Negative expected value suits in a signaling model.

Farmer, Amy ; Pecorino, Paul

1. Introduction

Bargaining failures can be extremely costly when a plaintiff and defendant find themselves in trial after failing to negotiate a settlement. The presence of asymmetric information is the most prominent explanation of this type of bargaining failure. Reinganum and Wilde (1986) is one of the two canonical information-based models of pretrial bargaining. Theirs is a signaling model in which an informed plaintiff makes an offer to an uninformed defendant. Reinganum and Wilde (RW) assume that all plaintiffs have a credible threat to proceed to trial. We relax this assumption and in the process endogenize the plaintiff's filing decision. This is a realistic extension of the RW model, which has some very important implications.

First, if we add plaintiffs with negative expected value (NEV) suits to the RW model and make no other changes, the equilibrium of the model will require that all submitted offers be rejected at a rate of 100%. (1) To restore an equilibrium with settlement, it is necessary to assume that the plaintiff incurs a fee at the time that suit is filed. In this amended model, we find that all plaintiff offers are rejected with a higher probability when compared to the model without NEV suits. This higher rate of rejection is a necessary part of the equilibrium as it discourages potential plaintiffs from filing NEV suits. Thus, no NEV suit is filed in the equilibrium of the model, but the potential presence of these suits causes more trials to occur. Further, the increase in the dispute rate may be substantial.

While the analysis of our model implies that the presence of NEV suits can cause a dramatic increase in dispute rates, this analysis will not apply to all potential lawsuits. There may be a substantial number of cases where, because of the nature of the case, it is common knowledge between the plaintiff and defendant that the plaintiff's suit has a positive expected value at trial. In these situations the results of the RW model will apply. However, we believe there are a substantial number of civil actions in which the distribution of plaintiff types will include NEV suits. This may be particularly true when the plaintiff's injury claims are difficult to verify (e.g., back pain).

In our model the filing decision by the plaintiff is endogenous, and the plaintiff must incur a positive fee in order to file suit. In the presence of potential NEV suits, the probability of settlement is increasing in the filing fee paid by the plaintiff. This provides a possible justification for increasing these fees as a policy measure designed to lower the dispute rate. In addition, we find that the effects of fee shifting on the incidence of trial work entirely through their effects on the incentives for plaintiffs to file suit. When RW add fee shifting to their baseline model, they find no effect on the incidence of trial. (2) However, when the plaintiff expects to shift more fees to the defendant than the defendant expects to shift to the plaintiff, we find that more plaintiffs file suit and that, holding plaintiff type constant, there is a higher rate of rejected offers. Conversely, if the defendant expects to shift fees to the plaintiff on net, then fewer plaintiffs file suit, and, holding plaintiff type constant, there is a lower rate of rejected offers. Thus, consideration of NEV suits is important in any policy evaluation of the merits of fee shifting.

2. Related Literature

Along with the signaling model of Reinganum and Wilde, the other canonical information-based model of pretrial bargaining is Bebchuk (1984). In his screening model, an uninformed plaintiff makes an offer to an informed defendant. Bebchuk assumes (as do RW) that all plaintiffs have a credible threat to proceed to trial. Nalebuff (1987) extends Bebchuk's model by allowing for the presence of defendant types against whom the plaintiff lacks a credible threat to proceed to trial; that is, some plaintiffs have NEV suits. If a sufficiently low offer were rejected by the defendant, plaintiffs would learn they had a NEV suit and drop their case. As a result, if the plaintiff's credibility constraint is binding, she submits a higher offer to the defendant relative to the offer she would submit if this constraint does not bind. This limits the bad news communicated to the plaintiff when a defendant rejects her offer and results in a greater incidence of trial. Nalebuff finds that the comparative static results of Bebchuk's model are reversed when the credibility constraint is binding.

Other extensions of the Bebchuk (1984) model include Bebchuk (1988) and Katz (1990). Both authors model plaintiffs who know they have NEV suits in the context of a screening model in which the defendant makes the offer. In Bebchuk (1988) there is no filing cost for the plaintiff, whereas Katz does introduce such a cost. Regardless, in both models NEV suits increase litigation. The Katz model involves a mixed strategy equilibrium in which a plaintiff with a potential NEV suit files it with some probability. (3)

Another important related work is Sobel (1989). He develops a model with two-sided asymmetric information, but the solution to his model retains an important signaling component. (4) In his model plaintiffs and defendants can each take on one of two types. Throughout much of his analysis, plaintiffs are assumed to have positive expected value suits, but Sobel (1989, pp. 148-9) does consider a case where one plaintiff type has a NEV suit and finds that this causes rejection rates to rise to 100%. Sobel does not analyze the effects of positive filing fees on plaintiff behavior.

In this paper we extend the RW model by allowing for the inclusion of plaintiffs with NEV suits. We solve for the equilibrium of this model in the absence of filing costs and find that the presence of NEV suits causes the plaintiff's offer to be rejected at a 100% rate. This is analogous to Sobel's result discussed above. Next, we introduce filing costs and find an equilibrium that does result in some settlement, but at a lower rate than in the absence of NEV suits. The reduction in settlement caused by the presence of NEV suits is potentially quite large. Last, we analyze the effects of fee shifting at trial and find that the effects of fee shifting on settlement operate through the filing decision of the plaintiff.

