# The Price of Punishment

## A Game Theoretic Approach to Valuing Punitive Damages Risk

Author: Robert J. Parnell, CFA

(Originally published May 14, 2013)

### Introduction

In this paper we examine the “shadow” of punitive damage risk in settlement bargaining. The prospect of punitive damages poses a particularly difficult challenge for parties in litigation. In the first place, there is generally so much uncertainty about their award and scale that parties have great difficulty in estimating them. For corporate defendants there are important questions about materiality and appropriate accounting disclosure. For general counsel and financial managers, the possibility of a large punitive damage award represents a unique valuation and risk management challenge. In extreme cases, punitive damage risk may become a corporate governance issue involving questions of financial distress, solvency and stock valuation impact. And through all of these issues, there is the question of how the prospect of punitive damages should influence the settlement bargaining process. In short, given so much uncertainty, how are we to put a price on punishment?

While few cases involving punitive damage risk ever proceed to court, research has suggested a “shadow effect” of punitive damages in settlement bargaining. In this essay, we examine some of the difficulties associated with pricing punitive damages and provide an overview of our own quantitative approach.

### Stark Unpredictability

Courts and scholars have long recognized difficulties with the application of punitive damages: raising questions as to how they should be determined and whether or not they achieve their purpose. At issue is whether the inherent uncertainty surrounding punitive damages undermines their purpose. In Exxon Shipping Co. v Baker [1], the Supreme Court of the United States refers to the “stark unpredictability of punitive awards” as “the real problem”. However, this problem is not just a matter of legal policy, but a very real economic concern for parties seeking to settle disputes involving the potential for punitive damage awards.

If punitive damages are so completely uncertain, how do litigants currently reach a settlement over their potential award? And more importantly, how should they? To understand this issue we must first examine the nature of punitive damages and the uncertainty attending them.

There are two dimensions to the unpredictability referred to by the Supreme Court. The first is the question of whether or not punitive damages will be awarded. The second is the quantum of damages if they are. Neither of which offers any measure of certainty for litigants. The overall infrequency with which punitive damages are awarded might provide initial comfort for fearful defendants, but then the potential for their surprise application can be equally vexing.

The uncertainty in the finding of a punitive award is then only compounded by a very wide and uncertain range in the quantum of damages themselves. While the average ratio of punitive damages to compensatory damages is usually close to unity, outlier judgments can provide for enormous variance. Hersch and Viscusi (2010) observe that in a sample of “blockbuster” punitive damage cases (those over $100 million) the median ratio of punitive damages to compensatory damages was a staggering 8.85.

### The Shadow Effect

The potential for punitive damage awards at trial and the tremendous uncertainty attending their application and scale inevitably impacts settlement negotiations. Sometimes referred to as the “shadow effect”, this is the tendency of settlement agreements to reflect or “discount” the risk of a potential punitive award.

We might reasonably ask whether this shadow effect can be at all significant given the relatively rare incidence of both trials and punitive damage awards themselves. However, in their 1998 paper, “Punitive Damages: An Economic Analysis”, Polinsky (1997) cautions that “It would be a mistake to conclude that because punitive damages are not a significant factor in cases that go to trial, they also are not a significant factor in the settlement process.” That punitive damage cases are rarely pursued to adjudication does not mean that such damages are not frequently present in settlement negotiations.

Surprisingly, very little empirical research has been done to quantify the scale of the punitive shadow. In his study of insurance adjustor data, Koenig (1998) refers to the “largely unexplored shadow” effect of punitive damages. Nevertheless, Koenig was able to show that 10.6% of settled insurance claims were affected by the threat of punitive damages.

Doherty and Eckles (2003) note that, “’Threats’ of punitive damages loom over the bargaining process, and both parties to litigation do, in fact, bargain in the ‘shadow’ of punitive damages.” Taking a financial economic approach to punitive damages research, Rhee (2012) describes how the potentially catastrophic corporate liability associated with punitive damage risk provides the plaintiff with an arbitrage opportunity in settlement negotiations. In effect, the plaintiff can bargain strategically to capture some consideration of an anticipated punitive damages award in the settlement agreement.

