Endogenous Choice of a Mediator: Inefficiency of Bargaining

Transcription

1 Endogenous Choice of a Mediator: Inefficiency of Bargaining Jin Yeub Kim October 9, 2013 Abstract In this paper, I build a theory of how mediators are chosen by privately informed parties, which has not been well developed in the existing literature. I consider a simple bargaining problem in which two parties with private information about their types negotiate over choosing a mediator. A mediator, in my context, is equivalent to a communication mechanism that respects the private information. Not surprisingly, it is not obvious which one should be chosen from the very large set of interim incentive efficient mediators. My main insight is that the selection of a mediator is endogenous and is driven by the presence of information leakage. I show that the ex ante incentive efficient mediator will never be chosen. To fully explore the issue, I take a two-prong approach: cooperative and noncooperative. With a cooperative approach in line with Myerson (1984b), I find that there exists a unique neutral bargaining solution an incomplete information version of the Nash bargaining solution. With a noncooperative approach using a similar concept in Cramton and Palfrey (1995), I show that there exists a unique threat-secure mediator, who survives a vote against any alternative mediators. I establish that, remarkably, the selected mediators in both approaches are the same in the class of environments. Moreover, the selected mediator represents, of all the interim incentive efficient mediators, the most ex ante incentive inefficient and the least peaceful one. In this sense, I argue that the very process of selecting a mediator itself exhibits an inherent inefficiency of bargaining. The novelty of my paper is that it represents a first cut at building a bridge between a cooperative and a noncooperative game theory in the context of bargaining with incomplete information. Keywords: Cooperative game, Noncooperative game, Bargaining, Mechanism design. JEL Classification: C71, C72, C78, D82. I am greatly indebted to Roger Myerson, Lars Stole and Ethan Bueno de Mesquita for valuable advices and continuous support. I also appreciate Yeon-Koo Che, Peter Cramton, Alex Frankel, Jong-Hee Hahn, Jinwoo Kim, Heung Jin Kwon, Cheng-Zhong Qin, Hugo Sonnenschein, Richard van Weelden, and seminar participants at the University of Chicago Economic Theory Working Group Meeting, the International Conference on Game Theory at Stony Brook 2013, the Yonsei Economic Research Institute Seminar 2013, and the Asian Meeting of Econometric Society at Singapore 2013 for helpful suggestions and discussions. Department of Economics, University of Chicago, Chicago, IL jinyeub@uchicago.edu, webpage: jinyeub. 1

2 1 Introduction Two parties with private information often employ mediators to help them reach agreements as one of the primary tools of dispute resolution. In many economic, political, and social situations, however, two parties in potential conflict often fail to reach a consensus in their selection of a mediator that resolves conflict peacefully. Despite a significant amount of the theoretical work on mediation, the theory of how mediators are chosen by privately informed parties is not well developed. My interest is in understanding the endogenous selection of mediation. Through this paper, I attempt to build a theory of how parties with private information might agree on a mediator and provide a richer understanding in the failure of efficiency in mediation. I consider a simple bargaining problem in which the two players with private information about their own types strong or weak can each choose war or peace; there are also mediators that players can negotiate over which to choose. In this paper, I conceive of a mediator as a person who is not informed about the players types but who is trying to negotiate settlement between the players while respecting their private information. This allows us to consider a mediator to be equivalent to a communication settlement device, or a mechanism. 1 Therefore, by the revelation principle, I take the space of mediators that are available to the players to be synonymous with the space of incentive compatible and individually rational 2 mechanisms that the players can agree upon. Which mediator should the players choose? One might be tempted to think that the players would bargain to an ex ante Pareto dominant solution. That is, we might imagine that the players should select the mediator that is incentive efficient given the other s type, and maximize the ex ante payoffs of all the players. Indeed, in my setting, there is a unique ex ante incentive efficient mediator which both players might find it focal to choose. However, this naive idea that the ex ante incentive efficient mediator would be chosen is problematic. In particular, if players already know their types at the time they bargain over choosing a mediator, there is a problem of information leakage : the fact that a player is expressing his preference for a particular mediator conveys information about his type. For this reason, the issue of which mediator would get selected is far from trivial. My main insight is that the selection of the mediator is endogenous and the selection reflects this informational concern. Not surprisingly, it is not at all obvious that the players amongst themselves would be able to get to the ex ante incentive efficient mediator; and if they do not, it is also not clear which one should be chosen. Taking into account the endogenous information leakage issue, be either directly or indirectly, we can think about what are the desiderata for the solution concept. This paper is about understanding how this information leakage issue impacts 1 A mediator is a person or machine that can help the players communicate and share information. (Myerson, 1991, 250) 2 No player could gain by being the only one to lie to the mediator about his type or to not participate in the mediation. 2

