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Feuding, arbitration, and the emergence of an independent judiciary

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Abstract

Anthropologists, historians, and political economists suggest that private violence—feuding—provides order and enforces agreements in the absence of a state. We ground these accounts in a series of formal models that shows the relationship between feuding, informal arbitration, and formal judicial resolution. Feuding enables cooperation by deterring exploitative behavior, but its ability to do so is conditioned by two credible commitment problems that affect both militarily weak and strong actors. These commitment problems can be partially ameliorated through arbitration, even in the absence of coercive authority, by providing information that makes the wronged party’s threat to feud more credible. Transitioning to a formal, coercive justice system, however, represents a qualitative change to the nature of disputing—a change that can be universally beneficial. We therefore provide a new explanation for the creation of independent courts rooted in the logic of dispute resolution and illustrate this explanation with reference to the creation of the Imperial Chamber Court of the Holy Roman Empire.

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Notes

  1. The historical details are scarce, but see https://de.wikipedia.org/wiki/Hirschhorn(Adelsgeschlecht).

  2. See, e.g., Halsall (1999) for a discussion of the relationship between feud and arbitration or mediation in medieval Europe and Evans-Pritchard (1940, 163) for the centrality of a prominent mediator, discussed below, prior to the outbreak of violence among the Nuer of Sudan.

  3. Stone Sweet is focused on the dynamic interaction between resort to third-party dispute resolution and the normative expectations that parties in dyadic relationships hold, an issue that does not bear directly on the central concern of this paper.

  4. This kind of “constitutional analysis” is, of course, at the heart of the constitutional political economy paradigm (Buchanan & Tullock, 1962; Buchanan, 1990; Vanberg, 2018).

  5. But see White (2016) for a somewhat contrasting view.

  6. Halsall (1999) contrasts what he calls faida—the type of limited feuding that was common in early medieval Europe—with “true” feud, or “blood-feud,” that required multiple cycles of revenge. Scholars typically use the term “feud” to refer to both phenomenons, a terminological confusion growing out of competing disciplinary perspectives between historians and anthropologists (Carroll, 2017). Our analytical concern is specifically with this limited type of “legitimate” violence, though the extent to which the different ideal types represent qualitatively different types of interactions rather than existing on a spectrum is a matter of debate.

  7. For examples, see Gluckman (1955) or Reinle (2017).

  8. Boehm (1983) notes in his study of feuding in Montenegro that Montenegrins take special care to avoid offending members of strong groups, including unintentionally (88). This can take the shape of avoiding some types of interactions entirely, unless one is “certain that his adversary ... would still refrain from killing him” after a supposed affront (106).

  9. As we discuss below, in addition to the immediate parties to a dispute, there are other affected parties that we ignore for purposes of the model because they have no direct bearing on the central strategic logic we focus on. Most importantly, feuding often imposed costs on third-party bystanders.

  10. Of course, there may be circumstances in which it is possible to demonstrate credibly that E has not in fact exploited V; such circumstances fall outside the scope of our model.

  11. Substantively, our interest is in the strategic problem created by the fact that sequential exchange potentially exposes vulnerable parties to exploitation. For this reason, in the analysis we assume that \(\theta >\frac{f_E}{b+f_E}\), which assures that the Exploiter is willing to engage in an interaction. If this is not the case, the substantive problem of interest does not arise.

  12. A similar principle applies in the context of crime prevention more generally (Tsebelis, 1989). If deterrence is costly, it cannot be perfect since perfect deterrence would induce elimination of the (costly) measures that give rise to deterrence—but then the deterrent effect vanishes.

  13. Note that this conclusion is potentially attenuated in a model in which players become concerned not only about the present interaction, but also worry about maintaining a reputation for “toughness.” In such a context, players may be willing to feud even if the expected payoff from doing so is negative if the value from establishing or projecting a reputation for future interactions is significant enough. Because this is a separate strategic logic that is not of immediate relevance to the aims of the current paper, we defer it to future work.

