The Plaintiff s Attorney in the Liability Insurance Claims Settlement Process: A Game Theoretic Approach

The Plaintiff s Attorney in the Liability Insurance Claims Settlement Process: A Game Theoretic Approach Lisa L. Posey * Abstract: The decision of a claimant to obtain legal counsel, the timing of this decision, and the impact on the settlement amount an insurer successfully can offer are the focus of this study. This paper draws on strategic models of litigation/settlement with endogenous settlement amounts in the law and economics literature, but differs by endogenizing the decision to hire an attorney and its timing. The main results are that under symmetric damage information the claimant gains bargaining power over the insurer and a higher net payoff by retaining an attorney on a contingency fee basis before the insurer makes an offer. Hourly fees never appear in equilibrium under symmetric information. Under asymmetric information when only contingency fees are available, there exists a separating equilibrium where the insurer can deduce the expected damage award of each claimant by his attorney-hiring strategy and offer a settlement appropriate for each type. There is no separating equilibrium if the possibility of hourly fees is included. L INTRODUCTION iability insurers settle over 98% of all third-party claims out of court (Litan, Swire, and Winston, 1988). Claimants may enter the settlement process with an attorney or deal directly with the insurer. What can those who hire an attorney at the beginning of the process gain from doing so * Lisa L. Posey is Assistant Professor, Department of Insurance and Real Estate, Penn State University. 119 Journal of Insurance Issues, 1998, 21, 2, pp. 119 137. Copyright 1998 by the Western Risk and Insurance Association. All rights reserved.

120 LISA L. POSEY and can insurers infer anything about claimants from their decision to hire an attorney? These questions are addressed in this paper using a game theoretic approach to the settlement process. The decision of a claimant to obtain legal counsel, the timing of this decision, the type of fee arrangement used, and the impact that all these factors have on the settlement amount an insurer can offer and expect to have accepted are the focus of this study. The main results are that under symmetric damage information the claimant gains bargaining power over the insurer and a higher net payoff by retaining an attorney on a contingency fee basis before the insurer makes an offer. Hourly fees never appear in equilibrium under symmetric information. Under asymmetric information when only contingency fees are available, there exists a separating equilibrium where the insurer can deduce the true damage type of each claimant by his attorney-hiring strategy and offer a settlement appropriate for each type. There is no separating equilibrium if the possibility of hourly fees is included. This paper draws on strategic models of litigation versus settlement with endogenous settlement amounts developed in the law and economics literature (see Bebchuck, 1984; Nalebuff, 1987; Salant, 1984; Reinganum and Wilde, 1986). The litigation versus settlement issue was first analyzed in the 1970s by Landes (1971) and Gould (1973). 1 These early papers focus on why some cases end up in court and some settle out of court. They do not model the pre-trial bargaining process explicitly and are unable to make predictions about settlement amounts. A second generation of papers (e.g., Bebchuck, 1984; Nalebuff, 1987; Salant, 1984; Reinganum and Wilde, 1986) uses game theoretic bargaining models to address the litigation versus settlement issue. In these models, informational asymmetries between the parties are made explicit and equilibrium settlement amounts can be derived. One party makes a take-it-or-leave-it settlement offer to the other party, who must decide whether to accept or reject the offer. If the offer is rejected, the plaintiff decides whether to pursue the case in court. It is the threat of litigation and the expected payoffs at trial that determine the optimal settlement amount. If the parties are symmetrically informed about the level of damages and the probability of winning at trial, and the defendant makes a settlement offer, then the plaintiff will accept any offer that is greater than or equal to his or her expected payoff at trial net of litigation costs. Therefore, the defendant s optimal offer is the plaintiff s expected net payoff at trial. In symmetric information models, the equilibrium concept used is subgame perfect Nash equilibrium. When the parties are asymmetrically informed about either the level of damages or the probability of winning at trial, it is important whether it is the informed party or uninformed party that makes the offer. The