3 the endogenous selection of a mediator and which mediator is most likely to arise in the negotiation over mediation. To fully explore this problem, I am going to take two different approaches: cooperative and noncooperative. Before describing my specific set up and findings, I motivate the framework of my analysis by briefly listing the solution concepts and the results. In a nutshell, a noncooperative approach is about choosing a mediator in an extensive form ratification game in which the players vote for one mediator over the other. The cooperative approach, on the other hand, is about putting some structure on what we think a good solution, or a selection of a mediator, must satisfy. Cooperative Approach A Non-Strategic Analysis A cooperative approach asks: what is a reasonable set of mediators we should expect to see arise as an outcome of an undefined communication process within the set of incentive feasible mediators? In the cooperative approach, I explicitly take into account the information leakage problem through imposing a set of relevant properties or axioms that must be true about a reasonable solution that captures exactly the tradeoff between different types of players. I refine the solution set in the negotiating process with possibly various notions of reasonableness, following the seminal work of Holmström and Myerson (1983) and Myerson (1984b). As a first cut, I use the notion of interim incentive efficiency, which is a minimal requirement in a setting with incomplete information. In my model, there are an infinite number of interim incentive efficient mediators but a single unique ex ante incentive efficient one. In order to select from within this large set of interim incentive efficient mediators, I pose a set of arguably reasonable axioms. These axioms, in particular, imply a solution concept following Myerson (1984b) called a neutral bargaining solution. The neutral bargaining solution can be thought of as an incomplete information generalization of the Nash bargaining solution. Surprisingly in my setting, not only does there exists a unique neutral bargaining solution, but it is also not the ex ante incentive efficient one. Noncooperative Approach A Two-Stage Ratification Game The cooperative approach, while illuminating why we should expect to see an ex ante inefficient mediator to arise endogenously in the mediator selection process, may not by itself be entirely convincing without also considering a noncooperative game. To address that, I now turn to a more explicit noncooperative approach. Basically, I am cutting the solution set differently than the previous approach by considering an extensive form ratification game, using an approach similar to Cramton and Palfrey (1995). Take some candidate mediator and call it a status quo mechanism. The noncooperative approach is to ask: Given some status quo mechanism, would the players both agree to switch to another mediator in some sequential equilibrium, noting that information might be revealed by the players votes which could affect the continuation payoffs of the players? Importantly, I must allow for the fact that if one of them votes in favor and one of them votes 3

4 against, then they are going to learn from disagreement and this will have consequences for their payoffs. This is a pairwise test in a ratification game where some uninformed third party can announce an alternative mechanism and players can switch. From the noncooperative approach, a reasonable restriction to put on the selection of a mediator is that such a mediator should survive a vote against any alternative options. It is not clear such a mediator exists, but if it does exist, then we would have a compelling reason to select it. The notion of survivability against all alternatives in a pairwise voting game will be formalized as a notion of what I call threat-security. My innovations over Cramton and Palfrey (1995) are that the status quo is designed and, by assuming public voting with two players, both players would learn about each other after disagreement. What I find in this context is that there exists only one threat-secure mediator that will always survive the vote against any alternatives, and it is the one that is ex ante Pareto dominated by any alternative mediator. Equivalence of the Two Approaches In both approaches, the selected mediator will not be the ex ante incentive efficient one. Both approaches not only give us ex ante incentive inefficiency, but also lead to the same inefficient mediator. In the stylized game in this paper, the suboptimal choice of such a mediator will in fact be the most war-inducing mediator, whereas the ex ante incentive efficient one is the most peaceful. Surprisingly, the threat-secure mediator is exactly the same as the neutral bargaining solution. In general, the neutral bargaining solution and the threat-secure set are not nested. However, in the class of bargaining games I consider, two distinct ways of looking at the problem lead to the same solution which represents, of all the interim incentive efficient mediators, the most ex ante inefficient 3 and the least peaceful. In this sense, I argue that the very process of selecting a mediator itself has an inherent inefficiency of bargaining. We can get the intuition underlying the problem coming at it from both the cooperative and noncooperative directions. The strategic intuition as to why the ex ante incentive efficient mediator cannot survive in the noncooperative game underscores precisely the intuition about the information leakage problem behind the cooperative game. In the cooperative sense, players pick a mediator, effectively pooling in a way that no information is revealed. But the nature of that pooling is exactly like signaling in the noncooperative game. In a sense, this paper provides a first attempt to connect a cooperative bargaining theory to a noncooperative ratification game. 1.1 Literature on Mediation I conclude the introduction by discussing a few papers on mediation in the context of international bargaining situations. A significant amount of the theoretical work on mediators and mediation has focused on when and how mediation improves upon unmediated communication. (Kydd, 2003; 3 Precisely stated, contrary to the ex ante incentive efficient mediator, the most ex ante inefficient mediator is a solution to the program of minimizing the players ex ante expected payoffs over the set of all interim incentive efficient mediators. 4

