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

Pecorino, Paul

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 Equation 2 may be solved to obtain (10)

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

where [psi]([O.sub.P]) = ([O.sub.P] - [[O.sub.P].bar])/([C.sub.P] + [C.sub.D]). In this case the equilibrium probability of settlement is 1 - [psi]([O.sub.P]), or [e.sup.[psi]]. Substitute the equilibrium offer from Equation 3 to write the equilibrium probability of settlement under the fixed fee as

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

where the subscript F indicates that this is a model with a fixed fee and the subscript P denotes that this is a game in which the plaintiff makes the offer. In equilibrium, higher plaintiff types must have their offers rejected more frequently in order 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. Note that the probability of settlement is increasing in the joint costs of trial, [C.sub.P] + [C.sub.D]. As seen in the following analysis, one important avenue by which contingency tee contracts affect pretrial settlement is through their effect on the joint cost of proceeding to trial.

The Bifurcated Contingency Fee

The introduction of a contingency fee into the model raises several issues relating to the nature of the contingency contract and the agency relationship between the lawyer and her client. We will make our modeling choices in such a way as to be consistent with the assumptions in the model with the fixed fee. As a result, we will analyze a bifurcated contingency fee first. However, we later analyze the model with a unitary contingency fee contract.

First, in Reinganum and Wilde, it is assumed that the client controls all aspects of the case. We will maintain that assumption and ignore any possible conflict between the lawyer and client. Second, Reinganum and Wilde assume that legal costs are independent of the size of the judgment and are incurred only at trial. Thus, we will assume that the plaintiff's lawyer incurs the cost [C.sub.P] only when cases reach trial and that the bifurcated contingency percentage, denoted [[theta].sup.B], is paid only if the case reaches trial and the plaintiff receives a positive award. It would not change the character of the results to assume that some court costs are incurred prior to trial and that the lawyer's contingency percentage is lower for cases that settle than for cases that proceed to trial. (11) Thus, step ii of the game described in the previous section is replaced by

(ii') The plaintiff hires a lawyer under a contract in which she pays the lawyer [[theta].sup.B]J if the case reaches trial and there is a finding for the plaintiff. The lawyer receives 0 if the case settles prior to trial or if the plaintiff loses at trial.

Finally, we assume that at the time the lawyer accepts the plaintiff's case, she can observe the distribution of types from which the plaintiff is drawn but not the plaintiff's exact type. (12) In order to keep expected attorney's fees equal to those paid in the fixed fee model, we assume the contingency percentage [[theta].sub.B] is set such that the lawyer earns [C.sub.P] on average for all cases that proceed to trial. This average depends on the distribution of plaintiff types and the equilibrium rejection function derived in the following analysis. (13)

When a trial occurs, a plaintiff of type J expects to receive p(1 - [[theta].sup.B])J. The plaintiff chooses an offer to maximize

(5) [V.sub.P] = [phi]([O.sub.P])[p(1 - [[theta].sup.B])J] + (1 - [phi]([O.sub.P]))[O.sub.P],

where, as before, [phi]([O.sub.P]) is the rejection function. The first-order condition from the maximization of Equation 5 is

(6) [phi]'([O.sub.P])[p(1 - [[theta].sub.B])J - [O.sub.P]] + (1 - [phi]([O.sub.P])) = 0.

To make the defendant indifferent between acceptance and rejection, the equilibrium offer must continue to satisfy Equation 3. Defendant beliefs are again correct in equilibrium, so the offers described by Equation 3 will be fully revealing of the plaintiff's type. A rejection function that satisfies the diffemntial equation in Equation 6 is

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

Equation 7 is a special case of general solution presented by Reinganum and Wilde (1986) in their Equation 10. (14) Since the equilibrium offers satisfy Equation 3, Equation 7 implies an equilibrium probability of settlement for a type J plaintiff under the bifurcated lee equal to

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

What we would like to do is compare the settlement function under the bifurcated fee in Equation 8 to the settlement function under the fixed tee in Equation 4. First note that under both settlement functions, the weakest plaintiff type settles with probability 1: [S.sub.BP]([J.bar]) = [S.sub.FP]([J.bar]) = 1. Next, taking the derivative of these functions with respect to J and converting to a percentage change (denoted by "^") we get the following:

(9) [[??].sub.FP] = -p/[C.sub.P] + [C.sub.D]

(10) [[??].sub.EP] = -p/[[theta].sub.B] [p.sub.J] + [C.sub.D].

