Judicial Detection Skill and Contractual Compliance

by Roland Kirstein and Dieter Schmidtchen Center for the Study of Law and Economics^{∗} Discussion Paper 9707

August 1997

Abstract

Mutually beneficial agreements might fail if the parties fear contractual opportunism. Litigation is supposed to be a remedy, but gives scope for another kind of opportunistic behavior which we call litigational opportunism: Even knowing that the opponent has fulfilled his obligations, a party might bring suit. We introduce a new concept, called judicial detection skill, and show that positive judicial detection skill is a prerequisite if the court system is to deter opportunistic suits and simultaneously induce bilateral contractual compliance. The traditional literature on litigation either assumes judges with zero detection skill, or simply neglects that opportunistic suits might be successful. We prove that those models are unable to provide an answer to the question of how to prevent both types of opportunism simultaneously.

JEL-Classification: C 72, K 12, K 41

Encyclopedia of Law and Economics:7300, 0790

Keywords: Economic analysis of procedural law, judicial detection skill, litigational opportunism, contractual opportunism.

∗ Universität des Saarlandes, Bldg. 31, D-66041 Saarbrücken, Tel.++681-302-3582, fax ++681-302-

3591, email csle@rz.uni-sb.de, homepage http://www.uni-sb.de/rewi/fb2/csle. We are indebted to Lucian A. Bebchuk, Omri Ben-Shahar, Jürgen Eichberger, Ronald A. Heiner, Christian Koboldt, Vai-Lam Mui, Eric Neumayer, Eric Rasmusen, Chris W. Sanchirico, Hans-Jörg Schmidt-Trenz, Volker Stein, to participants of the 13th Annual Conference of the European Association of Law and Economics in Haifa, August 1996, to participants of the Annual Meeting of the American Association of Law and Economics in Toronto, May 1997, and to three anonymous referees for their helpful comments.

Contents

1. Introduction 3

2. Contracts without litigation 5

3. Contracts with litigation 6 3.1 Judicial detection skill . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 The extended game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Conditions for contractual compliance . . . . . . . . . . . . . . . . . . . . 10 3.4 Litigation under the European rule . . . . . . . . . . . . . . . . . . . . . 11

4. Interpretation of our result 12 4.1 Models with zero detection skill . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Models with r > w and w = 0 . . . . . . . . . . . . . . . . . . . . . . . . 13

5. Conclusions 14

References 15

List of Figures

1 Basic contract game Γ in extensive form . . . . . . . . . . . . . . . . . . 5 2 Parameters of judicial detection skill . . . . . . . . . . . . . . . . . . . . 7 3 Contract game with litigation Γ_{a} . . . . . . . . . . . . . . . . . . . . . . 9

”...we are not surely bound to keep our word because we have given our word to keep it.” D. Hume, A Treatise of Human Nature

1. Introduction

Mutually beneficial agreements are doomed to fail if at least one of the parties fears that the other one is tempted to cheat. Litigation is supposed to prevent this contractual opportunism. However, it allows for a second type of opportunistic behavior, which we call litigational opportunism: Even if a party knows that her opponent has fulfilled his obligations, she might bring suit. This leads as well to inefficiency. In this paper, we derive the necessary conditions a litigation system has to fulfill in order to avoid both kinds of opportunism and to induce bilateral contractual compliance.

We introduce a new concept called judicial detection skill^{1}. This term refers to the ability of a judge to detect both litigational and contractual opportunism. Judicial detection skill is measured by two conditional probabilities: The probability that a plaintiff wins, given the suit is legitimate, and the probability of prevailing, given the suit is opportunistic. If the probability of winning a legitimate suit is higher than the probability of winning an opportunistic one, then the judge is said to have positive detection skill. If these two probabilities are equal, then the judge is said to have zero detection skill^{2}.