5 Hafer, 2008; Fey and Ramsay, 2009; Fey and Ramsay, 2010) 4 Including Fey and Ramsay (2009), there has been an ongoing agenda of research on mediation, that adopts mechanism design tools to study the effectiveness of mediation on preventing conflicts. Fey and Ramsay (2011) show the relationship between the kind of uncertainty states face and the possibility of peaceful resolution of conflict. In a simple model of conflict, Hörner, Morelli and Squintani (2011) prove that mediation improves the ex ante probability of peace with respect to unmediated peace talks when the intensity of conflict is high or when asymmetric information is significant. Also, in their working paper, Meirowitz et al. (2012) claim that mediation is just as effective as a hypothetical optimal institution with strong enforcement power and the ability to internalize the long-term consequences of its behavior. Despite an extensive literature on mediation, however, the issue of how mediators are selected is not sufficiently understood. Much of the literature has focused on the analysis of the effectiveness of mediation and not on the important question of how bargaining over mediation itself changes the trajectory of conflict. My goal is to build a theory of mediator selection that determines which mediator we should reasonably expect to see arising endogenously, by defining the smallest solution set for the given bargaining problem. I utilize the solution and refinement concepts in Myerson (1983), Holmström and Myerson (1983), Myerson (1984b), and Cramton and Palfrey (1995). 5 Section 2 introduces the basic model. Section 3 characterizes the interim incentive efficient set and the neutral bargaining solution under a cooperative approach. Then, in Section 4, I develop and illustrate a concept of threat-security under a noncooperative approach. Section 5 discusses the equivalence between the two approaches and the intuition behind the distortions. Section 6 demonstrates the results in a simple example and Section 7 concludes. An extension will be considered in the online Appendix. All proofs are in Appendix A. 2 The Model I consider a simple Bayesian bargaining problem as a mechanism design problem of mediators, in which two players with private information can bargain over selecting a mediator to help coordinate them incentive compatibly, or else remain in status quo if they fail to agree. In my context, I 4 Among literature on the information mediation problems, Kydd (2003) identifies sufficient conditions for effective mediation. He argues that in order for a mediator s communication to be credible, the mediator must be biased and endowed with some independent knowledge of the private information of the disputants. Fey and Ramsay (2010) endogenize the process by which information is gained and show that information mediation has no significant effect on the likelihood of ending a dispute if mediators do not have access to exogenous sources of information, beyond what is relayed to them by the disputants. These results hold in cases where the potential source of conflict is private information of private values. However, I am interested in private information of interdependent value. Hafer (2008), in her working paper analyzing a model of contested political authority, shows that neither cheap talk communication nor mediation promote peaceful resolutions or lead to improvements in social welfare. On the other hand, Fey and Ramsay (2009) identify conditions under which mediation achieve peace with probability one. 5 For classical bargaining literature, also see Nash (1950); Nash (1953) (the canonical solution concept for twoperson bargaining problems); Harsanyi and Selten (1972) (an extension of the Nash bargaining solution for two-person games with incomplete information); and Myerson (1979) (a modified version of Harsanyi and Selten (1972)). 5