Since the contingency fee percentage is based on an average of plaintiff types, for low values of J it will be the case that p[[theta].sup.B]J < [C.sub.P]. As a result, the settlement rate under the bifurcated lee initially declines in percentage terms more sharply than the settlement rate under the fixed fee. The rate of decline of settlement under the fixed tee does not exceed the rate of decline under the bifurcated fee until J > [C.sub.P]/p[[theta].sup.B]. For values of J above this, the rate of settlement declines faster under the fixed fee. At the point where J = [C.sub.P]/p[[theta].sup.B], the ratio of the settlement rates [S.sub.FP]/[S.sub.BP] > 1 is maximized; that is, at this point the incidence of trial under a bifurcated fee relative to the fixed lee is the greatest. Thus, it is only possible for [S.sub.BP] [greater than or equal to] [S.sub.FP] at some value of J such that J > [C.sub.P]/p[[theta].sup.B]. For a given upper support of the distribution [bar.J], there is no guarantee that there are any values of J such that [S.sub.FP]/[S.sub.BP] < 1. However, if [bar.J] is sufficiently large, the settlement functions will eventually cross so that there will be values of J such that [S.sub.BP] > [S.sub.FP]. The fact that the functions cannot cross until the expected contingency payment exceeds the fixed fee (p[theta]J > [C.sub.P]) suggests but does not prove that overall settlement rates will tend to be lower under the bifurcated contingency fee.

This analysis is summarized as follows:

RESULT 1: (i) Under a bifurcated fee, low-damage plaintiffs (J < [C.sub.p]/p[[theta].sup.B]) proceed to trial more frequently when compared to the fixed fee.

(ii) For a sufficiently large J, [S.sub.BP] > [S.sub.FP], but the crossing point must occur at a value of J such that J > [C.sub.P]/p[[theta].sub.B].

PROOF: This follows immediately from the fact that [S.sub.BP]([J.bar]) = [S.sub.FP]([J.bar]) = 1 and a comparison of Equation 9 with Equation 10. QED

Under a contingency fee system, low plaintiff types face a lower expected cost of pursuing trial than the same plaintiffs under a fixed fee. This reduces the cost of having offers rejected for these plaintiffs. Thus, as plaintiff type rises above [J.bar], the equilibrium rejection rate initially has to rise at a faster rate compared with the model with the fixed fee in order to induce the fully revealing offers described by Equation 3. As a result, a bifurcated contingency fee contract will produce more rejections (and more trials) among low-damage plaintiffs.

As type J increases, the expected cost of trial rises for the plaintiff, and the rate of increase in the rejection rate necessary for truthful revelation of type decreases. When J > [C.sub.P]/p[[theta].sub.B], the settlement rate under the contingency tee declines at a slower rate than under the fixed lee. At a sufficiently high value of J. the settlement rate under the bifurcated lee will be higher than under the fixed fee. The previous analysis suggests that less settlement will occur under the bifurcated fee, but it is not totally conclusive on this point. Importantly, the bifurcated contingency tee changes the selection of disputes since it causes more low-stakes cases to reach trial than under the fixed fee.

It is relatively straightforward to write down the set of equations that determine an equilibrium value of the contingency fee percentage [[theta].sup.B], that is, the value that compensates (in an expected value sense) the plaintiff's lawyer for the opportunity cost of her time. However, these equations will not yield an analytical solution. Therefore, we have omitted this analysis from the study. (15)

The Unitary Fee

While the use of bifurcated contracts is noted in the literature on contingency tees, most contracts are unitary in the sense that the contingency percentage is the same whether the case settles or the plaintiff wins at trial. In this section we assume that the contingency fee is paid whenever the plaintiff receives a payment, whether that be at trial or through a pretrial settlement. Denote the unitary contingency percentage as [[theta].sup.U].