In our model, judges are represented by these two conditional probabilities. We show that positive judicial detection skill is a necessary condition for inducing contractual compliance as well as preventing litigational opportunism. Our definition of litigational opportunism requires to distinguish between opportunistic und legitimate suits. Since this distinction depends on the behavior of the parties in the pre-litgation stage, we first introduce the contract game without litigation as a trust game. The equilibrium of this game is Pareto inefficient. Then we add a litigation stage, where one of the parties is given the option to sue. Litigation is called legitimate if the other party has chosen not to comply with the contract. It is called opportunistic if the potential defendant has actually fulfilled her obligation. Thus, the litigation stage consists of two subgames. If the judge has positive detection skill, then the probability that a plaintiff prevails differs in these two subgames.

The contract game with litigation hence contains three parts: the underlying behavior, the opportunistic litigation subgame, and the legitimate litigation subgame. Failure to integrate all of these three parts into one game would have two important drawbacks.

First, neglection of the underlying behavior would make it impossible to analyze how it is influenced by the litigation stage^{3}. Furthermore, according to our concept of judicial detection skill, the notion of positive detection skill requires that the parties have more than one path to enter the litigation stage. The proper way to analyze the relation between potential future interaction and earlier behavior is to solve the complete game for an equilibrium.

1

This approach is based on Ronald Heiner’s theory of imperfect decision-making. For an introduction

into this theory, see Heiner (1983), (1985), (1986) and Heiner/Schmidtchen (1995).

2

In this paper, we use the term ”judge” to denote any institution that makes decisions on the merits

of a case, e.g. judges, justices, collegial courts or juries.

Second, not to distinguish expressly between the two different subgames in the litigation stage would imply a unique probability that the plaintiff prevails^{4}. Such an approach either assumes zero judicial detection skill or simply neglects that opportunistic cases might be won with positive probability^{5}. Note that perfect detection skill, zero detection skill, and the neglection of opportunistic suits can be seen as special cases of our judicial detection skill model.

Even though the assumption of zero judicial detection skill clearly contradicts reality, it has its merits, e.g. for analyzing the decision between settlement and trial, whenever the underlying behavior between the parties can be taken as given. Yet, as we will show, this type of models never allows to derive an answer to the question of how to prevent contractual and litigational opportunism simultaneously. On the other hand, the papers that work with a zero-probability to win an opportunistic suit seem to neglect the maximum fine result^{6}. If a potential defendant can expect with certainty that he will not be punished in case he obeys the law, then a maximum fine for non-compliance would be the simplest way to create the desired incentives. Applied to our analysis, if a fee for being sued is greater than the potential gain from non-compliance, this would be sufficient to induce bilateral contractual compliance. In such a setting, there would be no need to charge the potential plaintiff in order to set incentives right. However, the existence of other cost allocation rules, such as the American or the European rule, requires an explanation which can be derived from the fact that the probability of success in opportunistic suits is positive.

This paper is organized as follows:

In section 2 we present the basic contract game without third party enforcement. This game describes the underlying behavior and allows us to define the term contractual opportunism. The inefficiency of this game’s equilibrium provides the motivation to introduce litigation as a second stage in the parties’ interaction, which opens up the possibility of litigational opportunism.

3

E.g., see the literature that only focusses on the decision between settlement and trial; see Posner

(1986), Cooter/Rubinfeld (1989), and the papers on the ”selection hypothesis”, starting with

Priest/Klein (1984). Rosenberg/Shavell (1985), Bebchuk (1988), and Miceli (1993),

who particularly focus on frivolous suits, disregard the impact the enforcement system might have

on potential litigants’ behavior. Priest/Klein (1984), Katz (1990) and Polinsky/Rubinfeld

(1993) assume an exogenously fixed quota of frivolous plaintiffs. Such an approach would not take

into account that a potential plaintiff’s decision to bring a frivolous suit depends on his incentives

rather than on his exogenously given type.

4

Among the many see e.g. Gould (1973), Landes (1971) and also Shavell (1995) for trials of the

first instance.

5See Katz (1990) and Polinsky (1989) as examples for the latter approach.