  14. Consider the worst case scenario from E’s perspective: V always feuds in response to potential exploitation. In this case, the expected payoff from exploiting V is given by \(sb+(1-s)\alpha -f\), while the payoff from being honest (and thus avoiding the feud) is b. If f falls below the deterrence threshold, the payoff from feuding always exceeds the payoff of being honest.

  15. Or, more accurately, our argument is restricted to those (common) situations in which it is not easy for E to credibly demonstrate to V that he has acted in good faith.

  16. Reacting to the Prisoners’ Dilemma tournaments conducted by Axelrod (1984), Vanberg and Congleton (1992) explore a related logic, demonstrating that introducing an “exit option” (i.e., the ability to refuse to play) significantly changes the incentives in the repeated Prisoner’s Dilemma, and that strategies that exercise this exit option in response to defection can be highly effective.

  17. This aspect is also raised by Shapiro (1981) with respect to the “logic of the triad” in the context of (formal) judicial resolution. We return to this point below.

  18. To ease exposition in the main body of the paper, we assume that \(\epsilon =0\). Allowing \(\epsilon >0\) generates an additional equilibrium that is substantively not relevant. For a full derivation of equilibria including this case, see the appendix.

  19. Naturally, extensions of our model could focus on modeling this information transmission process directly.

  20. Holler and Lindner (2004) examine a separate informational aspect of informal mediation: The fact that the decision by a party to offer or accept mediation may reveal information regarding the relative strength of its position to the other party. This aspect is highly relevant in settings like modern commercial mediation (which Holler and Lindner focus on) in which mediation is voluntary. It is less relevant in our setting, in which strong social norms typically compel disputants to seek arbitration, thus removing the signalling aspect of a decision to do so.

  21. However, if caught in an exploitation attempt, the exploiter persists in not honoring his obligations, and a feud follows.

  22. Feuding previously occurred with positive probability whenever \(f<sv\); it now only occurs when \(f < (1-s)(\alpha -b)\), and with reduced probability in the Partial Deterrence equilibrium compared to the Exploitation and Feuding equilibrium. The threshold on Strategic Avoidance has shifted upward, resulting in more cooperation. The equilibrium likelihood of exploitation is lower in the Arbiter Authority equilibrium than in the Deterrence equilibrium whenever \(\epsilon <f/s\), which is true whenever using the arbitrator is not unreasonably costly.

  23. One can also think of informal courts with high punishment capacity—for example, French Huguenot church courts were effective at limiting feuds because they could threaten excommunication for stubborn disputants (Carroll, 2003). At the same time, some formal “courts” might have very low enforcement capacity and instead resemble the informal arbitrator depicted above. While our model puts strong and weak justice systems on a continuum, our shorthand does not capture the variety of enforcement capacities across them.

  24. We assume that \(0<f < \theta v (\alpha -b-k-\phi )/(\theta v + a - b)\), in order to capture all four equilibria.

  25. For a fascinating illustration, see Kuran and Rubin (2018), who demonstrate that the fact that courts in Ottoman Istanbul in the 17th century were biased in favor of individuals connected to the Sultan made it more risky to extend loans to these individuals—as a result of which they faced significantly higher costs of credit.

  26. Note that this point is closely related to Douglass North’s argument that the emergence of institutions that reduce transactions costs expand opportunities for exchange and are thus a central driver of economic development (North, 1993). Moreover, the argument also related to the empirical literature that has investigated the connection between the presence of independent judiciaries and economic growth (Feld and Voigt, 2003; Feld et al., 2015).

  27. The Diet of Worms of 1495 is not to be confused with two other prominent meetings at Worms: the Concordat of Worms in 1122, which resolved the Investiture Controversy of the 12th century, and the Diet of Worms in 1521, which resulted in the condemnation of Martin Luther.

  28. The court also had some appellate jurisdiction and original jurisdiction regarding disputes between “immediate” Imperial subjects, i.e., individuals who had no lord other than the Emperor himself (Müßig, 2018).

  29. There is an affinity between this conclusion and Olson (1993)’s argument that “stationary bandits” have an incentive to create some protections for their subjects’ property rights in order to encourage productivity—in both cases, the strategic nature of the interaction, which allows weaker players to limit their losses by anticipating the actions of stronger players, generates a credible commitment problem that is to the detriment of those who are strong.