Under this assumption, the plaintiff's objective function (Equation 5) now becomes

(11) [V.sub.P] = [phi]([O.sub.P])[p(1 - [[theta].sup.U])J] + (1 - [phi]([O.sub.P]))(1 - [[theta].sup.U])[O.sub.P].

This model is solved in a manner analogous to the earlier model. The plaintiff chooses an offer to maximize Equation 11. The equilibrium offer to the defendant continues to be described by Equation 3. The first-order condition from the maximization of Equation 11 plus Equation 3 implies the following rejection function in equilibrium:

(12) [phi]([O.sub.P]) = 1 - [e.sup.-[psi]] where [psi]([O.sub.P]) = p(J - [J.bar])/[C.sub.D].

Equation 12 can be derived in a relatively straightforward way from Equation 9 in the Reinganum and Wilde article. (16) The equilibrium probability, of settlement under the unitary fee is

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

A comparison of Equations 13 and 4 reveals that for all J > [J.bar] we have [S.sub.FP] > [S.sub.UP]. Take the derivative of Equation 13 with respect to J and convert to a percent change to obtain

(14) [S.sub.UP] = - P/[C.sub.D]

This leads to the following result:

RESULT 2: A unitary contingency tee increases the incidence of trial relative to both (i) a fixed fee contract and (ii) the bifurcated contingency fee contract.

PROOF: Part (i) follows directly from a comparison of Equation 13 to Equation 4. For part (ii), note that at J = [J.bar], [S.sup.UP] = [S.sub.BP] = 1. A comparison of Equations 14 and 10 shows that Sup declines at a faster rate than [S.sub.BP] for all Y. Thus, for all J > [J.bar], [S.sub.BP] > [S.sub.UP].

Note further that since the fixed fee [C.sub.P] in Equation 4 and the expected contingency payments at trial p[[theta].sup.B]J in Equation 8 should be sizable, the increase in the incidence of trial may be quite large. Since she now pays her lawyer regardless of whether the case proceeds to trial, the cost of proceeding to trial is lower for the plaintiff. For plaintiffs of type J to be discouraged from making an offer associated with type J + [DELTA] J, the rate of rejection must rise more steeply than in either of the two earlier models. As a result, for all J > [J.bar], the probability of acceptance is lower when a unitary contingency fee contract is used.

Fee Shifting

Both Donohue (1991) and Smith (1992) have analyzed the interactions between contingency fees and fee shifting in models along the lines of Shavell (1982) in which trials result when exogenous beliefs of the parties to the dispute diverge sufficiently. Both authors analyze fee shifting under the assumption that the defendant pays the plaintiff's lawyer [[theta].sup.B]J in the event that the plaintiff is victorious at trial. (17) We will follow this assumption but note some issues that it raises as well. We will also assume that the plaintiff rather than her attorney pays the defendant's attorney lees [C.sub.D] in the event the defendant wins at trial. (18) Fee shifting under which the loser at trial pays the legal fees of the victorious party is sometimes referred to as the English Rule.

We will analyze only the case of the bifurcated fee, but the analysis of the unitary fee yields similar conclusions. The general nature of the equilibrium is as described previously. As before, the plaintiff's offer must make the defendant indifferent between acceptance and rejection. Thus, the offer must reflect the probability of shifted fees as follows:

(15) [O.sub.P] = p(J + [C.sub.D] + [[theta].sup.B]J).

In equilibrium, a type J plaintiff will make the offer described in Equation 15. Thus, offers are fully revealing in equilibrium. The plaintiff's expected wealth may now be written

(16) [V.sub.P] = [phi]([O.sub.P])[pJ - (1 - p)[C.sub.D]] + (1 [phi]([O.sub.P]))[O.sub.P],

where, as before, [phi]([O.sub.P]) is the probability the offer [O.sub.P], is rejected by the defendant. Equation 16 reflects a bifurcated fee under which the plaintiff's lawyer is paid in the event of a plaintiff victory at trial but not in the event of a pretrial settlement. The plaintiff chooses [O.sub.P] to maximize Equation 16, taking the rejection function as given. Proceeding as before, the first-order condition to Equation 16 is solved by a differential equation of the form

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substitute Equation 15 into Equation 17 to see that the settlement rate with lee shifting (i.e., the English Rule) is

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A comparison of Equation 18 with Equation 8 implies the following:

RESULT 3: For a given value of the bifurcated contingency fee, [[theta].sup.B], fee shifting under which the loser at trial pays the winner's court costs unambiguously increases the incidence of trial.