6See Becker (1968).

Section 3 analyzes the two stage game. The first stage consists of the basic contract game introduced in section 2. In the second stage one party is given the opportunity to sue the other. We describe the theory of judicial detection skill in more detail and prove as a propostition that positive judicial detection skill is a necessary condition for bilateral contractual compliance.

In section 4 we discuss our results in comparison to the literature that models judges with zero detection skill (section 4.1) or neglects the chances of winning an opportunistic suit (section 4.2). Section 5 concludes the paper.

2. Contracts without litigation

In this paper, a contract means an agreement between the players Charlie and Lucy as follows: Charlie promises to work for Lucy, which costs him an effort he values with X. She promises to pay Y in return. Her gross monetary benefit from Charlie’s work is Z. We assume that the parties are risk neutral payoff maximizers and that Z > Y > X > 0. Hence the parties would mutually benefit if both of the promises were fulfilled. However, this condition is not sufficient to guarantee that the parties will actually act as agreed: The contract is not self-enforcing.

The agreement leads to a one-shot game we denote as Γ. Figure 1 represents the extensive form of Γ, where Charlie’s payoff is the first entry in the brackets, Lucy’s payoff is the second. Even though ”distrust game” seems to be the more appropriate term, this game is usually called ”trust game”^{7}.

Figure 1: Basic contract game Γ in extensive form

[figure: C; out; (0, 0); in; L; cheat; (−X, Z); honor; (Y − X, Z − Y)]

Lucy is tempted to cheat instead of honor the contract. We call this behavior of Lucy contractual opportunism. The unique subgame perfect equilibrium of this game is the strategy combination {out; cheat}. This equilibrium is pareto-suboptimal: Both parties were better off by playing the path {in, honor}. It is the anticipation of Lucy’s contractual opportunism that makes Charlie choose out. Hence both parties would agree to employ a device that makes the option in the more attractive choice for C, as long as the gain from cooperation Z − X is larger than the costs of this device.

7See e.g. Kreps (1990) or Güth/Kliemt (1993).

3. Contracts with litigation

In this section we add a litigation stage to the trust game of section 2. In the extended game, C is given the option to sue L for the payment Y, which is the value of the case. To give C the option to sue L may serve as a device against contractual opportunism, but opens up another source of opportunistic behavior: Charlie might bring suit even if Lucy has fulfilled her contractual obligation. In this case, we call the suit opportunistic. If, on the other hand, C brings suit after L has cheated, we call the case legitimate. Hence the distinction between the two possible types of litigation depends on the parties’ behavior in the pre-litigation stage of the game.

In section 3.1, we introduce our concept of judicial detection skill. The trust game with litigation is outlined in section 3.2, where we analyze the impact of the litigation stage on the underlying behavior, and derive a condition for bilateral contractual compliance under the American litigation cost allocation rule. In section 3.3, we show that our result also holds for the European litigation cost allocation rule.

3.1 Judicial detection skill

Even if the parties do not disagree about the value of the case Y, third parties in general have difficulties to determine whether Lucy actually has honored the contract. Therefore it makes sense to assume that potential litigants do not expect the judge to be free of errors. A variety of reasons for judicial errors exists even if judges are seen as benevolent^{8}. We only focus on errors about the true facts of the case, but not on errors concerning the interpretation or the choice of legal rules^{9}. Let us define r as the conditional probability that a judge decides in favor of the plaintiff, given the suit is actually legitimate, and w as the probability of making this decision, given the suit is opportunistic. The probability that a legitimate plaintiff does not prevail is 1 − r, whereas the probability that a judge rejects an opportunistic claim is 1 − w. An overview of these conditional probabilities is given in figure 2.

8

We want to point out that we do not analyze in this paper the judge’s decision-making process itself.

Therefore the notion of benevolence plays no role in our model. On judicial decision-making see e.g.

