probability of trial derived from the model used.

The example and the formulas in the table indicate that the efficiencylosses predicted by the screening and signaling models, as compared withthe efficient solutions in the imperfect information model, differ from oneanother. More generally, as long as p meets the simple screening condition(SSC) of Section 12.3.1, the signaling model predicts less of an efficiency lossthan the screening model. This is because while a low-damage P settles out ofcourt in both models, a high-damages P always goes to court under ascreening model while they only go to court with a fractional likelihood underthe signaling model. Note also, however, that when p does not satisfy (SSC),then the screening model predicts full efficiency while the signaling modelstill predicts an inefficient outcome.

A similar type of comparison could be performed for the ultimatum gamesinvolving asymmetric information about the likelihood of liability (withdamages commonly known). Typically, such analyses assume that D hasprivate information about the true likelihood of being found liable. In thescreening model, the higher-likelihood Ds settle and the lower likelihood Dsreject P's offer and proceed to trial. In the signaling model, the higherlikelihood D makes an offer that P accepts while the lower likelihood D makesa lower offer that P rejects with an equilibrium rejection probability. Mostnotably, in the case wherein there is a continuum of types, the distribution ofcases that go to trial include essentially all the D-types (with thepreponderance of types being less likely to be held liable).

This is worth contrasting with a vast literature which has grown out of apaper by George Priest and Benjamin Klein (1984) . Reviewing the literature inthis area (mainly empirical studies with a variety of predictions) would take ustoo far from our main purpose, but a few words are appropriate. The Priest-Klein model employs an inconsistent priors approach to examine the selectionof cases that proceed to trial. Their results imply that, as the likelihood ofproposal rejection by parties becomes small, then among those cases thatdon't settle, the likelihood that either P or D wins at trial approaches 50%. As Shavell (1993) shows for the two-type screening case, and as is clearly alsotrue for the signaling case (as described above) and the continuum screeningcase, by varying parameters one can get essentially any prediction

about case selection that is desired.

* * **C. Variations on the Basic Models **

This Part locates and briefly reviews a number of recent contributions to thesettlement literature. Two cautions should be observed. First, no effort will bemade to discuss unpublished work. This is primarily motivated by the factthat such work is, generally, not as accessible to most readers as are thejournals that published work has appeared in. There are some classicunpublished papers (some of which have significantly influenced the existingpublished papers) that are thereby slighted, and my apologies to theirauthors. Potentially, such a policy also hastens the date of the succeedingsurvey.

Second, the selection to be discussed is a subset of the existing publishedpapers: it is not meant to be comprehensive. Instead, the selection is meant toshow ideas that have been raised, or how approaches have been revised. Alimited number of papers that address relevant issues, but which are notfocused on settlement itself, are also mentioned.

Following the outline of Part B, papers will be grouped as follows: 1)players; 2) actions and strategies; 3) outcomes and payoffs; 4) timing and 5)information. Not surprisingly, many papers could conceivably fit in a numberof categories, and a few cross-references will be made.

Alison Watts (1994) adds an attorney for P (denoted A _{P }) to the set of playersin a screening analysis of a P-proposer ultimatum game; D is privatelyinformed about expected damages at trial (for a discussion of agencyproblems in contingent fee arrangements, see Geoffrey P. Miller, 1987) . Themain role of A _{P } is expertise: A _{P } can engage, at a cost, in discovery effortswhich release some predetermined portion of D's information. The cost to A _{P }is lower than the cost of obtaining the same information would be to P.Moreover, more precise information about D's likely type costs more to obtain(for either P or A _{P }) than less precise information (precision is exogenouslydetermined in this model). Before bargaining with D, P can choose whether ornot to hire A _{P }, and attorneys are paid on a contingency basis. If hired, A _{P }obtains information about D's type and then makes a settlement proposal toD. Given the precision of obtainable information and the expertise of A _{P } (thatis, A _{P }'s cost of obtaining information as a fraction of P's cost of obtaining thesame information), Watts finds a range of contingency fees that P and A _{P }could agree upon (a settlement frontier for __P and A _{P } __ to bargain over), and thattheir concession limits decrease as the expected court award in the settlementproblem with D increases.

As mentioned earlier, J is generally modeled in this literature as learning thetruth and making awards equal to the true damages. Early models haveallowed for unsystematic error on the part of J. Since the informed playerusually computes payoffs at trial on the basis that their true type will be fullyrevealed in court, something here is needed to indicate how J learns the truetype, or if J doesn't learn the true type, what J does in that event. Thus, whatJ will know could influence the settlement strategies and outcomes.