PROOF: For a given value of the contingency lee [[theta].sup.B], Equation 18 < Equation 8 for each plaintiff type J > [J.bar].

Reinganum and Wilde show that lee shifting does not affect settlement in the model with the fixed fee. (19) The reason is that adjacent plaintiff types shift the same expected fee p[C.sub.P] to the defendant. To understand why fee shifting affects settlement rates when contingency tees are used, consider two adjacent plaintiff types J and J + [DELTA]J. Type ,I expects to shift p[[theta].sup.B]J and type J + [DELTA]J expects to shift p[[theta].sup.B](J + [DELTA]J). This difference in the expected value of shifted tees is reflected in the offer the plaintiff submits to the defendant. This increases the incentive of a plaintiff to "bluff" by submitting a higher offer than is associated with her type. Thus, with fee shifting, the rejection rate must rise more steeply in order to induce revealing otters as described by Equation 15.

To the extent that a lower dispute rate is considered a policy goal, this analysis suggests that the introduction of the English Rule may be undesirable in a system that makes extensive use of contingency fees (or vice versa). It is straightforward to show that the use of fee shifting also raises the incidence of trial when a unitary contingency fee contract is used. (20)

There are several issues raised by the interaction of contingency fees and fee shifting that may affect how such a system would be implemented in practice. First, under a bifurcated fee, the plaintiff pays her lawyer a small contingency in the event the case settles but never pays her own lawyer at trial. (If she wins at trial, the defendant pays her lawyer.) Thus, the plaintiff has the incentive to negotiate a contingency percentage at trial that is the highest allowable by law so as to maximize the work incentives of her lawyer. (21) It is important, therefore, that shifted tees be reasonable and perhaps limited to the expenditures by the opposing party at trial. (22)

Second, under a bifurcated tee, the contingency percentage at trial [[theta].sup.B] embeds within it the probability the plaintiff will lose at trial. Suppose the plaintiff wins with probability 0.5, the expected judgment is $100,000, and a lawyer working under a fixed fee would earn $16,666.67 if the case went to trial. The corresponding contingency payment to this lawyer would be [[theta].sup.B] = 1/3. Thus, if the entire contingency payment is shifted when the plaintiff wins, the defendant pays $33,333.33 compared with $16,666.67 under a fixed tee. The defendant is, in an expected value sense, paying the plaintiff's lawyer for the state of the world in which the plaintiff loses. Thus, under a bifurcated contingency fee, a fee shifting rule that shifts the entire contingency payment is strongly proplaintiff.

On the other hand, if a unitary contingency fee is used, it will overpay the lawyer for cases that settle and underpay (in an expected value sense) the lawyer for cases that proceed to trial. Since the overwhelming majority of cases settle, a unitary fee shifted at trial to the defendant may be a very significant underestimate of the true cost of the plaintiff's lawyer (most of which is paid out in the state of the world in which the case settles).

Since shifting [theta]J appears problematic under either contingency fee contract, it might be more desirable to shift fees based on the number of hours put in on the case by the lawyer using prevailing hourly rates to value this time.

3. A Signaling Model Where the Defendant Has Private Information

To fully understand how contingency fees affect the outcome of a signaling game, it is important to analyze the alternative information structure under which an informed defendant makes a single offer to an uninformed plaintiff. To make our results comparable to section 2, we will assume that the defendant knows the value of the judgment J in the event that the plaintiff prevails at trial. (23) We will examine both the bifurcated and the unitary contingency fee in the context of this model. The derivation of the model solutions follows the previous analysis very closely. (24)

For comparison, we first consider the following game in which the fee is fixed:

(i') Nature determines the defendant's type J. The plaintiff does not observe J but knows the distribution f(J) from which it is drawn, where [bar.J] and [J.bar] are the upper and lower supports off (J).

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

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

(iv') If the plaintiff accepts the offer, she receives a payoff of [O.sub.D], and the defendant receives a payoff of -[O.sub.D]. If the offer is rejected, 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 a payoff of -[C.sub.P], and the defendant receives the payoff -[C.sub.D].