Anderson/Shugart/Tollison (1989), Cohen (1991), Posner (1990), (1993), Rasmusen

(1994) or Macey (1994) with a comment by Alexander (1994). Sanchirico (1995) explicitly

models the utility function of benevolent judges. Miceli/Coşgel (1994) and Rasmusen (1994)

tried to explain why benevolence might be rational for judges.

9

This follows the distinction in Posner (1990, 197, 203). Rasmusen (1995) focusses on a differnt kind

of judicial error than we do: He analyzes that judges might err in estimating the correct damage.

See also Tullock (1994)

Figure 2: Parameters of judicial detection skill

Judge’s decisionSuit is  legitimate  opportunistic in favor of C in favor of L  r  w 1 − r  1 − w

As mentioned in the introduction, we use the conditional probabilities r and w to describe judicial detection skill^{10}. The basic idea of this (r, w) approach is adapted from Heiner (1983), (1985), who refers to the literature in experimental psychology about imperfect detection of signals^{11}. In such experiments a person must decide repeatedly whether a signal is present or not. The judge is in a very similar situation, when he has to decide whether suits are opportunistic or not. Concerning the parameters r, w ∈ [0, 1] we distinguish four cases:

• If (r −w) = 1, the parties expect the judge to decide in favor of the plaintiff only if the case actually is legitimate. In case of opportunistic litigation, they expect the judge to always decide in favor of the defendant. Hence they perceive the judge to have perfect detection skill^{12}, which clearly contradicts reality, since judges are likely to make errors.

• With (r − w) = 0, the parties expect the judge to decide no better than by pure chance. The perceived probability of a decision in favor of the plaintiff does not depend on whether the case is legitimate or opportunistic. The parties expect the judge to have zero detection skill.

• (r − w) < 0 means negative detection skill^{13}.

10

Note that this approach substantially differs from the approach in Priest/Klein (1984) or Polinsky

(1989), who also use two different probabilites. In their models, the two probabilities represent the

plaintiffs and the defendants beliefs concerning the chances of the plaintiff’s victory. Applying our

terminology, the authors distinguish between a plaintiff’s r_{C} and a defendant’s r_{L}, but disregard

the w.

11

Swets (1988) gives an introduction to signal detection theory. Heiner (1986) was the first who

applied basic insights of signal detection theory to a law and economics problem, when giving an

economic rationale for the ”stare decisis” doctrine.

12

As the whole literature on contract theory assumes. See as well the concept of sanction matrices in

Schmidt-Trenz (1990, 195).

13

If a benevolent judge is aware of being of this type, he could improve his performance by consequently

sentencing against the opinion he has actually formed.

• In case of (r − w) ∈]0, 1[, the judge is perceived as having positive detection skill. The parties expect the judge to be able to distinguish between legitimate and opportunistic cases better than by pure chance, but not being free of errors.

The observation that judges acquire many years of training and experience in their profession may induce potential litigants to perceive that judges have positive detection skill. Shavell (1995, 394) makes such an assumption - without using our terminology - at least for appeals courts, but the way he models courts of the first instance implies zero detection skill. The judges in Katz (1990) and Polinsky (1989) are modelled - if we apply our terminology - with r > 0 and under the implicit assumption w = 0. Polinsky/Shavell (1989) and Polinsky/Rubinfeld (1993) assume r ≥ w. However, the approach chosen by these authors needs to be distinguished from ours. They assume that a legitimate plaintiff can easier produce evidence. Hence in their models frivolous plaintiffs prevail at trial with a lower probability than legitimate ones, whereas in our model this effect is caused by the detection skill of the judge.

3.2 The extended game

The model starts with the contract stage as described by the game Γ, which we call the underlying behavior. In the second stage Charlie, after having chosen the option in and observed Lucy’s reaction, now has to decide whether to sue Lucy in order to make her honor the contract (i.e. pay Y as they had agreed upon)^{14}. Hence Y is the value of the case or the value at stake, if litigation takes place. The extensive form of this game with litigation is shown in Figure 3, where Charlie’s decision nodes are labelled as C_{1}, C_{2} and C_{3}. Lucy’s node is labelled as L.