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Correspondence to Georg Vanberg.

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Appendices

Appendix

See the main body of the paper for a description of the model, including the sequence of play and payoffs. We impose the restriction that the environment is favorable with probability \(\theta >\frac{f_E}{b+f_E}\). This condition ensures that the potential exploiter is always willing to engage, even if he faces a potential feud, thus giving rise to the substantive problem posed by non-simultaneous exchange that we are interested in. To see this, note that the exploiter can always act honestly, in which case the expected payoff to the interaction, even if V always feuds in response to potential exploitation, is at least \(\theta b-(1-\theta )f_E\), which is positive if the constraint is met.

1.1 Some notation

Let the probability that the environment is favorable be \(\theta \in (\frac{f_E}{b+f_E},1)\), the probability that E is dishonest in the favorable environment be \(\delta \in (0,1)\), and the probability that V feuds if he sees a payoff of 0 be \(q\in (0,1)\). Then V’s updated belief that the environment is favorable (or, equivalently, V’s updated belief that E has engaged in exploitation) is given by

$$\begin{aligned} \gamma =\frac{\theta \delta }{\theta \delta +1-\theta } \end{aligned}$$
(1)

1.2 V’s feuding decision

The expected utilities for V of accepting and feuding are given by:

$$\begin{aligned} EU_V(Acc)= & {} 0 \end{aligned}$$
(2)
$$\begin{aligned} EU_V(Feud)= & {} \gamma (sv-f_V)+(1-\gamma )(-f_V)=\frac{\theta \delta s v}{\theta \delta +1-\theta }-f_V \end{aligned}$$
(3)

Note that for \(f_V\ge sv\), accepting dominates feuding; if the cost of feuding is too high, V will never pursue the feud regardless of E’s action. If \(f_V<sv\), V will engage in a feud whenever the following condition is met:

$$\begin{aligned} \delta >\frac{f_V(1-\theta )}{\theta (sv-f_V)} \end{aligned}$$
(4)

This threshold is always positive for \(f_V<sv\), and it is less than 1 (i.e., it can be met) if \(\theta >\frac{f_V}{sv}\). If \(\theta \le \frac{f_V}{sv}\) (i.e., \(s\le \frac{f_V}{\theta v}\)), the threshold is greater than 1 and V will never feud. Thus, we have the following best response function for V:

$$\begin{aligned} BR_{V}(\delta )=\left\{ \begin{array}{lr} \text {Don't Feud (q=0) for } \delta <\frac{f_V(1-\theta )}{\theta (sv-f_V)}\\ \text {Feud (q=1) for } \delta >\frac{f_V(1-\theta )}{\theta (sv-f_V)}\\ \text {Any mix for } \delta =\frac{f_V(1-\theta )}{\theta (sv-f_V)}\\ \end{array} \right. \end{aligned}$$
(5)

1.3 E’s initial decision

Note that E cannot be playing a pure strategy of being honest, since V would then never feud in response to a payoff of 0, and this would then lead E to be dishonest.

Case 1 (\(\theta \le \frac{f_V}{sv}\)): In this case, V never feuds, so the only best response for E is to be dishonest.

Case 2 (\(\theta >\frac{f_V}{sv}\)): The expected utilities for E of being honest and dishonest are given by:

$$\begin{aligned} EU_E(Honest)= & {} b \end{aligned}$$
(6)
$$\begin{aligned} EU_E(Dishonest)= & {} q(s(b-f_E)+(1-s)(\alpha -f_E))+(1-q)\alpha \end{aligned}$$
(7)

E will be honest whenever the following condition is met:

$$\begin{aligned} q\ge \frac{\alpha -b}{s(\alpha -b)+f_E} \end{aligned}$$
(8)

This condition is always positive, and it is less than 1 if the following condition is met: \(f_E>(\alpha -b)(1-s)\). Thus, if \(f_E\le (\alpha -b)(1-s)\), E will always be dishonest.