Again, we assume that [C.sub.P] < p[J.bar], so that the plaintiff will proceed to trial even against the defendant with the strongest case.

The defendant will make an offer [O.sub.D] to maximize

(19) [V.sub.D] = [empty set]([O.sub.D])[-pJ - [C.sub.D]] + (1 - [empty set])([O.sub.P]))(-[O.sub.D]),

where [empty set]([O.sub.D]) is the probability the plaintiff rejects an offer [O.sub.D].

Since plaintiffs pursue a mixed strategy in equilibrium, they must be indifferent between accepting and rejecting an otter. As a result, defendants of type p make an offer that satisfies

(20) [O.sub.D] = pJ - [C.sub.P].

Proceeding as before, the first-order condition from Equation 19 combined with Equation 20 and the boundary condition [empty set]([[bar.O].sub.p]) = 1 together imply the following rejection function in equilibrium:

[empty set]([O.sub.D]) = 1 - [e.sup.[psi]],

where [psi]([O.sub.D]) = ([[bar.O].sub.D] - [O.sub.D])/([C.sub.P] + [C.sub.D]). Use Equation 20 to find an equilibrium probability of settlement equal to

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the subscript FD denotes a fixed fee model in which the defendant makes the offer. Defendants with stronger cases (lower values of J) have their offers rejected more in equilibrium so as to discourage "bluffing" by defendants with weaker cases. The defendant with the weakest case, [bar.J]. has his offer accepted with probability 1. Take the derivative of Equation 21 with respect to .I and convert to percentage terms to find

(22) [S.sub.FD] = P/[C.sub.p] + [C.sub.D].

The positive sign on [S.sub.FD] implies that the settlement rate falls as J falls below [bar.J].

Contingency Fees

The previous game will be modified so that the plaintiff's lawyer is paid based on a bifurcated contingency fee contract. Steps ii' and v' replace those in the game developed in the previous section.

(ii') The plaintiff hires a lawyer under a bifurcated contingency fee contract that pays her lawyer [[theta].sup.B]J if the case proceeds to trial and 0 if the case settles prior to trial.

(v') At trial, there is a finding for the plaintiff with probability p, in which case she receives the payoff J(1 - [[theta].sup.B]), 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 a payoff of 0, and the defendant receives the payoff -[C.sub.D].

In this game, the defendant's optimization problem remains unchanged from Equation 19. However, the offer that leaves the plaintiff indifferent between acceptance and rejection becomes

(23) [O.sub.D] = p(l - [[theta].sup.B])J.

This offer reflects the contingency percentage [[theta].sup.B] paid at trial in the event of a plaintiff victory. Again, solving the first-order condition from the maximization of Equation 19 while making use of Equation 23 yields the following equilibrium probability of settlement under a bifurcated contingency fee:

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking the derivative of Equation 24 with respect to J and convening to percent change yields

(25) [S.sub.BD] = (1 -[[theta].sup.B])p/[[theta].sup.B]pJ + [C.sub.D]

This analysis leads to the following:

RESULT 4: In the model where the informed defendant makes the offer, a bifurcated fee will result in a higher settlement rate than the fixed fee for all values of J < [bar.J] if

(26) (1 - [[theta].sup.B])[C.sub.P] < [[theta].sub.B](pJ + [C.sub.D] [for all]J.

PROOF: At [bar.J] we have [S.sub.BD] = [S.sub.FD] = I. If the previously stated condition holds, then a comparison of Equation 25 with Equation 22 shows that as p falls, the rate of decrease in the settlement rate under the fixed fee always exceeds the rate of decrease under the bifurcated contingency fee.

Even if the condition in Equation 26 does not hold, it is still quite possible that the contingency fee will result in greater settlement overall, though the settlement functions will cross at some point.

Furthermore, for a reasonable set of parameter values, the condition in Equation 26 will hold for all values of J. For example, when [C.sub.P] = [C.sub.D] and [[theta].sup.B] = 1/3, the condition in Equation 26 collapses to [C.sub.P] < pJ. Earlier we imposed [C.sub.P] < p[J.bar] to guarantee that the plaintiff had a credible threat to proceed to trial against all defendant types. Thus, for these parameter values, Equation 26 holds for all defendant types, and there is an unambiguous reduction in the incidence of trial relative to the model where the fixed fee is utilized. By contrast, in the model where the informed plaintiff makes the offer, there is an ambiguous effect on the settlement rate and, if anything, a presumption of an overall increase in disputes.