6 consider a mediator to be represented as a mechanism. The definitions of what follows are done in full generality, although I later come to a specific problem in the context of international conflict in Section 2.2 in which I will be interested in the case of two players, each with two types, and two outcomes. 2.1 The Base Setup The two-person bargaining problem is characterized by the following structures Γ = (D, d 0, T 1, T 2, u 1, u 2, p 1, p 2 ), whose components are interpreted as follows. D is the set of feasible bargaining outcomes available to the two players from which that they can choose. d 0 D denotes the disagreement outcome, which the two players get if they fail to coordinate on choosing a mediator. For each i {1, 2}, T i is the set of possible types t i for player i. 6 The players are uncertain about the other player s types, where the players types are unverifiable. For each i {1, 2}, u i is player i s utility payoff function from D T 1 T 2 into R, such that u i (d, t), i = 1, 2, denotes the payoff to player i if d D is the outcome and t T is the true vector of players types. The payoffs are in von Neumann- Morgenstern utility scale. Each p i is a conditional probability distribution function that represents player i s beliefs about other players types as a function of his own type. That is, p i (t i t i ), i = 1, 2, denotes the conditional probability that player i of type t i would believe about the other player s type being t i. Let T = T 1 T 2 denote the set of all possible type combinations t = (t 1, t 2 ). For mathematical simplicity, I assume D and T are finite sets. Without loss of generality, I assume that utility payoff scales are normalized so that u i (d 0, t) = 0 for all i and for all t. That is, each player could guarantee himself a payoff of zero by refusing to involve a mediator. For simplicity, I assume that the players beliefs are consistent with some prior probability distribution p in (T ), which is common knowledge, under which the players types are independent random variables. That is, I assume that for every i there exists a probability distribution p i in (T i ) such that p i (t i ) is the prior marginal probability that player i s type will be t i and p i (t i t i ) = p i (t i ), i N, t i T i, t i T i. Since two players have incomplete information, the players in a bargaining problem do not have to agree on a specific outcome in D. Instead, two players may agree on some communication mechanism which is a decision rule specifying how the outcome d D depends on the players types. As the players choose a mechanism and not an outcome, they can conceal their true types 6 Each t i T i represents player i s characteristics such as preferences, strengths, capabilities, or endowments. 6

7 in order to get a better bargaining deal. In my context, they would bargain over the selection of a mediator who mediates according to a mechanism µ, which is a communication mechanism of the following form: Each player is asked to confidentially report his type to the mediator; then, after getting these reports, the mediator recommends an outcome to the players. 7 Allowing randomized strategies, a mediation mechanism can be defined as a function µ : D T R such that µ(c t) = 1 and µ(d t) 0, d D, t T. c D That is, µ(d t) is the probability that d is the bargaining outcome chosen by the mediator with the mechanism µ, if t 1 and t 2 are the players types. Given any mechanism µ, for any i {1, 2} and any t i T i, U i (µ t i ) = p i (t i t i )µ(d t)u i (d, t) t i T i d D is the conditional expected utility for player i given that he is of type t i, if the mechanism µ is implemented. The expected utility for type t i of player i if he lies about his type and reports s i while the other player is honest is Ui (µ, s i t i ) = p i (t i t i )µ(d t i, s i )u i (d, t). t i T i d D A mediation mechanism µ is incentive compatible if and only if it satisfies the following informational incentive constraints: U i (µ t i ) U i (µ, s i t i ), i N, t i T i, s i T i. A mediation mechanism µ is individually rational if and only if it satisfies the following participation constraints: U i (µ t i ) p i (t i t i )u i (d 0, t), i N, t i T i. t i T i Since u i (d 0, t) = 0, i N, t i T i, the participation constraints reduce to U i (µ t i ) 0, i N, t i T i. 7 In the existing literature of third party intervention, the essence of intermediaries is in conveying information and mediated communication involves a nonstrategic communication device that receives and transmit messages. Then, a mediator recommends a bargaining outcome to the players and his recommendations will depend on the players reports. I consider setting in which the players can follow the chosen mediator s recommendation or go to disagreement outcome. Although this might seem limiting, it is the simplest way to find out which mediator the players are likely to choose in negotiation over mediators. This approach allows us to analyze the outcomes of bargaining games and characterize the probability of peace obtainable with different mediators while leaving potentially endless variation in the extensive form of Bayesian bargaining games unspecified. 7