We have the following best response function for E:

$$\begin{aligned} BR_{E}(q)=\left\{ \begin{array}{lr} \text {Dishonest }(\delta =1)\text { for } q<\frac{\alpha -b}{s(\alpha -b)+f_E}\\ \text {Honest }(\delta =0)\text { for } q>\frac{\alpha -b}{s(\alpha -b)+f_E}\\ \text {Any mix for } q=\frac{\alpha -b}{s(\alpha -b)+f_E}\\ \end{array} \right. \end{aligned}$$
(9)

1.4 Equilibria

There are three equilibria implied by these best response functions:

Equilibrium 1: For \(f_V \ge \theta sv\), E will always be dishonest. V will never feud in response to potential exploitation, and will never engage in the interaction.

To see the logic of this equilibrium, suppose the players were to engage in an interaction. Then the probability of a feud is 0. The expected utility of the two players is given by:

$$\begin{aligned} EU_E[EQ1]=\theta \alpha \end{aligned}$$
(10)

This payoff increases in \(\theta\) and \(\alpha\).

$$\begin{aligned} EU_V[EQ1]=0 \end{aligned}$$
(11)

Since we assume that players will only engage in the interaction if their expected payoff is positive, V will never engage in this equilibrium. Thus, along the path of play, no interaction takes place, and both players have an expected payoff of 0.

Equilibrium 2: For \(f_V< \theta sv\) and \(f_E<(\alpha -b)(1-s)\), E will always be dishonest, and V will always feud in response. The equilibrium probability of a feud is 1, and the expected utilities of the players are given by:

$$\begin{aligned} EU_E[EQ2]=\theta (sb+(1-s)\alpha )-f_E \end{aligned}$$
(12)

This payoff increases in b, \(\alpha\), and \(\theta\). It declines in s and \(f_E\).

$$\begin{aligned} EU_V[EQ2]=\theta sv-f_V \end{aligned}$$
(13)

This payoff increases in \(\theta\), s, and v, and decreases in \(f_V\).

Given the conditions for this equilibrium, and the restriction on \(\theta\), both of these expected payoffs are positive, so both players will engage in the interaction.

Equilibrium 3: For \((\alpha -b)(1-s)<f_E\) and \(f_V<\theta sv\), we have a mixed strategy equilibrium in which E is dishonest with probability \(\delta ^*=\frac{f_V(1-\theta )}{\theta (sv-f_V)}\) and V engages in a feud with probability \(q^*=\frac{\alpha -b}{s(\alpha -b)+f_E}\) if he faces a payoff of 0. In this case, E finds feuding too expensive to act dishonestly all the time and V finds feuding cheap enough to engage in it. This offers the possibility of deterrence.

The equilibrium probability that E is dishonest increases in \(f_V\) and \(\theta\), and decreases in v and s. The equilibrium probability that V will feud if he observes a payoff of 0 decreases in \(f_E\) and s, and increases in b and \(\alpha\).

The expected utilities of the players are given by:

$$\begin{aligned} EU_E[EQ3]=\theta ((1-\delta ^*) b + \delta ^*(q^*(sb+(1-s)\alpha -f)+(1-q^*)\alpha ))+(1-\theta )(q^*(-f)+(1-q^*)0) \end{aligned}$$
(14)

which reduces to:

$$\begin{aligned} EU_E[EQ3]= & {} \frac{b(f_E+s\alpha \theta )-f_E\alpha (1-\theta )-b^2s\theta }{s(\alpha -b)+f_E} \end{aligned}$$
(15)
$$\begin{aligned} EU_V[EQ3]= & {} \theta ((1-\delta ^*) v + \delta ^*q^*(sv -f))+(1-\theta )q^*(-f)=\frac{v(\theta s v-f_V)}{sv-f_V} \end{aligned}$$
(16)

Both of these payoffs must be positive, given the equilibrium conditions and the restriction on \(\theta\), so in equilibrium there will be an interaction.