We denote this game form, which is common knowledge among the parties, as Γ_{a}, where the subscript a indicates that the litigation costs are allocated according to the American rule, i.e. each party has to bear its own costs. The payoffs in Γ_{a} do not only depend on the contract parameters (X, Y, Z), but also on the paramters r and w which describe the judge’s detection skill, and on the litigation costs. With P, we denote the costs a plaintiff has to bear if litigation takes place, whereas D denotes the defendant’s costs.

In general, litigation costs would consist of two parts: First the parties might spend ressources in order to influence the expected outcome of the suit. In addition to these expenses, the court might impose fees. In this paper we assume that the parties’ investments, e.g. the hiring of lawyers, have no impact on the probability that the plaintiff prevails. Therefore the optimal decision is not to make expenses voluntarily, and the parties’ litigation costs are equal to the fees they have to pay according to the litigation cost scheme.

At C_{3}, where litigation is legitimiate, C would prevail with probability r. The expected

14

For simplicity, we disregard that C might also bring an opportunistic suit after having chosen the

option out at his first node, since taking this into account would not lead to different insights. For

the same reason, we disregard that L also might bring suit.

Figure 3: Contract game with litigation Γ_{a}

[figure: C_{1}; out; (0, 0); L; cheat; C_{2}; ((1 + w)Y − P − X, Z − (1 + w)Y − D); C_{3}; (rY − P − X, Z − rY − D); (−X, Z); not; sue; in; honor; not; sue; (Y − X, Z − Y)]

gross gain is rY, whereas the costs are P. Hence the expected net effect for C of bringing suit at this node is rY − P. Since he did already invest X to reach this node, his total expected payoff from sueing at C_{3} is rY −P −X. L would have to pay Y with probability r, and to bear her costs D. The expected net effect of litigation to her payoff thus is −rY − D. Since she has already received X from C, which provides a value of Z to her, the total expected payoff for her is Z − rY − D. If C did not bring suit, the payoffs to the parties are similar to those in the basic contract game Γ.

In the same way we derive the payoffs at the node C_{2}, where the probability of prevailing in an opportunistic case is w. The aim of the subsequent analysis is to derive the condition under which bilateral contractual compliance occurs between risk-neutral players. Bilateral contractual compliance occurs if the parameters (r, w, P, D) and the cost allocation rule in a contract game with litigation are such that {in, honor, not} is the unique subgame perfect equilibrium path. In such an equilibrium, the parties both fulfill their contractual obligations and Charlie does not bring an opportunistic suit. Therefore, neither contractual nor litigational opportunism occurs. To play this path makes the two parties together best off since it provides the maximum common payoff to them: The gain from cooperation, Z − X, is being created and litigation fees (P, D) have not to be spent.

3.3 Conditions for contractual compliance

First we define the following expressions:

• The Selection Condition^{15}, hereafter denoted as SC_{a}: w < \frac{P}{Y} < r

• The Compliance Condition, denoted as CC_{a}: r > \frac{Y −D}{Y}

The subscript a again indicates that these expressions refer to the American cost allocation rule. Using these expressions, we state our first results in the following propositions:

Proposition 1: If the selection condition SC_{a} holds in the game Γ_{a}, then C brings suit if, and only if it is legitmate.

Proof: SC_{a} is equivalent to wY − P < 0 < rY − P, which is necessary and sufficient to motivate C not to sue at his node C_{2}, but to sue at his node C_{3}, where litigation is legitimate: Q.E.D.

Proposition 2: If in game Γ_{a} the selection condition SC_{a} holds, then L honors the contract if, and only if, CC_{a} holds.

Proof: If SC_{a} holds, then L can expect that C does not bring suit at C_{3}, but does so at C_{2}. Thus, she expects to receive Z − Y as payoff from choosing th eoption honor, and Z − rY − D from choosing the option cheat. To honor the contract is more attractive, if the former payoff is greater than the latter. This condition is equivalent to CC_{a}: Q.E.D.

