«JOURNAL OF LAW, ECONOMICS & POLICY VOLUME 10 SPRING 2014 NUMBER 2 EDITORIAL BOARD 2013-2014 Steve Dunn Editor-in-Chief Crystal Yi Meagan Dziura Sarah ...»
70 Backward induction requires us to first determine the optimal strategy for the last-moving players in time, i.e., the genuine and piggyback plaintiff’s decisions of whether to accept or reject the defendant’s take-it-or-leave-it settlement offer in period 4. Once we have ascertained both plaintiffs’ optimal strategies in period 4, we work backwards in time to period 3 to identify the defendant’s optimal take-it-or-leave-it settlement offer. With this information, we continue working backwards in time, first
B. Settlement–Trial Subgame We begin by describing the outcome of the settlement–trial subgame, which begins at the point where a plaintiff files a lawsuit. Consider first the plaintiff’s decisions regarding whether to accept or reject the defendant’s take-it-or-leave-it settlement offer in the final period. If the plaintiff is genuine, it will only accept the defendant’s offer if it is at least as large as the expected value of trial. The expected value of trial is equal to the magnitude of the plaintiff’s damages less the plaintiff’s cost of trial.71 We assume that this is a positive amount, meaning that the plaintiff’s losses exceed the cost of trial.72 Any offer smaller than the plaintiff’s expected value of trial will cause a genuine plaintiff to reject the offer and proceed to trial. In contrast, piggyback plaintiffs, because they will lose at trial with certainty, will accept any positive settlement offer, and will drop the suit if offered zero.
We now move backwards in time to the point where the defendant makes its settlement offer. Since backward induction implies that the defendant has strategic foresight, it knows how the two types of plaintiffs will behave when faced with a given offer. Thus, if the defendant had perfect information and could distinguish genuine from piggyback plaintiffs, she would rationally offer genuine plaintiffs a settlement amount equal to their expected value of trial (as defined above), and zero to piggyback plaintiffs.
When faced with these offers, genuine plaintiffs will accept the offer and settle, while piggyback plaintiffs will drop their suits—which is to say, they will accept the offer of zero. As a result, with perfect information, the defendant is able to avoid a costly trial with genuine plaintiffs via settlement, and to pay nothing to piggyback plaintiffs. And because rational plaintiffs will anticipate this outcome, only genuine plaintiffs will ever file suit.
The outcome will be quite different in the uncertainty case, however, because the defendant will not be able to distinguish between the two types of plaintiffs at the settlement stage and therefore must make a single offer.
One possible strategy for the defendant is to offer the minimum amount that a genuine plaintiff will accept, in which case both types of plaintiffs will accept the offer, and no cases will ever go to trial. While this strategy saves on trial costs, it has the downside of paying off all piggyback suits as if they were genuine. And anticipating this, all potential piggyback plaintiffs will file suit. Alternatively, the defendant could offer zero to all plaintiffs—that is, refuse to settle any suits. While this strategy ensures that no piggyback deriving whether plaintiffs will file suit in period 2, and finally concluding with the injurer’s optimal care and activity choices in period 1. The presumption here is that players have strategic foresight.
71 For example, suppose the plaintiff’s damages are $100,000, and the cost of trial is $20,000.
Then, if the plaintiff goes to trial and wins, she will receive compensation of $100,000, but will have to pay trial costs of $20,000. Thus, the plaintiff will not settle for any amount less than $80,000.
72 Note that the plaintiff’s filing costs do not matter for the settlement decision because, once the case has been filed, filing costs are a sunk expenditure.
2014] THE IMPACT OF FRIVOLOUS LAWSUITS ON DETERRENCE 315 plaintiffs would be able to extract a positive settlement amount—and hence would be deterred from filing suit—it would cause all genuine plaintiffs to opt for trial, thus requiring the defendant to incur litigation costs in addition to paying damages.
It is important to note that it will always be optimal for the defendant to make one of these two offers—either the minimum amount a genuine plaintiff will accept (equal to the genuine plaintiff’s expected value of trial), or zero. It would obviously never pay for the defendant to offer more than a genuine plaintiff would accept (for both types would accept the lower amount), nor would it pay to offer an amount between zero and the minimum acceptable offer (for that offer would be rejected by genuine plaintiffs and would overcompensate piggyback plaintiffs).
The defendant’s optimal choice between the two offers will depend on its beliefs about the likelihood that a given plaintiff is genuine or piggyback. Those beliefs will depend on three factors: (1) the level of care the defendant exercised (which determines the likelihood that a genuine accident will occur); (2) the total number of potential piggyback plaintiffs who might file suit (which we treat as exogenous);73 and (3) the decision of piggyback plaintiffs about whether or not to file suit. If, taking all of these factors into account, the defendant perceives that the probability a plaintiff is genuine exceeds a certain threshold, then its optimal strategy will be to settle the claim—that is, to offer the minimum amount that a genuine plaintiff will accept. Although the defendant knows that by adopting this strategy there is a chance it will be paying off a piggyback plaintiff, the strategy is a worthwhile venture given the relatively high probability that the plaintiff is genuine and therefore would be willing to go to trial if offered nothing. Of course, potential piggyback plaintiffs will anticipate this behavior and will therefore file suit. Thus, in equilibrium, it must be the case that the total number of such plaintiffs is small relative to the number of genuine plaintiffs in order to fulfill the defendant’s expectations. We will refer to this “pure strategy” equilibrium,74 in which all potential piggyback plaintiffs file suit and the defendant settles all cases, as a Type 1 equilibrium.
