Can Decoupling Punitive Damages Deter an Injurer’s Harmful Activity?

2.1 Period 3: the trial

Suppose that the defendant must pay both compensatory and punitive damages when the plaintiff wins the case. We define R(>0) as the amount of compensatory damages, which is equal to the amount of harm the plaintiff suffered, and α(≥1) as the multiplier of punitive damages. In this case, the defendant’s total payment is αR and the amount of punitive damages is (α−1)R. However, under the decoupling liability system, the plaintiff cannot receive the entire amount the defendant pays. For simplicity, we assume that the entire amount of punitive damages awarded is confiscated, as suggested by Chief Justice Rehnquist. This means the plaintiff receives only compensatory damages.

We further assume that the adversarial system is adopted at the trial. The adversarial system, previously modeled by Tullock (1975, 1980), Parisi (2002), and Froeb and Kobayashi (2012), is a procedural system where the trial process is party-controlled. The judges or jurors decide which parties win based on the evidence and arguments they present. This system contrasts with the inquisitorial system, where the judges independently search for the true facts in the case. ^[5] While the procedural systems in most countries have elements of both systems, we adopt a pure version of the adversarial system to highlight its main feature. Further, we assume that the probability of prevailing in litigation is determined solely by the relative efforts of the parties at the trial.

Following the previous studies noted above, we model the adversarial system as follows. During the trial, the plaintiff and defendant present their arguments and evidence. The efforts expended on this task by the plaintiff and defendant are x≥0 and y≥0, respectively. ^[6] Based on their efforts, the jury or judge decides the winning side. As a result, the plaintiff’s probability of winning the case is determined by both parties’ efforts. ^[7] The probability function is defined as follows. ^[8]

The plaintiff wins the case and the defendant bears strict liability with the probability of p(x,y). ^[9] We can easily obtain the derivatives of p(x,y); px>0, pxx<0, py<0, and pyy>0 as well as the cross derivative pxy=(x−y)/(x+y)3.

Moreover, it is assumed that the plaintiff incurs fixed cost k∈[0,kˉ]. ^[10] We assume that cost k is positive since it includes a variety of costs during the trial. For instance, it includes the opportunity costs of time for the plaintiff to file a suit and the psychological costs or mental burden on the plaintiff. It is also assumed that kˉ>0 and that k varies across plaintiffs. Further, k is uniformly distributed. ^[11]

The plaintiff decides her effort level x to maximize her expected trial payoffs:

Since the first-order condition is pxR−1=0, the plaintiff’s reaction function is

Likewise, the defendant’s optimization problem and reaction function are as follows:

Figure 2:

Reaction curves of the plaintiff and defendant.

The reaction curves depicted in Figure 2 have distinctive shapes. They have not only strategic substitute parts, which means that the curves slope downward, but also strategic complement parts, which means that the curves slope upward. The Nash equilibrium, which is the intersection of the two curves, is located in the strategic substitute part of the plaintiff’s reaction curve and the strategic complement part of the defendant’s reaction curve. ^[12]

We can find the following Nash equilibrium (x∗,y∗) by solving simultaneous eqs (3) and (5):

We examine the comparative statics of this equilibrium. The defendant increases his effort as the multiplier rises since

By contrast, the plaintiff reduces her effort as the punitive damages multiplier increases, where α>1, since

Thus, we can derive the following lemma:

The rise in the punitive damages multiplier forces the defendant to expend more effort since he wants to avoid a huge amount of damages. This indirectly affects the plaintiff’s effort level. That is, the increase in the defendant’s effort level discourages the plaintiff from competing.

This holds because the cross derivative is negative (pxy=(x∗−y∗)/(x∗+y∗)3<0). ^[13] It is negative because the defendant always expends more effort than the plaintiff in the equilibrium (y∗>x∗). Since the defendant’s economic stake is greater than that of the plaintiff, y∗ is greater than x∗.

We obtain the plaintiff’s probability of winning the case in the equilibrium as follows:

We check the comparative statics of this equilibrium:

This lemma is the consequence of the adversarial system. As we show in Lemma 1, when the punitive damages multiplier rises, the defendant increases his efforts at the trial and this discourages the plaintiff from competing under the adversarial system. Thus, the plaintiff’s probability of winning the case decreases.

2.2 Period 2: plaintiff’s decision to sue

In period 2, the plaintiff anticipates the trial result (in period 3) and calculates her expected payoff of filing a suit. Based on these factors, she decides whether to sue. In our model, since her expected payoff of suing is p(x∗,y∗)R−x∗−k and her payoff of not suing is zero, the plaintiff sues only when

In condition (11), T is defined as the plaintiff’s expected payoff of suing other than the fixed cost of the trial. Since k is assumed to be uniformly distributed from 0 to kˉ and T=R/(1+α)2 is greater than 0, the condition (11) can be depicted as shown in Figure 3.