If the Selection Condition SC_{a} and the Compliance Condition CC_{a} are fulfilled simultaneously, this is necessary and sufficient for bilateral contractual compliance, as stated formally in the following proposition:

Proposition 3: In Γ_{a}, bilateral contractual compliance is equivalent to CC_{a} ∧ SC_{a}.

Proof: For convenience we represent the term bilateral contractual compliance by the acronym bcc. We have to show CC_{a} ∧ SC_{a} ⇔bcc, which is equivalent to (CC_{a} ∧ SC_{a} ⇒ bcc) ∧ (CC_{a} ∧ SC_{a} ⇐ bcc).

SC_{a} ∧ CC_{a} ⇒ bcc follows from the propositions 1 and 2, if an an additionional step is made to complete the backward induction solution of Γ_{a}. If the propositions 1 and 2 hold, C will sue if, and only if it is legitimate, and L will honor the contract. Then C has to choose between his options in and out at his first node. To choose the option in would lead to the path {honor, not}, hence to Y − X as the payoff for C. This is greater than zero by assumption. Thus, CC_{a} ∧ SC_{a} is sufficient for bcc.

15

The term ”selection” must not be confused with the one used in the literature on the ”selection hypothesis”; see Priest/Klein (1984). In Schmidtchen/Kirstein (1997) we developed the selection effect which judicial detection skill and litigation costs may have on the behavior of potential litigants.

Additionally, CC_{a} ∧SC_{a} ⇐ bcc is to be shown. This is equivalent to ¬CC_{a} ∨ ¬SC_{a} ⇒ ¬bcc. This is equivalent to (rY ≤ Y − D ∨ wY ≥ p ∨ rY ≤ P) ⇒ ¬bcc by definition.

If, e.g., rY ≤ P, then C would not be deterred from bringing suit at his node C_{3}. In this case, {in, honor, not} would not be the unique subgame equilibrium path, as it is required for bcc. Hence, bcc is sufficient for SC_{a} ∧ CC_{a} or, equivalently, SC_{a} ∧ CC_{a} is necessary for bcc: Q.E.D.

If the parties are risk-neutral and the condition SC_{a} ∧ CC_{a} holds, then bilateral contractual compliance occurs^{16}. Both types of opportunism are prevented, the contractual as well as the litigational opportunism.

The follwoing proposition represents the main result of our paper:

Proposition 4: In Γ_{a}, positive judicial detection skill is necessary (but not sufficient) for bilateral contractual compliance.

Proof: We have to show bcc ⇒ r > w. According to Proposition 3, bcc ⇔ CC_{a} ∧ SC_{a} holds. Hence bcc implies SC_{a}. SC_{a} implies, by definition, r > w: Q.E.D.

Without positive judicial detection skill, the court system is unable to motivate the parties to contractual compliance. Positive judicial detection skill is a prerequisite for this motivational impact, but might be still insufficient. If r > w, then it is possible to design the parties’ litigations costs (P, D) in a way that SC_{a} and CC_{a} hold simultaneously and bilateral contractual compliance is induced^{17}. One can think of other ways to select opportunistic from legitimate suits, such as raising a higher plaintiff’s fee for the former ones. However, to find out whether this higher fee applies or not would require positive detection skill as well.

3.4 Litigation under the European rule

According to the European type of litigation cost allocation rules, the loser of a trial has to bear both parties’ costs. We denote the total costs as G = P + D. Let Γ_{e} stand for the trust game with litigation, given the European rule. Whereas the sequence of actions and the payoffs for the paths without litigation are the same as in Γ_{a}, the payoffs for the two paths that end with litigation have to be modified.

16

If the parties were assumed to have a different risk-attitude, this would require to modify the condition

for bilateral contractual compliance. However, this affects our results only quantitatively, but not

qualitatively.