Suppose alternatively that when all piggyback plaintiffs file suit, the defendant perceives a relatively low probability that a given plaintiff is genuine. In this case, one might suppose that the optimal strategy would be 73 This number will depend on various factors, including the total number of persons with whom the defendant interacts when engaged in its activity, the ease with which they feign injury, and their inherent propensity to falsely file a lawsuit.
74 A player adopts a pure strategy when it chooses a particular action with a probability of 1. In other words, a player chooses a single action with certainty. The intersection of players’ pure strategies is a pure strategy equilibrium. For readers without a background in economics, see MARTIN J.
OSBORNE, AN INTRODUCTION TO GAME THEORY 107–08 (Oxford Univ. Press 2004) (discussing the concept of a pure strategy and the resulting equilibrium briefly). In the context of our theoretical model, a Type 1 equilibrium is a pure strategy equilibrium because piggyback victims file suit with a probability of 1 and the defendant chooses to settle with a probability of 1.
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for the defendant to refuse to settle any cases—i.e., to offer a zero settlement amount—as a way of inducing all piggyback plaintiffs to drop their cases. Of course, this would entail going to trial with genuine plaintiffs, but given the defendant’s initial beliefs, she calculates that this strategy is cheaper than paying off all of the piggyback plaintiffs. Piggyback plaintiffs will correctly anticipate this outcome and therefore will not file suit. But notice that this contradicts the defendant’s initial beliefs that piggyback plaintiffs constituted a relatively large fraction of all plaintiffs who actually file suit. If, in consequence, the defendant revises his beliefs to suppose that all plaintiffs are genuine, its optimal strategy would then be to settle all cases. But then all potential piggyback plaintiffs would be induced to file suit, and again the defendant’s beliefs would be contradicted.
This circular reasoning reveals that a pure strategy equilibrium in which the defendant offers a zero settlement amount to all plaintiffs cannot exist in this case. There does, however, exist a second type of equilibrium that involves “mixed strategies” by the defendant and piggyback plaintiffs.
Generally speaking, a mixed strategy exists when a player randomly chooses between two or more pure strategy options.75 In the current context, the defendant randomizes between offering a positive settlement amount and zero, while piggyback victims randomize between filing suit and not filing.
(All genuine victims will file suit with certainty no matter what strategy they expect the defendant to play.) For these mixed strategies to constitute an equilibrium, it must be the case that defendants are indifferent between their two options, and piggyback victims are indifferent between their two options. Thus, the probabilities attached to each option for the two players must adjust to ensure that this is true.
As noted, this mixed strategy equilibrium in the settlement–trial subgame, which we will refer to as a Type 2 equilibrium, arises when the fraction of genuine plaintiffs in the population of all potential plaintiffs is below a threshold. In the resulting equilibrium, the defendant randomly settles with some plaintiffs and refuses to settle with the remaining ones (though remember, it cannot determine which plaintiffs are genuine), and piggyback victims randomly choose between filing and not filing suit.
Among the piggyback plaintiffs who file, the ones who receive a positive settlement offer accept and settle, while the ones who receive an offer of zero are forced to withdraw their suits. At the same time, the defendant necessarily incurs some litigation costs when those genuine plaintiffs to whom she offers zero opt for trial.
In summary, the outcome of the settlement–trial subgame involves two equilibria. The first is a pure strategy equilibrium in which all piggyback victims file suit and the defendant settles with all plaintiffs—both genuine and piggyback. This Type 1 equilibrium emerges when the fraction of gen
75 See id.2014] THE IMPACT OF FRIVOLOUS LAWSUITS ON DETERRENCE 317
uine plaintiffs in the population of all plaintiffs is “sufficiently large.” The second is a mixed strategy equilibrium in which only some piggyback victims file suit, and the defendant settles a fraction of these cases. Among the cases that the defendant chooses not to settle—that is, to whom she makes a settlement offer of zero—genuine plaintiffs go to trial while piggyback plaintiffs drop their suits. This Type 2 equilibrium emerges when the fraction of genuine plaintiffs in the population of all plaintiffs is “sufficiently small.” Notice, however, that at least some piggyback plaintiffs succeed in obtaining settlements no matter which equilibrium emerges.
C. Injurer’s Care and Activity Choices
Now that we have identified the two equilibria that can arise from the settlement–trial subgame, we shift our attention to the injurer’s care and activity choices in the initial period. The injurer’s problem at this point is to maximize the net return from the activity in question. In terms of our example, the owner of a supermarket will choose the number of stores to operate, or equivalently, the number of hours to be open (its activity), and the amount of time and effort devoted to maintaining a safe environment (its care), to maximize profit. In making these choices, the injurer rationally anticipates the liability risk that it faces, which is reflected in the outcome of the settlement–trial subgame as just described. The injurer’s expected return will therefore incorporate the expected liability and litigation costs that emerge from that subgame.
We will examine the impact of the threat of piggyback suits on the injurer’s choice of activity and care in two ways. First, we will compare the injurer’s optimal choices in the model with piggyback suits (the imperfect information model) to its choices in a model of perfect information—that is, where the injurer can perfectly distinguish between genuine and piggyback plaintiffs. After that, we will compare the outcome in the imperfect information model to the socially optimal choices of activity and care—that is, the choices that a social planner would make.
D. Comparison to the Perfect Information Model
Recall that when the injurer has perfect information regarding the plaintiff’s type, it will rationally settle with all genuine plaintiffs and offer zero to any piggyback plaintiffs, who would then drop their cases. Thus, at the point where the injurer makes its care choice, the injurer expects to face only genuine plaintiffs, with whom it will settle for their expected value of trial. The injurer’s optimal care choice therefore minimizes the sum of the costs of care and expected liability, which equals the plaintiff’s expected value of trial multiplied by the probability of an accident.
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