8 The revelation principle (Myerson, 1979) implies that a mechanism cannot be implemented, by any equilibrium of a communication game induced by any communication system, unless the mechanism is incentive compatible and individually rational. Thus, there is no loss of generality in focusing on such incentive compatible, individually rational direct revelation mechanisms. 8 In my context, therefore, taking relevant incentive constraints into account, I define the incentive feasible mediator to be whoever mediates according to the incentive compatible, individually rational mediation mechanism for the above Bayesian bargaining problem. That is, by the revelation principle, I can naturally assume that players bargain over the set of incentive feasible mediators. As I take a mediator to be synonymous with a mechanism, I use the terms mediator and its corresponding mediation mechanism interchangeably throughout the paper. 2.2 The Benchmark Model Suppose a simple toy version in the context of international relations with two symmetric players, each with two discrete types, that must make a decision d D. For example, two states are involved in a dispute, in which conflict is occurring over a divisible item, area of territory, or an allocation to resources, which could lead to war. There are two possible decisions called d 0 and d 1. Let D = {d 0, d 1 }, where d 0 can be interpreted as the players going to war, and d 1 as peace. Each i = {1, 2} has private information t i T i = {s, w}, where s denotes the strong type and w denotes the weak type. I assume that types are independent, i.e., t i is drawn from the distribution p i, independently of t i and this is common knowledge. For the sake of simplicity and tractability, I assume symmetry in probability that the prior marginal probability of the strong types are the same for both players, i.e., p 1 (s) = p 2 (s) p(s). I can relax this assumption, however, and I still obtain interesting theoretical implications. There are known specialists in mediation who essentially announce incentive feasible mechanisms. It is important to note that there are several unique features of international conflict that I must incorporate into my analysis and that prevent a direct application of the standard results in the literature. It is a central facet of international conflict that no enforcement body exists to permit binding contracts. Thus, a country can, at any time, unilaterally choose to initiate a war and there is no way a country can commit not to do so. I call this feature the unilateral war assumption: War can be forced by either player. With two outcomes war or peace that the players can choose from, a disagreement outcome is clearly just war and the following common theoretical assumption is necessary. 9 8 See Holmström and Myerson (1983, 1804) and Myerson (1991, 487). 9 This may be unrealistic in international relations as there is always a stalemate option for countries. As it would be interesting to say something more about the more realistic possibility of multiple threats (either going to war and staying in an unproductive stalemate), I will consider the extension of three-option model in the online Appendix. However, since the results qualitatively do not change in the extension and this toy model with two outcomes is enough to get interesting results, I am going to focus on this benchmark model. 8

9 Assumption. War is the only threat in bargaining. With this assumption, I can designate d 0 D to be the disagreement outcome. Since I can normalize the disagreement payments such that u i (d 0, t) = 0 for all i and for all t, I can write the normalized payoffs, for each i, as u i (d 0, t) = 0, u i (d 1, t) = π i (t) ω i (t), where π i (t) is the value of a peace treaty (e.g., negotiated settlement, transfer of money, etc) that results from settling for a peace outcome and ω i (t) is the benefit of war with costs incorporated. In the context of international conflict, the payoffs are natural examples of interdependence. Assumptions. I maintain the following assumptions on the payoffs: (A1) i u i(d 1, t) > 0 for all t (A2) u 1 (d 1, sw) < 0 (A3) u 1 (d 1, t) = u 2 (d 1, t) when t 1 = t 2 ; u 1 (d 1, sw) = u 2 (d 1, ws) and u 2 (d 1, sw) = u 1 (d 1, ws). (A1) implies that war is socially wasteful and is a destructive option, and the peace outcome is socially better than war. Although the war outcome is always socially Pareto inferior to some peaceful settlement, by (A2), a stronger country can take most of the stake of the dispute when war occurs and get a payoff larger than he can expect to get from a negotiated settlement. That is, a strong type prefers going to war when he is sure of facing a weak type. 10 (A3) simply assumes that payoffs are symmetric, only depending on type combinations. Because of (A3), we can just focus on i = 1 in all the analysis that follows. 3 Cooperative Approach At some level, there is potentially endless variation of extensive form communication game for the negotiation process over selecting mediators, which I am not explicitly modeling. Rather, I take payoffs and the existence of incentive feasible mediators as primitives. The revelation principle implies that if all incentive compatible direct mechanisms have some property, then every equilibrium 10 Because of these assumptions on the payoffs, together with interdependence, I can assume away any further restrictions on π i(t) or ω i(t) by itself alone. That is, only the relative payoffs from peace compared to war matter. For example, it could even be the case that a weak type fares better in war when she meets a strong type, i.e., ω 1(w, s) > ω 2(w, s), as long as the weak type earns relatively much more from a peace treaty, i.e., π 1(w, s) π 2(w, s). In other words, as long as π 1(w, s) ω 1(w, s) > 0 such that (π 1(w, s) ω 1(w, s))+(π 2(w, s) ω 2(w, s)) > 0 (by (A1)), where π 2(w, s) ω 2(w, s) < 0 (by (A2) and (A3)). In international relations literature, it is often assumed that a strong type fares better in war than a weak type, a weak type incurs larger costs than a strong type, or a strong type gets a weakly better division of a negotiated settlement. In this sense, my model renders much more generality. 9