3. The Reinganum and Wilde Model

We first summarize the signaling model presented in Reinganum and Wilde (1986) and then consider how the presence of NEV suits affects their model. In their model the plaintiff has private information concerning the damages, J. In particular, the plaintiff knows the value of J, which will be awarded in the event of a finding for the plaintiff at trial. The defendant knows that J is distributed by f(J), where [J.sub.L] and [J.sub.H] are the lower and upper supports of this distribution. The probability, p, that the plaintiff will prevail in trial is common knowledge, as are [C.sub.P] and [C.sub.D], the fees paid to attorneys of the plaintiff and defendant. The informed plaintiff makes a single take-it-or-leave-it offer, [O.sub.P], to the defendant.

We assume that [pJ.sub.L] > Cp so that all plaintiffs have a credible threat to proceed to trial. Thus, in the RW model, [J.sub.L] = [J.bar], where [J.bar] denotes the lowest plaintiff type to file suit. In later sections, we consider a model in which [pJ.sub.L] < [C.sub.P]; that is, it is possible that [J.bar] > [J.sub.L]. The game is summarized as follows:

1. Nature determines the plaintiff's type, J. The defendant does not observe J, but knows the distribution, f(J), from which it is drawn.

2. The plaintiff decides whether to hire a lawyer who is paid [C.sub.P] if the case proceeds to trial and 0 if the case settles prior to trial. If the plaintiff hires a lawyer, she then files a suit and pays a fee [C.sub.0] [greater than or equal to] 0.

3. The plaintiff makes a single take-it-or-leave-it offer, [O.sub.P], to the defendant.

4. If the defendant accepts the offer, the plaintiff receives a payoff of [O.sub.P] - [C.sub.0], while the defendant receives - [O.sub.P]. If the defendant rejects the offer, the plaintiff decides whether or not to drop the case.

5. If the plaintiff drops the case, she receives a payoff of -[C.sub.0] and the defendant receives a payoff of 0. Otherwise, the case proceeds to trial.

6. At trial, there is a finding for the plaintiff with probability p, in which case she receives the payoff J - [C.sub.P] - [C.sub.0], while the defendant receives the payoff -(J + [C.sub.D]). With probability 1 - p, the finding is for the defendant; in this case, the plaintiff receives the payoff -([C.sub.P] + [C.sub.0]), and the defendant receives the payoff -[C.sub.D].

We will initially follow RW by assuming that the filing fee [C.sub.0] = 0. The assumption [pJ.sub.L] > [C.sub.P] ensures that no suits are dropped at step 5. These two assumptions together ensure that all potential plaintiffs will file suit.

There are potentially many equilibria in this signaling game, but RW use the refinement arguments of Banks and Sobel (1987) to eliminate all but a separating equilibrium. The equilibrium refinement places structure on out-of-equilibrium beliefs. Because out-of-equilibrium beliefs play an important role in the equilibrium of the model, we will discuss them and how they relate to the equilibrium refinement concept in greater detail both in section 4 and in the Appendix.

In the separating equilibrium, the plaintiff's offer is perfectly revealing of her type, and the defendant plays a mixed strategy under which he rejects the offer [O.sub.P] with probability [phi]([O.sub.P]). The equilibrium rejection function must be such that optimizing plaintiffs reveal their type through their offer. Given the rejection function [phi]([O.sub.P]), the plaintiff will make an offer to maximize her expected wealth, [V.sub.P], which can be written

[V.sub.P] = [phi]([O.sub.P])[[p.sub.J] - [C.sub.P]] + (1 - [phi]([O.sub.P]))[O.sub.P]. (1)

Maximization of Equation 1 by the plaintiff yields the following first-order condition:

[phi]'([O.sub.P])[[p.sub.J] - [C.sub.P]- [O.sub.P]] + (1 - [phi]([O.sub.P])) = 0. (2)

The function B([O.sub.P]) describes the defendant's beliefs about the plaintiff's type as a function of her offer to the defendant. In a perfect Bayesian equilibrium these beliefs must reflect the equilibrium actions of the plaintiff. Thus, beliefs are correct in equilibrium: B([O.sub.P](J)) = J. Since the defendant pursues a mixed strategy in equilibrium, the plaintiff's offer must make him indifferent between acceptance and rejection. The equilibrium offer by a type J plaintiff equals the defendant's expected payoff at trial against this plaintiff:

[O.sub.P] = pJ + [C.sub.D]. (3)

There exists a family of solutions that solves the differential equation in Equation 2. The appropriate boundary condition is [phi]([[O.sub.P].bar]) = 0, where [[O.sub.P].bar] is the settlement demanded by the least damaged plaintiff. (5) Using this boundary condition yields the following solution to the differential equation:

[phi]([O.sub.P]) = 1 - [ae.sup.[psi]], (4)

where [psi]([O.sub.P]) = ([O.sub.P] - [[O.sub.P].bar])/([C.sub.P] + [O.sub.P]) and a = 1. The parameter a is included to ease the comparison to the solution for [phi]([O.sub.P]) when we analyze NEV suits in section 4. As it turns out, in the model with NEV suits and a filing fee, the solution for [phi]([O.sub.P]) will take the same general form as in Equation 4, but we will have a < 1.

While noting a = 1, substitute the equilibrium offers from Equation 3 to write the equilibrium probability of settlement found in RW as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where the B subscript indicates that this is the baseline case. In equilibrium, higher plaintiff types must have their offers rejected more frequently to discourage lower plaintiff types from "bluffing" by submitting an offer higher than the one associated with their type. The mapping between higher offers and an increased probability of rejection is exactly sufficient to induce fully revealing offers.

