Pretrial bargaining with asymmetric information: unilateral versus bilateral payoff relevance.

Farmer, Amy ; Pecorino, Paul

1. Introduction

Asymmetric information is a leading explanation for bargaining failure, and the role of asymmetric information has been extensively analyzed in the civil litigation literature. (1) Most of this literature concerns information that has bilateral payoff relevance in the sense that the information in question affects the expected payoff at trial of both the plaintiff and defendant. When there is bilateral payoff relevance, trials may be predicted in the equilibrium of the bargaining game, regardless of whether the informed or uninformed party makes the offer. In this article, we analyze information that has unilateral payoff relevance, meaning that it affects the expected payoff of one of the two parties to the dispute, but not the other. As an example, suppose the plaintiff has private information on her risk preferences. This affects her expected payoff at trial but not the defendant's. When there is unilateral payoff relevance, there are never inefficient trials in the equilibrium of the game where the informed party makes the offer. (2) However, there may still be costly disputes in the equilibrium of the game in which the uninformed party makes the offer.

Examples of information with bilateral payoff relevance include information on the probability that the plaintiff will prevail at trial, and information on the amount of the judgment to be awarded at trial in the event the plaintiff is victorious. In both cases, the information clearly affects the expected payoff of both the plaintiff and defendant at trial, and these are the types of informational asymmetry considered most often in the literature. What type of information has only unilateral payoff relevance? One example, mentioned above, is information on risk preferences. Farmer and Pecorino (1994), Swanson and Mason (1998), and Heyes, Rickman, and Tzavara (2004) all examine asymmetric information on risk preferences within the context of the screening model in which the uninformed party makes the offer. (3) In each case, trials are predicted in the equilibrium of these models, as long as trials are not too costly for the participants. However, other types of information also have only unilateral .payoff relevance. A large literature on the ultimatum game has provided convincing evidence that under certain circumstances individuals will express a preference for being treated fairly. (4) This preference may be expressed specifically by the percentage of her own court costs a player is willing to concede to her opponent via his settlement offer. (5) Since this preference is not observable and presumably differs across individuals, the proposer in a screening game must decide how much of the joint surplus of settlement he will attempt to extract without knowledge of how low he can go before triggering a rejection.

Suppose the judgment is $100,000 and the plaintiff's court costs are $30,000. A plaintiff without a taste for fairness will accept an offer of $70,000 rather than proceed to trial. A plaintiff with a taste for fairness might (as an example) accept no less than $90,000. The defendant then must choose between an offer of $90,000 and $70,000 without knowledge of whether the other party has a taste for fairness or not. The defendant will make the low offer if he believes it will be accepted with a sufficiently high probability; if the plaintiff does indeed have a taste for fairness, she will reject the low offer and a trial will result. Presumably a mechanism along these lines explains the persistence of disputes in ultimatum game experiments.

Another example of information with unilateral payoff relevance is the degree of litigiousness, which can be modeled as differences in perceived court costs on the part of the plaintiff (Eisenberg and Farber 1997). A litigious plaintiff perceives lower trial costs than a nonlitigious plaintiff who may incur psychological or other intangible costs from pursuing trial. Since the degree of litigiousness is not directly observable, the defendant would have to choose between a low offer that only the nonlitigious would accept and a higher offer acceptable to both plaintiff types. Also note that the plaintiff's degree of litigiousness does not affect the defendant's payoff at trial.

A fourth example concerns self-serving bias. Individuals suffering from a self-serving bias may interpret the facts of a case in a way which is favorable toward themselves. This phenomenon has been documented in the experimental literature, and an excellent survey of this literature is provided by Babcock and Lowenstein (1997). The extent of an individual's self-serving bias is not directly observable. Moreover, if the plaintiff has a self-serving bias, this affects her perceived payoff from trial but not the payoff for the defendant. Self-serving bias has been addressed in theoretical models by Farmer and Pecorino (2002), Bar-Gill (2006), and Langlais (2008). Langlais models the extent of self-serving bias as a form of asymmetric information that can lead to trials in equilibrium. As with the work on risk preferences, this is done in a model in which the uninformed player makes the offer.

In each of the examples above, the information in question directly affects the payoff of the individual who holds the information. (6) For example, the plaintiff's risk preferences affect the plaintiff's expected payoff at trial but not the defendant's. When unilateral payoff relevance takes this form, we find that disputes can occur when the uninformed party makes the offer but not when the informed party makes the offer. When there is a two-sided informational asymmetry, where each piece of information has only unilateral payoff relevance, the solution to the game involves only screening elements. When there is bilateral payoff relevance, the corresponding solution involves both signaling and screening elements. As we will demonstrate, solving a model with a two-sided asymmetry is much easier under unilateral payoff relevance than under bilateral payoff relevance. (7) In addition, in this setting disputes may occur regardless of which party makes the offer.

Our last example differs from those above, because it involves a situation where the defendant holds information that affects the plaintiff's expected payoff at trial but not his own. In particular, the defendant may know whether the plaintiff will incur high costs or low costs in enforcing a judgment should she prevail at trial. For example, the defendant may know how difficult it will be for the plaintiff to uncover the defendant's assets so as to force payment of the judgment. Kaplan and Sadka (2008) analyze data from Mexican labor courts and find that many plaintiff awards go uncollected because of enforcement costs. (8) This suggests that enforcement costs are both uncertain and potentially large. When the defendant has information that affects the plaintiff's payoff but not his own, we find that there will be 100% settlement regardless of who makes the offer.

It should be clear from this discussion that there are a wide variety of circumstances in which the distinction between unilateral and bilateral payoff relevance is of importance. Furthermore, the full implications of unilateral payoff relevance are not well understood in the literature. In particular, although models have been worked out for several of the examples mentioned, these typically have the uninformed party making the offer. The models in which the informed party makes the offer have not been analyzed with one exception. In a nontechnical discussion of the model with asymmetric information on risk preferences, Daughety (2000, p. 145-6) notes that trial will not occur if the informed party makes the offer. Thus, Daughety is the first to make this point in regard to the model with asymmetric information on risk preferences. We provide a more formal analysis and extend this insight to an entire information class, that is, to the class of information with unilateral payoff relevance. In addition, we also consider informational structures, which (to our knowledge) have not been previously analyzed in a model in which there is unilateral payoff relevance. These include two-sided informational asymmetries and a model in which private information is held by one player but affects only the payoff of the other player.

