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The Priest-Klein Hypotheses: Proofs, Generality, and Extensions

Yoon-Ho Alex Lee and Daniel Klerman*

Draft: July 14, 2014

ABSTRACT

Priest and Klein’s 1984 article, “The Selection of Disputes for Litigation,” famously

hypothesized a “tendency toward 50 percent plaintiff victories” among litigated cases.

Despite the passage of thirty years and the article’s enduring influence, their hypothesis

has never been formally proven, and doubts remain about its validity and the

assumptions needed to sustain it. This article makes four contributions. First, it

distinguishes six hypotheses plausibly attributable to Priest and Klein. Second, it

rigorously proves five of these hypotheses, shows that two of them are valid with

greater generality than Priest and Klein assumed, and shows that one of them is false.

Third, it shows that some of the hypotheses remain valid even when an incentive-

compatible bargaining mechanism is employed. And finally, it shows that some of the

hypotheses remain valid when parties use Bayes’ rule and information about the

distribution of all disputes to refine their estimates of the plaintiff’s probability of

prevailing.

* Yoon-Ho Alex Lee is Assistant Professor of Law, USC Gould School of Law.

alee@law.usc.edu. Daniel Klerman is Charles L. and Ramona I. Hilliard Professor of Law and History,

USC Gould School of Law. Charles L. and Ramona I Hilliard Professor of Law and History, USC Gould

School of Law. www.klerman.com. dklerman@law.usc.edu. The authors thank Ken Alexander, Andrew

Daughety, Dan Erman, Jonah Gelbach, Bart Kosko, Jennifer Reinganum, as well as the participants at the

Twenty-Fourth Annual Meeting of the American Law and Economics Association for their helpful

comments and suggestions.

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INTRODUCTION

Priest and Klein’s 1984 article about the selection of disputes for litigation famously

hypothesized there will be a “tendency toward 50 percent plaintiff victories” among litigated

cases (p.20). Their article has been one of the most influential law review articles of all time,

and its influence is growing as empirical work on law becomes more prominent. Compare

Shapiro & Pearse (2012) to Shapiro (1996). Even with the introduction of more rigorous models

of settlement—notably game-theoretic models involving asymmetric information—Priest and

Klein’s article continues to be cited by sophisticated empiricists and in the most respected peer-

reviewed journals. See, e.g., Hubbard (2013); Gelbach (2012); Atkinson (2009); Bernardo,

Talley & Welch (2000); Waldfogel (1995); Siegelman & Donohue (1995).

Despite the passage of nearly thirty years since the publication of Priest and Klein’s

original paper, their results have never been rigorously proved, and doubts remain about the

assumptions needed to sustain their conclusions. Because Priest and Klein supported their

argument only with simulations and an informal, graphical proof, the precise statement and

scope of their claims have not been entirely clear. Subsequently, Waldfogel (1995) formalized

Priest and Klein’s model, following carefully Priest and Klein’s original set-up and notation but

without formally proving any results. Shavell (1996 p. 499, nn. 19-20) provided a sketch of a

more rigorous proof. Part of the challenge is that the formalization involves a double integral

over a region of integration that is only implicitly defined. (Waldfogel (1995), p.237). Although

Hylton and Lin (2012) also formalize and prove some of Priest and Klein’s claims, they do so

using a model substantially different and in many way’s less plausible and less general than

Priest and Klein’s.1

In this paper, we provide the first rigorous set of proofs of various hypotheses that can be

attributed to Priest and Klein, while remaining faithful to Priest and Klein’s original set-up as

well as generalizing and extending it. Through our proofs, we establish both exact mathematical

statements of Priest and Klein’s hypothesis and the precise extent to which their statements are

true.

Specifically, we think it helpful to distinguish between six different hypotheses plausibly

attributable to the Priest and Klein (1984):

Trial Selection Hypothesis. This hypothesis states that “the disputes selected for

litigation (as opposed to settlement) will constitute neither a random nor a representative

sample of the set of all disputes” (p.4). This proposition is probably the most important

contribution of their article, and it is universally accepted. It is true even under more

recent asymmetric information models of settlement. See Klerman & Lee (2014).

Fifty-Percent Limit Hypothesis. This hypothesis states that “as the parties’ error

diminishes” there will be a “convergence towards 50 percent plaintiff victories” (pp.18).

This hypothesis is often called the Priest-Klein hypothesis. Priest and Klein acknowledge

1 Hylton and Lin’s model differs from Priest and Klein’s in that Priest and Klein assume that case

strength is measured by a real number, , where plaintiff prevails if , where

represents the decision standard or legal rule. In contrast, Hylton and Lin assume that case strength is a

probability. Although this difference might seem small, it has major modeling implications. In addition,

Hylton and Lin make use of an extrinsic “censoring function” -- the function determining which cases

will be litigated – to arrive at their result.

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that “the fifty percent success rate will actually be achieved only near the limit” (p.22).

In this paper we provide the first formal proof that the Fifty-Percent Limit Hypothesis is

valid, both under the assumptions in Priest and Klein’s1984 article and under a broader

set of conditions and models.

Asymmetric Stakes Hypothesis. This hypothesis states that, if defendant would lose

more from an adverse judgment than plaintiff would gain, then plaintiff will win less than

fifty percent of the litigated cases; conversely, if plaintiff has more to gain, then plaintiff

will win more than fifty-percent (see pp.24-26). This hypothesis is most plausibly a

statement about the limit percentage of plaintiff victories as the parties become

increasingly accurate in predicting trial outcomes. In this paper, we show that, under the

divergent expectations model used by Priest and Klein, the asymmetric stakes hypothesis

is generally valid. But we also find that even the hypothesis that plaintiffs win less often

when defendants have more to lose is not generally true under an incentive-compatible

mechanism.

Irrelevance of the Dispute Distribution Hypothesis. This hypothesis claims that as the

parties become increasingly accurate in predicting trial outcomes, the limit value of the

plaintiff trial win rate will be “unrelated to the position of the decision standard or to the

shape of the distribution of disputes” (pp. 19 & 22). This hypothesis is closely related to

the Fifty-Percent Limit Hypothesis, but more fundamental. It is also more general,

because it holds even when the stakes are unequal.

No Inferences Hypothesis. This hypothesis states that because selection effects are so

strong, no inferences can be made about the law or legal decisionmakers from the

percentage of plaintiff trial victories, because “the proportion of observed plaintiff

victories will tend to remain constant over time regardless of changes in the underlying

standards applied.” (p. 31). As a result, if one is looking at plaintiff trial win rates,

differences between legal standards or legal decisionmakers “will always be concealed by

the parties’ selection of disputes for litigation.” (p. 49). If the stakes are equal, plaintiff

will prevail fifty-percent of the time (or close to that percentage), regardless of the law,

legal institutions, or the biases of judges or juries. To the extent that there are deviations

from the fifty-percent prediction, they will be the result of asymmetric stakes or factors

other than the legal standard or characteristics of the judge or jury. In a companion article,

Klerman & Lee (2014), we show that the No Inferences Hypothesis is generally false.

Under the divergent expectations model used by Priest and Klein and under more recent

asymmetric information models, changes in the legal standard and differences in legal

decisionmakers may result in predictable changes in the percentage of plaintiff trial

victories.

Fifty-Percent Bias Hypothesis. This hypothesis states that regardless of the legal

standard, the plaintiff trial win rate will exhibit “a strong bias toward . . . fifty percent” as

compared to the percentage of cases plaintiff would have won if all cases went to trial (pp.

5 & 23). In this paper we show that the Fifty-Percent Bias Hypothesis is true less often

than the Fifty-Percent Limit Hypothesis. More specifically, unlike the Fifty-Percent

Limit Hypothesis, the validity of the Fifty-Percent Bias Hypothesis will depend on the

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symmetry and the log-concavity of the distribution of disputes. So the Fifty-Percent Bias

Hypothesis will be true for normal distributions, but not for many other distributions.

Of course, analytical proofs (and disproofs) of these hypotheses, by themselves, do not imply

their empirical validity. Nonetheless, analytical proofs can guide empirical research by

elucidating under what conditions the hypotheses should be valid.

In addition to proving these hypotheses using Priest and Klein’s original model, we

consider two possible extensions of Priest and Klein’s model. First, the model has been

criticized for lacking an incentive-compatible mechanism. Priest and Klein implicitly assumes

that as long as the plaintiff’s subjective expected net trial recovery is greater than the defendant’s

subjective expected net loss, the parties will settle. Nevertheless, much has been written about ex

post inefficiency arising in strategic bargaining. See Myerson and Satterthwaite (1985); but see

McAfee and Reny (1992). Our paper addresses the possibility of bargaining inefficiency by

coupling Priest and Klein’s model to an incentive-compatible mechanism. Our approach is

similar to Friedman and Wittman (2007) in that we employ the Chatterjee-Samuelson

mechanism, but different in that our model retains Priest and Klein’s original set-up. By

remaining faithful to Priest and Klein’s original model, we can identify the extent to which their

hypotheses are robust to an incentive-compatible mechanism. Under this set-up, we identify one

class of plausible Nash equilibria and show that several of Priest and Klein’s hypotheses remain

valid for this class, including the Trial Selection, Fifty-Percent Limit, and Irrelevance of Dispute

Distribution Hypotheses. Somewhat surprisingly, the Fifty-Percent Limit result appears to hold

for some cases with asymmetric stakes, which strengthens the Fifty-Percent Limit Hypothesis.