10 of every game form has this property. With a cooperative approach, I am explicitly taking into account the information leakage problem by imposing relevant properties that must be satisfied for a reasonable solution. The cooperative component of my analysis asks: Within the set of all incentive feasible mediators, what must be true about a mediator that we should expect for it to arise in the two-person bargaining problem that satisfies (A1) through (A3)? My goal is to develop a formal argument of endogenous choice of a mediator that determines the smallest possible set of solutions, so as to get strongest possible predictions. Therefore, the cooperative approach is to write down a reasonable set of requirements or certain axioms that must be true about those selected mediators. In this sense, I consider how small a solution set I can define for this bargaining problem and still satisfy these properties. In this section, I refine the solution set in the negotiating process that gives us a reasonable prediction, with possibly various notions of reasonableness, 11 characterizing the interim incentive efficient set and the neutral bargaining solution using the solution concepts in Holmström and Myerson (1983) and Myerson (1984b); and I show the inefficiency of bargaining. All proofs are in Appendix A. 3.1 The Interim Incentive Efficient Set As a first cut, I use the notion of interim incentive efficiency. This is a minimal requirement in a game with incomplete information, in which each player knows his own type but does not know the other s type at the beginning of the game when plans and strategies are first chosen. Thus, in a bargaining problem, each player would be actually concerned with his own conditionally expected payoff given his own true type at the initial decision making stage. 12 Of all the mediator that players can choose, it should be that, once the players know their types, the mechanism that a mediator is associated with is interim incentive efficient. Otherwise, a third party can offer an interim Pareto superior one instead and players will all be better off. As players are rational, they would pick something from the interim incentive efficient set. To briefly recall the definition of efficiency, an incentive feasible mediator associated with a mechanism µ is interim incentive efficient if and only if there exists no other incentive feasible mediator with a mechanism that is interim Pareto superior to µ. If every player would surely prefer µ over µ when he knows his own type, whatever his type might be, then µ is interim Pareto superior to µ. Interim welfare is evaluated after each player learns his own type but before he learns the other player s type. An incentive feasible mediator associated with a mechanism µ is ex ante 11 We have lots of solution concepts because we do not have a formally defined game. These solutions sets can be useful both for prescriptive purposes, providing predictions, and for descriptive purposes, if I assume that players tend to reach agreements in which each gains as much as he contributes to the other. 12 Holmström and Myerson (1983) have argued that,..., the most appropriate concept of efficiency for Bayesian games with incomplete information is interim efficiency in the set of incentive feasible mechanisms. (Myerson, 1991, ) 10

11 incentive efficient if and only if there exists no other incentive feasible mediator with a mechanism that is ex ante Pareto superior to µ. Ex ante welfare is evaluated before any types are specified. If a mediator is incentive efficient, then another mediator who does not know any player s actual type could not propose any other incentive feasible mediation mechanism that every player is sure to prefer according to an appropriate welfare criterion. 13 Therefore, I am interested in characterizing the interim incentive efficient set of mediators, or mediation mechanisms, within the set of incentive feasible mechanisms F. Nonetheless, a mediator does not know each player s type, so it would be useful to identify an ex ante incentive efficient mechanism. Note that for any given set of incentive feasible mechanisms, the set of ex ante incentive efficient mechanisms is a subset of the set of interim incentive efficient mechanisms. One might naively think about ex ante incentive efficiency as a requirement. But if a player expresses a preference for the ex ante incentive efficient outcome, it would be giving information to the other side that might be detrimental. Therefore, there is no reason a priori to expect that the ex ante incentive efficient outcome would be negotiated with privately informed parties. Let S(Γ) denote the set of incentive feasible mediators for Γ each of whom is ready to offer to mediate by some incentive compatible and individually rational mediation mechanism that is on the interim incentive efficient frontier. The following lemma gives three thresholds that separate four cases along the range of p(s) in characterizing S(Γ). Lemma 1 (Thresholds). Let p, p, and p be such that p p p Then, p > p and p > p. u 1 (d 1, sw) u 1 (d 1, ss) u 1 (d 1, sw), u 1 (d 1, ww) u 1 (d 1, ww) + u 1 (d 1, ws), u 1 (d 1, ss)u 1 (d 1, ww) u 1 (d 1, sw)u 1 (d 1, ws) u 1 (d 1, ss)u 1 (d 1, ww) u 1 (d 1, sw)u 1 (d 1, ws) + u 1 (d 1, ss)u 1 (d 1, ws). The interpretations of these thresholds are given in Appendix A. For now, only note that I must distinguish the two instances: p p and p > p. Proposition 1 fully characterizes S(Γ) depending on the probability of the strong types when p p. Corollary 1 is for when p > p. In fact, each interim incentive efficient mediator can be entirely defined by the probability weight he puts on the outcome of war and peace for each type realization. Therefore, it is useful to set the following structure which I am going to use routinely throughout the paper. 13 If a mediator is interim incentive efficient and each player knows only his own type, then it cannot be common knowledge that the players unanimously prefer some other incentive feasible mediator over the mediator. This does not mean that the players could never reach a unanimous agreement to change; It only means that if a unanimous agreement is reached then each player must know more than just his own type; communication must have occurred. This is considered in Section 4 with an extensive form game defined. 11