PLAINTIFF S ATTORNEY IN LIABILITY CLAIMS SETTLEMENT PROCESS 121 informed party s offer may convey some information to the other party, who must then decide whether or not to accept. Bebchuck (1984) and Nalebuff (1987) develop models where the uninformed party makes the settlement offer, while Salant (1984) and Reinganum and Wilde (1986) have the party with private information about the level of damages making the settlement offer. In these models of asymmetric information, the equilibrium concepts used are the perfect Bayesian equilibrium or the sequential equilibrium. Three more recent papers explore the effects of relaxing the assumption (made in the previous bargaining models of settlement and litigation) that only one party makes a take-it-or-leave-it offer and the other party must accept it or reject it and face costly litigation. Cheung (1988) develops a sequential bargaining model under symmetric damage information, where each party has the option to make a counteroffer after rejecting the previous offer and the plaintiff has the outside option of going to trial. He finds that the unique subgame perfect Nash equilibrium in this model is the same as that under a model where only the defendant makes an offer and the plaintiff can either accept it or reject it and go to court. In particular, the parties will settle immediately for the plaintiff s expected judgment at trial less litigation costs. Wang, Kim, and Yi (1994) consider a model where the defendant and plaintiff can exchange an infinite number of offers and counteroffers and the plaintiff has private information about actual damages. They find that there is a unique sequential equilibrium where the defendant makes an offer that is accepted by plaintiffs with expected net payoffs from litigation less than the offer and rejected by any other plaintiffs, who then proceed to trial rather than making counteroffers. This too mirrors the outcome under the single take-it-or-leave-it offer models of previous research. Finally, Daugherty and Reinganum (1993) study a model with endogenous sequencing of settlement offers. They allow either the plaintiff or the defendant or both to make the first offer, and it is endogenously determined who actually decides to make an offer. They show that the equilibrium payoffs that arise in their model coincide with those in the standard models of litigation and settlement where one party or the other is assumed to make an offer. These results support the use of the more simple static models of the settlement negotiation process in previous work. This growing body of literature is directly applicable to the study of insurance claims settlement practices. Many of the above studies cite statistics on insurers settlement versus litigations ratios to motivate their analysis. Although these researchers intend for their models to represent the insurance claims settlement process, insurance researchers can add to

122 LISA L. POSEY this literature by incorporating additional characteristics of the insurance claims settlement process into the existing models. That is one of the goals of the current paper. This paper adds the role of the plaintiff s attorney into the settlement process. In the present setting, the plaintiff is referred to as the claimant and the defendant is represented by the insurer. The claimant has the option of either hiring an attorney at a competitively determined fee at the beginning of the settlement process or delaying such action until and unless the case proceeds to trial. An attorney is necessary for the trial phase but the settlement negotiations may be performed by either the claimant or the attorney. The model is analyzed under various assumptions about the type of fee arrangements available to the claimant, first under symmetric information and then under asymmetric information where only the claimant and his or her attorney (if one is hired) know the true expected value of the damages to be awarded at trial. It is assumed that the insurer makes a settlement offer and the claimant must either accept the offer or reject it and proceed to trial. 2 Therefore, under asymmetric information, it is the uninformed party that makes the settlement offer but it is the informed party that moves first in the game by choosing whether or not to hire an attorney. In the typical asymmetric information models of litigation and settlement, either it is assumed that the informed party makes the settlement offer, in which case the settlement offer signals some information to the other party, or else it is assumed that the uninformed party makes the settlement offer. A goal of this paper is to see if something other than a settlement offer, namely the decision to hire an attorney, can be used by a claimant to signal information about expected damages and whether insurers can in fact infer anything about a claimant from his or her decision to hire an attorney. This can be analyzed best with a model in which the claimant makes the first move (deciding whether to hire an attorney) and the insurer makes the settlement offer. Although this paper is unique in endogenizing the decision to hire an attorney, there is a body of literature on plaintiffs attorneys in the litigation process with an emphasis on the role of various fee structures such as contingency fees and hourly fees (see, for example, Clermont and Currivan, 1970; Schwartz and Mitchell, 1978; Danzon, 1983; Miller, 1987; Halpern and Turnbull, 1983; Miceli and Segerson, 1991; Miceli, 1994; Hay, 1997). A few studies have looked at plaintiffs attorneys as providing specific expertise. Watts (1994) assumes that attorneys are able to learn part of the defendant s private information at a lower cost than plaintiffs can and Dana and Spier (1993) consider the case where attorneys can more accurately determine