The denominator in Equation 25 is analogous to that in Equation 10, and the mechanism described in section 2 operates in this model as well. In particular, the plaintiff has a weak case when J takes on a low value. This group of plaintiffs has a low expected payment to their lawyer if the case proceeds to trial. Thus, the joint cost of trial is lower for the plaintiff and defendant than it is for plaintiffs with stronger cases (i.e., those matched against defendants with higher values of J). Thus, plaintiffs with weak cases proceed to trial more often relative to a fixed fee. As noted previously, this mechanism gives an ambiguous prediction as to how the incidence of trial will change as we move from a fixed fee regime to a bifurcated contingency fee.

However, in this version of the model, there is an additional mechanism not present in the earlier model. This mechanism is reflected by the 1 - [[theta].sup.B] term in the numerator of Equation 25, which has no analog in Equation 10. The contingency fee pushes the offers of adjacent defendant types closer together relative to the fixed fee. Under a fixed fee, two adjacent defendant types will make the offers pJ - [C.sub.D] and p(J - [DELTA]J) - [C.sub.D]. Under a bifurcated contingency fee, the same two defendant types will offer p(1 - [[theta].sup.B])J and p(1 - [[theta].sup.B])(J - [DELTA]J). Because the offers are closer together under the bifurcated contingency fee, there is less incentive for the defendant to "bluff" by submitting an offer lower than the offer indicated by Equation 23. As a result, compared to the fixed fee, the rejection function need not rise as steeply as the defendant's offer falls.

Now briefly consider the use of a unitary contingency fee. The defendant's objective function Equation 19 remains unaffected, but a larger offer is necessary to make the plaintiff indifferent between acceptance and rejection. In equilibrium, the defendant's offer must satisfy

(27) [O.sub.D] = [p.sub.J].

Since the plaintiff pays her lawyer, regardless of whether the case settles, the equilibrium offer to the plaintiff must rise to the expected judgment she would receive at trial. Solving the model as before using Equation 27 yields an equilibrium probability of settlement of

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

Taking the derivative with respect to J and converting to percent changes yields

(29) [[??].sub.UD] = p/[C.sub.D]. Ct)

Equations 28 and 29 lead to the following result:

RESULT 5: A unitary contingency lee unambiguously reduces settlement rates relative to (i) a fixed fee and (ii) a bifurcated lee.

PROOF: Result (i) follows from a direct comparison of Equation 28 to Equation 21. For result (ii), note that at [bar.J], [S.sub.UD] = [S.sub.BD] = 1. A comparison of Equation 29 with Equation 25 shows that as J falls, the rate of decrease in the settlement rate under the unitary fee always exceeds the rate of decrease under the bifurcated contingency fee.

Since the defendant cannot extract any of the plaintiff's expected trial costs with his offer, the cost of having an offer rejected by the plaintiff is lower than in the case of either the fixed fee or the bifurcated fee. As a result, the rejection rate needs to rise more quickly as the defendant's offer falls in order to induce the revealing offers indicated by Equation 27.

With the unitary contingency fee, the offer to the plaintiff rises as reflected by a comparison of Equation 27 with Equations 20 and 23. This increase in the offer reflects the insights of Bebchuk and Guzman (1996), who argue that this represents an advantage of a (unitary) contingency fee contract relative to an hourly fee arrangement. (25) The increased dispute rate under a unitary contingency fee contract may offset this advantage, as the contingency percentage will have to rise to reflect the increased hours put in by lawyers on the additional cases that reach trial.

The predicted increase in the incidence of trial under the unitary contingency fee is quite robust. It occurs in the signaling model regardless of whether it is the plaintiff or the defendant who is modeled as the informed party. In addition, this result would clearly generalize to a screening model along the lines of Bebchuk (1984).