Second, the original version of Priest and Klein’s model assumes (implicitly) that parties

estimate the plaintiff’s probability of prevailing without taking into account information about

the underlying distribution of all disputes. More precisely, Priest and Klein assume that the

defendant’s degree of fault can be represented by a real number, , and plaintiff and defendant

estimate the defendant’s fault with error. So plaintiff’s point estimate is , where

has mean zero and standard deviation . Similarly, defendant’s point estimate is ,

where also has mean zero and standard deviation , and and are independent. Parties

then estimate the plaintiff’s probability of prevailing by calculating the likelihood that true is

greater than under the assumption that the true is distributed with mean or and

standard deviation . This means that the parties are not using Bayes’ rule and the information

they may have about the distribution of to estimate the likelihood of plaintiff victory. Later

scholars, including Waldfogel (1995), have similarly assumed that the parties are naïve and non-

Baysean in this regard. Priest and Klein’s model can, however, be modified to include the

assumption that parties are sophisticated Bayseans who take into account the underlying

distribution of disputes. This modification, while plausible, has its own weaknesses, and we

think it is potentially of value only for limiting behaviors. We show that several of Priest and

Klein’s hypotheses remain valid if the parties use Bayes’ rule, including the Trial Selection,

Fifty-Percent Limit, Asymmetric Stakes, and Irrelevance of Dispute Distribution Hypotheses.

The rest of the paper proceeds as follows. Part I explores a formalized version of Priest

and Klein’s model. Part I is faithful to the original model in that, although we generalize or relax

certain assumptions, the set-up is entirely consistent with Priest and Klein’s article. Parts II

explore the validity of some of the hypotheses when parties use the Chatterjee-Samuelson

mechanism. Like Part I, Part II assumes the parties do not use Bayes’ rule and information about

the distribution of disputes in calculating the plaintiff’s probability of prevailing. Finally, Part III

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explores how things change when parties do use Bayes’ rule in calculating the plaintiff’s

probability of prevailing. Part III does so both under the assumption that parties settle whenever

plaintiff’s subjective expected litigation payoff is lower than defendant’s subjective expected

litigation and under the Chatterjee-Samuelson mechanism.

I. FORMALIZATION OF PRIEST-KLEIN’S ORIGINAL MODEL

This Part assumes familiarity with Priest and Klein (1984) and follows Waldfogel’s

(1995) formalization. Though there have been other attempts to formalize Priest and Klein’s

model (see Wittman (1985) and Hylton and Lin (2012)), Waldfogel offers the formalization that

is most faithful to the model in Priest and Klein’s original paper. See Hylton & Lin (2012), n.5.

Priest and Klein (1984) represent the merits of a case by a real number , and the

decision standard as , where defendant prevails if , and plaintiff prevails if .

For example, in a negligence case, might be the efficient level of precaution expenditures

minus defendant’s actual precaution. Under these assumptions, Y* would be zero. Priest and

Klein, for simulation purposes, assume is distributed according to a standard normal

distribution. We do not impose that restriction but instead assume is distributed according to

a probability density function that is bounded above everywhere, and locally continuous

and nonzero at . Note that if all disputes were to be litigated, the overall plaintiff win rate

would be ∫

If cases go to trial, courts observe the true and give verdicts based on its location as

compared to . Plaintiff and defendant themselves make unbiased estimates of ,

and , respectively, where and are have mean 0, standard deviations and

, respectively, and are distributed according to the joint probability distribution ( )

Priest and Klein assume that and are independent and that ( ) is bivariate

normal with We do not impose these restrictions. Instead we assume that and

are distributed with mean zero, not necessarily independent, and distributed with standard

deviations and , respectively, according to a joint probability density function ( )

such that ( ). In other words,

( ) belongs to a family

of mean-zero probability density functions that vary parametrically with and and can be

standardized with proper scaling. The reason why we make this assumption is that our analysis

requires taking the limit as and approach zero, and therefore, there needs to be a well-

defined family of distributions as the standard deviation parameter varies. A bivariate normal

distribution with mean zero certainly satisfies this condition, but a host of other distributions also

satisfy this condition.2 We assume the support of

( ) is the entire , but our proof

can easily be generalized to cover distributions with finite support.

2 A partial list of probability distributions that satisfy this condition include bivariate distributions

composed of: uniform distributions, triangular distributions, and raised cosine distributions for finite

support; and generalized normal distributions, hyperbolic secant distributions, Laplace distributions,

logistic distributions (if we scale by

√ ), Pearson Type IV distributions. Given any univariate probability

density function with mean zero and standard deviation 1, one can always construct such a family of

bivariate density function by letting ( )

(

) (

)

.

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In order to estimate the probability with which the plaintiff will prevail, both the plaintiff

and the defendant need to take into account the fact that their estimates of case merit,

and , are not wholly accurate. Therefore, they must estimate both the mean and standard

deviation of their estimates of . Following Priest and Klein (1986) and Waldfogel (1995), in

this section we assume that plaintiff estimates the mean of sampling distribution of to be

and the standard deviation to be .3 As a result, plaintiff’s subjective estimate of the

probability it will prevail, ( | ), will simply be [

], where is

the univariate standardized cumulative distributions of . Similarly, defendant estimates the

mean of to be , and the standard deviation to be . So defendant estimates the

probability that the plaintiff prevails to be [

], where and is the univariate

standardized cumulative distributions of Priest and Klein assume that the parties litigate if , where

is the damages that defendant pays plaintiff if the case is litigated and plaintiff prevails,

and are litigation costs for plaintiff and defendant respectively, and and are settlement

costs for plaintiff and defendant respectively. This condition for litigation makes sense, because

settlement can only happen if both parties perceive the payoffs to settlement to be higher than the

payoffs to litigation. The settlement condition can be rewritten as , where

and . is known as the Landes-Posner-Gould

condition for litigation, after the three scholars who formulated it. Priest and Klein simulate their

results with

. We assume .

Priest and Klein are silent about how the parties bargain to arrive at a settlement.

Technically, the Landis-Posner-Gould condition is merely a sufficient condition for litigation,

but not a necessary one. Litigation might happen even if the condition is violated, because

parties might not be able to agree on the settlement amount, even if there is a range of settlement

amounts that would be in their perceived mutual interest. That is, as modern mechanism design

research as shown, bargaining is frequently inefficient. See Myerson and Satterthwaite (1983);

but see McAfee and Reny (1992). Nevertheless, Priest and Klein (1984) and others using the

divergent expectations model have assumed that the Landes-Posner-Gould condition is necessary

as well as sufficient for litigation, and, in this Part, we proceed with that assumption.4

Priest and Klein allow for the possibility that parties may have asymmetric stakes and

suggest there will be a deviation from fifty-percent in such cases. For example, the defendant

may be more concerned about its reputation or an adverse precedent, so it may lose more from an

adverse judgment than the plaintiff gains from prevailing. As Priest and Klein point out,

asymmetric stakes can be formalized by assuming that, the plaintiff would win if it prevailed

and the defendant would lose if the plaintiff won. Under these assumptions, parties litigate if

if . If we assume , the trial condition becomes

.5 If is greater than 1, then plaintiff faces a greater stake in the litigation than the

defendant, and vice versa. It is often easier to think of cases in which , as in cases in

3 See Part III for an alternative way of estimating the mean and standard deviation of

4 We relax that assumption in Part II.

5 As Waldfogel notes, this condition assumes that parties have symmetric trial and settlement

costs even as they face asymmetric stakes. See Waldfogel (1995), p.236.

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which defendants has a lot more to lose—such as reputation and future loss of profits—from

litigation than plaintiffs. Nevertheless, in this paper, we place no theoretical limitation on

other than that it is positive. From the trial condition, since it follows that if

plaintiff will not bring the case, because it has no credible threat to take the case to trial.

Without a credible threat to go to trial, defendant has no incentive to settle, so plaintiff derives no

benefit from suing. Thus, we assume throughout

In addition, since plaintiff and

defendant make their decision to go to trial based on their expectation of judgment amount, the

analysis with respect can readily translate to cases where plaintiff and defendant disagree

about the amount of damages.

A. TRIAL SELECTION HYPOTHESIS AND IRRELEVANCE OF THE DISPUTE DISTRIBUTION

HYPOTHESIS

Let be the probability that a case goes to trial when the decision

standard is , where parties predict their case merits with errors and that are distributed

with mean zero and standard deviations and . When , we will simply denote

this probability as .

can be written as the probability that

[

] [

]

6

The fraction of cases litigated is ∫

. This value clearly approaches

zero as and approach zero. Recall that if all disputes were litigated, the overall plaintiff

win rate would be ∫

The plaintiff trial win rate is

∫

∫

Since this is different from ∫

, we can readily see that the set of litigated cases is not

simply a random set of all disputes. The Trial Selection Hypothesis is therefore clearly correct.