PLAINTIFF S ATTORNEY IN LIABILITY CLAIMS SETTLEMENT PROCESS 123 the merits of the case than plaintiffs can. In none of these models has the timing of the decision to hire an attorney or the decision itself been endogenized, as is the case in the current model. These previous studies take the decision to hire an attorney and the time the attorney enters the process as given. BASIC MODEL WITH SYMMETRIC DAMAGE INFORMATION Contingency Fee Arrangements Only Assume that an injury has occurred and that the victim has filed a claim with the insurer s liability insurance company. The insurer has routinely performed a preliminary investigation and determined that the policyholder is liable and that the expected value of the damages to be awarded at trial is d. All parties are assumed to observe d in this section. 3 At this point the claimant faces the decision of whether or not to hire an attorney for the settlement process or delay such action until and unless it is necessary to proceed to trial. The claimant s attorney-hiring decision is denoted A; A = 0 indicates that the claimant does not hire an attorney for the settlement phase and A = 1 indicates that the claimant does hire an attorney. It is assumed that an attorney must be hired if a trial ensues and that those who hire an attorney earlier in the process continue with this attorney through trial if settlement is not reached. The market for attorneys is assumed to be competitive. It is first assumed that claimants can afford an attorney only if a contingency fee arrangement is used (i.e., if they had to pay out of their own wealth before they received a judgment, they would not be able to afford an attorney). F S is the cost incurred by an attorney for providing services to a claimant during the settlement phase and F T is the cost incurred by an attorney for providing services to a claimant during a trial. If an attorney is hired before the settlement phase begins and ends up in a trial, the total costs incurred by the attorney will be F S + F T. The market for attorneys is competitive, so that the equilibrium fees are a function of these costs, and the costs are assumed to be known by all. If an attorney is hired, the claimant and attorney agree on a contingency fee rate at the onset of their relationship. In equilibrium, the zero expected profit contingency fee rate, determined at the onset of the relationship, will incorporate the expectation of trial versus settlement as well as the insurer s optimal settlement offer strategy. Once the claimant has entered the settlement phase, the insurer can observe whether an attorney has been hired. The

124 LISA L. POSEY Figure 1. insurer then makes a settlement offer to the claimant and the claimant may accept or reject the offer. If the offer is rejected, the claimant takes the case to trial using either a newly hired attorney or one hired at the earlier phase. The amount of damages awarded at trial is a random variable with an expected value equal to d. Figure 1 presents a game tree representing the situation considered here. 4 At the first node, the claimant decides whether or not to hire an attorney. If so, the claimant and attorney set their zero expected profit contingency fee rate. At the second set of nodes, the insurer decides what settlement offer to make. At the third set of nodes, the claimant either accepts or rejects the offer. If it is rejected, the claimant hires an attorney and sets a fee if he has not hired one earlier, and then proceeds to trial. The equilibrium concept for the game in this section is subgame perfect Nash equilibrium. Beginning at the bottom of the tree, consider first the zero expected profit contingency fee if an attorney is hired just before trial (node 3B). The cost of trial to the attorney is F T and the contingency fee rate, α 0, is set so that the attorney s share is α 0 d = F T or α 0 =F T /d, where 0 signifies that an attorney was not hired at the beginning of the settlement process (A = 0). So the claimant will obtain an expected net payoff of (1 α 0 )d = d F T if he goes to trial. Therefore, the claimant will accept any settlement offer from the insurer that is greater than or equal to d F T, and the insurer s optimal strategy is to make the lowest acceptable offer, S 0 = d F T. The insurer prefers this amount to going to trial since the insurer s expected net payout at trial is d + C (C is the insurer s litigation cost), which is greater than d F T. Next consider the claimant s decision to accept or reject the offer if an attorney has been hired earlier in the process. The claimant and attorney have set the contingency fee rate, α 1, at the onset of their relationship, before an offer has been made (the 1 signifies that A = 1, i.e., that an attorney has been hired for the settlement phase). When such a claimant considers a