Fee Shifting

In this section we analyze fee shifting with a bifurcated fee using a modeling approach similar to that presented in section 2. Specifically, the defendant pays the plaintiff's lawyer [[theta].sup.B] J in the event that the plaintiff is victorious at trial, and the plaintiff pays the defendant's attorney fees [C.sub.D] in the event the defendant wins at trial.

In equilibrium, the defendant's offer must make the plaintiff indifferent between acceptance and rejection. Thus, the offer must satisfy

(3) [O.sub.D] = [p.sub.J] - (1 - p)[C.sub.D],

which reflects the probability (1 - p) that the defendant's court costs will be shifted to the plaintiff. The defendant's expected wealth is now

(31) [V.sub.D] = [phi]([O.sub.D])[-p(J + [[theta] J + [C.sub.D])] + (1 - [phi]([O.sub.D]))(-[O.sub.D]),

where, as before, [phi]([O.sub.D]) is the probability the defendant's offer is rejected by the plaintiff. The defendant chooses [O.sub.D] to maximize Equation 31, taking the rejection function as given. Proceeding as before, the first-order condition is solved by a differential equation of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Substitute from Equation 30 to see that the settlement rate with fee shifting is

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

A comparison of Equation 32 with Equation 24 yields the following:

RESULT 6: For a given value of the bifurcated contingency fee, [[theta].sup.B], fee shifting under which the loser at trial pays the winner's court costs unambiguously increases the incidence of trial.

PROOF: For a given value of the contingency fee [[theta].sup.B], Equation 32 < Equation 24 for each plaintiff type J < [??].

As in the model where the plaintiff makes the offer, fee shifting unambiguously increases dispute rates when it is combined with an existing contingency fee arrangement. (26) The robustness of this result across information structures lends some insight into why American-style contingency fees contracts have been banned in Britain. They also suggest that some care be taken to address the interaction between contingency fees and fee shifting if fee shifting is to be introduced into the United States on a widespread basis.

4. Conclusion

The effects of a contingency fee on settlement depends both on the nature of the contingency fee contract and on the way in which the informational asymmetry is modeled. One robust result is that under a unitary contingency fee, the incidence of trial is unambiguously higher compared with either a fixed fee or a bifurcated contingency fee. When the informed plaintiff makes the offer, a bifurcated fee changes the selection of disputes for litigation but has an ambiguous effect on the overall incidence of trial. In particular, under such a fee structure, more low-stakes cases will proceed to trial than under the fixed fee contract. When an informed defendant makes the offer, there is a presumption that a bifurcated contingency fee reduces the incidence of trial relative to the fixed fee.

Fee shifting may have important interactions with contingency fees. In particular, when fee shifting is introduced into a model with a bifurcated contingency fee, we find (for a given value of the contingency fee) an unambiguous increase in the incidence of trial. By contrast, when the lawyer is compensated with a fixed fee, Reinganum and Wilde show that fee shitting has no effect on the incidence of trial.

The analysis in this study assumes that the plaintiff controls all aspects of the case. We view the results of this study as complementary, to the results of earlier work that analyzed contingency fees under the assumption that the plaintiff's lawyer effectively controlled the settlement decision.

(1) See Gravelle (1998) and Rubinfeld and Scotchmer 11998). See Helland and Tabarrok (2003) for a recent empirical analysis of the effects of contingency tees.

(2) In addition, there was considerable debate in Britain in the early 1990s over whether the use of contingency tees should be allowed. While there were some changes in British law, U.S.-style contingency fees are still banned them. Lawyers are now allowed to enter in conditional fee arrangements, Under these arrangements, no fee is paid by the plaintiff if she loses, and a markup over the usual lawyer tees is paid if she wins. See Gravelle (1998).

(3) For more on the selection of disputes for trial see Priest and Klein (1984) and Shavell (1996), among others.

(4) This is the approach taken by Miceli (1994), who develops a model with nuisance suits. See also Gravelle and Waterson (1993). who develop a model in which the lawyer controls the case and maximizes a weighted average of her income and her client's income. In both of these articles, the uninformed party makes an offer to the informed party. In our study, the informed party makes the offer: that is, ours is a signaling model. Reinganum and Wilde (1986) analyze contingency fees in the signaling model. Their work is discussed extensively in the following text.