Note that , as written, depends on many parameters, including the shape of

the distribution of disputes. In the limit as and approach zero, however, the plaintiff trial

win rate will not vary with the distribution of disputes. The following Proposition provides the

specific functional form for the limit of the plaintiff trial win rate and establishes the Irrelevance

of the Dispute Distribution Hypothesis. All proofs are in Appendix.

PROPOSITION 1 (IRRELEVANCE OF THE DISPUTE DISTRIBUTION HYPOTHESIS). The limit value of

the plaintiff trial win rate under the Priest-Klein model will not depend on the distribution of

disputes if and only if the plaintiff’s stake in not significantly larger than defendant’s stake.

More specifically, suppose for a fixed , is locally continuous and

nonzero around and bounded above everywhere, and and are distributed with mean

6 As discussed above, this formulation assumes that parties do not take the underlying distribution

of disputes into account.

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zero according to ( ) such that

( ) with full support

over . Then the plaintiff trial win rate is given by

∫

∫

where ∬

( )

and

{( ) | [

] [

] }

For

, the limit of this value as approach 0 reduces to an expression

independent of :

∫

∫

For

, the limit value

will always be 1 and thus also independent of When

, whether the limit value

will depend on and will depend on the shape of [ ] and [ ].

Before we go on, we make a few observations. First, although is well-

defined for all positive values, it is undefined at .7 This is because when , each

party knows whether it will win or lose with 100% certainty and no cases will go to trial.

Proposition 1 is therefore a statement about the limit value of a function at a point at which it is

not continuous (since undefined). Second, the result indicates that for

, the

limit value of the overall plaintiff win rate does not depend on the shape of the distribution of

disputes, , but only on the shape of the distribution of errors, , which in turn

depends on the shape of as well as the region of integration (which is affected by ).

Third, by contrast, when

, the limit value of plaintiff’s win rate will also not depend

on the distribution of disputes because it will be 1 regardless of or values and regardless of

. This is somewhat surprising and might be labeled as the “One Hundred Percent Limit

Hypothesis.” In this case, there is a threshold case estimate for the plaintiff, above

which the defendant will be unable to make an attractive settlement offer even for disputes the

defendant considers as sure-win cases (e.g., and even as the range of error becomes

really small. Thus, as goes to zero, all disputes that are intrinsically stronger cases for

plaintiffs than will always litigate, but no cases below the threshold value will get

litigated. Therefore, in the limit, plaintiffs will all cases litigated, and the limit value is 1.

The formal proof is included in the Appendix. We include only a portion of the proof

here. Since for a fixed as approaches zero, necessarily approaches

zero. We begin by normalizing the variables. For a given , let

. Then we have

7 This does not affect integrability over since the point of discontinuity has measure zero.

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( )

( )

(

)

Meanwhile, { | [ ] [ ]

} . Therefore, for each

∬ (

)

This normalization of the variables is useful, because before normalization, as a

function of , was a double integral of a fixed bivariate distribution over a region of integration

in the –plane that depended on both and and . After the change of variables,

, as a function of , is a double integral of a bivariate

distribution (whose center has been translated) over a region of integration in the –plane that

does not depend on depend on either or but only on . Figure 1 represents the region of

integration when . Note that in this case, the region of integration in the -plane,

call it , is bounded by a horizontal asymptote above and a vertical asymptote on the left

side. In other words, it will be contained in a translated fourth quadrant. This will generally be

true for

, although depending on the value the location of the asymptotes will vary.

The integral in the numerator of is evaluated by centering the bivariate distribution

of errors at each point along the line in the first quadrant and integrating it over the shaded

area. Two such center points are illustrated in Figure 1.

So, for all we can rewrite as Therefore,

∫

∫

∫

∫

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Figure 1. Litigation Set in the -Space when

For the main result when

, we are done if we can take the limits under

the integral by applying Lebesgue’s Dominated Convergence Theorem. Then

∫

∫

∫

∫

since is locally continuous at and since . Application of the Theorem,

however, requires existence of a Lebesgue-integrable function that dominates

. Lemma A1 from the Appendix shows when

such construction is possible by

taking the same integral but over a region of integration that properly contains . To

show this upper-bound function is Lesbesgue-integrable—in other words, that the integral of the

constructed function is indeed finite—we make use of bivariate Chebyshev’s Inequality. Note in

particular from Figure 1 that is calculated as follows: for each point , along

the line , the integral of centered at and integrated over .

Since is bounded above, we need only show that is dominated by a Lebesgue-

integrable function. But notice that after some threshold point in either direction,

will always be properly contained in the set which is the complement of a square box centered at

, whose length increases as increases. Therefore, , eventually, will always

be strictly smaller than the integral of centered at but integrated over the

complement of the square box with an increasing length. Because Chebyshev’s Inequality gives

us an inverse quadratic relation between the distance from the mean and the integration of any

probability distribution away from the mean by that distance, the integral of centered

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at over the complement of the square box must eventually decrease at least as fast as the

speed of , which must converge to a finite value. Hence, a Lebesgue-integrable dominating

function is constructed, and we can take the limits under the integral.

In contrast, when

, it is easy to see that is characterized as a region

to the left of a graph that is characterized by two vertical asymptotes (shown in Figure 2). In this

case, will properly contain a region defined by for some . To get the

limit, we employ Chebyshev’s Inequality to the complement of for the numerator, and

we can show that the numerator and the denominator must diverge to infinity for positive at the

same pace, and thus the limit will equal 1. Therefore, we have the One Hundred Percent Limit

Hypothesis in this case.

Figure 2. Litigation Set in the -Space when and

B. FIFTY-PERCENT LIMIT HYPOTHESIS

The fact that we are able to derive the functional form of the limit value is useful. This

means that the limit value will equal fifty percent only when ∫

∫

is equal to fifty

percent, and not otherwise. Therefore, the limit value of the overall plaintiff win rate will be

determined entirely by the shape of . For example, if is symmetric around

zero, the limit value will equal exactly fifty percent; and if is not symmetric around

zero, the limit value will generally not equal exactly fifty percent except by coincidence. The

rest of this section makes clear that the assumptions made by Priest and Klein satisfy this

corollary and explores some of the other parameters which would also lead to a fifty-percent win

rate at trial.

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LEMMA 1.1. If plaintiff’s and defendant’s stakes are symmetric and their prediction errors,

and , are distributed with a common standard deviation and according to a joint probability

density function that is symmetric around 0 and symmetric with respect to each other, then the

limit value of the overall plaintiff win rate will be fifty percent. More specifically, if

( ) ( ), which will be necessarily true when ( )is symmetric in ,

in , and with respect to and (that is, ( ) ( )), then is

symmetric around . That is,

for all .

We make a few observations. In particular, if ( ) is symmetric (as in Lemma

1.1), the parties’ stakes are symmetric and the parties are equally competent in their ability to

estimate the merit of their case, then the limit value will indeed equal fifty-percent.

COROLLARY 1.2 (FIFTY-PERCENT LIMIT HYPOTHESIS). If plaintiff’s and defendant’s prediction

errors, and , are distributed with a common standard deviation (i.e., and

according to a joint probability density function that is symmetric around 0 and symmetric with

respect to each other, and if the stakes are equal (i.e., , then the limit value of the plaintiff

trial win rate is fifty percent.

This Corollary formalizes Priest and Klein’s statement that “[t]here will always persist a

tendency toward 50 percent plaintiff victories regardless of the shape of the distribution or the

position of the standard,” (p.22), as long as we understand the phrase “tendency toward 50

percent” as a statement about the limit value. As stated, Corollary 1.2 is more general than Priest

and Klein’s original version, because Priest and Klein assume that ( ) is independent

bivariate normal and the distribution of disputes, is normal. These latter assumptions,

while reasonable, are not strictly necessary. Meanwhile, the specific functional form of the limit

value shows that under skewed distributions, the fifty percent limit hypothesis will be

systematically violated. If ( ) is skewed toward the first quadrant in the -space, the

limit value of the plaintiff’s overall win rate will be greater than fifty percent, and likewise, if it

is skewed to the third quadrant, the limit value of the plaintiff’s overall win rate will be less than

fifty percent.

C. ASYMMETRIC STAKES HYPOTHESIS AND ASYMMETRIC INFORMATION

In this Part, we discuss the case of . This can be interpreted as plaintiff and

defendant facing asymmetric stakes, or disagreeing about the amount of damages. The fact that,

in Theorem 1, there is an upper bound on the asymmetry of the stakes, is intuitive but also

significant. As gets high, plaintiff is more likely to litigate the case regardless of its merit. For

example, if then every single case will be litigated since the trial condition

will always be satisfied. In that case, the overall plaintiff trial win rate will simply be

∫

. In this extreme case, plaintiff’s win rate will depend on the shape of but

will be independent of . In other words, it will be independent of the joint probability

distribution of the errors and the region of integration. For an intermediate range of , the

overall plaintiff win rate will depend on all three factors.