12 Definition 1. A mediation mechanism µ y,z is defined to have the following structure: µ y,z (d 0 t) = 0, µ y,z (d 1 t) = 1 for t {ww} µ y,z (d 0 t) = y, µ y,z (d 1 t) = 1 y for t {sw, ws} µ y,z (d 0 t) = z, µ y,z (d 1 t) = 1 z for t {ss}, where y 0 and z 0. That is, µ y,z implements d 0 (war) with probability zero when t = {ww}, with probability y 0 when t = {sw, ws}, and with probability z 0 when t {ss}. Remark 1. Every interim incentive efficient mechanisms will have the structure of µ y,z as defined in Definition 1, i.e., S(Γ) = {µ y,z y 0, z 0}. Therefore, from now on, I can think about each interim incentive efficient mediator as being indexed by these two numbers y and z. The following Proposition 1 tells us what further restrictions on y and z s are depending on the different regions of p(s). Proposition 1. When p p, the set of interim incentive efficient mediators S(Γ) = {µ y,z } can be completely described such that: Case 1. p(s) < p : S(Γ) = { µ y,z y [ y( p(s)), 1 ], z = 0 }, where y( p(s)) = 1 + which is decreasing in p(s); Case 2. p(s) [p, p ]: S(Γ) = {µ y,z y [0, 1], z = 0}; p(s)u 1(d 1,ss) (1 p(s))u 1 (d 1,sw) Case 3. p(s) (p, p ): S(Γ) = {µ y,z y [0, 1], z := z(y, p(s)) [0, z( p(s))]}, where z(y, p(s)) = [ ] y 1 (1 p(s))u 1(d 1,ww) p(s)u 1 (d 1,ws) and z( p(s)) z(1, p(s)) which is increasing in p(s); and Case 4. p(s) p : S(Γ) = {µ y,z y = 0, z = 0}. The key point to notice in Proposition 1 is that each interim incentive efficient mediator can be identified by a two-dimensional object µ y,z with restrictions on y (the probability on war for t = {sw, ws}) and z (the probability on war for t = {ss}), where z is pinned down entirely by y. For example, when p(s) [p, p ] (Case 2 ), µ 0,0 always puts probability 1 on peace; µ 0.3,0 puts probability 1 on peace when both players are of the same types, but puts positive probability 0.3 on war when players are of different types; and µ 1,0 puts probability 1 on peace when both players are of the same types, but puts probability 1 on war when players are of different types. These mediators cannot be interim Pareto dominated by any other incentive feasible mediator, and so all of these mediators are interim incentive efficient. 12

13 I point out three things to note for Case 1, Case 3, and Case 4, respectively, before going further. First, when p(s) < p (Case 1 ), the lowest probability that an interim incentive efficient mediator can possibly put on war for t = {sw, ws} must be bounded below by y( p(s)). This is because, if some mediator puts even lower probability than y( p(s)) on war, then it would not be incentive feasible. Therefore, when the strong type is very rare, it must put some minimum probability on war in order to induce a strong type to participate and not deviate to unilaterally forcing war. Otherwise, always war would give the strong type strictly a higher expected payoff than when participating in mediation, since he expects the other player to be weak with high enough probability. Second, when p(s) (p, p ) (Case 3 ), an interim incentive efficient mediator must also put some positive probability on war when both players are of the strong type, in order to prevent the weak type from reporting dishonestly. Lastly, when p(s) p (Case 4 ), any mediator who puts positive probability on war regardless of type realization would be interim Pareto dominated by µ 0,0. The following Corollary 1 characterizes the set of interim incentive efficient mediators when p > p. The only difference from Proposition 1 is that when p(s) (p, p ) (Case 2 ), an interim incentive efficient mediator has to be associated with a mechanism which puts some positive probability on war for t = {sw, ws} with a lower bound requirement, along with some corresponding positive probability on war for t = {ss}. Corollary 1. When p < p, the set of interim incentive efficient mediators S(Γ) = {µ y,z } can be completely described such that: Case 1. p(s) p : S(Γ) = { µ y,z y [ y( p(s)), 1 ], z = 0 } as in Proposition 1. Case 1; { [ ] Case 2. p(s) (p, p ): S(Γ) = µ y,z y y( p(s)), 1, z := z(y, p(s)) [ z( p(s)), z( p(s)) ]}, where z( p(s)) > 0 and y( p(s)) > 0 are defined in proof; Case 3. p(s) [p, p ): S(Γ) = {µ y,z y [0, 1], z := z(y, p(s)) [0, z( p(s))]} as in Proposition 1. Case 3; and Case 4. p(s) p : S(Γ) = {µ y,z y = 0, z = 0}. Proposition 1 and Corollary 1 imply that there exists multiple interim incentive efficient mediators when p(s) < p (Case 1, 1, 2, 2, and 3 ). When p(s) p (Case 4 ), only one mediator µ 0,0 exists in S(Γ). Therefore, the intriguing cases would be only when p(s) < p in which there are an infinite number of interim incentive efficient mediators that players can select from. In fact, within each case, I can order the mechanisms according to ex ante welfare criterion. Lemma 2. For each Case 1, 1, 2, 2, and 3, µ y,z is ex ante Pareto superior to µ y,z if y < y with a corresponding z z. 13 if and only