(5) Daughety and Reinganum (2003) develop both a signaling model and a screening model in which the plaintiff pays her lawyer via contingency lee. As in our section 3, they consider a model with an informed defendant (see pp. 141-7). The locus of their study is split-award statutes under which punitive damage awards are split between the plaintiff and the state. They do not analyze the effect of contingency fees on settlement. In addition, they consider the unitary tee but not the bifurcated fee introduced later in this article.

(6) We are following Hay's (1997) terminology to describe these fee structures.

(7) Based on survey data, Hensler et al. (1991, p. 136) find that 87% of plaintiffs hired lawyers on contingency, with 67% using a fixed contingency fee and 20%, using a varied fee that would include the type of bifurcated lee analyzed in this study. See also Miller (1987, p. 201).

(8) See Miller (1987). Polinsky and Rubinfeld (2002) argue that this traditional result may not hold once the lawyer's incentive to under provide effort at trial is taken into account. Polinsky and Rubinfeld (2003) propose a mechanism that aligns the interest of the client and the lawyer under a contingency fee. See also Santore and Viard (2001).

(9) The concept they use is "universally divine equilibrium." See Banks and Sobel (1987). Using this refinement, all pooling and semipooling equilibria can be eliminated. See Reinganum and Wilde (1986, p. 566).

(10) For a discussion on why this is the correct boundary condition, see Reinganum and Wilde (1986, p. 565). An analogous condition applies in all the models analyzed in this article.

(11) This is the assumption made by Miceli (1994, p. 215). See also Hay (1997).

(12) This is the assumption made by Miceli and Segerson (1991, p. 383).

(13) As noted in section 1, we are treating the form of the contract between the plaintiff and her lawyer as exogenous. An alert reader may be concerned that plaintiffs with strong cases should (in the context of our setup) prefer fixed lee contracts because for these plaintiffs the expected contingency payment is large. This issue can be addressed by changing the tinting of the model. In particular, we can assume that the plaintiff learns her exact type through consultations with her lawyer that take place after a contract has been entered into.

(14) To see this, set [beta] = 1, [alpha] = 1 - [[theta].sub.B], and [C.sub.P] = 0 in their equation 9.

(15) The equations which determine the endogenous value of [[theta].sup.B] are provided in the working paper version of this article (Farmer and Pecorino 2003).

(16) Set [c.sub.p] = 0 and [alpha] = [beta] = 1 - [0.sup.u] in their equation 9. Solve the resulting dilferential equation to obtain our Equation 12.

(17) In contrast, Reinganum and Wilde (1986, pp. 562-3) assume that only the noncontingent portion of court costs are subject to shifting. They show that allocation of court costs can affect the probability of trial if this interacts with a contingency fee arrangement under which the plaintiff does not retain all the pretrial settlement.

(18) Smith and Donohue analyze both cases.

(19) But see note 17.

(20) Fee shifting has been shown to increase total expenditures at trial (Braeutigam, Owen, and Panzar 1984). An increase in expenditure should lead to a reduced incidence of trial. This presumption must be weighed against the results we present here that suggest the opposite effect.

(21) See the discussion in Smith (1992, pp. 2167-71).

(22) Such a limitation is likely to affect high type J plaintiffs but not low types. As long as there are some low plaintiff types for which such a limitation is not binding, the rejection rate under fee shifting will exceed the rejection rate without tee shifting for all plaintiff types J > [J.bar]. However, for the range of plaintiff types for which the cap on shifted fees is binding, Equation 18 will provide an underestimate of the settlement rate.

(23) An example (suggested to us by an anonymous referee) would be a chemical company acting as a defendant that knows more about the expected damages it has inflicted on the plaintiffs than the plaintiffs do. Assuming instead that the defendant is better informed about p. the probability the plaintiff prevails at trial will not affect any of our results.

(24) See note 5.

(25) See also Rickman (1999).

(26) But see note 20.

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Amy Farmer * and Paul Pecorino ([dagger])

* Department of Economics, University of Arkansas. Fayetteville, AR 72701, USA: Email: afanner@walton.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. We would like to thank Richard Boylan. Akram Temimi, and two anonymous referees for providing helpful comments on this article.

Received August 2003: accepted May 2004.