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Nevertheless, it is somewhat striking that the upper bound on should be relatively

small—that if the plaintiff’s stake is greater than twice the defendant’s stake, the plaintiff trial

win rate depends on the shape of the distribution of disputes. As discussed in Klerman and Lee

(2014), it seems most plausible to assume

⁄ . Therefore, since Theorem 1’s requirement

is that

, this implies an upper-bound of around ⁄ for , before the shape of the

dispute distribution has an effect on the limit value.

Meanwhile, if is lower than the upper bound (but the stakes are still asymmetric), then

the region of integration will not be symmetric with respect to the line , and thus the

limit value will not generally equal to fifty percent (except by coincidence). More specifically,

within a given range of , we have the following corollary.

COROLLARY 1.3 (ASYMMETRIC STAKES HYPOTHESIS). If the plaintiff’s stake in not significantly

larger than defendant’s stake, then the overall plaintiff win rate will be increasing in the ratio of

the plaintiff’s stake to the defendant’s stake. Specifically, for

, the limit value

of plaintiff’s win rate is increases strictly in . In particular, if the limit value is exactly equal to

fifty percent when , then the limit value will be less than fifty percent if the defendant has a

greater stake than the plaintiff, and more than fifty percent if the plaintiff has a greater stake

than the defendant. When

, the limit value will always be 1.

The result is consistent with Priest and Klein’s observation that “where the stakes are

greater to defendants than to plaintiffs, relatively more defendant than plaintiff victories ought to

be observed in disputes that are litigated. The results are reversed where the stakes are greater for

the plaintiff.” (p.25). Nevertheless, when the plaintiff has more at stake than the defendant,

Priest and Klein’s statement must be qualified. The plaintiff will win more than fifty percent

when the stakes are only moderately asymmetric. If the plaintiff has much more at stake, that is,

if

, then the plaintiff win rate in the limit will be 1 and therefore, here again, the

plaintiff will indeed win more than fifty percent.

In addition, although the Priest and Klein did not discuss asymmetric information, it is

interesting to note that by assuming that the parties differ in their ability to estimate trial

outcomes, the Priest-Klein model can generate results that are consistent with modern

asymmetric information models. If plaintiff and defendant differ in their ability to predict the

merit of the case, then, even when ( ) is bivariate normal, the joint probability

function will not be symmetric in and and thus the limit value will not generally equal to

fifty percent (except by coincidence). Instead, the plaintiff’s win rate will vary predictably with

his comparative accuracy (over the defendant) of estimating the case’s merit.

COROLLARY 1.4 (ASYMMETRIC INFORMATION). The plaintiff trial win rate is (comparatively)

higher the more (comparatively) accurately plaintiff can estimate the true , and vice versa. In

other words, given

Furthermore, as approaches

infinity (i.e., plaintiff has full-knowledge), the overall plaintiff trial win rate rate will approach

one; and as approaches zero (i.e., defendant has full-knowledge), the overall plaintiff trial win

rate will approach zero.

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14

Note that this corollary is similar to results under the standard screening and signaling

models. The proportion of litigated cases won by the party with the informational advantage will

be larger than the proportion of cases it would have won if all cases were litigated. See Hylton

(1993), Klerman & Lee (2014). The prediction of Corollary 1.4, however, is more extreme, as a

result of Priest and Klein’s assumption that courts decide without error. Under the assumptions

usually made by asymmetric information models – that one knows its type, while the other

knows only the distribution of all types – Corollary 1.4 predicts that the informed party will win

100% of the time, whereas that will not generally be true under asymmetric information models.

This extreme prediction follows from the fact that, in Priest and Klein’s model, defendant type is

a real-number representation of the merits of the case, and the real line is divided into two halves,

a part where the plaintiff wins with certainty, and a part where the defendant wins with certainty.

Thus, if a party knows the defendant’s type, it can predict the outcome with certainty. In contrast,

in most asymmetric information models, party type is the probability that the plaintiff will

prevail, which ranges from zero to one and can take on all intermediate values. As a result,

under most asymmetric information models, if a party knows its type, it usually knows a

probably that is neither zero nor one.

D. FIFTY-PERCENT BIAS HYPOTHESIS

Thus far we have limited our discussions to hypotheses relating to the limit as parties

become increasingly accurate in predicting trial outcomes. Proposition 1, however, also offers

some insight into the Fifty-Percent Bias Hypothesis. That hypothesis says that the plaintiff trial

win rate will be closer to fifty percent than the percentage of cases that plaintiff would have won

if all cases had been litigated and none had settled.

Note, however, that this hypothesis is only plausible if the following two conditions are

met: first, the stakes are symmetric ( ) and both parties are equally well informed ( ;

and second, the plaintiff trial win rate if all cases were litigated isn’t itself 50%. If the first

condition is not satisfied, Proposition 1 tells us that the limit value of the plaintiff win rate will

not itself be fifty percent in general, and therefore, the Bias Hypothesis is likely to be false for

sufficiently small values of . On the other hand, if the first condition is satisfied, the plaintiff

trial win rate will converge to 50%, and thus it is reasonable to think that the plaintiff trial win

rate in the real world (where most cases settle) will be closer to its limit value than the plaintiff

trial win rate in a counterfactual world where no cases settle. The second condition is also

necessary (and plausible) because if the plaintiff win rate if no cases were settled happens to be

fifty percent, it is impossible for plaintiff trial win rates to be closer to fifty percent.

It turns out that, even if the two conditions above are satisfied, the bias hypothesis is not

generally true unless we make more restrictive assumptions about . We show that if

is symmetric (although not necessarily symmetric around ) and is logarithmically concave—

conditions which are satisfied for normal distributions and Laplace distributions—the Fifty-

Percent Bias Hypothesis is true. However, neither symmetry nor logarithmically concave

cumulative distribution by itself is sufficient. See Appendix.

PROPOSITION 2 (FIFTY-PERCENT BIAS HYPOTHESIS). If plaintiff’s and defendant’s prediction

errors, and , are distributed with a common standard deviation (i.e., and

according to a joint probability density function that is symmetric around 0 and symmetric with

respect to each other, and if the stakes are equal (i.e., , and is symmetric and

Preliminary Draft

15

logarithmically concave, then the overall plaintiff trial win rate will be closer to fifty percent

than the plaintiff win rate among all disputes. In other words, | |

|∫

| for all , and all > 0. In addition, the inequality is strict unless

∫

.

E. NO INFERENCES HYPOTHESIS

Priest and Klein asserted that “plaintiff victories will tend toward 50 percent whether the

legal standard is negligence or strict liability, whether judges or juries are hostile or sympathetic”

(p.5). One implication is that it will not generally be possible to make inferences about the legal

standard based on the observed plaintiff trial win rate. As long as parties are aware of the legal

standard and judicial preferences, their settlement behavior will take those factors into account,

and the plaintiff trial win rate will not vary. (p.36). If this hypothesis were about limit values,

then it would be true under the Priest-Klein model since the limit value under Proposition 1 is

independent of and . Nevertheless, the limiting result holds only as goes to zero. But

this result is not necessarily relevant to empirical work, because, goes to zero, the number of

litigated cases also goes to zero. Thus, whenever one is doing empirical work on litigated cases,

one is necessarily dealing with a situation in which is strictly positive. Therefore, we think the

No Inferences Hypothesis can only be understood in the context of a fixed and positive . In a

companion article, Klerman & Lee (2014), we show that despite strong selection effects,

inferences about the legal standard of liability will be possible under a fairly broad set of

assumptions and parameter values. For the sake of completeness, we include our result for

inferences under Priest and Klein’s model but refer the reader to our companion piece for

exposition as well as the proof.

PROPOSITION 3 (NO INFERENCES HYPOTHESIS). The No Inferences Hypothesis is generally false.

Instead, inferences from plaintiff trial win rates to the legal standard are possible. Specifically,

under the assumptions of the Priest-Klein model, where

and parties predict the case’s

true merit with errors distributed according to a bivariate distribution with positive standard

deviation that is the same under the two legal standards and where the distribution of disputes is

log-concave, the plaintiff trial win rate is strictly higher under the more pro-plaintiff legal

standard. In other words,

| ̅

In fact, the proof shows that a natural inference is possible as long as the region of

integration is not empty and is invariant with respect to . This will be true under the model’s

current set-up where the Landes-Posner-Gould condition does not take into consideration the

specific distribution of disputes. Therefore, as long as there is a positive probability that

[ ] [ ]

—a condition violated when

–an inferences

arepossible. ( ) is assumed to have full support, but we need not assume that

( )or that the errors are distributed according to mean zero.

II. BARGAINING UNDER THE PRIEST-KLEIN’S MODEL

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Priest and Klein’s model has been criticized because it does not include an incentive-

compatible bargaining mechanism. Instead, Priest and Klein assume that whenever settlement

would be in the parties’ perceived mutual interest, they will successfully bargain to a settlement.