Daughety and Reinganum (1995) consider a J whose omniscience isparametric (that is, with an exogenously specified probability, J learns thetruth; if not, J must infer it based upon P and D's observable actions) in acontinuum-type ultimatum game signaling model, wherein P is informed aboutdamages and D is not. J is a second "receiver" of a signal. If all that isobservable to J is the failure of settlement negotiation, then when J observesthat a case comes to trial, he can infer the distribution of such cases (usingthe posterior model shown in Figure 3) and pick a best award (note that thismeans that all the elements of the settlement game are common knowledge toJ, as is this fact to P, D and J). If J can also observe P's settlement demand,then he uses that information, too. The result is that this feeds back into thesettlement process, resulting in P making demands to influence J. As J'sdependence on such information increases (omniscience decreases),revelation via the settlement demand disappears as more and more types of Ppool by making a high demand (P "plays to the judge"). The result can bethat, for sufficiently high reliance on observation instead of omniscience, Jhas even less information than if he couldn't observe P's settlement demandat all (and must rely on the posterior distribution of unsettled cases).

Influencing J is also the topic of Rubinfeld and David Sappington (1987) ,which while not focused on settlement __per se __, does model how effort byplayers can inform J. The setting is nominally a criminal trial, but the pointpotentially applies to civil cases, too: if innocent Ds should be able to (morereadily than guilty Ds) obtain evidence supporting their innocence, then theamount of effort so placed can act as a signal to J of D's innocence or guilt.This is not a perfect signal, in the sense that the types of D are not fullyrevealed. As in much of the literature dealing with criminal defendants (this isdiscussed in more detail in Section 17.4 below), J here maximizes a notion ofjustice that trades off the social losses from punishing the innocent versusfreeing the guilty and accounts for the costs incurred by D in the judicialprocess.

One final note on this topic. There is an enormous literature on jury andjudicial decision-making spread across the psychology, political science,sociology and law literatures that has yet to have the impact it deserves onformal models of settlement bargaining.

Many settings involve multiple litigants (e.g., airline crashes, drug side-effects, etc.). Yeon-Koo Che and Jong Soo Yi (1993) and Bill Yang (1996) consider games with two Ps who D faces sequentially; partly these aremodels of precedent and partly these are models analyzing game-to-gameinformational links. Damages are correlated over plaintiffs and thusinformation obtained in bargaining with the first P (P _{1 }) influences bargainingwith the second P (P _{2 }). Che and Yi concatenate two D-proposer ultimatumgames, where D faces informed P's in the two games. Note that when D playsthe second ultimatum game, the correlation of levels of damages overplaintiffs means that learning in game one affects D's strategy in game two,thereby feeding back into D's game one strategy choice. By allowing for effortapplied by all parties to influence the probability of winning, Che and Yi get a"front-loading" effect to influence the precedent-setting case.

In a two-P model, the outcome of the settlement process with P _{1 } (whichcan include trial) influences P _{2 }'s decisions. Employing a model similar to thatof Che and Yi, Yang includes the decision by both informed Ps to initially filetheir respective cases. Thus, D's actions with respect to P _{1 }, and the likelyoutcome from going to trial, may deter P _{2 } from filing. Yang finds conditionsunder which this causes D, in dealing with P _{1 }, to be more or less aggressivethan the one-P model would find. While more aggressive play (being"tough") against P _{1 } may seem intuitive, less aggressive play is alsoreasonable if going to trial will reveal information (such as, that P _{1 } had highdamages) that encourages P _{2 } to file. Why would this encourage P _{2 }, whoknows his damage? Assume that P _{2 } is a low-level-of-damages plaintiff. Bymaking a pooling offer to P _{1 }, D does not learn P _{1 }'s type, which ignorance P _{2 }would be aware of. Thus, P _{2 } knows that D is still uninformed, and cannotcapitalize on D having received "bad news" that P _{1 } is a high-level-of-damagesplaintiff, shifting his prior assessment about P _{2 } up (recall that D takesdamages as correlated). There is a strategic advantage to not being informedif the knowledge of the information places you at a disadvantage. If Dremains uninformed then if P _{2 } is low he can't expect D to overestimate him ashigh and make a second pooling offer.

Bruce Kobayashi (1992) considers a two-D model in a plea-bargainingsetting; because of the change in payoffs, this will be discussed in moredetail in Section 17.4 below.

