Dispute rates and contingency fees: an analysis from the signaling model.

• Author: Farmer, Amy

1. Introduction

   Contingency lee contracts, under which the plaintiff pays her lawyer a percentage of the judgment if she wins at trial and nothing if she loses, are very common in the United States but are banned or severely restricted in many other countries. (1) Their use in the United States generates great controversy, and while their use is restricted in some states, further calls for restrictions on contingency fee contracts are often heard. (2) Although contingency fee contracts are quite simple, their effects on the litigation process are complex and wide ranging. Previous studies have considered the effect of contingency fees on the frequency of litigation (Danzon 1983: Miceli and Segerson 1991) and nuisance suits (Miceli 1993, 1994) and on various aspects of the lawyer-client relationship (Miller 1987: Dana and Spier 1993; Rubinfeld and Scotchmer 1993; Watts 1994: Hay 1996, 1997). In this article, we focus on the effects of contingency fees on the selection of disputes for trial and the overall dispute rate in the signaling model of Reinganum and Wilde (1986). (3) Along with Bebchuk (1984), this is one of the two canonical models of pretrial settlement. In addition, since the solution to a model with two-sided informational asymmetries has a strong signaling element (Daughety and Reinganum 1994), it is particularly important to understand how contingency fees affect dispute rates in the signaling model.

   As in the standard literature on pretrial settlement (e.g., Bebchuk 1984; Reinganum and Wilde 1986), we assume that the plaintiff controls all aspects of the case. (4) In the Reinganum and Wilde model, an informed plaintiff makes a take-it-or-leave-it offer to an uninformed defendant who rejects these offers with some probability in equilibrium. In order to conduct a complete analysis of the contingency fee in a signaling model, we also consider the case in which an informed defendant makes a single offer to the uninformed plaintiff.

   Reinganum and Wilde first develop a signaling model in which the plaintiff's lawyer is paid via a fixed fee and then go on to present (pp. 562-3) a very general solution to the model in the presence of a contingency contract. We use the general framework of Reinganum and Wilde to focus on specific forms for the contingent lee contracts. Our contribution lies not in the derivations of the model solutions, as these follow from the Reinganum and Wilde framework rather closely. Instead, our contribution lies in the analysis of how these specific contingency fee contracts affect pretrial settlement patterns. We analyze the effects both on overall dispute rates and on the selection of disputes for litigation. In section 2, we will further note the relationship between our work and the earlier analysis provided by Reinganum and Wilde.

   We examine two types of contracts in this article, a bifurcated contingency fee and a unitary contingency fee. Under a unitary contract, the same contingency payment is made by the client to her lawyer regardless of whether there is an out-of-court settlement or a victory at trial. (5) Under a bifurcated fee, the contingency percentage at trial is generally higher than the percentage for cases that settle. (6) While unitary fees appear to be more common, a significant use of bifurcated fees has been noted in the literature. (7) In a model with attorney moral hazard, Hay (1997) notes that bifurcated fees are generally optimal from the perspective of the plaintiff. On the other hand, Bebchuk and Guzman (1996) argue that a unitary fee can lead to a larger net settlement for the plaintiff. Since both types of contracts appear to be relevant empirically, we examine both in this article.

   One robust finding from our article is that unitary fees lead to an unambiguous increase in the incidence of trial relative to both the fixed fee contract and the bifurcated contingency fee contract. Since the plaintiff's lawyer is paid the same percentage fee whether there is a plaintiff victory at trial or a pretrial settlement, the joint cost of proceeding to trial for the plaintiff and defendant is lower under this unitary contingency tee. It is well known that under a unitary fee, the lawyer has an excessive incentive to settle relative to the interests of the client. (8) By contrast, when the client controls the case, she has an excessive incentive to bring the case to trial.

   Under a bifurcated contingency contract, our results depend on the nature of the signaling model. In the model where an informed plaintiff makes the offer, the use of the contingency fee contract has an ambiguous effect on the overall incidence of trial but does have a clear effect on the selection of disputes for litigation. The use of a bifurcated contingency fee contract tilts the rejection function so that more low-stakes cases proceed to trial compared with a fixed fee. In addition, there may be some presumption that the use of the bifurcated fee raises the overall incidence of trial. In the model where an informed defendant makes the offer, we find that for reasonable parameter values, the use of the bifurcated fee contract unambiguously reduces the incidence of trial.

   In our article, we treat the terms of the contract between the plaintiff and her lawyer as exogenous and compare settlement rates across different contracts. This type of comparison is valid in a policy context, where there are proposals to ban contingency fee contracts. Such a ban would (exogenously) force plaintiffs to switch to fixed fee contracts. An even deeper understanding of the effects of the contingency fee on settlement will ultimately require a model that derives these contracts as the solution to an optimization problem while embedding the analysis in a model of settlement. The analysis of these and other aspects of the lawyer-client relationship are beyond the scope of this article.

2. A Signaling Model with an Informed Plaintiff

   We present the derivation of the model solutions for the convenience of the reader, but it should be noted that these derivations either exactly follow the Reinganum and Wilde analysis (the fixed fee case), are special cases of their solutions (the bifurcated fee), or could be derived in a relatively straightforward way from their analysis (the unitary fee). Our contribution is to focus on specific forms of the contingency fee contract and to analyze how these contracts affect dispute rates.

   The Fixed Fee

   We start with the fixed fee analysis of Reinganum and Wilde. In this model, the plaintiff has private information concerning the damages, J. The defendant knows that J is distributed by f(J), where [??] and [bar.J] 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 attorney fees of the plaintiff and defendant, respectively. It is assumed that p[J.bar] > [C.sub.P] so that all plaintiffs have a credible threat to proceed to trial. The informed plaintiff makes a single take-it-or-leave-it offer to the defendant. The model is summarized as follows:

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

   (ii') The plaintiff hires a lawyer under a contract in which she pays [C.sub.P] if the case proceeds to trial and 0 if the case settles prior to trial.

   (iii') The plaintiff makes a single take-it-or-leave-it offer [O.sub.P] to the defendant.

   (iv') If the defendant accepts the offer, the plaintiff receives a payoff of [O.sub.P], while the defendant receives -[O.sub.P]. If the defendant rejects the offer, the case proceeds to trial.

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

   There are potentially many equilibria in this signaling game, but Reinganum and Wilde use refinement arguments to eliminate all but a separating equilibrium in which 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 the probability [phi]([O.sub.P]). (9) In equilibrium, the 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 in order to maximize his expected wealth [V.sub.P], which can be written

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

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

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

   The function B([O.sub.P]) describes the defendant's beliefs about plaintiff's type as a function of his 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:

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

   An out-of-equilibrium offer [O.sub.P] < [[O.sub.P].bar] = p[J.bar] + [C.sub.D] is believed to be made by player type [J.bar] and is accepted with probability 1. An out-of-equilibrium offer Op > p) + Co is believed to be made by player type [bar.J] and is rejected with probability 1. Analogous out-of-equilibrium beliefs and actions apply to all the models we analyze in this article.

   Using the boundary condition that [phi]([[O.sub.P].bar]) = 0, the differential equation in...