That is, they assume that the Landes-Posner-Gould litigation condition, , is a

necessary as well as sufficient condition. Put another way, Priest and Klein implicitly assume

the existence of an ex post efficient8 bargaining mechanism through which the parties would

always be able to settle when doing so was in their perceived mutual best interest.

Nevertheless, modern research in bargaining and mechanism design has reached the

pessimistic conclusion that, when there is asymmetric information, such an efficient mechanism

may not exist. See Myerson and Satterthwaite (1983). On the other hand, Myerson and

Satterthwaite’s theorem does not apply directly to the Priest-Klein model, because the parties’

estimates are not independent and type spaces are infinite. Instead, McAfee and Reny (1992)

suggests that in such cases an incentive-compatible ex post efficient trading may indeed be

possible if we have an outside broker—a budget balancer. Nevertheless, settlement negotiations

seldom if ever employ an outside broker who may contribute his or her own money, nor has

anyone implemented any other kind of efficient mechanism for settlement.Consequently, it is

worth investigating whether the validity of the Priest-Klein hypotheses would be affected by

relaxing the assumption that the Landes-Posner-Gould litigation condition is both necessary and

sufficient, and instead assuming a more plausible (albeit inefficient) bargaining mechanism.

Following Friedman and Wittman (2007), we investigate the implications of the

Chatterjee-Samuelson mechanism. Under that mechanism, plaintiff and defendant each submit

secret offers to a neutral party or computer. If the plaintiff’s offer is greater than the defendant’s

offer, the case goes to trial. If the plaintiff’s offer is less than or equal to the defendant’s offer,

then the case settles for the average of the two offers.9 As explained by Friedman and Wittman

(2007), even though the Chatterjee-Samuelson mechanism is seldom used in actual litigation, it

can be seen as a “reduced form of a more complicated but unspecified haggling between the

plaintiff and defendant lawyers.” (p. 110). We diverge from Friedman and Wittman (2007) in

that we graft the Chaterjee-Samuelson mechanism onto the Priest-Klein model rather than

starting from a completely different model.

We begin by making a few simplifications. First, we assume settlement costs are zero,

.10

Second, we assume , in that the plaintiff and the defendant are equal

in their ability to estimate . Third, we assume and are distributed independently

according to joint probability distribution ( ) ( ) ( ) such that

and is the standard cumulative distribution.

8 The literature uses “ex post inefficiency” to refer to cases where there is a mutually beneficial

agreement but the parties fail to reach it. In the litigation-settlement model, there is always ex post

inefficiency, because the parties would have been better off settling for the judgment amount without

incurring litigation costs. Thus, the relevant inefficiency must be relative to the parties’ ex ante

evaluation of the situation. That is, there is inefficiency if a settlement would have made both parties

think they were better off, given their ex ante evaluations of the case. 9 An alternative mechanism is to consider a sequential bargaining. But such a set-up will present a

“model duality” problem. The outcome may depend on whether the opening offer is made by the plaintiff

or the defendant. See Daughety & Reinganum (1994). 10

Positive settlement costs raise a question as to how to distinguish between cases that are simply

dropped by the plaintiffs and cases for which the two parties agree to settle at zero dollars.

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17

The mechanism proceeds as follows. The plaintiff receives a signal and the

defendant receives a signal . Simultaneously, the plaintiff makes a settlement

demand, , and the defendant makes a settlement offer, . If , the parties settle at

; otherwise, parties litigate.

In this Part, we continue to assume that parties do not use Bayes’ rule to estimate the

mean and standard deviation of the true value of . That is, plaintiff estimates the mean of to

be and the standard deviation to be , and defendant estimates the mean of to

be and the standard deviation to be .11

The rest of the set-up follows Friedman and Wittman (2007) closely. We consider pure

strategies contingent on the realized signal. Thus, a plaintiff’s strategy is a measurable function

that assigns the demand ( ) [ when he observes signal .

Similarly, a defendant’s strategy is a measurable function that assigns the demand

when he observes signal . The objective of the plaintiff is to maximize

the expected net payments, conditioned on his realized signal and the defendant’s strategy

. The defendant’s object is to minimize expected net payments.

There are several important differences, however, from Friedman and Wittman (2006).

First, the support for the strategy functions is the entire real line. Therefore, we cannot work

with uniform distributions. Second, the signals and are not independent. Instead, the

plaintiff makes inferences about the distribution of based on and likewise for the defendant.

The plaintiff estimates using a compound distribution: he first figures out the conditional

distribution of given his , and then conditions the expected distribution of on his

expected conditional distribution of . The defendant does likewise.

The payoff function for the plaintiff is

( )

∫ (

)

{ | }

( | )

∫ ( ( | ) ){ | }

( | )

The first-term in the right-hand side is the expected value of settling and the second-term is the

expected value of litigating. Therefore, ( | ) and ( | ) represent the conditional

distribution of the other party’s signal based on the party’s own signal. Likewise, the payoff for

the defendant is

( ( ) )

∫ ( ( )

)

{ | }

( | )

∫ ( ( | ) ){ | ( ) }

( | )

11

See Part III for relaxation of that assumption.

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We define the Nash equilibrium of the litigation game as follows.

Definition. A Nash equilibrium (NE) of the litigation game is a strategy pair

( ( ) ) such that ( ) ( ) and

( ( ) ).

As Friedman and Wittman (2007) note, existence of such a Nash equilibrium is not guaranteed

because we are limiting our analysis to pure strategy Nash equilibria. Note, however, that there will

always be trivial equilibria such as ( ( ) ) in which every dispute

goes to trial and neither party has any incentive to deviate from its strategy. Because such

equilibria predict that parties never settle, which is inconsistent with the reality of litigation, we

consider only equilibria in which the parties sometimes settle.

In their analysis, Friedman and Wittman (2007) focuses on symmetric Nash equilibria.

Their definition of symmetry would translate to our set-up as follows.

Definition. The strategies ( ) and are symmetric around if there exists

some such that, for all , we have , or equivalently,

.

The equilibrium we identified in Proposition 4 is indeed symmetric around in the limit. It

turns out that all Nash equilibria that are symmetric around will satisfy the Fifty Percent Limit

hypothesis. This can be seen as follows. The region of integration is defined as follows:

{ | } This region will be

symmetric around the line if and only if we can show that whenever ,

we must also have . But if the strategies are symmetric around , we

must have and .

Therefore, we must have if and only if we

have . So we have proved the following

Lemma.

LEMMA 4.1 (SYMMETRIC NASH EQUILIBRIUM). For all families of Nash equilibria that are

symmetric in the limit, the Fifty-Percent Limit Hypothesis holds true and the Asymmetric Stakes

Hypothesis will be false.

The following proposition (proved in Appendix) shows that for all is locally

continuous and nonzero around and bounded above everywhere, there exists at least one

family of non-trivial Nash equilibria which converges to a symmetric Nash equilibrium.

Although this is a family of equilibria (as parameter and vary), the specification in fact

exhibits a single equilibrium for given and .

PROPOSITION 4 (BARGAINING UNDER THE PRIEST-KLEIN MODEL). Suppose is locally

continuous and nonzero around and bounded above everywhere, and and are

distributed with mean zero according to ( ) ( ) ( ) such that

with full support over and is continuously differentiable. Then there exists and

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19

some suitable value such that for each the following condition

holds: there exists such that for all , there will generally12

exist a pair of

continuous functions ( ) such that

and for each , the following ( ( ) ) is a Nash equilibrium:13

( ) {

{

Moreover, for all such , the limit value of the plaintiff’s trial win rate will be

fifty-percent. This means simultaneously that for this equilibrium, the Fifty-Percent Limit

Hypothesis will hold but also the Asymmetric Stakes Hypothesis will be false for such values of .

Figure 3. Litigation Set in the -Space under Bargaining

12

We say “generally” because the proof makes use of the Implicit Function Theorem and thus

will depend on a particular Jacobian not taking on the value of zero at the particular equilibrium value.

Because the particular Jacobian is not identically zero, this will generally be the case, although it may be

possible to construct an example in which the Jacobian can take on the value of zero at the particular

equilibrium point. 13

The stability of this Nash equilibrium was checked for normal distributions using Excel

simulations.

Preliminary Draft

20

Figure 3 depicts, the set of case estimates litigated in the -plane when the

parties use the Chatterjee-Samuelson mechanism as approaches zero. In the case depicted

above, is properly contained in , and the set can be

considered the region of ex post inefficiency. What is surprising about Proposition 4 is that

because a similar set can properly contain even when (imagine a very small

change in which changes the shapes of only slightly from ), the Fifty Percent

Limit Hypothesis will continue to hold even with asymmetric stakes.

We cannot, however, claim the results for nonsymmetric equilibria. We are currently in

the process of running Matlab simulations to look for other families of equilibria. For the

moment, we conclude only that under the family of equilibria we have identified and under any

symmetric equilibria, the 50% Limit Hypothesis will be true.