2. Some Preliminaries

In most of the literature on pretrial bargaining, private information has bilateral payoff relevance. To fully understand the implications of unilateral payoff relevance, we will cover a number of cases in what follows. Here we introduce a relatively simple but general notation that characterizes all the cases we discuss. To keep the analysis simple there will be no more than two player types for both the plaintiff and defendant. The plaintiff's expected payoff at trial is denoted [PI], and the defendant's expected cost at trial is denoted C. Note that we are defining C to be the defendant's total expected cost at trial (inclusive of the expected judgment), and not simply court costs as is often the case in this literature.

A plaintiff may either have a high expected payoff at trial [[PI].sup.H], or a low expected payoff [[PI].sup.L] > 0. (9) These are referred to as type H and type L plaintiffs, respectively. In some of our games there will be two defendant types. Those with a high expected cost at trial [C.sup.H] will be called type H defendants, and those with a low expected cost [C.sup.L] will be referred to as type L. In the case of bilateral payoff relevance, there is a 100% correlation in player types. In other words, when the plaintiff is type H and therefore expects a high payoff at trial, this implies that the defendant is also type H and has a high expected cost at trial. Similarly, a type L plaintiff implies a type L defendant under bilateral payoff relevance. Note that a player need not know his or her own type but at a minimum will know the unconditional distribution of both player types.

When there is unilateral payoff relevance, player types are independent of one another. For example, in one version of the model discussed below, the plaintiff's expected payoff may either be [[PI].sup.H] or [[PI].sup.L], but the defendant's expected cost is always C. The expected payoffs and costs are measured in monetary terms, but the framework is flexible enough to capture elements such as risk aversion. If we interpret the asymmetric information to be over risk preferences, then we will interpret [[PI].sup.H] to be the expected payoff of a risk neutral plaintiff and [[PI].sup.L] to be the certainty equivalent that a risk-averse plaintiff is willing to accept rather than proceed to a risky trial. When there is two-sided asymmetric information, similar interpretations may be given to [C.sup.H] and [C.sup.L], where [C.sup.L] is the expected cost of a risk-neutral defendant. In the Appendix, we explicitly show how [[PI].sup.L] and [C.sup.H]can be computed as the certainty equivalents of trial.

In the model with bilateral payoff relevance, the payoff received by the plaintiff at trial is equal to the cost incurred by the defendant net of the trial costs of both parties to the dispute. This setting naturally gives arise to the parameter restrictions [C.sup.L] > [[PI].sup.L] and [C.sup.H] > [[PI].sup.H], which we assume hold in the model with bilateral payoff relevance. (10) With unilateral pay off relevance, we will consider the restriction C > [[PI].sup.H] to be the "usual" case. However, violations of this condition are possible under certain circumstances. For example, if the type H plaintiff is risk loving rather than risk neutral (and assuming a sufficiently low cost of trial), we may have [[PI].sup.H] > C. (11) This could also arise if the plaintiff exhibited an extreme degree of litigiousness such that she experienced a negative cost of proceeding to trial. Finally, this inequality could arise if the plaintiff were a repeat player with an eye on future litigation. For example, if the plaintiff is pursing intellectual property violations, the publicity of a trial might be valued positively if it discouraged future possible violators.

For the model with unilateral payoff relevance we will assume that C > [[PI].sup.L] when there is one defendant type and that [C.sup.H] > [[PI].sup.L] when there are two defendant types. When a result requires stronger conditions than these, we will state the condition explicitly and discuss what happens when the needed condition fails to hold. In the absence of these minimal conditions trials are efficient in the sense that the sum of the payoffs of the plaintiff and defendant are higher at trial than when the case settles prior to trial. When trials are efficient, all plaintiffs proceed to trial even under complete information because of the absence of a contract zone between the defendant and both plaintiff types. By contrast, when the conditions stated above hold, trials are inefficient in the sense that the sum of the payoffs of the two parties are higher under settlement than under a trial. In this article, we are primarily concerned with inefficient trials which arise as the result of asymmetric information.

Given below is a general form of the game that is flexible enough to capture the individual cases analyzed in this article. The stages of the game are as follows:

0. Nature determines the general structure of the game to be played. This includes whether information has unilateral or bilateral payoff relevance and the identity of the player who makes the games only settlement offer.

1. Both player types are determined. The information each player has regarding their own type and the other player's type is also determined.

2. A single offer O is made by one of the players.

3. The player receiving the offer chooses to accept or reject it. If the offer is accepted, the game ends. The plaintiff receives a payoff of O and the defendant receives a payoff of--O. If the offer is rejected, a trial occurs.

4. The payoffs at trial are determined by each player's type.

We will first review some standard results in games where information has bilateral payoff relevance and then proceed in sections 4 and 5 to analyze games where information has unilateral payoff relevance.

3. The Model with Bilateral Payoff Relevance

We will first analyze models with bilateral payoff relevance (determined by Nature in stage 0) so that we can later compare these standard results to those derived from a model with unilateral payoff relevance. In this model there is a 100% correlation between the plaintiff and defendant types. We analyze both the game where the uninformed party makes the offer and the game where the informed party makes the offer. The game in which the uninformed party makes the offer is a simplified version of Bebchuk (1984), and the game in which the informed party makes the offer is a simplified version of Reinganum and Wilde (1986). These results are well known, and an excellent discussion can be found in Daughety (2000).

The plaintiff's expected payoff at trial is [[PI].sup.H] with probability q and [[PI].sup.L] with probability 1 - q. A type H plaintiff is always associated with a type H defendant, and a type L plaintiff is always associated with a type L defendant. Also, we assume that [C.sup.H] > [[PI].sup.H], and [C.sup.L] > [[PI].sup.L]; this reflects the legal expenditures incurred at trial by both parties to the dispute. The defendant does not directly observe whether he is type H or type L, but he knows the prior probability q that he is type H.