Priest (1996) observes that, “It is obvious and indisputable that a punitive damages claim increases the magnitude of the ultimate settlement and, indeed, affects the entire settlement process, increasing the likelihood of litigation.”

While most of this research relates to punitive damages in particular, much of it could equally apply to the shadow effect of outlier judgment awards in general – punitive or compensatory. (In the remainder of this essay we will refer to punitive and outlier awards interchangeably.) At issue more generally, are the normative and descriptive theories as to the shadow effect of what Rhee refers to as “low frequency, high severity awards”. In our view, the characterization of the award is incidental. From a financial economic point of view, we might better think of the problem as one of discounting the statistical “fat tail” that the prospect of outlier punitive or compensatory damages creates in an otherwise normal distribution of expected judgement awards.

It is interesting to speculate how litigants actually price outlier award expectations in practice and what thought processes govern the size and shape of the shadow effect in their settlement bargaining. For example, how would a litigant price their expectation of a 10% chance of a $100 million punitive damage award? Clearly there is a punitive damage expected value of $10 million at issue. But what thought processes take the litigant from his belief about the expected value at trial to their consideration of this amount in the settlement price? There is no doubt that litigants navigate this path from expected value to settlement, but because of the largely private nature of settlement agreements, the size and shape of the punitive shadow may never be fully understood empirically.

The more important question from our perspective is the normative one: how *should* particular opposing litigants price the potential for a punitive damage award in their individual settlement negotiations? To explore this question, we will first review some research on normative theories and later discuss our own approach to solving this problem.

### Normative Theories

Research is scant on normative approaches to settlement bargaining over punitive damage awards. However there is some evidence of what we might call a “full accounting” expected value approach in which the probability weighted amount of the expected punitive damages award is simply added to the settlement price.

In their model of punitive damages in insurance claims settlement, Doherty and Eckles find that where punitive damage awards are available asymmetrically, the insured will “only be indifferent between going to court and settling if he can obtain the expected value of that [punitive damage] lottery in the settlement offer.” Other research sketches a similar “full accounting” expected value approach to punitive damages pricing.

This research syncs with a simple heuristic solution to the settlement pricing problem in which litigants price punitive damages in settlement at their full undiscounted expected value. In this framework it would be argued that a 10% probability of a $100 million punitive damage award would contribute $10 million to the settlement price. Such an approach is certainly tractable and it has intuitive appeal. The only problem is that it very likely overstates the normative value of punitive damage risk.

### The Flaw of Expected Values

This method of pricing punitive damages reflects a decision theory approach to the problem that assumes the offering litigant can determine the settlement price in isolation – as if he or she can unilaterally decide to buy or sell settlement as a good. This approach reflects a one-player perspective of the settlement game. But in reality we know that claims are settled in a (typically) bilateral bargaining process.

The bargaining process has the effect of discounting all trial expected values in a way that reflects, among other things, litigant uncertainty about the trial outcome and a division of the bargaining surplus. Just as compensatory damage expected values are discounted in settlement bargaining, so then are punitive damages. Setting aside issues of risk aversion for a moment, we argue that bargaining over the punitive shadow is fundamentally no different from bargaining over a compensatory shadow.

From a bargaining economics point of view, punitive and compensatory damages are simply contingent, probabilistic cash flows — some are just more uncertain and improbable than others. While it is unlikely for a compensatory damage award to have the same degree of uncertainty as a punitive damage award, the normative approach to pricing these two categories of judgment award should be fundamentally the same.

To illustrate this point, we can imagine a simplified distribution of expected judgment awards where the potential for a punitive damage or other outlier award is represented by a “fat tail” in the distribution. [2] See Fig. 1.

In this admittedly stylized expected judgment distribution, it is believed that there is a 10% chance that punitive damages will be awarded equal to 3x compensatory damages.

Our point is simply that the appropriate shadow of punitive damage risk in settlement bargaining is nothing more or less than the shadow of a fat tailed award distribution. The characterization of the award is irrelevant in terms of settlement pricing.