** ** *16.1. Credibility of Proceeding to Trial Should Negotiations Fail * ** **

In the screening examples in Section 12.3 above, an uninformed D made anoffer to an informed P with liability commonly known but the level of damagesthe source of the informational asymmetry. For convenience of exposition,consider the reversed setting with D informed about the likelihood of beingfound liable at trial, P uninformed, but both commonly knowledgeable aboutthe level of damages (this is the original Bebchuk example). Thus, a screeninganalysis means a P-proposer ultimatum model. In this context Barry Nalebuff(1987) examines the assumption that P is committed to proceeding to trial afterbargaining fails (alternatively put, Nalebuff relaxes the assumption that thereis a minimum positive likelihood of liability that, multiplied by the level ofdamages, would exceed P's court costs; this is the analogy to our earlierassumption that d _{L } > k _{P } holds).

Nalebuff considers cases that, initially, have "merit": E _{P }(d) > k _{P }. Nalebuffappends a second decision by P (concerning whether or not to take the caseto trial) to the P-proposer ultimatum game screening model. After observingthe response by D to the screening demand made by P, P recomputes E _{P }(d)using his posterior assessment; denote this as E _{P }(d| D's response), meaningP's expectation of the level of damages which will be awarded in court givenD's choice to accept or reject the offer. Since all types whose true likelihoodof liability implied levels of expected damages in excess of that associatedwith the demand (that is, the more likely-to-be-liable types of D) haveaccepted the demand, the collection of types of D who would reject thedemand by P are those with stronger cases (that is, those that are less likelyto be held liable). Thus, P's new expected payoff from proceeding to trial,E _{P }(d| D's response) - k _{P }, is lower than before the bargaining began (E _{P }(d) - k _{P }).The decision by P as to whether or not to actually litigate __after __ seeing theoutcome of the screening offer results in a reversal of some of the predictionsmade by the original screening model about the impact that changes in thelevels of damages and court costs will have on settlement demands and thelikelihood of trial.

For example, Nalebuff shows that the settlement demand in the relaxedmodel is __higher __ than that in the model with commitment. Why would thishappen? To see why, consider what happens in stages. When P is making hissettlement demand, he is also considering the downstream decision he will bemaking about going to trial and must choose a settlement demand that makeshis later choice of trial credible. If P can no longer be committed to going totrial with any D who rejects his screening demand, this means that he will useE _{P }(d| D's response) to decide about going to trial, and this is now heavilyinfluenced by the presence of "tough" types; those types that are "weak,"that is, who have high likelihoods of liability, have accepted P's offers. Thus,if P raises his settlement demand, some of the intermediate types will poolwith the tough types, resulting in E _{P }(d| D's response) - k _{P } > 0, making thethreat of trial credible. Thus, the effect of relaxing the assumption that Pnecessarily litigates any case that rejects his settlement demand actuallyresults in an increased demand being made.

One potential effect of a cost associated with filing a suit is to provide adisincentive for a P pursuing what is known as a "nuisance" suit. Thedefinition of what a nuisance is varies somewhat, but generally, such a suithas a negative expected value (NEV) to P; that is, E _{P }(d) < k _{P }; such a suit is notone that P would actually pursue to trial should negotiations fail (note thatthis ignores any psychic benefits that P may derive from "having his day incourt" which might make such a suit have a positive expected utility). Clearly,the reason for consideration of NEV suits is the perception that Ps can pursueNEV suits and obtain settlements: the asymmetry of information between aninformed P and an uninformed D allows Ps with NEV suits to mimic PEV suits(positive expected value suits) and extract a settlement. Two questions havearisen in this context: what contributes to the incentive for plaintiffs topursue NEV suits and what attributes of the process might reduce or eliminateit.

An example may be of use at this point. Consider a D facing three possibletypes of P, with possible damages d _{N } = 0, d _{L } = $10,000 and d _{H } = $50,000,respectively and with associated likelihoods p _{N } = 0.1, p _{L } = 0.3 and p _{H } = 0.6 (Nhere stands for "nuisance"). Further assume both P and D have court costs of$2,000. The screening equilibrium involves the firm offering $8,000 andsettling with the nuisance and the low-damages type and going to courtagainst the high-damages type: the nuisance-type benefitted from thepresence of the low-damage type and the expected costs to the firm are higher(an average expected cost of $34,400 per case) than would obtain if thenuisance type had not been present (an average expected cost of $33,600 percase).