The Uninformed Party Makes the Offer

In this case, we assume that the defendant makes an offer [O.sub.D] to the plaintiff, which the plaintiff chooses to accept or reject. If the offer is accepted, the game ends with the plaintiff receiving a payoff of [O.sub.D] and the defendant paying a cost of [O.sub.D]. If the offer is rejected, trial occurs. If the plaintiff is type H, her expected payoff is [[PI].sup.H], and the defendant's expected cost is [C.sub.H] > [[PI].sup.H]. If the plaintiff is type L, her expected payoff is [[PI].sup.L], and the defendant's expected cost is [C.sup.L] > [[PI].sup.L].

The plaintiff will accept any offer that leaves him at least as well off as the expected outcome at trial. In other words, a type i plaintiff will accept any offer such that [O.sub.D] [greater than or equal to] [[PI].sup.i], i = H, L. In making his offer OD, the defendant will choose either a high pooling offer [O.sup.H.sub.D] = [[PI].sup.H] that both plaintiff types will accept or the low screening offer [[O.sup.L.sub.D] = [[PI].sup.L] that only a type L plaintiff will accept. The low offer is rejected with probability q (the probability the plaintiff is type H). The defendant offers OLD if and only if(1 - q) [[PI].sup.L] + q[C.sup.H] [less than or equal to] [[PI].sup.H]. Rearranging, this may be expressed as

q [less than or equal to] [[PI].sup.H] - [[PI].sup.L]/[C.sup.H] - [[PI].sup.L] (1)

The defendant makes a low screening offer if the probability q of encountering a high-damage plaintiff is sufficiently small. When the screening offer is made, trials will occur with probability q. If the condition in Expression 1 fails to hold, the defendant will make the pooling offer under which all cases settle.

The Informed Party Makes the Offer

In this game the plaintiff makes an offer [O.sub.p] to the defendant who chooses to accept or reject the offer. For this game we adopt the convention that if the informed player is indifferent between two offers, she will make the offer associated with the weaker player type. In this case, that would be the offer associated with the type L plaintiff.

The relevant solution concept for this form of the game is a perfect Bayesian equilibrium. This solution concept requires, among other things, that agents update their equilibrium beliefs in accord with Bayes' rule in a manner that accurately reflects the equilibrium actions of the players. We also need to specify the out-of-equilibrium beliefs of the defendant. The equilibrium refinement concept D1 (Cho and Kreps 1987) places structure on out-of-equilibrium beliefs. (12) Under DI, there is a unique separating equilibrium in this game under which type H plaintiffs make the offer [O.sup.H.sub.L] = [C.sup.H] and type L plaintiffs make the offer [O.sup.L.sub.P] = [C.sup.L] Since the plaintiff follows a pure strategy, these offers are revealing of the plaintiff's type. In other words, in equilibrium a defendant receiving an offer [O.sup.H.sub.P] believes with probability 1 that this offer has been made by a type H plaintiff. Likewise an offer [O.sup.L.sub.P] is believed with probability 1 to have been made by a type L plaintiff.

The low offer will be accepted by the defendant with probability 1. Type L plaintiffs will reveal their type via the offer [O.sup.L.sub.P], only if [O.sup.H.sub.P] is rejected by the defendant with a sufficiently high probability 4. Note that the revealing offer [O.sup.H.sub.P] leaves the defendant indifferent between acceptance and rejection. Thus, the defendant is free to respond to this offer with a mixed strategy. The equilibrium offer [O.sup.H.sub.P] will be rejected by the defendant with probability (13)

[phi] [C.sup.H] - [C.sup.L]/[C.sup.H] - [[PI].sup.L]. (2)

The probability of a trial is q[phi], the probability that the plaintiff is type H times the probability that the high offer is rejected.

To complete the description of the equilibrium, we need to specify out-of-equilibrium beliefs and actions. It is a dominant strategy for the defendant to reject any offer greater than [C.sup.H] and to accept any offer less than [C.sup.L]. Any offer such that [C.sup.L] < [C.sup.P] < [C.sup.H] is believed to be made by a type L plaintiff and is rejected with probability 1.

For our purposes, the key result from the standard models with bilateral payoff relevance is that disputes can potentially occur regardless of which party makes the offer. We would also reach a similar conclusion if there were a two-sided informational asymmetry. Note that even in a fairly simple setting, models with two-sided asymmetries are considerably more difficult to work through than models with one-sided asymmetries.

4. Unilateral Payoff Relevance

Next, we consider a model with unilateral payoff relevance. Models with asymmetric information on risk preferences, a taste for fairness, or the degree of litigiousness are all consistent with the analysis in this section. (14) We first reanalyze the basic litigation games, assuming a one-sided informational asymmetry that has unilateral payoff relevance. First, we analyze the game where the uninformed party makes the offer and then the game in which the informed party makes the offer. Later we consider a model with a two-sided informational asymmetry, where each player has private information that has only unilateral payoff relevance.

In the next two subsections, the game has the following form: It is determined that the defendant's expected cost at trial is C, and this is common knowledge. With a probability q, the plaintiff is type H and receives an expected payoff of [[PI].sup.H] at trial, and with probability 1-q, the plaintiff is type L and receives an expected payoff of [[PI].sup.L], where C > [[PI].sup.L]. The plaintiff knows her own type, but the defendant knows only the unconditional probabilities with which each plaintiff type is encountered.

The Uninformed Party Makes the Offer

We first consider the game in which the uninformed defendant makes the offer. As summarized in Proposition 1, the equilibrium of this game is very similar to the game with bilateral payoff relevance.