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### A Game Theoretic Approach

Based on this thinking we have developed a proprietary method of valuing punitive or outlier award expectations using a game theoretic analysis of the settlement bargaining problem. Our approach explores the rational optimal settlement bargaining position of each litigant given the combination of known and uncertain economic variables in the case. We extend the application of this model to punitive damage award expectations by using a fat-tailed distribution of the expected judgment award as an input to the analysis.

Our approach is fundamentally different to the full accounting expected value approach as it captures the essential characteristics bargaining over litigation settlement, that is: the bilateral nature of the settlement bargaining problem, the probabilistic nature of the response, uncertainty, strategic interaction of the disputants, and the contingent nature of legal claims. The game theoretic model we have developed is based on the canonical screening game of litigation and settlement and it captures a great deal of the reality of settlement negotiations:

- It assumes that litigants make rational choices to optimize their expected outcome.
- Each party determines their offer in the context of imperfect information and where the acceptance of an offer or demand is not certain.
- The optimal settlement decision balances the two countervailing economic forces in the bargaining process: making a more aggressive offer improves a litigant’s economic outcome if the offer is accepted, but a more aggressive offer is less likely to be accepted and thus increases the probability of a suboptimal trial.

Our model appropriately discounts the influence of the trial expected value in the settlement price by compounding the improbability of trial with the improbability of a plaintiff trial victory and the improbability of a punitive damage award － all in the context of a bilateral strategic bargaining game. As such it provides a more realistic assessment of punitive damage risk than the full accounting expected value method and goes some way to describing the effective size and shape of the punitive shadow.

### Hypothetical Case Analysis

To illustrate this approach we examine here a typical personal injury lawsuit in which liability of the defendant is certain and the dispute is only over the extent of damages. The cost structure of the litigation is assumed to be fairly typical and for simplicity both litigants are deemed to have the same cost of capital. The case is tried in a jurisdiction with English cost shifting rules and the indemnification rate is estimated to be 60%. There is uncertainty about the expected value of the compensatory damages award and both litigants understand there to be the potential for punitive damages. The damage award distribution is modeled as truncated, quasi-normal between $5 million and $30 million, with the potential for an outlier combined compensatory and punitive damage award at approximately $27 million. We analyze the relationship between the plaintiff’s settlement offer and their expected wealth from the bargaining process (wealth function) for different levels of punitive damage risk (probability of occurring) between 0% and 45%. See Fig 2.

### Visualising the Price of Punishment

The results of this analysis can be graphically illustrated by using the Settlement Dynamics Chart application below, which allows the reader to examine the impact of increasing punitive damage risk on the plaintiff’s wealth function. Sliding the blue control button from left to right demonstrates the effect of increasing punitive damage probability from 0% to 45%.

Our solution to pricing punitive damages in settlement illustrates how punitive damages and outlier award expectations in general have a much more subtle and complex effect on settlement bargaining than previously realized.

### Legal Dynamics Chart

#### Punitive Damages Risk

Punitive Risk + Baseline %: 1

In general we find that for only modest levels of punitive damage risk, the expectation of a punitive damage award has only a gentle influence on the optimum settlement offer associated with the compensatory award. In essence, the weight of the punitive damage award in the optimum settlement offer is heavily discounted by the improbability of trial, the improbability of a plaintiff victory and the improbability of the punitive award itself. Our model demonstrates that the confluence of these discounting mechanisms greatly mitigates the normative effect of outlier awards in a bilateral bargaining game. For these reasons, increasing the risk of punitive damages at this low level will in general pull only slightly at the maximum in the wealth function and only result in a modest increase to the equilibrium settlement offer.

At the same time, we can see that the possibility of a punitive damage award risk creates a second local maximum in the wealth function at the higher end of the rational settlement offer range. This is because the fat tail in the award distribution gives plaintiffs a greater incentive to make more aggressive demands, but only up to a point. We will refer to this feature in the structure of the litigant’s wealth function as ‘the punitive cliff’. Beyond the demand consistent with the location of the fat tail in the award distribution, the wealth response falls precipitously as might be expected.

However, the most interesting result of this analysis is the tipping point that occurs when the risk of a punitive damage award becomes extreme. In this particular case, where the probability of a punitive damage award rises to over 40%, the local maximum in the wealth function at the edge of the punitive cliff becomes dominant. At this point there is a stepwise increase in the litigant’s optimal settlement offer as the punitive award comes to dominate the litigants’ bargaining economics.