P'ng (1983) , in one of the earliest papers to consider strategic aspects ofsettlement bargaining, endogenizes both the choice by P to file a case and thechoice to later drop the case should bargaining fail; NEV suits are considered,but the level of settlement is exogenously determined in this model. Rosenberg and Shavell (1985) found NEV suits can occur if filing costs for Pare sufficiently low when compared with D's defense costs and if D mustincur these costs before P must incur any settlement or trial costs. Bebchuk(1988) provides an asymmetric information (P is informed), D-proposerultimatum game that specifically admits NEV suits (that is, no assumption ismade that all suits are PEV). Filing costs are zero. Bebchuk shows how courtcosts and the probability assessment of the possible levels of damagesinfluences settlement offers and rates. He finds that when NEV suits arepossible, a reduction in settlement offers in PEV cases and an increase in thefraction of PEV cases that go to trial, when compared with an analysisassuming only PEV suits, is predicted.

Avery Katz (1990) considers how filing and settlement bargaining affectsincentives for frivolous law suits. By a frivolous suit, Katz means one withzero damages and positive court costs; the numerical example above involveda frivolous suit. Katz appends a filing decision made by an informed P to a D-proposer ultimatum game. This is a two-type model (the paper also includes acontinuum-type extension) with d _{L } = 0. The modification of the D-proposermodel is that the offer is either the high-damage P's concession limit or zero,so the strategy for D becomes the probability of making the high-damageconcession limit offer. P's filing strategy is whether or not to file; if he files Pincurs a cost f _{P }. Of course, once incurred this cost is sunk. Under theassumption that d _{H } - k _{P } - f _{P } > 0, and if a condition similar in notion to (SSC) isviolated, then in equilibrium both types of Ps file cases, D pools the Ps byoffering the high-damage P's concession limit, d _{H } - k _{P }, to all Ps and all Psaccept. Katz also addresses a case selection issue: such a policy by D willattract frivolous cases, changing the likelihood that a randomly selected caseis a frivolous. Thus, the profits from filing a frivolous case will be driven tozero. In this "competitive" equilibrium, Katz shows that the likelihood of trialis not a function of p or k _{D }, but it is increasing in f _{P } and decreasing in E _{D }(d).

Using an imperfect information model, Landes (1994) shows thatcounterclaims (suits filed by D against P as part of the existing action by Pagainst D, rather than filed as a separate lawsuit) do not always reduce P'sincentive to sue, and may (by raising the stakes in the game) actually increasethe likelihood of the case going to trial. This is based on mutual optimismabout each player's own claim (mutual optimism involves each playerexpecting to win the action that they initiated; this need not involveinconsistent priors, as discussed in Section 6). Under these conditions thecounterclaim reduces the size of the settlement frontier (and possiblyeliminates it).

Most of the earliest analyses allowed for risk aversion by assuming thatpayoffs were in utility rather than monetary terms. The Nash BargainingSolution can be applied to such problems (now all four axioms come intoplay), again yielding a unique solution, though not necessarily where the 45¡-line crosses the frontier. The solution is efficient (due to axiom 2; see Section8.2) in perfect and imperfect information cases. The divergence of the solutionfrom the 45¡-line reflects the relative risk aversion of the two players, with themore risk averse player receiving a smaller share of the pie (see, for example, Binmore (1992) , pps. 193-4).

There is a similar analysis in the perfect information strategic bargainingliterature, where risk is introduced into an infinite horizon game by ignoringthe time value of money but incorporating a probability of negotiationsbreaking down. Once again, for players whose preferences over outcomesreflect aversion to risk, the less risk averse player gets the greater share of thepie (see Binmore, Rubinstein and Asher Wolinsky (1986) ).

In the settlement context, Amy Farmer and Paul Pecorino (1994) view aplayer's risk preferences as private information. While trial outcomes areuncertain, the likelihood of the outcomes themselves is common knowledge.P is taken to be risk averse (that is, privately informed of his risk preferences)and D is risk neutral and uninformed; the model allows for two types of P(extension to three types is also considered). This is a D-proposer ultimatummodel; if the roles of proposer were reversed, then P's risk aversion would notinterfere with an efficient settlement solution, so the order here is crucial toobtaining the possibility of trial. A standard screening condition is found(not unlike (SSC)), but more interesting is the result that increases in theuncertainty of the trial outcome result in the screening condition being morereadily met, thereby increasing the likelihood of trial (a result consistent withthe earlier analysis involving risk aversion). This occurs because it is themost risk averse Ps that settle, and the greater the uncertainty the more theyare prepared to accept a settlement in lieu of court, which encourages D tomake tougher offers.