PROPOSITION 1. In a game with unilateral payoff relevance and one-sided asymmetric information in which the uninformed defendant (who knows only that plaintiffs are type H with the unconditional probability q) makes the offer to the informed plaintiff (who knows her own type, H or L):

i. If q [less than or equal to] [([[PI].sup.H] - [[PI].sup.L])/ (C - [[PI].sup.L])], the defendant offers FlL, which is accepted by type L plaintiffs and rejected by type H plaintiffs. Trials occur with probability q.

ii. If q > [([[PI].sup.H] - [[PI].sup.L])/(C - [[PI].sup.L])] , the defendant offers [[PI].sup.H], which both plaintiff types accept. Trials occur with probability 0.

PROOF.

Stage 3: A type i plaintiff will accept any offer such that [O.sub.D] [greater than or equal to] [[PI].sup.i] and reject any offer [O.sub.D] < [[PI].sup.i], i = H, L.

Stage 2: If [O.sub.D] = [[PI].sup.H], then both plaintiff types will accept yielding a cost to the defendant of [[PI].sup.H]. If [O.sub.D] = [[PI].sup.L], the type L plaintiff will accept, and the type H plaintiff will reject, yielding an expected cost to the defendant of (1 - q) [[PI].sup.L] + qC. Thus, the defendant offers [[PI].sup.L] if and only if (1 - q) [[PI].sup.L] + qC [less than or equal to] [[PI].sup.H] or q [less than or equal to] [([[PI].sup.H] - [[PI].sup.L])/ (C - [[PI].sup.L])]. If [[PI].sup.L] is offered, it is accepted by type L plaintiffs and rejected by type H plaintiffs. If q > [([[PI].sup.H] - [[PI].sup.L])/ (C - [[PI].sup.L])], the defendant offers [[PI].sup.H], which both plaintiff types accept.

Thus far, the work in the law and economics literature that has considered formal models where information has unilateral payoff relevance has done so in the context of the screening game in which the uninformed party makes the offer. As Proposition 1 shows, the outcomes of these games are very similar to the standard model in which information has bilateral payoff relevance.

The Informed Party Makes the Offer

In this subcase of the unilateral payoff, one-sided asymmetric information game, the informed party (the plaintiff) makes the offer. We will also initially impose the additional restriction that C > [[PI].sup.H]. In the model with bilateral payoff relevance, the defendant's decision to reject or accept depends upon his beliefs regarding the plaintiffs type, because his payoff at trial depends on the plaintiffs type. When there is unilateral payoff relevance, the defendant's payoff is independent of the plaintiffs type. The defendant will be willing to accept the offer [O.sub.p] = C, regardless of which plaintiff type makes this offer. This is the offer made in equilibrium, and as a result, trials do not occur in the signaling version of the game.

PROPOSITION 2. In a game with unilateral payoff relevance and one-sided asymmetric information in which the informed plaintiff (who knows her own type, H or L) makes the offer to the uninformed defendant (who knows only the unconditional probability q that the plaintiff is type H):

i. Both plaintiff types offer C, which the defendant accepts. Trials occur with probability 0. PROOF.

Stage 3: The defendant will accept any offer [O.sub.P] [less than or equal to] C and will reject any offer [O.sub.P] > C.

Stage 2: An offer less than C is accepted but yields the plaintiff less than an offer of C. An offer greater than C is rejected and yields the type H plaintiff [[PI].sup.H] < C and the type L plaintiff [[PI].sup.L] < C. Thus, both plaintiff types offer [O.sub.P] = C, which is accepted with probability 1. (15)

Thus, when the informed party makes the offer, there is a sharp difference between a model with bilateral payoff relevance and a model with unilateral payoff relevance. The model with bilateral payoff relevance predicts disputes, while the model with unilateral payoff relevance does not. By contrast, when the uninformed party makes the offer, both models predict disputes.

Note that Propositions 1 and 2 are not a function of the fact that the plaintiff is the informed party. We would obtain the same results in a model where the defendant is the informed party; that is, the model would predict disputes when the uninformed plaintiff made the offer and would predict 100% settlement when the informed defendant made the offer.

Suppose the condition C > [[PI].sup.H] were violated. In this case, type L plaintiffs would offer [O.sub.P] = C and settle, while all type H plaintiffs would make a demand [O.sub.P] > C. This offer would be rejected, and type H plaintiffs would proceed to trial with a 100% probability. There are a couple things to note about this case. First, trials are bilaterally efficient since [[PI].sup.H] > C. (16) Second, trials would occur even if the plaintiff's type were common knowledge, because there is no contract zone between the type H plaintiff and the defendant. Importantly, we can conclude that inefficient trials never occur in the equilibrium of the game in which the informed party makes the offer.

Two-Sided Asymmetric Information with Unilateral Payoff Relevance

If disputes never occur in the model in which the informed party makes the offer, then institutions might evolve to take advantage of this fact in order to avoid the joint costs of a dispute. Specifically, when private information has unilateral payoff relevance, all disputes would be avoided if the parties arrange it so that the informed party makes the final offer prior to trial. (17) However, it is possible to have unilateral payoff relevance with a two-sided informational asymmetry. For example, each player could have private information on their own degree of litigiousness, where this information has no impact on the other players expected payoff from a trial. As we shall see, this model is consistent with disputes regardless of which party makes the offer. Also, solving this model is trivial compared to solving the corresponding model of bilateral payoff relevance. When there is bilateral payoff relevance, the solution to a model with a two-sided asymmetry contains both signaling and screening elements. However, when there is unilateral payoff relevance, the solution involves only a minor variation on the screening model.

Up to now, our notation has been flexible enough to capture both risk aversion as well as other forms of asymmetric information with only unilateral payoff relevance. In what follows, this is no longer so. For the analysis below, we will assume that all agents are risk neutral. In the Appendix, we consider the case of risk aversion. Although consideration of risk aversion does change the exact expressions found below in Proposition 3, the general flavor of the result is the same whether we are considering risk aversion or some other form of informational asymmetry.