4. Negative Expected Value Suits

Zero Filing Fee

Suppose that [J.sub.H] > [C.sub.P/p], so that some plaintiffs have positive expected value (PEV) suits, but [J.sub.L] < [C.sub.P/p], so that there also exist plaintiffs with NEV suits. Plaintiffs of type J < [C.sub.P/p] do not have a credible threat to proceed to trial and will drop their claim if they face a rejection by the defendant. We show that the equilibrium described in section 3 is not robust to the addition of plaintiffs with NEV suits.

First, plaintiffs of type J < [C.sub.P/p] will not make the revealing offer in Equation 3 in equilibrium. Such offers would always be refused by the defendant given the knowledge that the plaintiff would later drop the case. Thus, such offers give these plaintiffs a payoff of 0. Second, under the equilibrium described in section 3, these players will receive a positive payoff by making an offer associated with a plaintiff of type J [greater than or equal to] [C.sub.P/p], since these offers are accepted with a positive probability. If the offer is accepted, the plaintiff receives a positive payoff, while if it is rejected, the plaintiff drops the case and receives a payoff of 0. This will destroy the equilibrium from section 3, which is predicated on the defendant's indifference between accepting and rejecting the plaintiff's offer from Equation 3. This indifference depends on all plaintiffs "truthfully" revealing their type by making the offer in Equation 3. If plaintiff types J < [C.sub.P/p] bluff by submitting an offer associated with a higher plaintiff type, the defendant will strictly prefer to reject this offer. As a result, the equilibrium described in section 3 will come unraveled.

How is the outcome of the model affected by the presence of NEV suits? Consider the following equilibrium:

PROPOSITION 1. When [C.sub.0] = 0, there exists a perfect Bayesian equilibrium for the game described by steps 1-6 with the following properties:

(i) Plaintiffs file suit, iff they are of type J [greater than or equal to] [C.sub.P/p].

(ii) Plaintiffs who file suit make the offer given by Equation 3.

(iii) Defendants reject offers [O.sub.P] [greater than or equal to] [O.sub.P] with the probability 1.

(iv) Offers [O.sub.P] < [O.sub.P] = [C.sub.P] + [C.sub.D] only occur out of equilibrium. Defendants believe with probability 1 that such offers are made by type J < [C.sub.P/p] players and always reject such offers.

(v) Offers [O.sub.P] > [pJ.sub.H] + [C.sub.P] occur only out of equilibrium and are always rejected.

The proof is omitted because a zero filing fee represents a special case of the model developed below. Thus, the proof of Proposition 1 is contained in the proof of Proposition 2 given below.

With a zero filing cost, NEV plaintiffs can be discouraged only by 100% rejection rates. If any offer were accepted with positive probability, NEV plaintiffs would be attracted to that offer. Thus, the inclusion of plaintiffs with NEV suits causes the rejection rate to rise to 100%. As noted earlier, this is analogous to a result in Sobel (1989). As Sobel notes (p. 149), this result indicates a discontinuity in the model. As the fraction of NEV plaintiffs approaches 0, the outcome of the game does not approach the outcome of the game where the probability of a NEV suit is 0.

Note that other equilibria exist in which the rejection rate is 100%. For example, it would also be an equilibrium strategy for NEV players to make revealing offers consistent with Equation 3. The defendant would reject this offer, and the NEV players would drop the case and receive a payoff of zero. Since not filing also yields a payoff of zero, it is consistent with equilibrium for NEV players to make the offer in Equation 3. This equilibrium has the same qualitative characteristics as the equilibrium presented in Proposition 1 in that all offers are rejected in equilibrium. Using the same arguments as RW (1986, p. 566) it is possible to show that pure pooling and semipooling equilibria require out-of-equilibrium beliefs that are not consistent with DI. (6) Thus, equilibria that satisfy D1 must be along lines of the separating equilibrium presented in Proposition 1.

Note that when all plaintiffs have a credible threat to proceed to trial (as in the RW model), it is not an equilibrium strategy for all offers to be rejected. In particular, the boundary condition discussed earlier requires that the lowest plaintiff type have her offer accepted with 100% probability.

Equilibrium with a Positive Filing Fee

The key to restoring an equilibrium with settlement is to assume that the plaintiff incurs a positive filing cost, [C.sub.0] > 0. This parameter may reflect more than just filing costs as the plaintiff may incur significant legal expenditures prior to an actual trial. For the purpose of the following analysis, we assume that [C.sub.0] [less than or equal to] [C.sub.P] + [C.sub.D]. (7) When some potential plaintiffs have NEV suits, the boundary condition [phi]([O.sub.P]) = 0 no longer holds, and we need to consider the entire family of rejection functions in which a [member of] [0, 1]. Also note that adding a filing fee, [C.sub.0] < [C.sub.P] + [C.sub.D], will have no effect on the equilibrium of the RW model in the absence of NEV suits.

With [C.sub.0] > 0, we have the following proposition:

PROPOSITION 2. When [C.sub.0] > 0, there exists a perfect Bayesian equilibrium for the game described by steps 1-6 with the following properties:

(i) Plaintiffs file suit, iff they are of type J [greater than or equal to] [C.sub.P/p].

(ii) Plaintiffs who file suit make the offer given by Equation 3.

(iii) Defendants reject offers [O.sub.P] [greater than or equal to] [O.sub.P] with the probability given by Equation 4, where a = [C.sub.0]/([C.sub.P] + [C.sub.D]).

(iv) Offers less than [O.sub.P] = [C.sub.P] + [C.sub.D] occur only out of equilibrium. Defendants believe with probability 1 that such offers are made by type J < [C.sub.P/p] players and always reject such offers.