** ** *17.2. Offer-Based Fee Shifting Rules * ** **

Many settlement papers consider the allocation of court costs (fee shifting)as part of their overall analysis. Typically, comparisons are made between the"American" system (each player pays their own costs) and the "British"system (the loser pays all costs). The very common association in theliterature of loser pays with Britain potentially understates the contrast; seePosner, 1992 who uses the term "English and Continental" to emphasize thata significant portion of the world uses loser pays. All the discussions in thissurvey have employed the pay-your-own system. The allocation of courtcosts is an extensive topic, with a typically suggested result that the loser-pays system discourages low-probability-of-prevailing plaintiffs more thanthe pay-your-own system (see Shavell (1982) ), but other observations are thatit may (or may not) adversely affect the likelihood of settlement (see Bebchuk(1984) and Reinganum and Wilde (1986) ). A recent extension of thebasic fee-shifting discussion to making fee-shifting dependent upon themagnitude of the outcome is discussed in Bebchuk and Howard Chang(1996) . Since the general area of fee shifting is a separable topic of its owninterest (which is likely to be addressed in a number of other surveys in thesevolumes), this survey will not attempt to cover it.

A related issue is recent work on Rule 68 of the U.S. Federal Rules of CivilProcedure, as an example of a variety of __offer-based __ fee-shifting rules whichdirectly address settlement offers made by defendants and rejected byplaintiffs. First, it should be noted that, under long-standing practice, andalso under many state rules of evidence and U. S. Federal Rule of Evidence408, information on settlement proposals and responses is not generallyadmissible as evidence at trial; a similar type of restriction usually applies incriminal cases to information about plea bargaining. Rule 68 includesrestrictions on the use of settlement proposals at trial.

Thus, in the case of offer-based fee-shifting, offers are not used in courtto infer true damages or actual liability; rather they influence the final payoffsfrom the game __after __ an award has been made at trial. Rule 68, for example, linkssettlement choices to post-trial outcomes by penalizing a plaintiff for certaincosts (court costs and, sometimes, attorney fees) when the trial award is lessfavorable than the defendant's "final" proposal (properly documented). As Spier (1994a) points out, the stated purpose of such a rule is to encouragesettlement (Spier also provides other examples of offer-based rules similar innature to Rule 68). Spier employs screening in a D-proposer ultimatum game.In comparison to the likelihood of settlement without Rule 68, she finds thatunder Rule 68: 1) disputes by P and D over damages are more likely to settle;and 2) disputes over liability or the likelihood of winning are less likely tosettle. Spier also finds that the design of a bargaining procedure and fee-shifting rule that maximizes the probability of settlement yields a rule thatpenalizes either player for rejecting proposals that were better than the actualoutcome of trial, providing some theoretical support for offer-based feeshifting rules such as Rule 68.

In previous sections of this summary the award at trial has been the level ofdamages associated with the plaintiff who goes to court. This, minus courtcosts, becomes P's threat. This is based on J choosing an award that bestapproximates P's damages. Other criteria for choosing awards are alsoreasonable. For example, J might choose awards that maximize overall socialefficiency or that minimize the probability of trial.

Polinsky and Che (1991) study decoupled liability: what the defendantpays need not be what the plaintiff receives. By decoupling, incentives forplaintiffs to sue can be reduced, while incentives for potential injurers toimprove the level of care can be increased; that is, both goals can be pursuedwithout necessarily conflicting. In particular, they show that the optimalaward to P may be less or more than the optimal payment by D. Spier (1994b) considers coupled awards in an asymmetric information setting, and findsthat the level of settlement costs influences the nature of the award thatminimizes social cost (precaution costs plus litigation costs plus harm). Notethat, here, precaution is one-sided: precaution on the part of a potential P isnot included. Spier considers a two-type screening D-proposer ultimatumgame and allows for two awards, a _{H } and a _{L }, for circumstances where the levelof damages is High or Low, respectively, and thus the payoff from trial is theaward minus court costs. Spier uses a condition such as (SSC) and showsthat, if total court costs are low enough the socially optimal award is equal tothe level of damages __plus __ P's court costs (as D will make a screening offer inthose circumstances), while if total court costs are sufficiently high, theoptimal award is the __expected __ damages plus P's court costs (as D will bemaking a pooling offer). Therefore, simply compensating for actual damagesis not socially optimal (recall also that court costs are fixed). Moreover, "finetuning" the award to reflect P's actual level of damages is socially optimalonly if total court costs are not too high.