Assume specifically then that in this variant of the general game, in stage 1 Nature chooses both the plaintiff and defendant types that are uncorrelated across the players. As before, the plaintiff's expected payoff is [[PI].sup.H] with probability q and [[PI].sup.L] with probability 1 - q, where these probabilities are independent of the defendant's type. Similarly, the defendant has expected cost CH with probability 1 - r and [C.sup.L] with probability r, where these probabilities are independent of the plaintiffs type and [C.sup.H] > [[PI].sup.L]. Each player knows their own expected payoffs but knows only the distribution of expected payoffs that apply for their bargaining partner. In stage 2, both players are uninformed. We allow the defendant to make the single offer.

The proposition below will make use of the following conditions on the parameters of the model:

q [less than or equal to] [[PI].sup.H] - [[PI].sup.L]/[C.sup.L] - [[PI].sup.L], (3)

q [less than or equal to] [[PI].sup.H] - [[PI].sup.L]/[C.sup.H] - [[PI].sup.L]. (4)

It is worth noting in regard to the results below that because [C.sup.L] < [C.sup.H], the condition in Expression 4 is more restrictive than the condition in Expression 3 in the sense that it is satisfied for fewer values of q.

The equilibrium of the model is summarized as Proposition 3.

PROPOSITION 3. In a game with unilateral payoff relevance and two-sided asymmetric information where each player knows their own type and the unconditional distribution of the other player's type (the plaintiff is type H with probability q and the defendant is type L with probability r) and where the defendant makes the offer:

i. When the conditions in Expressions 3 and 4 hold, both defendant types offer [[PI].sup.L]. This offer is accepted by type L plaintiffs and rejected by type H plaintiffs. Trials occur with probability q.

ii. When the condition in Expression 3 holds but the condition in Expression 4 does not, type L defendants offer [[PI].sup.L]. This offer is accepted by type L plaintiffs and rejected by type H plaintiffs. Type H defendants offer [[PI].sup.H]. This offer is accepted by both plaintiff types. Trials occur with probability qr.

iii. When the conditions in both Expressions 3 and 4 fail to hold, both defendant types offer [[PI].sup.H] and both plaintiff types accept. There is a 100% settlement rate. PROOF.

Stage 3: A type i plaintiff will accept any offer [O.sub.D] [greater than or equal to] [[PI].sup.i] and reject any offer [O.sub.D] < [[PI].sup.i], i = H, L.

Stage 2: If the defendant offers [[PI].sup.H], both plaintiff types accept, yielding a cost of [[PI].sup.H]. If the defendant offers [[PI].sup.L], it is accepted by type L plaintiffs and rejected by type H plaintiffs, yielding an expected cost of (1-q)[[PI].sup.L] + q[C.sup.j] or [[PI].sup.H], where j = H, L is the defendant's type. A type j defendant will offer [[PI].sup.L] iff (1 - q)[[PI].sup.L] + q[C.sup.j] [less than or equal to] [[PI].sup.H] or q [less than or equal to] [([[PI].sup.H] - [[PI].sup.L])/([C.sup.j] - [[PI].sup.L])]. Otherwise the offer is [[PI].sup.H].

i. If Expressions 3 and 4 hold, both types offer [[PI].sup.L]. Type L accept, type H reject.

ii. If Expression 3 holds and Expression 4 fails, type L offers [[PI].sup.L], which is rejected by type H, and type H offers [[PI].sup.H], which is accepted. Trials occur with probability qr.

iii. If both fail, both types offer [[PI].sup.H], which is accepted.

As in previous screening games, a low offer will be made if and only if the probability of rejection (i.e., the probability of encountering a type H plaintiff) is sufficiently low. However, now the screening condition itself is a function of the defendant's own type, which is independent of the plaintiff's type. Since Expression 3 is less restrictive than Expression 4, the type L defendant is more likely to make a low screening offer than a type H defendant.

The solution to the model is only slightly more complicated than the solution to a simple screening game in which the uninformed party makes the offer. Despite the two-sided asymmetric information, the private information of the party making the offer (the defendant) is irrelevant, because the plaintiff's willingness to accept an offer is independent of the defendant's type. By contrast, when there is a two-sided informational asymmetry, the bilateral payoff model retains signaling elements, which makes it considerably more difficult to solve.

It should be clear that if we redo the analysis and allow the plaintiff to make the offer, we will obtain conditions analogous to Expressions 3 and 4. Thus, the fact that the model predicts disputes (under certain conditions) is not sensitive to the bargaining structure. Note that if [[PI].sup.L] > [C.sup.L] , then type L defendants will make an offer [O.sub.D] < [[PI].sup.L], which both plaintiff types reject, resulting in 100% probability of trial for type L defendants. (18) Trial is bilaterally efficient in this case and would occur even in the absence of asymmetric information.

5. Information Regarding the Expected Payoff of Your Bargaining Partner

In this section we consider the variant of the general game in which the player with private information has knowledge regarding their bargaining partner's expected payoff at trial. For example, suppose the defendant has private information regarding the plaintiff's costs of enforcing a settlement on the defendant in the event the plaintiff is victorious at trial. If enforcement costs are low, then the plaintiff is type H, and if these costs are high, then the plaintiff is type L. The defendant knows the plaintiff's type, while the plaintiff merely knows that she is type H with probability q. The defendant's expected cost at trial, C, is independent of the plaintiff's type, where C > [[PI].sup.H]. The motivating example for this section (which involves the plaintiff's costs of recovering the judgment) leads naturally to the restriction C > [[PI].sup.H]. (19)

The Uninformed Party Makes the Offer

The results of this game are summarized as Proposition 4.

PROPOSITION 4. In a game with unilateral payoff relevance and one-sided asymmetric information in which the uninformed plaintiff (who knows only the unconditional probability q that she is type H) makes the offer to the informed defendant (who knows if the plaintiff is type H or L):

i. The plaintiff offers C, which the defendant accepts. Trials occur with probability 0. PROOF.

Stage 3: The defendant will accept any offer [O.sub.P] [less than or equal to] C, and reject any offer [O.sub.P] > C.