PLAINTIFF S ATTORNEY IN LIABILITY CLAIMS SETTLEMENT PROCESS 125 settlement offer, his expected damage award at trial is d, and he expects to retain (1 α 1 )d. The claimant likewise will retain (1 α 1 )S 1 of any settlement offer S 1 made. Therefore, he will not accept any offer less than the true expected damages d. So the insurer s optimal strategy is to offer S 1 = d. How will α 1 be set? The insurer s optimal strategy is to offer d, which will be accepted. The expectation is that they will settle out of court and the attorney s cost for participating in the settlement part of the process is F S. This implies that the zero expected profit contingency fee rate satisfies α 1 d = F S or α 1 = F S /d. The net payoff to the claimant in this case is (1 α 1 )d = d F S. Since F S < F T, the claimant obtains a larger net payoff by hiring an attorney earlier in the settlement stage, before an offer is made. So at node 1, the optimal strategy is to hire an attorney. Result: In equilibrium, the claimant hires an attorney on a contingency fee basis and is able to obtain a net payoff of d F S, which is greater than the net payoff of d F T, which would have been obtained if no attorney were hired. By retaining an attorney on a contingency fee basis before the insurer makes an offer, the claimant gains some bargaining power and a higher net payoff. The threat of trial governs the smallest amount the insurer can offer and avoid a trial. When a claimant does not have an attorney yet, the threat of trial is accompanied by a need to hire an attorney at a rate appropriate for trial costs; such costs can be subtracted from the expected judgment at trial to obtain an acceptable offer, with the insurer extracting all of the surplus. By hiring an attorney early on and pre-committing to a zero expected profit contingency fee rate with the expectation of settlement, the claimant has the ability to reject any offer that does not equal the true expected damage award at trial and the insurer cannot extract as much of the surplus. The attorney need not do anything but enter into a contingency fee arrangement with the claimant and make his or her presence known to the insurer to aid the claimant in increasing the net payoff obtained. Contingency Fee and Hourly Fee Arrangements Available In this section, all of the previous assumptions remain, except for those regarding the potential fee arrangements between the claimants and their attorneys. Here, claimants may enter into one of two types of contracts with their attorneys, either hourly fee contracts or contingency fee contracts where the rate is agreed upon at the onset of the relationship. If an hourly fee is used, the competitive total fee for the attorney is F S if the case is settled out of court, F S + F T if the case goes to court and the attorney had been hired at the beginning of the process, and F T if the case goes to trial with an attorney hired only for the trial phase. The zero expected profit contingency

126 LISA L. POSEY fee rate is set in a manner similar to that in the previous section. Once the claimant has entered the settlement phase, either with or without an attorney, the insurer can observe only whether an attorney has been hired, not the type of fee arrangement or the amount of the fee. Therefore, the insurer must specify beliefs about which fee arrangement the claimant has entered into, given that an attorney is present. The equilibrium concept used for the game in this section is the perfect Bayesian equilibrium, which, in this setting, is characterized by (1) a strategy for the claimant for hiring an attorney at the beginning of the settlement process: the claimant must specify whether he will hire an attorney under a contingency fee arrangement, hire an attorney under an hourly fee arrangement, or not hire an attorney, given the competitively determined fees; (2) a probability distribution representing the insurer s set of beliefs about which type of fee arrangement the claimant and attorney entered into if an attorney has been hired; (3) a settlement strategy for the insurer specifying two out-of-court settlement offers S A, A = 0,1, one for claimants with attorneys and one for claimants without attorneys; (4) an accept or reject strategy for the settlement offer for the claimant for each attorney strategy, where these strategies and beliefs satisfy the following requirements: (a) Pr(Contingency Fee Attorney) 0, Pr(Hourly Fee Attorney) 0, and Pr(Contingency Fee Attorney) + Pr(Hourly Fee Attorney) = 1; (b) the players strategies are sequentially rational; 5 (c) for each claimant strategy, given the fee structure, the insurer s beliefs at the information set corresponding to that strategy must follow from Bayes Rule and the claimant s strategy. Figure 2 presents a game tree representing the situation where both types of fee arrangements are available and information about the expected damage award is symmetric. At the first node, the claimant decides whether or not to hire an attorney and what type of fee arrangement to enter into if an attorney is hired. The claimant and attorney set their zero expected profit fees at this stage as well. At the second set of nodes, the insurer decides what settlement offer to make. The insurer knows if it