** ** *17.4. Other Payoffs : Plea Bargaining *

As an example of a significantly different payoff measure, considernegotiations between a defendant in a criminal action and a prosecutor. Thistype of settlement bargaining, called plea bargaining, has been addressed in anumber of papers over the last quarter-century. Early papers in this area are Landes (1971) and Gene Grossman and Michael Katz (1983) who modelsettlement bargaining between a prosecutor and a defendant. In Landes thepayoffs are expected sentence length for the prosecutor versus expectedwealth (wealth in two states: under conviction and under no conviction) forthe defendant, who is guilty. In Grossman and Katz some defendants may beinnocent; the defendants know their guilt or innocence (a two-type model).The defendants seek to minimize the disutility of punishment (they are riskaverse) while the uninformed prosecutor maximizes a notion of justice thattrades off the social losses from punishing the innocent versus freeing theguilty; the Grossman and Katz analysis is a screening model with innocentdefendants choosing trial.

More recent work includes Reinganum (1988) , (1993) and Kobayashi(1992) . Reinganum's 1988 paper involves two-sided asymmetric information:defendants know their guilt or innocence (two types), while the prosecutorknows the strength of the case; that is, the probability that the case will yielda conviction at trial (a continuum of types). Guilt and evidence are correlated,so there is a relationship between the two sets of types. A special case of thisrelationship appears in Grossman and Katz and was also employed inRubinfeld and Sappington, discussed in Section 15.2: innocent defendantscan more readily obtain supporting evidence than can guilty ones. Theprosecutor's payoff is social justice minus resource costs while thedefendant's payoff to be minimized is the expected sentence plus thedisutility of trial. Reinganum's 1993 paper uses a similar payoff for D but takesall Ds as guilty and therefore takes the prosecutor's payoff as expectedsentence length minus resource costs. In this latter model the probability ofconviction is determined in the equilibrium, influencing the initial choice toengage in criminal behavior by D.

Finally, Kobayashi considers conspiracies: there are two defendants (forexample, a price-fixing case) who face different (exogenously determined)initial probabilities of conviction based on the existing evidence. Each D can,however, provide information on the other D which increases that second D'slikelihood of conviction. Kobayashi assumes that the D with the higher initialconviction probability (the "ringleader") also has more information on theother D (the "subordinate"). Here the prosecutor makes simultaneous offersto each D so as to maximize the sum of the expected penalties from the twoDs. Litigation costs are taken to be zero so as to focus on plea-bargaining asinformation gathering.

Four papers have focused especially on the implications of changing thetiming assumption in the models used: Spier (1992) , Daughety andReinganum (1993) , Wang, Kim and Li (1994) and Bebchuk (1996) . The papersby Daughety and Reinganum and by Wang, Kim and Yi were discussed inSection 12.4 above. These two papers (along with Spier's) analyzed modelsthat contributed some support for the one-sided asymmetric informationultimatum games. The papers by Spier and Bebchuk examine what happens ifthe settlement interval is subdivided. We consider these two papers in moredetail in turn.

Spier considers a finite-horizon concatenation of P-proposer ultimatumgames, with D informed about damages for which he is liable (a continuum-type model). Spier finds a "deadline effect" in which some cases settle in thelast period, and that the distribution of settlements over time can be U-shapedin the sense that some cases settle immediately, some settle in the lastpossible period and a few settle in between. Spier's main analysis is differentfrom the other multiperiod models discussed in that the horizon is finite, notinfinite, and the model does not allow alternating offers (an infinite horizonmodel is also considered; see below). Moreover, P, the (initially) uninformedproposer, cannot choose to go to court during the horizon. P incurs twocosts: 1) each extra period incurs a negotiating cost and 2) going to trialincurs a court cost. D incurs neither cost, which is not a restrictiveassumption in this analysis. Both P and D discount money in future periodsat the same discount rate. Thus, the analysis involves subdividing thebargaining period into a sequence of periods and associating a delay cost foreach period that settlement is not reached. Since the pie is not shrinking, thedelay cost provides a clear incentive to P to settle sooner. On the other hand,P is uninformed and may need to use the approach of the end of the horizonto get D to reveal information. This trade off leads to some settlements beingmade immediately and some being made in the last possible period when Dfaces the imminent possibility of trial. Spier also considers an infinite-horizonextension, where P may now choose to go to court in each period; thisprovides a model that allows for an endogenous date for trial. The modelyields a range of equilibria (this is not unusual in strategic bargaining gameswith outside alternatives); the range runs from efficient to fully inefficient (allcases go to trial in the second period). Finally, Gary Fournier and ThomasZuehlke (1996) have recently tested the predictions of Spier's finite horizonmodel with data from a survey of civil lawsuits from 1979-1981 in U.S. federalcourts. They found results that were consistent with computer simulationpredictions based on Spier's analysis.