(v) Offers [O.sub.P] > [pJ.sup.H] + [C.sub.P] occur only out of equilibrium. Defendants always reject these offers.

PROOF. (i) First, we will show that all plaintiffs with a credible threat to proceed to trial (J [greater than or equal to] [C.sub.P/p]) will file suit by showing that the lowest plaintiff type to file suit is [J.bar] = [C.sub.P/p]. The lowest plaintiff type to file suit offers [[O.sub.P].bar] = p[J.bar] + [C.sub.D]. By Equation 4, this offer is rejected with probability 1 - a. For the borderline plaintiff type [J.bar] we have

[V.sub.P]([J.bar]) = a(p[J.bar] + [C.sub.D]) + (1 - a)(p[J.bar] - [C.sub.P]) - [C.sub.0] = 0. (6)

Recall that a = [C.sub.0]/([C.sub.P] + [C.sub.D]), and solve for [J.bar] to find [J.bar] = [C.sub.P/p].

Next consider plaintiffs with NEV suits (J < [C.sub.P/p]). What offer would these players make if they did file suit? If their offer is rejected, they will drop their case, so they choose their offer to maximize

[V.sup.NC.sub.P] = (1 - [phi]([O.sub.P]))[O.sub.P], (7)

where [psi]([O.sub.P]) is the probability an offer is rejected, and NC indicates that this is a plaintiff who does not have a credible threat to proceed to trial. Under the proposed equilibrium, 1 - [phi]([O.sub.P]) = aexp[-([O.sub.P] - [[O.sub.P].bar])/([C.sub.P] + [C.sub.D])I. Using this in Equation 7 and maximizing [V.sup.NC.sub.P] with respect to [O.sub.P] reveals that plaintiffs of type J < [C.sub.P/p] will offer (8)

[O.sup.NC.sub.P] = [C.sub.P] + [C.sub.D]. (8)

We now need to show that a plaintiff of type J < [C.sub.P/p] will choose not to file their case under the proposed equilibrium. If these plaintiff types file, they make the offer [O.sup.NC.sub.P] = [C.sub.P] + [C.sub.D], which is rejected with probability 1 - aexp[-([O.sub.P] - [[O.sub.P].bar])/([C.sub.P] + [C.sub.D])]. Since [O.sup.NC.sub.P] = [[O.sub.P].bar], the rejection probability simplifies to 1 - a. As a result, the expected payoff for filing a suit (inclusive of filing costs [C.sub.0]) for all plaintiffs such that J < [C.sub.P/p] is

[V.sup.NC.sub.P] = a([C.sub.P] + [C.sub.D]) - [C.sub.0]. (9)

To ensure that these plaintiffs will not file suit, we must have [V.sup.NC.sub.P] [less than or equal to] 0 for J < [C.sub.P/p]. Setting [V.sup.NC.sub.P] = 0 yields a = [C.sub.0]/([C.sub.P] + [C.sub.D]). Thus, the proposed equilibrium is consistent with plaintiffs of type J < [C.sub.P/p] not filing suit. This establishes part (i) of the proposition.

(ii) and (iii) Since only plaintiffs of type J [greater than or equal to] [C.sub.P/p], file suit in equilibrium, the actions specified for plaintiffs (under (ii)) and defendants (under (iii)) correspond to the equilibrium described in section 3, where it was assumed that J > [C.sub.P/p] for all plaintiffs. This behavior has already been verified as being consistent with equilibrium.

(iv) As shown in the Appendix, the out-of-equilibrium beliefs specified for the defendant in part (iv) are consistent with the refinement concept D1. It is optimal to reject offers [O.sub.P] < [[O.sub.P].bar] based on these beliefs, because plaintiffs of type J < [J.bar] do not have a credible threat to pursue the case to trial and will drop the case if their offer is rejected. (9)

(v) It is a dominant strategy to reject an out-of-equilibrium offer, [O.sub.P] > [pJ.sup.H] + [C.sub.P].

The analysis above confirms the equilibrium in Proposition 2. As shown in the Appendix, the value of a in our solution is the unique value that is consistent with the refinement concept D1. When [C.sub.0] = 0, the equilibrium described in Proposition 2 matches the equilibrium described in Proposition 1. Thus, while adding NEV suits introduces a discontinuity into the model (see the discussion above under Zero Filing Fee), no additional discontinuities are introduced when a filing fee is added to the model.

As discussed below, the settlement rate is lower in the model with NEV suits than in the RW model without NEV suits. First, however, we will discuss in more detail the role of out-of-equilibrium beliefs and how they relate to the refinement concept D1.

Out-of-Equilibrium Beliefs and Equilibrium Refinements

An offer [O.sub.P] < [[O.sub.P].bar] occurs only off the equilibrium path, and we specify beliefs such that these offers are believed to come from plaintiffs with NEV suits. On the equilibrium path of the game, once we reach the point at which a suit is filed, the probability that a player has a NEV suit falls to 0. This may tempt us to conclude that the defendant should place zero probability that an out-of-equilibrium offer made subsequent to filing comes from a player with a NEV suit; however, this belief is not ruled out in a perfect Bayesian equilibrium. Osborne and Rubinstein (1994, p. 236) note specifically that a perfect Bayesian equilibrium does not rule out reversal of zero-probability beliefs off the equilibrium path of the game. (10)