PLAINTIFF S ATTORNEY IN LIABILITY CLAIMS SETTLEMENT PROCESS 127 Figure 2. is at node 2C but cannot distinguish between 2A and 2B. At the third set of nodes, the claimant either accepts or rejects the offer. If it is rejected, the claimant hires an attorney and sets a fee if he has not done so earlier, and then proceeds to trial. Beginning at the bottom of the tree, consider the zero expected profit fee if an attorney is hired just before trial (node 3C). The total hourly fee will be F T since the cost to the attorney is F T. If a contingency fee is used, the rate, α 0, is set so that the attorney s share α 0 d = F T or α 0 = F T /d. So the payment to the attorney will be F T whether a contingency fee or hourly fee arrangement is used and the claimant will obtain an expected net payoff of d F T. Therefore, the claimant will accept any settlement offer from the insurer that is greater than or equal to d F T (regardless of the fee arrangement), and the insurer s optimal strategy (at node 2C) is to make the lowest acceptable offer, S 0 = d F T. The insurer prefers this to going to trial since the insurer s expected net payout at trial is d + C, which is greater than d F T. If the claimant hires an attorney under an hourly fee arrangement at the beginning of the settlement process, then once node 3B is reached, the claimant has paid the attorney F S and must pay an additional F T if they proceed to trial. Just as is the case where no attorney is hired until trial, the claimant will accept any offer greater than or equal to d F T if he has previously retained an attorney on an hourly fee basis, since this is the net payoff he can expect from the trial phase. If, on the other hand, the claimant hires an attorney under a contingency fee arrangement, as before he will only accept a settlement offer that is not less than the true expected damage

128 LISA L. POSEY award. The reasoning for this follows the previous model: the percentage of the insurer s payment given to the attorney is fixed at the beginning of the process and nothing less than the true expected value of the damages that will be awarded at trial will be an acceptable offer. The insurer must determine its optimal settlement offer strategy if the claimant hires an attorney. If the attorney is hired on an hourly fee basis, the optimal offer is d F T, and if the attorney is hired on a contingency fee basis, the optimal offer is d. Knowing only that either node 2A or node 2B has been reached, the insurer must specify beliefs about which node has been reached and then determine which of these two offers is the optimal offer, given those beliefs. Only the offers that are optimal for each of the two fee arrangements need to be considered by the insurer, since any offer between the two will not be accepted by the claimant with a contingency fee arrangement and will pay the claimant with an hourly fee arrangement more than is necessary to get him to accept the offer. Since the beliefs should follow Bayes Rule given the claimant s strategies in equilibrium, the various potential equilibria will be considered at this point. As will be shown, an equilibrium exists where the claimant hires an attorney on a contingency fee basis, there is no equilibrium where an attorney is hired on an hourly fee basis, but there is an equilibrium where no attorney is hired. The first potential equilibrium to be considered is the one that arises in the previous section: (i) the claimant hires an attorney at the beginning of the settlement process on a contingency fee basis; (ii) the insurer offers d to claimants with attorneys and d F T to claimants without attorneys; (iii) the claimant, who has an attorney, accepts the offer; (iv) the contingency fee is α 1 = F S /d, the claimant has a net payoff of d F S ; (v) the insurer s beliefs are characterized by putting probability = 1 on being at node 2A and probability = 0 on being at node 2B (this follows from Bayes Rule and the equilibrium strategies). The claimant has no incentive to deviate from this strategy, since he will obtain a net payoff of d F T without an attorney and d F S F T with an attorney on an hourly fee basis. So the equilibrium where the claimant hires an attorney on a contingency fee basis, which is the unique equilibrium in the previous section (with no possibility of hourly fees), also exists when the possibility of hourly fees is included.