III. ALLOWING PARTIES TO MAKE INFERENCES BASED ON THE DISTRIBUTION OF

DISPUTES

We mentioned that the original version of Priest and Klein’s model assumes (implicitly)

that parties estimate the plaintiff’s probability of prevailing without using Bayes’ rule or

information about the underlying distribution of all disputes. Consequently, thus far, we solved

the model under the assumption that ( | ) [

] and

| [

]. This means that the parties are not taking into

consideration the actual distribution of in estimating the likelihood of the plaintiff’s victory.

It also means that they are assuming that and , the standard deviations of and , are also

the standard deviations of the parties’ estimates .

It is not hard to see why this might be a problematic assumption. For example, consider

the simple discrete case where can only take on an integer value from 1 to 10 with each

integer value equally likely. Suppose each party observes within an error that is zero, +1, or -

1, each with probability one third. In this case, if a party observes 5, he is correct to presume that

the true will be 4, 5, or 6, each with probability one third. As a result, it would be rational for

the party to assume that the mean of the sampling distribution of is 5 and that same error

distribution (0, +1, -1) that generated the party’s observations will also describe the sampling

distribution. Suppose, however, that, with the same distribution of observation errors, never

takes the values 5 or 6. That is, { }, with each value equally likely, and each

party observes within an error that is 0, +1, or -1, each with probability one third. In this

situation, if a party observes 5 and knew about the underlying distribution of , he would know

with certainty that the true is exactly 4. It would be irrational to assume that the mean of the

sampling distribution was his observation,5, and that the true value of was distributed around

that value with errors zero, +1, or -1 with equal probability. This example suggests that a

rational party would use information about the distribution of disputes both in calculating the

mean of its estimate of and in calculating its standard deviation.

There are two potential modifications to Priest and Klein’s model to include the

assumption that parties do take the underlying distribution of disputes into consideration.

Unfortunately, neither of them is entirely satisfactory.

The first method is to interpret and not as the parties’ own estimates of but

merely informative signals the parties receive about defendant’s degree of fault. The parties then

Preliminary Draft

21

use the signals to make their estimates of . This approach is in significant tension with Priest

and Klein’s original model because it means that the parties point estimates of are not and

. This also means that, contrary to Priest and Klein’s model, the parties’ estimates will not be

distributed according to the stipulated functions, and . Rather, it is the signals

that are distributed in this way.

Another interpretation of this modification may be that the parties make mistakes and for

this reason, they observe different and . Suppose, for example, that the evidence (or signal)

is a series of numbers, that inferring the true requires adding those numbers together, and that

parties occasionally type the wrong number in their calculators. Then, even if the parties see the

same evidence (the numbers to be added), their calculations differ. Rational parties would do

their calculations, but would not fully trust the results. Instead, they would combine their

calculations with their knowledge of the distribution of all disputes and their knowledge of the

probably of key stroke error to make a corrected/conditional estimate of given the error-prone

calculation they actually made.

One conceptual difficulty with this approach is that the parties’ estimates are no longer

unbiased. For example, suppose the distribution of disputes is standard normal and true case

merit is . Although the parties will receive signals that average 1.5 (because the signals

are unbiased), they will take into account the fact that the standard normal is centered at zero, so

their estimates will, on average, be somewhat less than 1.5. This assumption is at variance with

a basic assumption of the Priest-Klein model.

Nevertheless, it turns out that this modification does not make a significant difference

when it comes to results pertaining to limits. This is because the new region of integration under

the modified approach, , while no longer independent of or , will coincide the

original region of the integration in the limit. Intuitively, it makes sense that even as

we allow the parties to make inferences based on the distribution of disputes, as we take the limit

as approaches zero, only the distribution of disputes that are very close to will significantly

affect the probability, and therefore the shape of on the whole will matter little. We

summarize the results in the following proposition, proved in Appendix.

PROPOSITION 5 (EQUIVALENCE OF THE LIMIT RESULTS UNDER THE MODIFIED BAYESIAN MODEL).

The results of Proposition 1 (together with Corollaries 1.2, 1.3, and 1.4) and the limit results

within Proposition 4 go through as stated under the modified Priest-Klein model. In other words,

the following results hold true under the modified Bayesian model of the Priest-Klein model:

IRRELEVANCE OF THE DISPUTE DISTRIBUTION HYPOTHESIS. The limit value of the plaintiff

trial win rate under the Priest-Klein model will not depend on the distribution of disputes

if and only if the plaintiff’s stake in not significantly larger than defendant’s stake.

FIFTY-PERCENT LIMIT HYPOTHESIS. If plaintiff’s and defendant’s prediction errors, and

, are distributed with a common standard deviation (i.e., and according to a

joint probability density function that is symmetric around 0 and symmetric with respect

to each other, and if the stakes are equal (i.e., , then the limit value of the plaintiff

trial win rate is fifty percent.

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22

ASYMMETRIC STAKES HYPOTHESIS. If the plaintiff’s stake in not significantly larger than

defendant’s stake, then the plaintiff trial win rate will be increasing in the ratio of the

plaintiff’s stake to the defendant’s stake.

ASYMMETRIC INFORMATION. The plaintiff trial win rate is (comparatively) higher the more

(comparatively) accurately plaintiff can estimate the true , and vice versa.

BARGAINING. Suppose is locally continuous and nonzero around and bounded

above everywhere, and and are distributed with mean zero according

to ( ) ( ) ( ) such that with full support over and

is continuously differentiable. Then there exists and some suitable value

such that for each the following condition holds: there

exists such that for all , there will generally14

exist a pair of

continuous functions ( ) such that

and for each , the following ( ( ) ) is a Nash

equilibrium:

( ) {

{

Moreover, for all such , the limit value of the plaintiff’s trial win rate

will be fifty-percent. This means simultaneously that for this equilibrium, the Fifty-

Percent Limit Hypothesis will hold but also the Asymmetric Stakes Hypothesis will be

false for such values of

Meanwhile, the proofs of other results—specifically those that are not about limits--do

not readily go through and require other assumptions. These include the Fifty-Percent Bias

Hypothesis and the No Inferences Hypothesis. The reason these results may be different is that

the disputes most heavily litigated may not be distributed close to the decision standard, . For

example, in the case of normal distributions centered at 0, given a decision standard , the

disputes most heavily litigated will be centered around rather than . This is another

way in which this modification leads to results contrary to Priest and Klein’s model. A central

aspect of Priest and Klein’s model was that disputes lying most closely to the decision standard

would be most heavily litigated. Under the modification, however, litigated disputes will tend to

be farther away from the mean than the decision standard. For large values, the center of

litigated disputes will in fact have little to do with the decision standard. This means that when

the legal standard favors the plaintiff, , litigated cases may more often be those the

14

We say “generally” because the proof makes use of the Implicit Function Theorem and thus

will depend on a particular Jacobian not taking on the value of zero at the particular equilibrium value.

Because the particular Jacobian is not identically zero, this will generally be the case, although it may be

possible to construct an example in which the Jacobian can take on the value of zero at the particular

equilibrium point.

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23

defendant will win. Conversely, when the legal standard favors the defendant, , litigated

cases may more often be those the plaintiff will win. For this reason, the Fifty-Percent Bias

Hypothesis is valid and inferences from litigated cases will only be possible when is sufficiently

small.

PROPOSITION 6. (NON-LIMIT RESULTS UNDER MODIFIED BAYSEAN MODEL) When is

standard normal distribution and sufficiently small, the results of Propositions 2 and 3 go

through. In other words,

FIFTY-PERCENT BIAS HYPOTHESIS. If plaintiff’s and defendant’s prediction errors, and

, are distributed with a common standard deviation (i.e., and according to a

joint probability density function that is symmetric around 0 and symmetric with respect

to each other, and if the stakes are equal (i.e., , and is symmetric and

logarithmically concave, then the overall plaintiff trial win rate will be closer to fifty

percent than the plaintiff win rate among all disputes.

INFERENCES FROM LITIGATED CASES. Under the assumptions of the Priest-Klein model,

where

and parties predict the case’s true merit with errors distributed according

to a bivariate distribution with positive standard deviation that is the same under the two

legal standards and where the distribution of disputes is log-concave, the plaintiff trial

win rate is strictly higher under the more pro-plaintiff legal standard.

An alternative modification is to assume that parties receive signals and

but the signals are distributed in such a way that when parties take into

consideration, they recover estimates and . If such signal

distributions can be constructed, then all of the results would go through. Unfortunately, it is not

clear that such distributions can always be constructed.

IV. CONCLUSION

This paper provides an in-depth analysis of Priest and Klein’s hypotheses by revisiting

the mathematics behind their model. We conclude that the Fifty-Percent Limit Hypothesis is

mathematically well-founded and is true under the assumptions made by Priest and Klein and

under a wider array of assumptions. For example, under Priest and Klein’s original model the

Fifty-Percent Limit Hypothesis, Trial Selection Hypothesis, and Irrelevance of Dispute

Distribution Hypothesis are true (a) for any distribution of disputes that is bounded and both

positive and continuous near the decision standard, (b) even if the parties’ prediction errors are

not independent. In addition, these hypotheses remain true if the parties use the Chatterjee-

Samuelson bargaining mechanism and even if parties use Bayes’ rule to calculate the mean and

distribution of true case merit. The Asymmetric Information and Fifty-Percent Bias Hypotheses

are often true, but are less robust to parametric and modeling changes. Finally, the No

Inferences Hypothesis is generally false.