In the Appendix we show that the out-of-equilibrium beliefs we specify are consistent with the refinement concept D1. Under D1 the defendant must believe that an out-of-equilibrium offer is made from the party most likely to benefit from such an offer. As we show in the Appendix, a low out-of-equilibrium offer is most likely made by a plaintiff with a NEV suit. Using the same arguments as Reinganum and Wilde (1986, p. 566) the refinement D1 can be used to eliminate all pooling and semipooling equilibria. (11) Thus, as in Reinganum and Wilde (1986), the separating equilibrium we present is consistent with this refinement, while all other possible equilibria are not. (12)

Discussion of the Model Results

Negative expected value suits are not filed in the equilibrium of this model. Nevertheless, the potential presence of these cases results in an equilibrium under which the highest feasible settlement rate is lower than when no potential NEV suits exist. Further, if Co is small relative to [C.sub.P] + [C.sub.D], the reduction in the maximum feasible settlement rate will be quite large. From Proposition 2, it is easy to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where [S.sub.N] denotes the settlement rate in the model when NEV suits are possible. From the baseline model, we can show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

where [S.sub.B] is the settlement rate.

In both models a reduction in [J.bar] lowers the settlement rate, but in the model with NEV suits, [J.bar] is determined endogenously. To allow a clear comparison of the dispute rates between the two models, consider the baseline model where [J.bar] = [C.sub.P/p], that is, where the weakest case is at the borderline between a credible and noncredible threat to trial. If NEV suits are then added to this distribution, we have [S.sub.N] = a[S.sub.B], recalling that a = [C.sub.0]/([C.sub.P] + [C.sub.D]) < 1. Thus, the addition of NEV suits unambiguously reduces the settlement rate. Furthermore, if [C.sub.0] is small relative to [C.sub.D] + [C.sub.P], the reduction in the settlement rate may be substantial.

There are several points worth discussing in a comparison of the settlement rates between the two models. First, Co may be substantial if significant legal expenses are incurred prior to the submission of the offer to the defendant. The later in the process this final offer occurs, the larger is [C.sub.0], and the smaller is [C.sub.P] + [C.sub.D]. (13) However, it is clear that the drop in settlement rates relative to the RW model may potentially be quite large.

This does not imply that the results of the model with NEV suits are relevant for all civil litigation. From the defendant's perspective, the distribution f(J), including the lower support [J.sub.L], is case specific. (14) For many cases, we may have [J.sub.L] > [C.sub.P/p], and NEV suits will not be an issue. For example, if the plaintiff was injured by the defendant in an auto accident and required extensive hospitalization, there may be uncertainty regarding the exact extent of the damages, but it will be common knowledge that this is not a NEV suit. As a result, the baseline model will apply to this case. It is when the injury is harder to verify (e.g., back pain), that NEV suits may be present, and our model predicts that these cases will have a much lower settlement rate than under the baseline. Thus, our NEV analysis will only apply to a subset of all lawsuits filed.

In addition, Farmer and Pecorino (2005a) show that informed plaintiffs in the signaling model are willing to make costly voluntary disclosures if they face a sufficiently high probability of having their offer rejected. (15) The high dispute rates predicted by the model with NEV suits should strongly encourage voluntary disclosures. To the extent that these disclosures occur, they will lead to lower dispute rates.

If it is believed that voluntary disclosures will not occur for some reason, then the model suggests that there may be benefits from raising the filing cost. In general, the filing cost may be set either above or below the social cost of a trial, but if we treat it as a policy variable, there is no reason to believe that social costs will rise when the filing fee is increased. On the other hand, increases in the filing fee will lower the dispute rate and produce some social savings. (16) If the filing fee is raised to [C.sub.0] = [C.sub.P] + [C.sub.D], dispute rates will fall to their levels in the RW model. Furthermore, all PEV suits (pJ - [C.sub.P] > 0) would file under this fee. Further increases in the filing fee would discourage some PEV suits and would presumably not be desirable. (17)

5. Fee Shifting

Under the so-called English rule, the loser at trial pays the attorney fees of the victorious party. In the baseline RW model, fee shifting has no effect on the incidence of trial. The probability that fees will be shifted to the defendant is simply the probability p that the plaintiff prevails at trial; since p is common knowledge, the expected value of shifted fees is common knowledge. Thus, while fee shifting affects the plaintiff's offer, it does not affect the distance between the offers of adjacent types J and J + [DELTA]J. As a result, the rate of rejection for a type J plaintiff remains unchanged when fee shifting is added to the baseline RW model. (18)

It is straightforward to add fee shifting to our model with NEV suits. In the interests of brevity, we omit the details, which are provided in Farmer and Pecorino (2005b). (19) The only change in the structure of the game is that step 6' replaces step 6.

 6'. At trial, there is a finding for the plaintiff with probability
 p, in which case she receives the payoff J - [C.sub.0], while the
 defendant receives the payoff -(J + [C.sub.D] + [C.sub.P]).

 With probability 1 - p, the finding is for the defendant; in this
 case, the plaintiff receives the payoff -([C.sub.P]+ [C.sub.0] +
 [C.sub.D]), and the defendant receives a payoff of 0.

Note that the filing fee [C.sub.0] is not subject to shifting.