PLAINTIFF S ATTORNEY IN LIABILITY CLAIMS SETTLEMENT PROCESS 129 Result: There is no equilibrium where the claimant hires an attorney on an hourly fee basis with symmetric information about the expected damage award. This can be shown as follows. If the claimant hires an attorney on an hourly fee basis, the insurer must place probability = 1 on the event of being at node 2B and will, therefore, offer d F T. The claimant will obtain net payoff of d F S F T and can improve upon this by not hiring an attorney and obtaining d F T, which is the insurer s optimal offer if no attorney is hired. Finally, consider the potential equilibrium where no attorney is hired. Here the claimant receives offer d F T and accepts and retains the entire offer. Whether the claimant has an incentive to deviate depends upon what the insurer offers if an attorney is hired. Because nodes 2A and 2B are off the equilibrium path, the insurer can place any probabilities on them under the concept of perfect Bayesian equilibrium, provided that they sum to one. In any event, only one of two offers will be made to a claimant with an attorney, either d or d F T. If the offer is d, the claimant will prefer to deviate and hire an attorney on a contingency fee basis, obtain a rate appropriate for the expectation of settlement, accept the offer and net d F S. Therefore, if the insurer s beliefs are such that the optimal offer to make to a claimant with an attorney is d, then an equilibrium where no attorney is hired cannot be sustained. But if the insurer s beliefs are such that it is optimal to offer d F T to a claimant with an attorney (in which case the claimant will net d F T F S if an attorney is hired), then the claimant will have no incentive to deviate from the strategy of not hiring an attorney. Result: An equilibrium where no attorney is hired and the net payoff to the claimant is d F T can be sustained when the possibility of hourly fee arrangements is included in the model even though such fee arrangements never appear in equilibrium. This net payoff is less than the net payoff of d F S that would be obtained under a contingency fee equilibrium. ASYMMETRIC DAMAGE INFORMATION Contingency Fee Arrangements Only Next, consider the case where there are two types of claimants high damage and low damage and a claimant is high damage with probability λ and low damage with probability 1 λ. A high (low) damage claimant has private information about the case and knows that the true expected value of the damage award at trial is d H (d L ), while the insurer has only the

130 LISA L. POSEY ex ante belief that the expected award at trial is the average expected award, λd H + (1 λ)d L. 6 We return, temporarily, to the assumption that the claimant can only afford to enter into a contingency fee arrangement and hourly fee arrangements are not offered by attorneys. This situation is depicted in Figure 3. The equilibrium concept used for this game with asymmetric damage information is again the perfect Bayesian equilibrium. In this context, the perfect Bayesian equilibrium is defined as: (1) an attorney-hiring strategy for the claimant for the settlement phase; for each type j the claimant may turn out to be, he must specify whether or not he will hire an attorney for the settlement phase, given the competitively determined contingency fees; (2) a probability distribution representing the insurer s set of beliefs about which type the claimant is, given whether or not he hired an attorney. Let µ(j A) be the insurer s belief about the probability of the claimant being of type j, given A, where A = 1 means that an attorney has been hired for the settlement phase and A = 0 means an attorney has not been hired; (3) a settlement strategy for the insurer specifying two out-of-court settlement offers, one for the case where an attorney is present, the other for the case where no attorney is present; (4) an accept or reject strategy for the settlement offer for each claimant type, where these strategies and beliefs satisfy the following requirements: (a) µ(j A) 0 and µ(h A) + µ(l A) = 1; (b) the players strategies are sequentially rational; (c) for each claimant strategy, given the fee structure, the insurer s beliefs at the information set corresponding to that strategy must follow from Bayes Rule and the claimant s strategy. The decisions of each claimant type regarding acceptance or rejection of the settlement offer at the fourth set of nodes are the same as was the case under symmetric damage information with only contingency fees (the third set of nodes in Figure 1). For a claimant of type j, j = L,H, the true expected damage award is denoted d j, and the contingency fee is denoted α j A.7 Without having hired an attorney in the earlier stage, the claimant must hire one to proceed to trial at contingency fee rate α j 0 = F T /d j and, therefore, will accept any offer greater than or equal to d j F T. On the other hand, if an attorney has been hired early on, the claimant will net (1 α j 1 )d j