Figure 3:

Distribution of the cost of filing a suit.

As shown in Figure 3, if T≤kˉ, then the plaintiffs of k∈[0,T] file a suit, whereas those of k∈(T,kˉ] do not. If T>kˉ, all the plaintiffs file suits. From now on, we focus on the situation when T≤kˉ, because in the real world it is natural that at least some people would choose not to file a suit.

We analyze the relationship between T and α:

It follows that the number of plaintiffs who file suits decreases as the punitive multiplier rises.

2.3 Period 1: defendant’s activity

In period 1, the defendant chooses whether to engage in a certain activity. This activity has a negative externality, which means it brings him a benefit β (>0) but inflicts harm R on the plaintiffs.

By anticipating the results of periods 2 and 3, the defendant calculates the equilibrium expected cost of this activity, denoted by W. Since the rate of plaintiffs’ suing is T/kˉ when T≤kˉ, the defendant calculates W as follows:

The derivative of W with respect to α is as follows:

The sign of Wα is negative since 4α2−α−1>0 when α≥1. This means that the expected cost W of the activity decreases as the punitive damages multiplier rises. ^[14]

As W continues to decrease, it is more likely that W becomes smaller than the benefit β from the activity. By contrast, when expected cost W is greater than benefit β, the defendant does not engage in the activity. ^[15] Thus, we can state the following proposition.

Although the deterrence effect is conventionally expected to increase to match the rise in the punitive damages multiplier, Proposition 1 reveals that the opposite is true under the decoupling liability and the adversarial system. ^[16] As we show in Lemma 2, the adversarial system decreases the plaintiff’s probability of winning the case as the punitive damages multiplier rises. This reduces the plaintiff’s incentive to sue and increases the defendant’s incentive to engage in a harmful activity. ^[17]

This proposition may have practical significance because some commentators point out that courts would impose higher damages on defendants if the state were to introduce the decoupling system. ^[18] In other words, the courts are likely to raise the punitive damages multiplier in such cases, resulting in an (admittedly unintended) reduction in the deterrence effect, as we show in Proposition 1.

2.4 Period 0: comparison with other liability systems

Finally, we compare the decoupling liability system with the coupling system that allows the plaintiff to receive both compensatory and punitive damages. The latter is a normal system in most common law countries that have punitive damages but do not have split-recovery statutes. Since the derivation process is the same, we describe it only briefly.

Under the coupling system, in period 3, the plaintiff maximizes p(x,y)αR−x−k and the defendant minimizes p(x,y)αR+y. By solving these optimization problems, we can derive a Nash equilibrium (xc,yc)=(αR/4,αR/4). Based on the equilibrium, the plaintiff’s probability of winning the case p(xc,yc) is calculated as 1/2. Thus, her expected gain from the trial is p(xc,yc)αR−xc−k=αR/4−k and the defendant’s expected cost for the trial is p(xc,yc)αR+yc=3αR/4.

In period 2, let Tc be the plaintiff’s expected payoff of suing other than the fixed cost of the trial. Tc is calculated to be p(xc,yc)αR−xc=αR/4. By comparing Tc with T in inequality (11), we can identify the system under which the plaintiff files a suit more easily. Since α≥1, we can obtain Tc≥T. Therefore, it is easier for the plaintiff to sue under the coupling system than under the decoupling liability system.

Under the coupling system, in period 1, we define Wc as the defendant’s equilibrium expected cost to engage in the activity that has negative externalities. Wc is calculated to be (Tc/kˉ)(3αR/4). By taking the derivative of Wc with respect to α, we obtain Wαc=3αR2/(8kˉ)>0. This result sharply contrasts with the one shown in Proposition 1. That is, the defendant’s expected cost of engaging in a harmful activity increases as the punitive multiplier rises under the coupling system, whereas it decreases under the decoupling liability system.

By comparing Wc with W in expression (13), we can identify the liability system under which the defendant engages in a harmful activity more easily. Since α≥1, we have Wc≥W. Based on this inequality and Wαc>0, we can say that it is easier for the defendant to engage in a harmful activity under the decoupling liability system than under the coupling system. It follows that the deterrence effect of the decoupling liability is weaker than that of the coupling liability.

We can summarize these results as the following proposition.

In addition, the deterrence effect of the decoupling liability system is even weaker than that of the compensatory damages system adopted by civil law countries, where the defendant pays and the plaintiff receives only compensatory damages. This fact can be easily elicited since α=1 in our model of the decoupling system corresponds to the compensatory damages system and the deterrence becomes weaker as α increases, as shown in Proposition 1.

This proposition may be important for countries that have adversarial systems and impose only compensatory damages on defendants. ^[19] By attempting to improve the deterrent effect without granting plaintiffs a windfall, such countries are likely to introduce a decoupling system. However, Proposition 3 notes that the introduction of a decoupling system may not improve the deterrent effect.