In the model with fee shifting, a plaintiff of type J who files suit makes the offer [O.sub.P] = p(J + [C.sub.P] + [C.sub.D]). The other key features of the equilibrium are that the rejection function is the same as Equation 4 with a = [C.sub.0]/([C.sub.P]+ [C.sub.D]), and that the borderline filer is the plaintiff type [J.bar] = (1 - p)([C.sub.P] + [C.sub.D])/p. (20) If we compare this to part (i) of Proposition 2, we see that more plaintiffs file suit under fee shifting if p[C.sub.P] > (1 - p)[C.sub.D]. When this inequality holds, the plaintiff expects to shift more fees to the defendant than the defendant expects to shift to the plaintiff. This encourages more plaintiffs to file. If p[C.sub.P]< (1 - p)[C.sub.D], the opposite is true and fewer plaintiffs file. (21)

Suppose, for example, that p[C.sub.P]> (1 - p)[C.sub.D] so that more plaintiffs file suit. The number of trials will increase because more plaintiffs file, and some of these plaintiffs will proceed to trial. However, there is another effect. From Equation 4, the probability of rejection increases in J - [J.bar], where [J.bar] is the borderline type. If [J.bar] is reduced because of fee shifting, then all plaintiffs who filed in the absence of fee shifting will face a higher rejection rate. If p[C.sub.P]< (1 -p)[C.sub.D], these conclusions are reversed, and the number of trials falls for both of the reasons discussed above.

Based on the figures cited in Spier (1992), [C.sub.P] equals 30% and [C.sub.D] equals 20% of the average trial award. Using this 3-to-2 ratio in expenditures, we can conclude that fee shifting will expand the set of plaintiffs who file suit if the probability of a plaintiff victory is greater than 40%. Conversely, it will reduce the set of plaintiffs who file suit if the probability of plaintiff victory is less than 40%. Expanding the set of plaintiffs who file suit will increase dispute rates (holding plaintiff type constant), assuming that [C.sub.P] and [C.sub.D] are held constant. However, it is well established in theory that fee shifting increases spending at trial by both parties to the dispute. (22) An increase in spending at trial would (other things held constant) reduce the set of plaintiffs who file suit and reduce dispute rates among the remaining set of plaintiffs who do file. While [C.sub.P] and [C.sub.D] are exogenous in our model, the effect of fee shifting on expenditure at trial needs to be taken into account in making a full evaluation of the policy.

6. Conclusion

Because it is one of the two canonical models of pretrial bargaining, we believe that is important to extend the Reinganum and Wilde (1986) model to allow for the presence of NEV suits. If it is to support an equilibrium with settlement in the presence of NEV suits, the RW model must be modified. In particular, we allow for the possibility of such suits and introduce positive filing costs paid by the plaintiff prior to trial. In the RW model, the lowest plaintiff type has her offer accepted with probability 1. When NEV suits are present in the distribution of plaintiff types, this probability of acceptance must fall sufficiently below 1 to discourage the filing of NEV suits. Thus, while no NEV suits are filed in equilibrium, the potential of such suits can significantly increase the dispute rate in a signaling model.

When NEV suits are present in the distribution of plaintiff types, an increase in the plaintiff's filing fee will reduce the dispute rate. The presence of NEV suits also affects the analysis of fee shifting in the signaling model. In the baseline RW model, fee shifting has no effect on the probability of reaching a pretrial settlement. By contrast, we find that fee shifting affects both the filing decision by the plaintiff and the probability that a case of a given quality settles. The direction of these effects depends on whether the plaintiff expects to shift fees to the defendant on net. If she does, then both filing and rejection rates increase. If the plaintiff expects to have fees shifted to herself on net, then both filing and rejection rates decline.

By allowing for the existence of NEV suits, our model represents an important extension of the signaling model of litigation. This represents one of the two canonical models of pretrial settlement in which asymmetric information plays a vital role. More generally, pretrial bargaining may be plagued by two-sided informational asymmetries. The solution to these models generally has a strong signaling element. Thus, the analysis in this paper should serve as an important input into future extensions of these models that consider the presence of NEV suits.

Appendix

The equilibria we propose in Proposition 2 satisfies the D1 refinement, and it is the only equilibrium that does so. The D1 refinement is due to Cho and Kreps (1987). Under D1 the defendant must believe that an out-of-equilibrium offer is made by the player type that is most likely to benefit from the offer. This applies to our equilibria in the following way: The out-of-equilibrium action is an offer such that [O.sub.P] < [[O.sub.P].bar]. The potential response by the defendant is the probability (not just the equilibrium probability) with which the defendant might choose to accept the offer. Denote this probability as q. What we are interested in is the set of all possible probabilities such that a given plaintiff is at least as well off by making the out-of-equilibrium offer as she would be had she chosen her equilibrium action. This set will have the form [[q.sup.*], 1], where [q.sup.*] is a critical acceptance probability that makes a given plaintiff type indifferent between her equilibrium strategy and a given deviation [O.sub.P] < [[O.sub.P].bar]. Under D1 if the critical value [q.sup.*] for type 1 is higher than the critical value for type 2, then the probability that the out-of-equilibrium offer comes from type 1 must be set to zero. For out of equilibrium offers, if the critical value of q is higher for plaintiffs with PEV suits than for plaintiffs with NEV suits, D1 requires that the defendant believe the offer is coming from a NEV plaintiff.

A.1. The Equilibrium in Proposition 2 Satisfies D1

In equilibrium NEV plaintiffs receive a payoff of 0. Given a probability q that an offer [O.sub.P] < [[O.sub.P].bar] is accepted, an NEV plaintiff's expected gain from deviating to [O.sub.P] is

B = q([O.sub.P] - [C.sub.0]) (1 - q)[C.sub.0] = q[O.sub.P] - [C.sub.0]. (A1)

When the offer is rejected, a NEV plaintiff drops the case and receives a payoff of - [C.sub.0], which is the filing cost. Note that all NEV plaintiffs will benefit in precisely the same fashion for a given offer since all types receive the same payoff if a given offer is accepted and they all drop the suit and receive -[C.sub.0] if it is rejected. The set of q for which any NEV plaintiff is willing to make an out-of-equilibrium offer [O.sub.P] < [[O.sub.P].bar] is ([C.sub.0]/[O.sub.P], 1].