PLAINTIFF S ATTORNEY IN LIABILITY CLAIMS SETTLEMENT PROCESS 131 at trial and (1 α j 1)S 1 through settlement and, therefore, only will accept offers at least as great as the true expected damage award, d j. Parameter conditions exist that support a pooling equilibrium with attorneys and a pooling equilibrium without attorneys. Our attention will be focused on the potential separating equilibrium in this setting. A separating equilibrium would allow the insurer to infer the claimant s damage type by the presence or absence of an attorney. In this setting of asymmetric damage information and contingency fees only, the following can be shown. Result: There exists a separating equilibrium where the low damage claimant hires an attorney on a contingency fee basis, the high damage claimant does not hire an attorney, and the insurer can deduce the expected damage award of each claimant and offer a settlement of d H F T to the high damage claimant and d L to the low damage claimant, both of which will be accepted. To see this, consider Figure 3. If the claimant s strategy is to hire an attorney if damages are low and not hire an attorney if damages are high, then the insurer must place probability = 1 of being at node 3B if an attorney is not hired and probability = 1 of being at node 3C if an attorney is hired. Therefore, the optimal settlement offer is d H F T if an attorney is not hired and d L if an attorney is hired. Both claimant types will accept their offers, the low damage type will have a net payoff of d L F S and the high damage type will have a net payoff of d H F T. When d H F T d L F S, the claimants have no incentive to deviate and this will be a separating equilibrium. In this case, the low damage claimant will have no incentive to deviate because he has a greater net payoff, d L F S, than if he deviated by not hiring an attorney and obtained d H F T. But why doesn t the high damage type deviate, given that the low damage type is obtaining a greater net payoff than he is? If the high damage type deviates and hires an attorney on a contingency fee basis, the attorney knows that the claimant will not accept the insurer s offer since he can do better by going to trial. Therefore, he cannot obtain a contingency fee based on the expectation of accepting the settlement offer d L. Instead, deviating to a contingency arrangement would give the high damage claimant a net payoff of d H F S F T, which is worse than remaining without an attorney. So the very pre-commitment to a contingency fee that gave the claimant additional bargaining power over the insurer with symmetric damage information prevents the claimant from credibly promising to settle for less than the true expected damage award, thereby leaving him with a high attorney s fee in this setting of asymmetric damage information. In this setting, it is the insurer who

132 LISA L. POSEY Figure 3. benefits from the lack of an hourly fee arrangement since the insurer can accurately deduce each claimant s type and reduce the high damage claimant s offer by the full cost of a trial. In the next section, the effect of adding an hourly fee arrangement under asymmetric information on the existence of the separating equilibrium, as well as the potential use of hourly fees in equilibrium, will be analyzed. Contingency Fee and Hourly Fee Arrangements Available The circumstances of the preceding separating equilibrium raise a few issues. The high damage claimant is locked into a fee appropriate for trial under contingency fees even though the payoff would be greater if the offer of d L < d H could be accepted and trial avoided. Including the potential of an hourly fee arrangement would allow the claimant to confine attorney s fees to those appropriate for settlement, since the claimant would have no incentive to proceed to trial and would accept d L. Result: There is no separating equilibrium if the possibility of an hourly fee arrangement is included in the model with asymmetric damage information, so the insurer cannot deduce the true damage type of the claimant.

PLAINTIFF S ATTORNEY IN LIABILITY CLAIMS SETTLEMENT PROCESS 133 To see this, consider Figure 4. If the claimant hires an attorney, the insurer does not know if node 3A, 3B, 3D, or 3E has been reached. The optimal strategies of the claimant at the fourth set of nodes are identical to the accept/reject strategies for the game in Figure 2. Now consider the potential separating equilibrium similar to the one noted above, where the low damage claimant hires an attorney on a contingency fee basis, the high damage claimant does not hire an attorney, the insurer offers d L to the claimant with an attorney and d H F T to the claimant without an attorney, and d H F T < d L F S. The high damage claimant now has an incentive to deviate, hire an attorney on an hourly fee basis, and settle out of court, accepting d L and getting net payoff d L F S. In addition, no separating equilibrium exists where one claimant hires an attorney on an hourly fee basis and the other does not hire an attorney. The best either claimant type can do by separating themselves, by hiring an attorney on an hourly fee basis while the other claimant type hires no attorney, is to obtain d j F S F T, which is less than the minimum they could obtain by not hiring an attorney earlier in the process, going to trial, and netting d j F T. So separation is not possible when hourly fees are possible. A logical next question is whether any claimant would hire an attorney on an hourly fee basis in equilibrium, if such fee arrangements were available. The answer is a qualified yes. More specifically, we have the following two results. Result: Under the assumptions of asymmetric damage information and availability of both continency fee and hourly fee arrangements, there is no equilibrium where hourly fees are used by both claimant types. To see this, note that if both claimant types hired attorneys on an hourly fee basis, then the best offer they could get from the insurer is d H F T, with a net payoff of d H F S F T. The high damage claimant would always prefer to deviate and not hire an attorney and obtain net payoff d H F T. Result: Under the assumptions of asymmetric damage information and availability of both continency fee and hourly fee arrangements, there are parameter conditions that support an equilibrium where one type of claimant hires an attorney on an hourly fee basis and the other type of claimant hires an attorney on a contingency fee basis. Consider the case noted earlier, where a separating equilibrium would be supported without hourly fees, but not when hourly fees are available (i.e., d H F T < d L F S ). An equilibrium with the following characteristics can be sustained if d L < λ[d H F T ] + (1 λ)[d L + C]: (i) the high damage claimant hires an attorney on an hourly fee basis and the low damage claimant hires