Stage 2: Since [O.sub.P] = C will be accepted, the plaintiff will not offer [O.sub.P] < C. Any offer [O.sub.P] > C is rejected, resulting in a payoff [[PI].sup.H] < C for type H plaintiffs and [[PI].sup.L] < C for type L plaintiffs. Thus, both plaintiff types offer C.

When there is unilateral payoff relevance where the defendant is informed about the plaintiff's expected payoff at trial, there is 100% settlement when the plaintiff makes the offer. This is in contrast to the model of bilateral payoff relevance, and the game with unilateral payoff relevance, where the plaintiff has private information about her own payoff. In both of these games, trials are predicted with a positive probability, if the probability of encountering a type H plaintiff is sufficiently low. Trials do not occur in this game, because the defendant's private information does not affect his willingness to accept an offer from the plaintiff.

It should be clear that there would also be 100% settlement if we switched the information structure so that the plaintiff was informed about the defendant's expected cost at trial, and the defendant made the offer to the plaintiff.

The Informed Party Makes the Offer

The game is as outlined above except that the defendant makes the single offer to the plaintiff.

Recall that in the game from section 4, there is 100% settlement when the informed party makes the offer. This is due to the fact that the informed plaintiff has nothing of value to signal to the defendant, because the defendant's expected cost at trial is independent of the plaintiff's type. In the current game, the defendant has something of value to signal to the plaintiff--her type. But in contrast to the traditional analysis, the information the defendant is attempting to signal has no effect on his own payoff.

The analysis of this game is summarized as Proposition 5.

PROPOSITION 5. In a game with unilateral payoff relevance and one-sided asymmetric information in which the informed defendant (who knows if the plaintiff is type H or L) makes the offer to the uninformed plaintiff (who knows only the unconditional probability q that she is type H):

i. There is no separating equilibrium;

ii. There exist a continuum of pooling equilibria such that trials occur with probability 0. PROOF.

Stage 3: The uninformed plaintiff will accept any offer that leaves her as well off as she expects to be at trial conditional on the updated beliefs she may have upon observing the offer. Let [gamma]([O.sub.d]) be the plaintiff's updated belief that she is type H conditional on the defendant's offer. (The functional dependency will be suppressed in the notation that follows.) The plaintiff will accept any offer [O.sub.D] [greater than or equal to] [gamma][[PI].sup.H] + (1 - [gamma])[[PI].sup.L] and reject any offer [O.sub.D] < [gamma][[PI].sup.H] + (1 - [gamma])[[PI].sup.L].

Stage 2: We will consider both a separating and a pooling equilibrium.

Separating equilibrium: Consider a separating offer such that defendants offer [[PI].sup.H] to a type H plaintiff and [[PI].sup.L] to a type L plaintiff. In equilibrium, a plaintiff receiving the offer [[PI].sup.H] has an updated belief [gamma] = 1, and a plaintiff receiving [[PI].sup.L] has the updated belief [gamma] = 0. For this equilibrium to be supported, [[PI].sup.L] must be rejected with a sufficiently high probability to discourage defendants who know that the plaintiff is type H from making the low offer. Specifically, a defendant who knows the plaintiff is type H will make a high offer iff [[PI].sup.H] [less than or equal to] [phi]C + (1 - [phi])[[PI].sup.L], where [phi] is the probability that the low offer is rejected. The lowest value of [phi] consistent with this behavior is [[phi].sup.*] = [([[PI].sup.H] - [[PI].sup.L])I(C - [[PI].sup.L])]. Suppose that the defendant knows that the plaintiff is type L and offers [[PI].sup.L]. This defendant also expects to pay [phi]C+ (1 - [phi])[[PI].sup.L], so that [[phi].sup.*] will also discourage this defendant from offering [[PI].sup.L]. (20) Because the defendant's cost is independent of the plaintiff's type, any rejection rate on the low offer either discourages all defendants from making the low offer, or it discourages none of them. It is straightforward to show that separation is also not possible for any intermediate offer [[PI].sup.L] < [O.sub.D] < [[PI].sup.H]. Thus, there is no separating equilibrium in this model.

Pooling: There exist a continuum of pooling equilibria on offers in the interval q[[PI].sup.H] + (1 - q)[[PI].sup.L] [less than or equal to] [O.sub.D] [less than or equal to] C. The plaintiff accepts the pooling offer, and there is a 100% settlement rate. The plaintiff rejects all offers below the pooling amount and believes all such offers are made by a defendant holding the information that she is type H. (21) In a pooling equilibrium, both defendant types make the same offer, so [gamma] = q. As a result, the plaintiff's expected payoff at trial is q[[PI].sup.H] + (1 - q)[[PI].sup.L]. Thus, the plaintiff is willing to accept an offer in the specified range. Since the plaintiff rejects all offers below the pooling amount, the defendant is willing to make the pooling offer, because [O.sub.D] [less than or equal to] C.

The reason there cannot be a signaling equilibrium in this game is that it is equally costly for the defendant to make the low offer regardless of what private information he holds regarding the plaintiff's expected payoff at trial. Note also that this is similar to the result in the game with unilateral payoff relevance, where the informed player has private information on her own payoff. It should be clear that an analog to Proposition 5 would apply to a model where the plaintiff makes an offer with information about the defendant's expected cost at trial but where the plaintiffs expected payoff is independent of this information.

The pooling equilibrium is efficient since there is 100% settlement. The sum of the player payoffs is 0 in the pooling equilibrium. If the signaling equilibrium existed, the sum of the payoffs would be q[[phi].sup.*]([[PI].sup.H] - C) < 0, reflecting the inefficient trials that would occur in a signaling equilibrium. Thus, from an efficiency standpoint, the pooling equilibrium is to be preferred.