Adding a supplementary institution

We now introduce a “supplementary institution” that provides access to arbitration prior to engaging in a feud. See the main text for a description of the extended model and payoffs. First, note that if the probability that the environment is favorable is too low, V will never appeal to the arbiter, even if she expects E to comply with a favorable ruling. Thus, in order for the arbiter to affect the interaction at all, it must be the case that \(\theta \ge \frac{\epsilon }{v}\), which we assume. Further, note that in this new game, the expected payoff for E must always be positive—by entering the interaction, E can always guarantee a payoff of at least \(\theta b\).

1.1 The final stages: compliance and feuding

Inspection of the payoffs reveals immediately that V will feud if E ignores a ruling in V’s favor if and only if \(sv>f_V\) (assuming that V chooses not to feud if indifferent). We refer to this condition as the arbiter feuding threshold:

$$\begin{aligned} T_{Feud}^{Arb}=s v \end{aligned}$$
(17)

Now consider E’s reaction to the arbiter’s ruling.

Case 1: If \(sv>f_V\), V will engage in a feud if E chooses to disregard the decision. In this case, the expected utilities are given by:

$$\begin{aligned} EU_E(Comply)= & {} b-\phi \end{aligned}$$
(18)
$$\begin{aligned} EU_E(Refuse)= & {} bs+(1-s)\alpha -f_E-\phi -k \end{aligned}$$
(19)

E will choose to comply if and only if \(f_E\) exceeds the following “compliance threshold:”

$$\begin{aligned} T_{Comp.}^{Arb}=(\alpha -b)(1-s)-k \end{aligned}$$
(20)

Case 2: If \(sv\le f_V\), V will not feud in response to noncompliance. In that case, inspection of the payoffs immediately reveals that E will never comply unless k is sufficiently large to overcome the benefit E gains from cheating—that is, if \(k>\alpha -\beta\). We refer to this as the unconditional compliance threshold:

$$\begin{aligned} T_{Unc. Comp.}^{Arb} = \alpha - \beta \end{aligned}$$
(21)

1.2 V’s information set: appealing to the arbiter

We are now in a position to consider V’s decision to appeal to the arbiter in response to a 0 payoff. Suppose that E is dishonest in the favorable environment with probability \(\delta\). Then V’s updated belief at his information set that E has cheated in the favorable environment is given by \(\gamma =\frac{\theta \delta }{\theta \delta +1-\theta }\). There are several cases to consider:

Scenario 1: \(k>\alpha -\beta\) In this case, the arbiter can inflict enough pain on E for refusing to comply with its judgment that E will accept the arbiter’s announcement regardless of V’s willingness to feud. The expected utilities of calling on arbitration and simply accepting potential exploitation are given by:

$$\begin{aligned} EU_V(Arbiter)= & {} \gamma (v-\epsilon )-(1-\gamma )\epsilon \end{aligned}$$
(22)
$$\begin{aligned} EU_V(Accept)= & {} 0 \end{aligned}$$
(23)

This implies the following best response function for V:

$$\begin{aligned} BR_{V}(\delta )=\left\{ \begin{array}{lr} \text {Accept (q=0) for } \delta <\frac{\epsilon (1-\theta )}{\theta (v-\epsilon )}\\ \text {Arbiter (q=1) for } \delta >\frac{\epsilon (1-\theta )}{\theta (v-\epsilon )}\\ \text {Any mix for } \delta =\frac{\epsilon (1-\theta )}{\theta (v-\epsilon )}\\ \end{array} \right. \end{aligned}$$
(24)

Note that—given the assumption on \(\theta\)—the cutpoint on \(\delta\) in this best response function is always a proper probability.

Scenario 2: \(k<\alpha -\beta\), \(f_V\ge sv\) In this case, the unconditional compliance threshold is not met, V will not feud and E will always defy the arbiter’s decision. Given that, there is no point in appealing to the arbiter in the first place and V will simply accede.