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24

APPENDIX

This Appendix contains all the proofs not already stated in the main text.

PROOF OF THEOREM 1. Based on the sketch of the proof included in the text, we begin by

showing Lemma A1.

LEMMA A1. ∫

is finite for all

and

∫

∫

15

Proof of Lemma A1 requires several steps. Our strategy in proving Lemma A1 is to construct a

Lebesgue-integrable function that dominates The function need not be

continuous. It need only integrate to a finite value. We construct by integrating (

) over an area that properly contains . Let us begin by considering Lemma A2, which

describes the shape of .

LEMMA A2.

is non-empty if and only if

.

For all

, is bounded left by a vertical asymptote at

.

For all

, is bounded above by a horizontal asymptote

.

When

, is not bounded above, and if

, it is characterized by a

region to the right of an increasing curve that has two vertical asymptotes, at

and

(

)

PROOF OF LEMMA A2. If

, is empty since [ ] [ ] for all . On the

other hand,

, for high enough and low enough , we can find some value such that [ ]

[ ]

Meanwhile, if [ ] [ ]

then [ ]

, and thus is bounded left by

. If

[ ] [ ]

Thus, is bounded above by

. Meanwhile, if

then the boundary curve [ ] [ ] must

continually increase as increases. Furthermore, if

, then the inequality must hold for all

value of (

) and all values of . Q.E.D.

Importantly, for

is properly contained by a translated fourth

quadrant in the –plane (with the origin at (

)

. This set is obviously contained

15

For

the value of depends on both the shape of [ ] and the value of .

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25

in all of . Moreover, for (

), the point (

) is at least

(

) away from

. For (

) (

) is at least (

) away from . Therefore, we have

the following two Corollaries.

COROLLARY A3. Let { (

) (

)} and

{ (

) (

)} Then for

is properly contained

in for every where

{ | |

{ || | | | |

| |

| } | |

COROLLARY A4. For

each ,

where

∬ (

)

We are now ready to prove Lemma A1.

PROOF OF LEMMA A1. For the first part, by Corollary A4, we need only show

∫

is finite.

∫

∫ ∫

∫

Meanwhile, for each

(| | | | |

| |

| ) (| | |

{ }|

|

| |

{ }| ). By bivariate Chebyshev’s Inequality,

(| | |

{ }| |

| |

{ }| )

√ ( )

(|

{ }| )

Therefore,

√ ( )

(|

{ }| )

which is quadratic in in the denominator and therefore integrates to a finite value over [ . The

integral over ] can likewise be shown to be finite. However, in this case we need not assume

since is always bounded by a vertical asymptote.

Meanwhile, for all

is defined by two vertical asymptotes. Therefore,

can be bounded below by which integrates (

) over the

complement of i.e., over a rectangle containing (

), which increases in . Therefore, again

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26

by bivariate Chebyshev’s Inequality, we can show that this lower bound integrand is bounded below by

( √ ( )

(|

{ }| )

), which clearly integrates to infinity over all . Q.E.D.

PROOF OF THEOREM 1 (CONTINUED). For

the rest of the proof is explained in

the text. For

, note that

∫

∫

∫

∫

∫

Here

∫

∫

as before, since is

bound by a left asymptote. Meanwhile, we cannot apply Lebesgue’s Dominated Convergence

Theorem to ∫

because contains all of the increasing large

rectangles. Instead, we can write

∫

∫ ( )

∫

∫

where ∬ (

)

where is the complement of

. Then we can apply Lebesgue’s Dominated Convergence Theorem to

∫

. Therefore,

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

Since

∫

, while all other terms are finite, the limit value

of the plaintiff win rate is 1. Q.E.D.

PROOF OF LEMMA 1.1. We show that for all ,

Notice

∬ ( )

where the region of integration is

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27

{( )| [

] [

]

}

{( )| ( [ ( )

]) ( [

])

}

{( )| [

] [

]

}

Since ( )is symmetric with respect to and , ( ) ( ). Thus we can

swap and without changing the value of the integral. The result follows Q.E.D.

PROOF OF COROLLARY 1.2. By Lemma 2, is symmetric around , which implies

is symmetric around zero. By the Theorem, the limit value must be . Q.E.D.

PROOF OF COROLLARY 1.3. The result is immediate since for

,

properly belongs to . Q.E.D.

PROOF OF COROLLARY 1.4. For

is properly contained by a translated fourth

quadrant in the –plane (with the origin at (

)

. affects the slope of the graph

along which to take the integral over . As increases, plaintiff is comparatively more accurate

in assessing the merit of the case than the defendant. But as increases, the slope of the line decreases

and the line becomes closer to in the first quadrant and farther from in the fourth

quadrant. Thus, plaintiff’s win rate increases and defendant’s win rate decreases. Furthermore, as

approaches infinity, the line will approximate the -axis and will eventually be properly

contained in for sufficiently large , and hence ∫

will be infinite. Meanwhile,

the line the line will lie wholly outside (and increasingly so) for negative . Thus,

plaintiff’s win rate will approach 1. Similarly, as approaches 0, plaintiff’s win rate will approach zero.

Q.E.D.

PROOF OF PROPOSITION 2. Since and are both symmetric. Without loss of

generality, assume is centered around 0. The inequality is clearly satisfied when is at . Since

is log-concave, (i) it is also single-peaked (including the possibility of a flat-top) and (ii) its

cumulative distribution, , is also log-concave. Without loss of generality, suppose Then we

need show | | ∫

First notice

This can be seen as

follows. over all of is greater than

over a symmetric image of |

around . The latter graph lies completely under because is symmetric and single-peaked.

And clearly, over a symmetric image of | around is ½. Thus, we need only

show ∫

This is equivalent to showing

∫

∫

∫

∫

∫

This inequality rearranges to

∫

∫

∫

∫

By changing the variable to for the right-side integral and for the left-side

integral and recognizing that is symmetric around , we have

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28

∫ (

∫

)

∫ (

∫

)

∫ (

∫

)

Since is strictly decreasing in , we need only show (

∫

) as a probability

density function defined over [ ∞ first-order stochastically dominates (

∫

). Equivalently,

we need to show ∫

∫

∫

∫

, or

( ) ( )

since for all . Since , it now suffices to show ( )

is

decreasing in . Under the Quotient Rule, this is true if

( ) ( )

which holds if

is decreasing in , or put differently, if ( )

. But this last

inequality holds since is log-concave (which must be true since is log-concave).

Meanwhile, for a general log-concave function or a single-peaked function that is not symmetric,

the inequality generally does not hold at the mean for all since does not always equal ½.

For a symmetric counter example, consider , which equals zero everywhere but takes on the value

of 1 on [

] and [

], where

This function is clearly symmetric around

. Then ∫

But in this case, plaintiff’s trial win rate is

∫

∫

∫

By Chebyshev’s Inequality, the numerator can get arbitrarily small as X increases, while the denominator

maintains a certain minimum value. Hence plaintiff’s win rate among litigated cases can get arbitrarily

small, even though the win rate over the entire sample would have been ½. Since this example has a

discontinuity at Y*, the limit hypothesis, too, will not hold. Instead, as prediction errors go to zero, the

plaintiff will lose zero percent of litigated cases. Nevertheless, it is possible to create a continuous

version of this counter-example which closely approximates it, under which the limit hypothesis is true,

but the bias hypothesis is false. The continuous version is the same as discussed above, except

takes on the value 1 on [

] and [

], where is much smaller than x-½. In

this continuous version, as the parties’ prediction errors go to zero, the proportion of cases won by the

plaintiff converges to fifty percent. Nevertheless, if prediction errors, are large compared to but

small as compared to x-½, then the proportion of plaintiff trial victories will be substantially smaller than

fifty percent, even though the proportion of plaintiff victories, if all cases went to trial, would be very

close to fifty-percent. Thus, the bias hypothesis fails, even though the limit hypothesis would hold.

Q.E.D.

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29

PROOF OF PROPOSITION 3. A version of this proof comes from Klerman & Lee (2014). Let be

the decision standard and that , where (

). Rewrite the plaintiff trial win rate is

∫

∫

where

∫ ( ) ( )

∫

is the ratio of plaintiff’s wins over defendant’s wins. It thus

suffices to show

Notice that since , we must also have

So if we make a change of variable ′ , then we can rewrite

as . So

∫

∫

By the Quotient Rule and Leibniz’s Rule,

if and only if

(∫

)(∫

)

(∫

)(∫

)

Since is log-concave, we have ( ) This in turn implies

( )

is

constantly (weakly) decreasing in . Therefore, we must have for all

and for all This indicates that the left-hand side is bounded

above by (∫

) (∫

), while the right-hand side is

bounded below by it. Moreover, the inequality must be strict since cannot be a constant function

when is a probability density function.