Bebchuk (1996) also examines the effect of subdividing the settlementinterval into parts. Costs accrue somewhat differently in his model than inSpier's. In particular, a player's costs of negotiation and court are fullysubdivided by the number of intervals. Thus, the level of costs incurred isinfluenced by when an agreement is reached. Bebchuk concatenates a seriesof settlement predictions using the average of the (one-period) concessionlimits under perfect information; filing costs for P are zero. Sequentialrationality does the rest: with costs subdivided so as to make it credible for Pto continue to bargain, Bebchuk finds that NEV suits can be successful ifthere are a sufficient number of periods and the difference between k _{P } and k _{D }is not too great. Note that D cannot commit not to respond to demands; if hecould, then P's NEV strategy would be unprofitable.

While the number of periods in the model is exogenously set, this issuggestive of a strategy for P: increase the number of settlement periods. Thenotion of credible commitments should work both ways, however. What if Dcould hire a busy lawyer? Delegation of bargaining authority or therequirement for advice, if credible, is one way for D to make the number ofperiods smaller. After all, each time P makes a proposal, D must have anattorney carefully evaluate the proposal, a costly and time-consumingactivity. This would create credible delay, which is beneficial from theviewpoint of D's direct monetary interests and now beneficial from theviewpoint of repelling some NEV cases. Thus, this suggests that a gamewherein the proposal/response period length and frequency is endogenouslydetermined would be of interest. To the degree that the overall horizon is sub-divided, however, Bebchuk's main point will still hold: there will be NEV suitsthat could be successful (again, assuming that filing costs are zero).

** ** *19.1. Acquiring Information from the Other Player : Discovery andDisclosure *

Shavell (1989) examines the incentives for informed players to voluntarilyrelease private information in a continuum-type, informed-P, D-proposerultimatum game. Before D makes a proposal, P can costlessly reveal hishidden information to D; he may also choose to stay silent. Shavell showsthat silence implies that P's information involves low types (for example, thatP's level of damages is low). Shavell considers two possibilities: claims by Pare costlessly verifiable by D or some types of P cannot make verifiableclaims; for convenience, call the first analysis an __unlimited verifiable claims __(UVC) model and the second a __limited verifiable claims __ (LVC) model (in theLVC model those types of P unable to make verifiable claims is anexogenously specified fraction u). In the UVC model, all Ps whose true type isless than or equal to a given value stay silent while all those above that valuemake their claims. Since the claims are verifiable, D settles with those types byoffering their concession limit and settles with the silent types by offering asettlement offer designed to reflect this group's types. Thus, there are notrials in equilibrium. Under discovery rights for D that provide mandatorydisclosure, all types of P reveal, resulting in a reduced expenditure for D: eachtype of P settles for their concession limit.

In the LVC model there will be trials. The reason is that the silent Ps nowinclude those types who cannot make verifiable claims; some of these willhave higher levels of damages than the offer made to the group of silenttypes, and will thus reject the offer and then go to trial (this relies on theassumption that u is independent of the level of damages). The rest of thesilent types will settle, as will those whose claim is both greater and verifiable.Discovery now means that D can settle with (1 - u) of the possible types of Pat their concession limit and must screen the silent types, all of whom haveunverifiable claims and, thus, some of whom will proceed to court (if thecontinuum-type version of (SSC) holds). Again, total D expenditures willgenerally be reduced from the original LVC payoff. Mandatory disclosure inthe LVC case raises the screening offer for the silent group, since those lowertypes with verifiable claims have settled at their concession limit. Thus, theprobability of trial will be reduced relative to the original LVC outcome.Moreover, those with verifiable claims would have settled with or withoutmandatory disclosure.

Sobel (1989) provides a two-sided asymmetric information model thatexamines the impact of discovery and voluntary disclosure on settlementoffers and outcomes; significantly, discovery generates costs and this affectsresults. He sandwiches one of two possible discovery processes between aninitial D-proposer ultimatum game and a final P-proposer ultimatum game.Settlement in the D-proposer model ends the game, while rejection leads tothe possibility of either mandatory discovery or no discovery of D's privateinformation by P. This is then followed by the P-proposer ultimatum game.The cost of disclosure to D is denoted c. In a voluntary disclosure setting,D's choice to disclose is costly, and therefore might be used by D to signalthat the information was credible. By making P the final proposer, P is able toextract all the surplus from settlement. Thus, D has no reason to voluntarilydisclose information if c > 0. This contrasts with results, for example, by PaulMilgrom and John Roberts (1986) who model costless voluntary disclosureand find that such disclosure can be fully revealing. As Sobel observes, thissuggests that such a conclusion is sensitive to the assumption that c = 0.Sobel also finds that mandatory disclosure reduces the probability of trial andmay bias the selection of cases that go to trial, generating a distribution inwhich P wins more often.