Scenario 3: \(k<\alpha -\beta\), \(sv>f_V\), \(f_E\ge (1-s)(\alpha -b)\) In this case, V is willing to feud and E complies with the arbiter’s decision. The expected utilities of calling on arbitration and simply accepting potential exploitation are given by:

$$\begin{aligned} EU_V(Arbiter)= & {} \gamma (v-\epsilon )-(1-\gamma )\epsilon \end{aligned}$$
(25)
$$\begin{aligned} EU_V(Accept)= & {} 0 \end{aligned}$$
(26)

The utilities are identical to Scenario 1 above; that implies the same best-response function:

$$\begin{aligned} BR_{V}(\delta )=\left\{ \begin{array}{lr} \text {Accept (q=0) for } \delta <\frac{\epsilon (1-\theta )}{\theta (v-\epsilon )}\\ \text {Arbiter (q=1) for } \delta >\frac{\epsilon (1-\theta )}{\theta (v-\epsilon )}\\ \text {Any mix for } \delta =\frac{\epsilon (1-\theta )}{\theta (v-\epsilon )}\\ \end{array} \right. \end{aligned}$$
(27)

Scenario 4: \(k<\alpha -\beta\), \(f_V<sv\), \(f_E<(1-s)(\alpha -b)\) In this case, V is willing to feud in response to noncompliance, but E will defy the arbiter anyway. The expected utilities for V are given by:

$$\begin{aligned} EU_V(Arbiter)= & {} \gamma (sv-f_V-\epsilon )-(1-\gamma )\epsilon =\frac{\theta \delta }{\theta \delta +(1-\theta )}(sv-f_V)-\epsilon \end{aligned}$$
(28)
$$\begin{aligned} EU_V(Accept)= & {} 0 \end{aligned}$$
(29)

In solving for the \(\delta\) cutpoint, we derive an additional condition: \(\epsilon < sv-f_V\), or \(f_V < sv-\epsilon\). If this condition is not met, then V will never pursue arbitration. This is because V is willing to feud if he knew that E had wronged him (\(f_V<sv\)) but is not willing to feud with the additional cost of \(\epsilon\).

If \(f_V<sv-\epsilon\), then V we have the following best-response function:

$$\begin{aligned} BR_{V}(\delta )=\left\{ \begin{array}{lr} \text {Accept (q=0) for } \delta <\frac{\epsilon (1-\theta )}{\theta (sv-f_V-\epsilon )}\\ \text {Arbiter (q=1) for } \delta >\frac{\epsilon (1-\theta )}{\theta (sv-f_V-\epsilon )}\\ \text {Any mix for } \delta =\frac{\epsilon (1-\theta )}{\theta (sv-f_V-\epsilon )}\\ \end{array} \right. \end{aligned}$$
(30)

Note that for the cutpoint on \(\delta\) in this best response function to be a proper probability, it must be the case that \(\epsilon <\theta (sv-f_V)\), or \(f_V<\frac{\theta (sv-\epsilon )}{\theta }\). If this is not the case, the required cut-off on \(\delta\) is greater than 1, and V will never pursue arbitration. The logic is similar to the point just mentioned. In this case, V is not willing to pursue arbitration and will accede regardless of E’s action.

1.3 E’s initial decision: to cheat or not to cheat?

We are now in a position to consider E’s initial decision on whether to be dishonest. Again, we must consider a number of subcases:

Scenario 1: \(k>\alpha -\beta\) The unconditional compliance threshold is met, so E will comply with the arbiter’s judgment regardless of V’s willingness to feud.

Case 1: Could we have an equilibrium in which E always cheats, i.e., \(\delta = 1\)? As long as \(\epsilon <\theta v\), V appeals to the arbiter and then E will comply with the arbiter’s ruling, resulting in a payoff of \(b-\phi\) instead of b by being honest. So there can be no equilibrium like this.

Case 2: Could we have an equilibrium in which E never cheats, i.e., \(\delta =0\)? If E never cheats, then \(\gamma =0\) and V will never pursue arbitration - but then E can profitably deviate to being dishonest. So there can be no equilibrium like this.