Now consider a plaintiff J [greater than or equal to] [C.sub.P/p]. These plaintiffs file suit in the equilibrium of the model. The net benefits of this player making an offer [O.sub.P] < [[O.sub.P].bar] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A2)

The first two terms represent the expected payout from submitting an offer [O.sub.P] < [[O.sub.P].bar], which is accepted with probability q and is rejected with probability 1 - q. When the offer is rejected, the plaintiff receives the expected trial payout based upon her true type J. The second two terms represent the benefits from submitting the equilibrium offer pJ + [C.sub.D], which will be accepted with probability a exp[-p(J - [J.bar])/([C.sub.P] + [C.sub.D])], where a = [C.sub.0]([C.sub.D] + [C.sub.P]).

Notice that for J = [C.sub.P/p], Equation A2 collapses to Equation A1 so there is no discontinuity in the expected benefits as J varies. The borderline type J = [C.sub.P/p] and all NEV types receive identical benefits from a given deviation; as a result, Equation A2 captures the benefits for all types as J varies where all NEV types receive the benefit associated with J = [C.sub.P/p]. From Equation A2 we can determine the values of q such that there are positive expected benefits from deviation to the offer [O.sub.P] < [[O.sub.P].bar]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A3)

Setting Equation A3 as an equality determines the critical value of q (denoted [q.sup.*]) for which type [J.bar] will deviate. Substitute a = [C.sup.0]/([C.sub.D] + [C.sub.P]) to write this critical value as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A4)

Equation A4 defines the critical probability for which a given type J would prefer to deviate to a given offer [O.sub.P] < [[O.sub.P].bar]; the set of such probabilities is ([q.sup.*], 1]. Differentiating Equation A4 with respect to J tells us if the set is expanding or shrinking as J rises. Performing this differentiation yields

[partial derivative][q.sup.*]/[partial derivative]J > 0 iff 1 > [O.sub.P] - pJ + [C.sub.P]/[C.sub.P]+ [C.sub.D]. (A5)

Since [O.sub.P] < [[O.sub.P].bar] and [[O.sub.P].bar] = p [J.bar] - [C.sub.P], this condition is guaranteed to hold for all J [greater than or equal to] [C.sub.P/p], that is, for all J who file in equilibrium. Recall that all NEV plaintiffs (who do not file in equilibrium) have expected benefits of deviation that are identical to the borderline type. As J rises above the borderline type, the set of probabilities for which placing a low out-of-equilibrium offer is beneficial falls. Thus, the set of probabilities that would induce a PEV plaintiff to make such an offer is a subset of probabilities that would induce a NEV or borderline person to submit the same offer. As a result, the refinement DI assigns a zero probability that this offer comes from a PEV plaintiff. Since the critical value of q is the same for NEV plaintiffs and for the borderline type, it is consistent with D1 for the defendant to believe that a low out-of-equilibrium offer is made by a NEV plaintiff.

A.2. The Value of a We Specify Is the Unique Value That Satisfies D1

The demonstration in A.I that our equilibrium is consistent with D1 relied heavily on the fact that in equilibrium all PEV suits (i.e., J [greater than or equal to] [C.sub.P/p]) file and all NEV suits (i.e., J < [C.sub.P/p]) do not. Clearly, values of a > [C.sub.0]/([C.sub.D] + [C.sub.P]) are not consistent with equilibrium, because they induce NEV plaintiffs to file. What we will now demonstrate is that values of a < [C.sub.0]/([C.sub.D] + [C.sub.P],) are not consistent with D1.

First, it is trivial to show that d[J.bar]/da < 0, which means that as a falls, the type of the borderline filer rises. When a = [C.sub.0]/([C.sub.D] + [C.sub.P]), the borderline filer is also the borderline between NEV and PEV lawsuits. If the value of a falls below [C.sub.0]/([C.sub.D] + [C.sub.P]), some PEV suits will fail to file. When there are PEV suits among the nonfilers, then it is these plaintiffs who are the most likely to benefit from an out-of-equilibrium offer [O.sub.P] < [[O.sub.P].bar]. If they do not file, their payoff is 0, so the net benefit from deviating to an offer [O.sub.P] < [[O.sub.P].bar] for such a plaintiff is

B = q[O.sub.P] + (1 q)(pJ - [C.sub.P]) - [C.sub.0]. (A6)

Since pJ > [C.sub.P] for these plaintiffs, the benefit from deviation always exceeds the benefit for NEV plaintiffs given in Equation A1. Further, the benefit is increasing in their type J. Thus, DI requires the defendant to believe an out-of-equilibrium offer [O.sub.P] < [[O.sub.P].bar] comes from the highest type that does not file. Denote this type [J.sup.HN]. Given the defendant's beliefs, any offer [O.sub.P] < p[J.sup.HN] + [C.sub.D] must be accepted with probability 1. This implies a discrete increase in the acceptance probability in the neighborhood of the borderline type. This would destroy a potential equilibrium, because low plaintiff types will want to deviate from their equilibrium offers to an offer just below p[J.sup.HN] + [C.sub.D]. Thus, values of a < [C.sub.0]/([C.sub.D] + [C.sub.P]) are not consistent with the refinement D1, because they imply that some plaintiffs with PEV suits do not file in equilibrium. (Our use of the term PEV is somewhat abusive, but we simply mean that J > [C.sub.P/p], which implies a credible threat to proceed to trial, conditional on having filed suit.)