Finally, to show that we can apply Leibniz’s Rule over the infinite interval, we need to show

|

| | | is bounded by an integrable function. Indeed, |

| | | | | is constantly (weakly) decreasing in . If

is always positive, then | | , which must integrate to a finite value

since , as a log-concave probability density function must approach zero in as goes to infinity. If

is negative at any point, then the integral of | | will be the difference of two

integrals of (over complementary regions), each of which must integrate to a finite value.

Hence, | | is an integrable function, and the result follows. Q.E.D.

PROOF OF PROPOSITION 4. We begin by making the usual change of variables. Rewrite the

plaintiff’s and the defendant’s strategies as follows: ( ) and . Our strategy

is to look for Nash equilibrium strategies in which both parties employ a step function strategy at same

values but at different cutoff points. When the case estimate is very low ( and are both highly

negative), then both parties settle at 0. When the case estimate is very high ( and are both highly

positive), then both parties settle at some . But the plaintiff and the defendant differ in terms of the

cutoff signals, which we write as for the plaintiff and for the defendant. For this

pair of strategies to be an equilibrium for a fixed , the plaintiff must be indifferent between

demanding 0 and demanding at , and the defendant must be indifferent between offering 0

and offering at . It is easy to check that under this set-up, these two indifference conditions

will be the necessary and sufficient condition for the stated pair of strategies to be a Nash equilibrium.

Preliminary Draft

30

For a given , the defendant is indifferent at if and only if

( | ) (

) ( | )

( | )

( | ) ( | )

Rewrite this as

( | ) (

)

( | ) ( ( | ) )

and call this Condition . For the plaintiff, we have

( | ) ( | ) (

)

( | ) ( ( | ) )

( | )

Rewrite this as

( | ) (

)

( | ) ( ( | ) )

and call it Condition .

Now take the limits of Condition and Condition as goes to zero. In the limit, we must

have the following

| (

) | |

| (

) | ( | )

Now, it is clear that | | and | | (assuming we have no probability masses). Furthermore, in the limit we will also have

| | . This can be seen as follows:

( | )

∫

∫

∫

∫

∫

∫

We can take the limits inside the integral since | |

which must integrate to over all . Similarly, we have

( | )

∫ ( )

∫

∫

∫

∫

∫

∫

∫

∫

∫

Preliminary Draft

31

Therefore, in the limit we have | | . This

means that if we have ( ∫

∫

) (∫

∫

) in the limit,

then the two conditions collapse into one. This will be true if ( )

where ∫

∫

. At this point, we show a suitable value exists that the two

conditions (now one) are satisfied:

| (

) | ( | )

Suppose so that . At , this equation is greater than (

) (

) (

) ,

which is positive for reasonable values of and . As approaches infinity, the first term approaches

0 and the second term approaches , which is strictly negative. Therefore, by the Intermediate Value

Theorem, there must be at least one value of for which the equation is true. By continuity, for close

to 1, the same argument shows that such must also exist generally.

Now we show that there exist a pair of continuous functions ( ) such that

and that ( ( ) ) is a Bayesian Nash equilibrium for

each . Again, by continuity, we need only show for . We rewrite Conditions and as follows:

(∫

) (

)

(∫

)( ∫

∫

)

(∫

) (

)

(∫

)( ∫

∫

)

Then ( ) is a continuously differentiable function from to

such that and the function is even definable for negative values of . Then by the

Implicit Function Theorem,16

as long as the Jacobian matrix is invertible at , there is a small

neighborhood around 17 for which we can find a unique ( ) for each such that

( ) and . Thus, we need only check that

the Jabobian matrix is invertible at . Looking at the equation, we see that

and the determinant cannot be identically zero since (

) (

)

(

) (

). The left-hand side has terms involving mostly ’s while the right-hand

side has terms involving ’s and ’s. Hence, the proposition is proved generally.

16

We thank Ken Alexander for suggesting the use of the Implicit Function Theorem to complete

this argument. 17

Although in this model, , as standard deviation, is necessarily positive, both and

simply as mathematical functions in three variables , and , are continuous in

around and well-defined for as well.

Preliminary Draft

32

The similar logic can be applied to show that when but close to it, for a suitable value,

there exists a similar equilibrium in which they settle at 0 for and far less than 0, settle at (

) for and far greater than 0, and seek to litigate when the signals are

sufficiently closer to 0. Notice this equilibrium settlement values equal the first equilibrium when .

Because under the set-up, the two strategies do converge to a pair of points symmetric around 0, in this

case even with asymmetric stakes, the Fifty-Percent Limit hypothesis will go through. Meanwhile, since

the region of integration in the limit is { | } symmetric around the line

, even for , the limit value will be 50% even for such , and the Asymmetric Stakes

Hypothesis will be false. Q.E.D.

PROOF OF PROPOSITION 5. For a given , we employ the same change of variables by letting

. Then we have

( )

( ) (

)

Meanwhile, { |

}

where

∫

∫

. Therefore, for each

∬ (

)

Now the region of integration in the –plane will continue to depend on and . But still, we have

∫

∫

∫

∫

Notice that the same logic as in the proof of Proposition 1 (and other limit results) will go through

as long as, there is an absolute lower limit ̅ and an absolute upper limit ̅ such that, for each

(where can be assumed to be sufficiently small) and for each , is bounded

above by ̅ and bounded on the left side by ̅ In this case,

∬ (

)

{ | ̅ ̅}, which will eventually decrease at least as fast as in | | . In

this case, we can take the limit inside the integral. Then

∫

∫

∫

∫

since is locally continuous and . Meanwhile

∬ (

)

∬ (

)

Since

{ |

∫

∫

∫

∫

}

Preliminary Draft

33

and since each integrand is bounded above by

, which is clearly Lebesgue-integrable, we can

take the limits inside the integral once again and factor out . And therefore,

{ |

∫

∫

∫

∫

}

{ | [ ] [ ]

}

It thus suffices to show that there is an absolute upper limit ̅ and an absolute lower limit ̅ for

, where is sufficiently small. We need show that, for sufficiently small , as

approaches infinity, the boundary of does not go to infinity, and as approaches

negative infinity, the boundary does not go to negative infinity. But this is obvious since

∫

∫

and ∫

∫

, purely as functions defined in terms of variable ,

are continuous in at . This establishes that the results of Proposition 1 and Corollaries 1.2, 1.3,

and 1.4 will go through.

To see that the result of Proposition 4 will go through, we note that, following the logic behind

the proof of Proposition 4, the two indifference Conditions will collapse into one under the limit and in

fact, this condition in the limit will be identical to the single condition from the proof of Proposition 4.

For the existence result to go through for , we apply the Implicit Function Theorem in the

following two functions in the similar manner:

(∫

∫

)(

)

(∫

∫

)( ∫

∫

)

(∫

∫

)(

)

(∫

∫

)( ∫

∫

)

Since the Jacobian is again not identically zero, the result will hold generally. Q.E.D.

PROOF OF PROPOSITION 6. We begin by assuming . Note that ( ′ | ′ )

∫ ( ( ′ ))

∫ ( ( ′ ))

. By making the change of variable ( ′ ) , rewrite this as follows:

∫ ( ( ′ ))

∫ ( ( ′ ))

∫ (( ′ ) ) ′

∫ (( ′ ) )

Note that

Preliminary Draft

34

∫ (( ′ ) ) ′

(

)∫

(

(( ′ ) )

)

′

Meanwhile,

(( ′ ) )

(

) [ (( ′ ) )

]

(

(

√ )

)

[

( ( ′ )

)

′

]

where ′ does not depend on . Factors not dependent on appropriately cancel when we

divide by ∫ (( ′ ) )

. So we are left with a CDF of normal distribution with

standard deviation

√ centered at

( ) evaluated at . The new litigation

condition is that a dispute ′gets litigated if and only if

[( )

( )

(

√ )

] [

(

√ )

]

[

√ ] [

√ ]

Notice that since , we must have So if we make a change of variable and rewrite as .

For the Fifty-Percent Bias Hypothesis, as with Proposition 2, we need to show

∫

∫

∫

∫

∫

∫

∫

Notice that at , the inequality to be shown translates to

∫

∫

∫

∫

which was shown in Proposition 2. Since the expressions are continuous in , for sufficiently small

, the inequality must continue to hold. For the result equivalent to Proposition 3, we again let

∫ ( ) ( )

∫

and we show

for sufficiently small . So

Preliminary Draft

35

∫

∫

By the Quotient Rule, Chain Rule, and Leibniz’s Rule,

if and only if

(∫

) ( ∫

)

is less than

(∫

) ( ∫

).

Again, at , the inequality translates to

∫

∫

∫

∫

which is equivalent to what we showed in the proof of Proposition 3. By the same logic as above, for

small values of , the inequality will continue to hold. Q.E.D.

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