Case 3: This implies that we must be in a mixed strategy equilibrium in which E cheats with some probability. That probability must be \(\delta =\frac{\epsilon (1-\theta )}{\theta (v-\epsilon )}\) in order to make V indifferent. In order for E to be willing to mix in this way, he must be indifferent between being honest and cheating, which requires that the following are equal:

$$\begin{aligned} EU_E(Honest)= & {} b \end{aligned}$$
(31)
$$\begin{aligned} EU_E(Dishonest)= & {} q(b-\phi )+(1-q)\alpha \end{aligned}$$
(32)

This will be the case for

$$\begin{aligned} q=\frac{\alpha -b}{\alpha -b+\phi } \end{aligned}$$
(33)

Scenario 2: \(k<\alpha -\beta\), \(f_V\ge sv\) In this case, E knows that V will not feud in response to noncompliance, and E prefers to ignore an adverse decision. As a result, V will not pursue arbitration. Given that V will not appeal to the arbiter, the best response for E is to be dishonest.

Scenario 3: \(k<\alpha -\beta\), \(f_V\le sv\), \(f_E<(1-s)(\alpha -b)\), \(\epsilon >\theta (sv-f_V\) In this case, V is willing to feud once he knows he has been cheated, E will not comply, and V is not willing to pay the total cost of feuding and paying the court so never uses it. Knowing this, the the best response is for E to be dishonest with probability 1.

Scenario 4: \(k<\alpha -\beta\), \(sv>f_V\), \(f_E\ge (1-s)(\alpha -b)\), \(\epsilon <\theta (sv-f_V\) In this case, V is willing to feud and E complies with the arbiter’s decision. The same set of considerations as in Scenario 1 applies here—a mixed strategy is the only feasible equilibrium. The outcomes are the same in both cases (E goes to the arbiter with some probability and V complies after an adverse ruling) resulting in the same set of expected utilities:

$$\begin{aligned} EU_E(Honest)= & {} b \end{aligned}$$
(34)
$$\begin{aligned} EU_E(Dishonest)= & {} q(b-\phi )+(1-q)\alpha \end{aligned}$$
(35)

And therefore the same \(q^*\):

$$\begin{aligned} q=\frac{\alpha -b}{\alpha -b+\phi } \end{aligned}$$
(36)

Scenario 5: \(k<\alpha -\beta\), \(f_V<sv\), \(f_E<(1-s)(\alpha -b)\) In this case, E anticipates that he will ignore a decision, and that V will feud.

Case 1: Could we have an equilibrium in which E always cheats, i.e., \(\delta =1\)? V appeals to the arbiter with certainty. Then the expected utilities for E of being honest and dishonest are given by:

$$\begin{aligned} EU_E(Honest)= & {} b \end{aligned}$$
(37)
$$\begin{aligned} EU_E(Dishonest)= & {} bs+(1-s)\alpha -f_E-\phi -k \end{aligned}$$
(38)

Being dishonest is a best response for E if the following condition is met:

$$\begin{aligned} f_E\le (1-s)(\alpha -b)-\phi -k \end{aligned}$$
(39)

Case 2: Could we have an equilibrium in which E never cheats, i.e., \(\delta =0\)? If E never cheats, then \(\gamma =0\) and V will never pursue arbitration - but then E can profitably deviate to being dishonest. So there can be no equilibrium like this.

Case 3: Suppose a mixed strategy equilibrium in which E cheats with some probability. That probability must be \(\delta =\frac{\epsilon (1-\theta )}{\theta (sv-f_V-\epsilon )}\) in order to make V indifferent. Suppose that V goes to arbitration with probability q. In order for E to be willing to mix in this way, he must be indifferent between being honest and cheating, which requires that the following are equal:

$$\begin{aligned} EU_E(Honest)= & {} b \end{aligned}$$
(40)
$$\begin{aligned} EU_E(Dishonest)= & {} q(bs+(1-s)\alpha -f_E-\phi -k)+(1-q)\alpha \end{aligned}$$
(41)

This implies the following best response function for E:

$$\begin{aligned} BR_{E}(q)=\left\{ \begin{array}{lr} \text {Dishonest }(\delta =1)\text { for } q<\frac{\alpha -b}{s(\alpha -b)+f_E+\phi +k}\\ \text {Honest }(\delta =0)\text { for } q>\frac{\alpha -b}{s(\alpha -b)+f_E+\phi +k}\\ \text {Any mix for } q=\frac{\alpha -b}{s(\alpha -b)+f_E+\phi +k}\\ \end{array} \right. \end{aligned}$$
(42)