Received January 2006; accepted December 2006.

We would like to thank Andrew Daughety, Jennifer Reinganum, and an anonymous referee for making helpful comments on this paper. We would also like to thank participants at the 2003 American Law and Economics Association Meeting in Toronto and the 2004 American Economic Association Meeting in San Diego.

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(1) We find this under the requirement that the equilibrium satisfy the refinement concept DI. This refinement concept is discussed extensively later in the paper. All of the model solutions we consider are consistent with D1. The refinement D1 is due to Cho and Kreps (1987).

(2) However, when contingency fees are used, they find that fee shifting generally will affect the probability of settlement. See RW (1986, pp. 562 3).

(3) Other papers in the NEV literature include Bebchuk (1996), who shows how a plaintiff with a NEV suit can have a credible threat to proceed to trial if litigation costs are spread out over time. Klement (2003) extends this work by adding asymmetric information to Bebchuk's model. In his model the defendant is informed, and this asymmetric information tends to undermine the credibility of the plaintiff to proceed to trial, even in the face of divisible costs. Farmer and Pecorino (1998) analyze N EV suits in a repeated game setting where an attorney can acquire a reputation for bringing such suits to trial.

(4) Other litigation models with two-sided informational asymmetries include Schweizer (1989) and Daughety and Reinganum (1994).

(5) It is a dominant strategy to accept any offer below [[O.sub.P].bar], so these offers will be accepted with probability 1. Thus, if [[O.sub.P].bar] were rejected with any positive probability, the lowest plaintiff type would deviate by offering slightly less than [[O.sub.P].bar]. See the discussion in RW (1986, p. 565).

(6) There can be no semipooling among plaintiffs who file suit. In the equilibrium described in Proposition 1, NEV plaintiffs pool on "do not file," but the beliefs supporting this equilibrium are consistent with D1.

(7) In note 13, we discuss the case [C.sub.0] > [C.sub.P] + [C.sub.D].

(8) Note from Equation 3 that this is the same offer that is made by the borderline plaintiff [J.bar] = [C.sub.P/p].

(9) This is a key difference with the RW model. Since the credibility constraint is never binding in their model, it is a dominant strategy for the defendant to accept offers such that J < [J.bar]; as a result, the model requires the boundary condition [psi]([[O.sub.P].bar]) = 0. This condition does not apply to the equilibrium of our model.

(10) Osborne and Rubinstein (1994, p. 236) note specifically that a perfect Bayesian equilibrium does not rule out reversal of zero-probability beliefs off the equilibrium path. They make this statement while discussing a sequential equilibrium. Since a sequential equilibrium places more restrictions on beliefs than a perfect Bayesian equilibrium, if these beliefs are not ruled out with a sequential equilibrium, they will not be ruled out by a perfect Bayesian equilibrium either.

(11) Reinganum and Wilde use the refinement "'universally divine equilibrium" (Banks and Sobel 1987). In the context of this model, equilibria that satisfy D1 will also satisfy universal divinity, but this is not true in general. It is a little easier to work with D1.

(12) Our use of the terms pooling, semipooling, and separating refers to the behavior of those plaintiffs who file suit. The equilibrium we describe is semipooling, in the sense that NEV players pool on the action, "do not file suit."

(13) As [C.sub.0] approaches [C.sub.P] + [C.sub.D] from below, the settlement rates approach those in the RW model. For [C.sub.0] > [C.sub.P] + [C.sub.D], some PEV suits (in the sense pJ - [C.sub.P] > 0) will fail to file, and settlement rates will exceed those found in RW. This would be true, even in the absence of NEV suits. When [C.sub.0] takes on such a high value, NEV suits are irrelevant, as it never pays to file such a suit.

(14) Thus, the distribution of judgments is itself drawn from a distribution. This could be indicated in the model by adding a superscript to f(J).

(15) Farmer and Pecorino (2005a) is an extension of earlier work by Shavell (1989) and Sobel (1989).

(16) In this model the defendant pays a plaintiff of type J, pJ + [C.sub.D] regardless of whether or not the case settles. Thus, the defendant's incentive for care will not be affected by changes in the dispute rate. If only the defendant's actions matter in determining the plaintiff's injury, we can conclude that reductions in the dispute rate will increase social welfare in this model.

(17) To the extent that Co reflects more than just filing costs, this would have to be taken into account in setting the fee.

(18) Behchuk (1984) finds that fee shifting raises the dispute rate in his screening model. In a screening model, Polinsky and Rubinfeld (1998) find that greater use of fee shifting causes more low probability of prevailing plaintiffs to proceed to trial. This is a small sample of what is a very large literature on fee shifting.

(19) Farmer and Pecorino (2005b) is the working paper version of this paper.

(20) This implies (once again) that no NEV suits are filed in equilibrium. In the model with tee shifting, plaintiffs have a credible threat to proceed to trial if pJ > (1 - p)([C.sub.P] + [C.sub.D]).

(21) These effects of fee shifting on the filing decision were first analyzed by Shavell (1982).

(22) See, among others, Braeutigam. Owen. and Panzar (1984).

Amy Farmer * and Paul Pecorino ([dagger])

* Department of Economics, University of Arkansas, Fayetteville, AR 72701 USA; E-mail amyf@wahon.uark. edu.

([dagger]) Department of Economics, Finance and Legal Studies, University of Alabama, Box 870224, Tuscaloosa, AL 35487 USA; E-mail ppecorin@cba.ua.edu; corresponding author.