Cooter and Rubinfeld (1994) use an analysis based on an axiomaticsettlement model with prior assessments that may be inconsistent; discoverymay or may not eliminate inconsistency. For example, discovery may reveal aplayer's private information or it may cause a player to adopt an alternativeperspective about what may come out of trial. Either way, a NBS is applied toa settlement frontier adjusted by the difference in expected trial payoffs. Oneof the main results is the proposal of an allocation of discovery costs so as toprovide disincentives for abuse by either player. The proposed allocationassigns discovery costs to each party up to a switching point, at which pointincremental costs are shifted to the requesting player.

** ** *19.2. Acquiring Information from Experts * ** **

As discussed in Section 15.1, Watts (1994) considers a model with an agentthat provides expertise in the sense that they can acquire information for aplayer more cheaply than the player can themselves. She shows how to viewthe problem as one of bargaining between the player and the agent andprovides some comparative statics about their settlement frontier.

In Daughety and Reinganum (1993) a model allowing simultaneous orsequential moves by both players (this is discussed in more detail in Section12.4) is embedded in a model which allows uninformed players to acquireinformation from an expert before settlement negotiations begin. Theinformation, which is costly, is what an informed player would know. Thus, aplayer may start the game already informed (called __naturally informed __) or startuninformed but able to acquire the information at a cost c > 0; forconvenience, assume that court costs are the same for the two players anddenote them as k. If both players are uninformed then, in equilibrium, neitherwill choose to buy the information. This is because informational asymmetryresults in some possible cases going to trial while symmetric uncertaintyinvolves no trials. With one of the players naturally informed and oneuninformed, then as discussed earlier, depending on the form of thecompromise function, the equilibrium involves either payoffs consistent withan ultimatum screening model or payoffs consistent with an ultimatumsignaling model. In those conditions which lead to the screening gamepayoffs, the uninformed player will choose to buy the information if c __< __ k.Alternatively, in those conditions which lead to signaling game payoffs, theuninformed player will __not __ acquire the information. Thus, uninformed playerswill not always choose to "re-level the playing field" by purchasinginformation; signaling will provide it if a revealing equilibrium is anticipated.

**D. Conclusions ** * *

The modeling of settlement bargaining has been influenced primarily by law,economics and game theory. In many ways it is still developing andexpanding, and hopefully deepening. The more recent analyses employ,primarily, a mix of information economics and bargaining theory (bothcooperative and non-cooperative) to examine, understand and recommendimprovements in legal institutions and procedures.

There has been a tug-of-war between the desire to address interestingbehavior and the current limited ability to relate seemingly irrational acts torational choice. As the development of technique has progressively allowedthis to be accomplished, and as the intuition as to why seeming irrationalitymay be rational has driven improvement in technique, a broader picture ofwhat elements contribute to, or impede, dispute resolution has evolved.

In this area, I think, it is fair to claim that issues have led techniques, agood thing. If there is an aspect that could use improvement, it is the factthat, to date, there are few empirical or laboratory studies of the details of thesettlement process. In an area where 93% of the outcomes are partially ortotally unobservable by researchers, empirical studies are hard to do, and thefew that have been done have undoubtedly involved hard work. Thedevelopment of improved summary data sources, some available on theInternet, is very exciting, but more studies of settlement bargaining at theactual process level (as in, for example, Farber and White) would also be mosthelpful.

Laboratory studies (experimental economics and related efforts insociology and psychology) are expanding but the more subtle predictions ofsome of these models means that laboratory studies have to walk a fine linebetween being a test of a particular model's prediction or ending up mainlygauging a subject's IQ. Such studies are also very labor-intensive (on the partof the researcher), though the recent increased entry of researchers into thisarea bodes well.

Most of the work in this area (covering the last quarter century) hasoccurred in the last dozen years (and most of that has occurred in the lasthalf-dozen years), indicating an accelerating interest and suggesting that thenext survey will have a lot more new, useful theory and detailed empirical andlaboratory tests to report, a good thing, too.

Support from NSF grants SBR-9223087 and SBR-9596193 is gratefullyacknowledged. I thank Jeremy Atack, A. Mitchell Polinsky, Richard Posner,Robert Rasmussen, David Sappington, Steven Shavell, John Siegfried andKathryn Spier for helpful comments and suggestions on (and corrections of)an earlier draft. I especially thank Jennifer Reinganum for many helpfulcomments and discussions.

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© Copyright 1997 Andrew F. Daughety