17

We derived the set of litigation costs for the American and the European rule that induce bilateral

contractual compliance in Kirstein/Schmidtchen (1996). This set is constrained by the condition

SC_{a} ∧ CC_{a}. It is non-empty, if the judge has positive detection skill.

If C brings an opportunistic suit at his node C_{2}, his payoff then is (1+w)Y −X−(1−w)G. In addition to the payoff without litigation, which is Y −X, he receives Y with probability w, and has to bear the total costs G only in case he looses, i.e. with probability (1 − w). L’s expected payoff is Z − Y − w(Y + G). If C sues legitimately, at his node C_{3}, his expected payoff is rY − (1 − r)G − X, whereas L expects to receive Z − r(Y + G).

Let us now define the selection condition and the compliance condition for the European rule:

• SC_{e} is w < \frac{G}{Y + G} < r

• CC_{e} is r > \frac{Y}{Y + G}

It is easy to see that the claims we made in the propositions 1 through 4 with respect to the American rule in the previous section do also hold for the European rule: If SC_{e} holds, then C brings suit if, and only if it legitimate. If SC_{e} and CC_{e} hold, L honors the contract and C chooses his option in, hence bilateral contractual compliance occurs. From the definition of SC_{e}, it is obvious that bilateral contractual compliance implies positive judicial detection skill, r > w.

4. Interpretation of our result

Modelling courts by a single probability does not allow for an analysis of the court’s influence on litigational or contractual opportunism. A model that represents judges by a single probability either assumes zero detection skill or it neglects that plaintiffs might win opportunistic suits. Let q denote this single probability that a plaintiff prevails. Using our terminology, the former assumption is then equivalent to q = r = w, whereas the latter one is equivalent to q = r and w = 0.

4.1 Models with zero detection skill

If zero judicial detection skill is assumed, then judges are not able to distinguish better than by pure chance between opportunistic and legitimate suits. The probability that a plaintiff prevails is equal in both cases, or r = w = q ∈]0, 1[. According to our proposition 4, bilateral contractual compliance cannot occur^{18}, since the court system does not set incentives to prevent contractual opportunism.

Whether a judge of zero detection skill at least prevents litigational opportunism in Γ_{a} depends on the value of q. If q is low, then C will choose hos option out, whereas a high C might motivate him to choose his option in, as stated in the following:

18The same holds true for the European rule.

Proposition 5: If q > \frac{P +X}{Y} holds in Γ_{a}, then the subgame perfect equlibrium path is {in; cheat; sue}. Proof: q > \frac{P +X}{Y} implies q > \frac{X}{Y}, which is necessary to make C sue at C_{3} and C_{2}. If L can expect to be sued whatever she does, she is better off by choosing cheat. q > \frac{P +X}{Y} then is necessary and sufficient to make the path {in; cheat; sue} more attractive than {out} for C: Q.E.D.

If q is high enough, it is attractive for C to bring suit whatever L does. Thus, litigational opportunism is not prevented. The gain from cooperation will be produced, yet it will not be distributed according to the terms of the contract. In addition to this, that path requires the parties to spend litigation costs. In this equilibrium, the expected payoffs are (qY − P − X) for C and (Z − rY − D) for L.

On the other hand, a q ≤ \frac{P +X}{Y} would prevent litigational opportunism^{19}. Yet in this case the parties do not even have an incentive to produce the gain from cooperation. The equilibrium payoff then is zero for both of them.

A similar result holds for the European cost allocation rule, according to which the loser has to pay both parties costs G = P + D. The condition for litigation at C_{3} and C_{2} is q > \frac{G}{Y +G}. To cheat makes L better off. The path \{in; cheat; sue\} is more attractive for C than \{out\} if, and only if, q < \frac{X+G}{Y +G} holds. If this condition holds, then C’s and L’s expected payoffy are qY − (1 − q)G − X and Z − Y − q(Y + G) respectively. Otherwise they both receive zero.