In general, our quantitative, game theoretic method helps to illustrate the flaw in using a full accounting expected value approach, showing in general how this approach tends to overstate the economic significance of outlier judgments and provides insufficient discounting of the outlier event. Using a game theoretic model of bargaining over outlier judgment awards, litigants can develop a more reasonable estimate of their rational effect on settlement bargaining. Except in extreme cases, we show that this effect will generally be somewhat muted. However, our approach demonstrates that as the prospect of punitive damages becomes more likely, the value of these legal claims can hit a tipping point wherein their effect can achieve a stepwise dominance, pushing the location of settlement bargaining to a focal point that is much more consistent with full accounting, undiluted impact of the prospective punitive award level.

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### Conclusions

Punitive damage award risk casts a shadow on the settlement bargaining process. Very little is known about the size and shape of this shadow in actual settlement negotiations. There is some evidence of a full accounting expected value approach in theoretical work and this may suggest a similar heuristic approach in actual settlement bargaining. Further empirical research is required to examine the effect of punitive damages on actual bargaining behavior.

However, using a game theoretic analysis of settlement bargaining combined with a fat-tailed distribution of judgment award expectations, we have developed a normative model that can more accurately cast the shadow of punitive damage awards in the settlement bargaining process. We believe it is the first fully specified model of this problem.

Our work demonstrates a more subtle and complex influence of punitive damages on the settlement bargaining process than has been previously realized. The model provides the first evidence of a distinctive structure to the influence of punitive damages on a litigant’s wealth function and the potential for punitive damages to cause a tipping point in the bargaining process as the probability of punitive damages becomes extreme.

This is not to say that punitive damage award risk can safely be ignored in all but the most extreme of circumstances. The impact of even modest punitive damage expectations can still be significant and should be thoroughly investigated. However, with careful analysis we can find their proper measure. Moreover, an understandable risk aversion will no doubt influence each litigant’s final decision, but a risk averse posture in the bargaining process should begin with a clear understanding of the rational, risk neutral perspective.

Where a credible punitive damages claim has been made, a heightened level of uncertainty is present in the dispute and in the settlement bargaining process. In such a situation, litigants need to ask important questions:

- What incremental risks are present for the defendant?
- What incremental opportunity is present for the plaintiff?
- What is the rational contribution of the punitive damage risk to the settlement price?
- What is the optimum settlement bargaining strategy for both parties?
- When do punitive damages become dominant in the bargaining process?
- How is the probability of settlement affected?

Quantitative modelling and game theoretic analysis of such disputes can greatly assist in answering these questions. When so much is at stake, it behooves both sides of the dispute to carefully analyze the price of punishment.

### Footnotes

- Exxon Shipping Company, et al., Petitioners v. Grant Baker et al. No. 07-219, Supreme Court of the United States (2008 U.S. LEXIS 5263).
- A fat tailed probability distribution has an unusually high probability of an extreme observation when compared to a normal distribution.

### References

Hersch, Joni, and W. Kip Viscusi. 2010. “Punitive Damages by Numbers: *Exxon Shipping Co. v. Baker.” Supreme Court Economic Review*, 18: 259-280.

Polinsky, A. Mitchel, 1997. “Are Punitive Damages Really Insignificant, Predictable, and Rational? A Comment on Eisenberg et al. , 26 J. LEGAL STUD. 663, 667 (1997).

Koenig, T., 1998. “The Shadow Effect of Punitive Damages on Settlements.” Wisconsin Law Review, 1, 169-209.

Doherty, Neil and Eckles, David, 2003. “Punitive Damage Effects on Post-Loss Bargaining and Settlement.” Insurance and Risk Management, University of Pennsylvania.

Rhee, Robert J., 2012. “A Financial Economic Theory of Punitive Damages”, Michigan Law Review, Vol. 111, 2012, pp. 1-56.

Priest, George L., 1996. “Punitive Damages Reform: The Case of Alabama”, 56 Louisiana Law Review.