What Propositions 4 and 5 show is that there are never disputes in a model with unilateral payoff relevance when the private information is held by one player, but that information affects the expected payoff of the other player. (22)

6. Conclusion

The distinction between unilateral payoff relevance and bilateral payoff relevance is noteworthy, because there are clearly important classes of private information that have unilateral payoff relevance, and this has significant implications for pretrial settlement. Examples of information with unilateral payoff relevance include private information on risk preferences, a taste for fairness, the degree of litigiousness, and the degree of self-serving bias. In each of these examples, the private information is held by the party who it affects. It is also possible to have unilateral payoff relevance but where the information affects the expected payoff of the player who does not possess the information. This would be the case if the defendant held private information on how costly it would be for the plaintiff to enforce a judgment post trial. For both sets of examples, we find that there are never inefficient trials when the informed party makes the offer. Disputes can occur when the uninformed party makes the offer only in the first set of examples in which information affects the payoff of the player in possession of that information. By contrast, when there is bilateral payoff relevance, disputes may occur regardless of which party makes the offer.

The results of this article may have important implications for the efficient structure of bargaining. Suppose, for example, that the plaintiff has an unknown taste for fairness but that the defendant (perhaps a corporation) is known not to have a taste for fairness. According to the results of the model, if the defendant makes the last offer before trial, some trials may result in equilibrium, while if the plaintiff makes the last offer, all cases will settle.

The kinds of information that are consistent with the analysis in section 4 can be thought of as an information class, because of the commonalities they exhibit in the way they affect pretrial bargaining. In addition to the differences noted so far, this information class exhibits one other difference when compared with the traditional examples used in models of bilateral payoff relevance. The examples of informational asymmetries used in models with bilateral relevance typically involve evidence. This may be evidence related to the probability of a finding for the plaintiff or evidence regarding the expected size of the judgment. If this is indeed evidence to be presented at trial, then there is some presumption that this information is verifiable and can be revealed via the discovery process or via voluntary disclosures prior to trial. Farmer and Pecorino (2005) show that if information disclosures are not too expensive, then most private information will be revealed prior to trial. (23) Thus, there should be a strong tendency for informational asymmetries of this sort to be eliminated prior to trial with the result being that most parties settle.

By contrast, the examples of unilateral payoff relevance that lead to trial--risk aversion, a taste for fairness, and the degree of litigiousness--are not easily verifiable and are not subject to discovery. If there is no credible way to communicate this information, then the informational asymmetry will not be eliminated prior to trial. As a result, these particular informational asymmetries may be especially important in explaining trial. This in turn may have important policy implications. If asymmetric information on risk preferences is driving bargaining failure, then this may have important implications for how we evaluate institutions such as fee shifting or contingency fees, because these institutions change the risk characteristics of trial. If asymmetric information is over the degree of self-serving bias, then efforts to debias the parties to the dispute may be particularly important in reducing disputes. Additional theoretical and empirical work exploring these issues will be crucial in providing sharper guidance for policy.

Appendix: A Further Treatment of Risk Aversion

In this Appendix, we will elaborate a bit on the interpretation of the model in which the source of asymmetric information is the risk preference of one or both parties. In this interpretation, type L plaintiffs and type H defendants are risk averse, while type H plaintiffs and type L defendants are risk neutral. Let H be the net monetary payment received by the plaintiff and [U.sub.P]([PI]) be the utility function of a risk-averse plaintiff: [U.sub.p]' > 0, [U.sub.p]" < 0, where the prime denotes a derivative. The expected utility at trial for this plaintiff is

E([U.sub.P]) = p[U.sub.P](J - [K.sub.P]) + (1 - p)[U.sub.P](- [K.sub.P]), (A1)

where p is the probability of a finding for the plaintiff, J is the judgment received by a victorious plaintiff', and [K.sub.P] are the plaintiff's court costs (including lawyer fees). The certainty equivalent of trial is the payoff, if received with certainty, which would give the plaintiff the same expected utility as trial. We can denote this payment as [[PI].sup.L], where [U.sub.P]([[PI].sup.L]) = p[U.sub.P](J - [K.sub.P]) + (1 - p)[U.sub.P](-[K.sub.P]). By risk aversion, [[PI].sup.L] < pJ - [K.sub.P].

We can make similar calculations for a risk-averse defendant. Let C be the total payment made by the defendant and [U.sub.D](C) be the utility function of a risk-averse defendant: [U.sub.D]' < 0, [U.sub.D]" < 0. Note that utility is decreasing in the payment, C. For the defendant, the expected utility of a trial is

E([U.sup.D]) = p[U.sub.D](-J - [K.sub.D]) + (1 - p)[U.sub.D](-[K.sub.D]), (A2)

where [K.sub.D] is the defendant's court costs. The certainty equivalent of trial for the defendant is the payment, which if made with certainty, that gives the defendant the same expected utility as trial. Denote this payment as CH, where UD(CH) = p[U.sub.D](-J - [K.sub.D]) + (1 - p)[U.sub.D](-[K.sub.D]). For the risk-averse defendant, [C.sup.H] > pJ + [K.sub.D].

Throughout section 4, it is sufficient to use [[PI].sup.L] to represent the certainty equivalent of trial for a risk- averse plaintiff. However, when we consider two-sided asymmetric information, it is not sufficient to use CH to represent the payoff of a risk-averse defendant. The reason is that such defendants face two types of risk. One stems from the trial and is captured by [C.sup.H]. The other stems from the risk that a low pretrial offer is rejected. Thus, the condition in Expression (4) must be modified for a risk-averse defendant.

A type H defendant will make a low screening offer [PI]L if the expected utility of such an offer is greater than the expected utility of the high offer [PI]H . The low offer is rejected with probability q, which is the probability that the plaintiff is type H. Thus, the type H defendant will make the low offer if

q[U.sub.D]([C.sup.H]) + (1 - q)[U.sub.D]([[PI].sup.L]) [greater than or equal to] [U.sub.D] ([[PI].sup.H]). (A3)

Keeping in mind that [U.sub.D]([C.sup.H]) - [U.sub.D]([[PI].sup.L]) < 0, this condition may be expressed as

q [less than or equal to] [U.sub.D]([[PI].sup.L]) - [U.sub.D]([[PI].sup.H])/[U.sub.D]([[PI].sup.L] - [U.sub.D]([C.sup.H]). (A4)

Although the expression in Expression A4 differs from that in Expression 4, the nature of the equilibrium is very similar when there is a risk-averse defendant. Since the type L defendant is risk neutral, there is no need to modify Expression 3.

