Negative expected value suits in a signaling model.

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.