The assumption r = w hence makes it impossible to discuss any impact of the court system on contractual opportunism. There might be an impact on litigational opportunism, but only with a sad result: If the probability to prevail is too low, this prevents not only opportunistic suits, but also the creation of the gain from cooperation. If, on the other hand, neither contractual, nor litigational opportunism is prevented, the parties at least can realize this gain, but have to spend litigations costs. Models with r = w might come to interisting insights, but fail to provide any insights on the influence of the court system on the underlying behavior.

4.2 Models with r > w and w = 0

We now discuss the case that the single probability q > 0 only refers to legitimate suits, whereas the possibility of opportunistic suits is simply excluded. Applying our notation, this means r = q and w = 0. Such a judge obviously has positive detection skill. In this case, it never makes sense to sue at the node C_{2} in the game Γ_{a}. The condition for the cooperative equilibrium path in the game Γ_{a} with r > 0 and w = 0 is stated in the next proposition:

Proposition 6: If r > 0 and w = 0 holds in Γ_{a}, then the subgame perfect equlibrium path is {in; honor; not}, if D > (1 − r)Y ∧ P < rY.

19A high P would have the same effect.

Proof: Due to w = 0 and P > 0, C will never sue at his node C_{2}. At C_{3}, he will sue if, and only if, rY > P. If this condition did not hold, then L would always choose cheat instead of honor. If the condition holds, then it is better for L to honor instead of cheat if, and only if, Z − Y > Z − rY − D or, equivalently, D > (1 − r)Y: Q.E.D.^{20}

Obviously, the case of w = 0 and r > 0 allows for the design of an extremely simple mechanism to implement bilateral contractual compliance. If the plaintiff’s fee P is set to zero and the defendant has to pay high enough a fee only for being sued, this provides an incentive for her not to cheat, but to honor the contract instead. Or, to state it formally, if w = 0 and P = 0, then any r > 0 fulfills the condition for bilateral contractual compliance in Γ_{a}, as long as D > (1 − r)Y. This is in accordance with the maximum fine result of Becker (1968): If the punishment on cheating, in our model D, is greater than the expected gain (1 − r)Y, then cheating is made inattractive.

Of course, this is a trivial result. However, taking this into account, it is quite surprising to see that models with w = 0 are used, e.g., to extensively discuss whether the European rule is better or worse than the American rule^{21}. If w = 0, it would not make much sense to discuss any of these complicated cost allocation rules, since the simple one outlined in the previous paragraph ensures optimal behavior.

In reality, litigation fee schemes do obviously not follow this simple proposal. The existence of these more complicated rules requires an explanation, which can be provided by the assumption that judges are likely to make errors of two types, hence r < 1 and w > 0. In such a setting, the simple mechanism would not always set the incentives right. To induce efficient behavior, the court system needs to have positive detection skill, and the litigation costs have to be adjusted carefully to provide sufficient incentives. This insight can only be derived by making use of our (r, w) approach.

5. Conclusions

In this paper, we presented a new apporach to model the behavior of judges. Our concept of judicial detection skill explicitly takes into account that judges can commit two kinds of errors: opportunistic suits might be successful, legitimate suits might fail. The paper shows that, with positive detection skill and appropriately designed litigation fees, contracting parties can be motivated to comply with their contract and carry out the promises they agreed upon.

The traditional literature on litigation implicitly assumes either zero detection skill or neglects the possibility that opportunistic litigants might prevail. Under these assumptions, it is impossible to show how contractual and litigational opportunism can be prevented simultaneously.

20

It is easy to prove that the equilibrium path is \{in; cheat; sue\}, if r < \frac{Y −D}{Y} ∧ r > \frac{P +X}{Y} holds, and

that C chooses \{out\} at his first node, if r < \frac{Y −D}{Y} ∧ r < \frac{P +X}{Y} or if r < \frac{P}{Y} holds. Again, for the

European rule a similar result can be derived; however, the path {in; cheat; sue} will never be an

equilibrium path in the game Γ_{e}.

21See, among others, Katz (1990).

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