134 LISA L. POSEY Figure 4. an attorney on a contingency fee basis, (ii) the insurer offers d L to those with an attorney, (iii) the attorney sets contingency fees based on acceptance of the offer for L types and going to trial for H types, but the H types do not use contingency fees, and (iv) both claimant types accept the offer. Returning to Figure 4, the insurer doesn t know whether it is at node 3B or 3D. The two settlement offers to consider are d H F T (appropriate for node 3B) and d L (appropriate for node 3D). If the insurer chooses to offer d L, which will be the case if d L < λ[d H F T ] + (1 λ)[d L + C], then neither H s nor L s will have an incentive to deviate. Their net payoff by accepting d L is d L F S, while the most they can obtain by not hiring an attorney is d H F T. L s are indifferent to hiring an attorney on an hourly fee basis and H s would do worse by deviating to a contingency fee. This differs from the result under symmetric damage information, where hourly fees would never be observed in equilibrium. There, contingency fees always give the claimant additional bargaining over the insurer relative to hourly fees. But in this setting, the asymmetry of damage information means that a claimant with an hourly fee arrangement may get a more attractive offer targeting the other claimant type who is using a contingency fee arrangement, leaving no incentive to deviate from an hourly fee arrangement.

PLAINTIFF S ATTORNEY IN LIABILITY CLAIMS SETTLEMENT PROCESS 135 SUMMARY The decision of a claimant in the liability insurance claims settlement process to obtain legal counsel, the timing of this decision, the type of fee arrangement used, and the impact that all these factors have on the settlement amount an insurer can offer and expect to have accepted are the focus of this study. This paper draws on strategic models of litigation versus settlement with endogenous settlement amounts developed in the law and economics literature, but differs from previous studies by endogenizing the decision to hire an attorney and the time the attorney enters the process. The main results are that under symmetric damage information the claimant gains bargaining power over the insurer and a higher net payoff by retaining an attorney on a contingency fee basis before the insurer makes an offer, and thus this is the unique equilibrium strategy for the claimant when only contingency fees are available and an equilibrium strategy when hourly fees are also available. An equilibrium where no attorney is hired and the claimant accepts a lower settlement from the insurer than under contingency fees can be sustained when the possibility of hourly fees is included in the model even though such fee arrangements never appear in equilibrium. Under asymmetric damage information and only contingency fees, there exists a separating equilibrium where the low (high) damage claimant hires an attorney on a contingency fee basis (does not hire an attorney), and the insurer can deduce the true expected damage award of each claimant and offer and have accepted a settlement appropriate for each type. There is no separating equilibrium if the possibility of hourly fees is included in the model with asymmetric information, so the insurer cannot deduce the damage type of the claimant. The possibility of an hourly fee breaks the separating equilibrium, and there are parameter conditions that support an equilibrium where one type of claimant hires an attorney on an hourly fee basis and the other type hires one on a contingency fee basis, but there is no equilibrium where hourly fees are used by both claimant types. NOTES 1 See Cooter and Rubinfeld (1989) for a detailed review of the litigation versus settlement literature. 2 As noted above, use of this take-it-or-leave-it offer model of the bargaining process is supported by analysis of more complex models by Cheung (1988), Daugherty and Reinganum (1993), and Wang, Kim, and Yi (1994). 3 In a later section, it will be assumed that the insurer cannot observed.

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