The required cutoff on q is a proper probability only if \(f_E>(\alpha -b)(1-s)-\phi -k\), the same condition as in Case 1. If f is below this threshold, the required cutoff on q is greater than 1, and E will always be dishonest. This is the “arbiter deterrence threshold:”

$$\begin{aligned} T_{Deterrence}^{Arb}=(\alpha -b)(1-s)-\phi -k \end{aligned}$$
(43)

1.4 Equilibria

This leaves us with six equilibria:

Equilibrium 1 (\(k>\alpha -\beta\)): Judicial Authority In this case, the arbiter has sufficient coercive power to inflict punishment on V for defying its judgment. V will therefore always comply, regardless of E’s willingness to feud. E cheats with probability \(\delta =\frac{\epsilon (1-\theta )}{\theta (v-\epsilon )}\), and V pursues arbitration with probability \(q=\frac{\alpha -b}{\alpha -b+\phi }\). The expected utility of V in this equilibrium must be positive, since E is honest with positive probability in the favorable environment.

Equilibrium 2 (\(k<\alpha -\beta\), \(f_V\ge sv\)): Strategic Avoidance In this case, E knows that V will not feud in response to noncompliance, and E prefers to ignore an adverse decision. As a result, V will not pursue arbitration. Given that V will not appeal to the arbiter, the best response for E is to be dishonest. Finally, this implies that V will not engage in the interaction in the first place.

Equilibrium 3 (\(k<\alpha -\beta\), \(f_V\le sv\), \(\epsilon > \theta (sv-f_V)\)): Strategic Avoidance In this case, although V is willing to feud once that node is reached, E will not comply with the arbiter’s ruling. V is unwilling to both feud and incur the cost of appealing to arbitration. As a result, V will not pursue arbitration. Given that V will not appeal to the arbiter, the best response for E is to be dishonest. Finally, this implies that V will not engage in the interaction in the first place.

Equilibrium 4 (\(k<\alpha -\beta\), \(sv>f_V\), \(f_E\ge (1-s)(\alpha -b)-k\)): Arbiter Authority In this case, V is willing to feud, and E complies with the arbiter’s decision. E cheats with probability \(\delta =\frac{\epsilon (1-\theta )}{\theta (v-\epsilon )}\), and V pursues arbitration with probability \(q=\frac{\alpha -b}{\alpha -b+\phi }\). The expected utility of V in this equilibrium must be positive, since E is honest with positive probability in the favorable environment.

Equilibrium 5 (\(k<\alpha -\beta\), \(f_V<sv\), \((1-s)(\alpha -b)-\phi \le (1-s)(\alpha -b)-k\), \(\epsilon < \theta (sv-f_V)\)): Partial deterrence E will ignore an adverse decision, and V will feud. E cheats with probability \(\delta =\frac{\epsilon (1-\theta )}{\theta (sv-f_V-\epsilon )}\) and V pursues arbitration with probability \(q=\frac{\alpha -b}{s(\alpha -b)+f_E+\phi +k}.\). The expected utility of V in this equilibrium must be positive, since E is honest with positive probability in the favorable environment. Likewise the expected utility of E in this equilibrium is positive since E receives (the equivalent of) b in the good environment and no longer receives a punishment in the bad environment.

Equilibrium 6 \(k<\alpha -\beta\), \(f_{V}<sv\), \(f_{E} < (1-s)(\alpha -b)-\phi -k\), \(\epsilon < \theta (sv-f_V)\): Exploitation and Feuding E will ignore an adverse decision, and V will feud. E always cheats, and V pursues arbitration. Given the assumption on \(\epsilon\), the expected payoff to V is positive, so he will engage in the interaction.

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Broman, B., Vanberg, G. Feuding, arbitration, and the emergence of an independent judiciary. Const Polit Econ 33, 162–199 (2022). https://doi.org/10.1007/s10602-021-09341-x

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