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Amy Farmer, Department of Economics, University of Arkansas, Fayetteville, AR 72701, USA; E-mail amyf@walton. uark.edu: corresponding author.

Paul Pecorino, Department of Economics, Finance and Legal Studies, University of Alabama, Tuscaloosa, AL 35487-0224, SA; E-mail ppecorin@cba.ua.edu.

We would like to thank Akram Temimi and two anonymous referees for providing helpful comments on the article.

Received October 2008; accepted September 2009.

(1) The key early papers are Bebchuk (1984) for the screening model and Reinganum and Wilde (1986) for the signaling model. A survey of the early literature may be found in Cooter and Rubinfeld (1989), while a more recent survey is presented by Daughety (2000).

(2) In this article, efficiency is defined narrowly by the sum of the payoffs of the two parties to the dispute. This issue is further discussed at the end of section 2. In a larger context, costly trials may be efficient via the effect they have on incentive for care. See Polinsky and Rubinfeld (1988). Trials may also provide positive externalities in the form of valuable legal precedent.

(3) In addition, in Curry and Pecorino (1993) and Faurot (2001) recourse to final offer arbitration results from asymmetric information on risk preferences.

(4) A small recent sampling of this literature includes Slonim and Roth (1998), Fehr and Schmidt (2000). and Andreoni, Castillo, and Petrie (2003).

(5) This is the approach taken by Farmer and Pecorino (2004), but they model the preference as observable.

(6) In the case of self-serving bias, the effect is on the perceived payoff rather than the actual payoff.

(7) For models with two-sided informational asymmetries, see Schweizer (1989), Sobel (1989), and Daughety and Reinganum (1994).

(8) Kaplan and Sadka (2008) is mainly an empirical investigation, but there is a brief theory section. In the theory section, they focus on a case that is not addressed in our article, namely, a situation where collection costs may be so high that the plaintiff fails to collect, even after a positive decision in her favor.

(9) Since [[PI].sup.L] > 0, the plaintiff will always have a credible threat to proceed to trial.

(10) If K is the sum of the plaintiff and defendant court costs, then in the standard model with bilateral payoff relevance, we typically have [C.sup.L] = [[PI].sup.L] + K and [C.sup.H] = [[PI].sup.K] + K.

(11) This case is considered in Faurot (2001).

(12) The refinement D1 places restrictions on out-of-equilibrium beliefs. In particular, it requires the recipient believe that an out-of-equilibrium offer is made by the player type most likely to benefit from such an offer. Suppose there are two player types, A and B, and the player A would benefit from a given out-of-equilibrium offer if it were accepted 20% of the time or more and that player B would benefit from the same offer if it were accepted 30% of the time or more. Under D1, the recipient of the offer would have to place 100% weight on the probability that the offer is made by type A. The refinement DI can be used to rule out pooling or semi-pooling equilibria. See Reinganum and Wilde (1986, p. 566).

(13) This is the smallest value of the rejection rate consistent with a separating equilibrium. Higher values are ruled out by DI. See Daughety (2000, pp. 133-4). Nothing in what follows crucially depends upon [phi] taking on its lowest possible value.

(14) With minor modifications, the analysis below could be applied to the case of self-serving bias as well. In this case, there is a need to distinguish between a player's perceived payoff at trial and their true payoff at trial.

(15) If this offer were rejected with a positive probability, the plaintiff would deviate to C - [epsilon]. It is a dominant strategy for the defendant to accept this offer, it will be accepted with probability 1. Note that under a perfect Bayesian equilibrium, any threat by the defendant to reject offers less than C is not credible and will not be believed in equilibrium. Thus, the equilibrium offer is unique.

(16) The sum of the payoffs at trial is [[PI].sup.H] - C > 0. If the case settles prior to trial, the sum of the payoffs is 0.

(17) In the examples we have examined thus far in section 4, the defendant is better off if he makes the final offer, but the sum of the payoffs of the plaintiff and defendant is higher if the plaintiff makes the final offer. Thus, the efficient outcome could be achieved if there were side payments to determine the order of the offers. This seems unlikely, but the efficient institution could also evolve through prior contracting. If the two parties enter a contract with the knowledge that a dispute could later occur, it might be possible for the contract to structure the pretrial bargaining so as to obtain the efficient outcome. This is analogous to the idea that contracting parties would include arbitration clauses at the time of contract, if arbitration was deemed the more efficient mechanism for resolving disputes. See, for example, Drahozal and Hylton (2003) and Dari-Mattiacci (2007). Of course, this argument would not apply to a dispute arising from a tort, since in this case, there is no prior contracting.

(18) The condition [[PI].sup.L] > [C.sup.L] could arise if, for example, the defendant were risk loving.

(19) Whether the recovery cost is high or low, this cost combined with the cost of the trial will ensure that the plaintiff receives less than what the defendant pays out in total.

(20) Recall the convention introduced in section 3 that when the informed player is indifferent between two offers, he will make the offer associated with the weaker type, in this case [[PI].sup.H]. In the absence of this convention it is possible to have a separating equilibrium, where defendants who know the plaintiff is type H offer [[PI].sup.H], defendants who know the plaintiff is type L offer [[PI].sup.L], and the low offer is rejected with probability [phi] in (5). However, the implied behavior seems implausible. All defendants are indifferent between the two offers, yet one type follows a pure strategy of making the high offer, and the other type follows the pure strategy of making the low offer.

(21) These are admissible out-of-equilibrium beliefs. In models with bilateral payoff relevance, the refinement D1 is used to eliminate pooling equilibria. Since, in our model, the defendant's payoff is independent of the plaintiff's type, the defendant's incentive to defect from equilibrium is independent of the information he holds. Thus, DI will not restrict out-of-equilibrium beliefs in this situation.

(22) However, this result does rely on the convention discussed in note 20.

(23) Whether this occurs via voluntary disclosure or discovery depends on whether it is a screening game or signaling game.