14 Proposition 2. The unique ex ante incentive efficient mediator in S(Γ) is When p p, (a) µ y( p(s)),0 for Case 1; (b) µ 0,0 for Case 2, 3 & 4. When p < p, (a) µ y( p(s)),0 for Case 1 ; (b) µ y( p(s)),z( p(s)) for Case 2 ; (c) µ 0,0 for Case 3 & 4. Proposition 2 gives the best mediator in an ex ante sense. The unique ex ante incentive efficient mediator is the one who puts the lowest probability on war among all the interim incentive efficient mediators. Note that µ 0,0 is such that µ 0,0 (d 1 t) = 1, t T, which can be interpreted as a peaceful mediator who always implements peace treaty regardless of the players types. Therefore, a peaceful mediator is the one for which there is never an impasse that prevent an agreement. The following Lemma 3 gives the interim welfare ordering in S(Γ). Lemma 3. For each Case 1, 1, 2, 2, and 3, µ y,z is ex ante Pareto superior to µ y,z if and only if µ y,z gives strictly lower interim expected utility for the strong type and strictly higher interim expected utility for the weak type. Corollary 2. Suppose that p(s) < p. Then, there exist multiple interim incentive efficient mediators in S(Γ) who are not ex ante incentive efficient. Moreover, the one that is ex ante Pareto inferior to any other mediators in S(Γ) is µ 1,o when p(s) p and µ 1, z( p(s)) when p(s) (p, p ), which gives the highest interim payoff for the strong type. Corollary 2 simply follows from the previous results. It suggests that there are multiple mediators in S(Γ) who will, if chosen, mediate players according to some incentive feasible mechanism that is not necessarily ex ante incentive efficient. The interim but not ex ante incentive efficient mediator can be considered bad in the sense that it sometimes allows players fail to reach a negotiated settlement and to go to war with higher probability than the ex ante incentive efficient mediator. The worst mediator, who puts a higher probability on war than any other interim incentive efficient mediators, is the one that can be described as being the farthest away from the ex ante incentive efficient mediator but giving the highest interim utility to the strong type. Alternatively, the worst one also can be thought of as the unique solution to the minimization problem of the ex ante expected payoffs. 3.2 The Neutral Bargaining Solution Before exploring on the results in this section, I invoke an important idea that needs to be clarified the notion of the inscrutability principle (Myerson, 1983). The inscrutability principle states that when we consider a mechanism selection game, in which individuals can bargain over mechanisms, there should be no loss of generality restricting our attention to equilibria in which there is one incentive feasible mechanism that is selected with probability 1, independently of anyone s type. 14

15 (Myerson, 1991, 504) It can be justified by viewing the mechanism selection process itself as a communication game induced from the original Bayesian bargaining game and by applying the revelation principle, as follows. At some level, there is potentially endless variation on the communication process for selecting a mediator. Without reference to a specific game form, we can think about the following. Once a noncooperative game of the negotiation process over mediators is well-defined, we can take a Bayesian Nash equilibrium of it. This would ultimately be an equilibrium mapping from some distributions of player s types into some distributions over mediators playing the mechanisms. For example, suppose that there is an equilibrium of some mediator selection game in which some mechanism δ would be selected if the profile of players types were in some set A T, and some other mechanism γ would be selected otherwise. Then, there should exist an equivalent equilibrium of the mediator selection game in which the players always select a grand mechanism µ, which coincides with mechanism δ when the players report a profile of types in A and which coincides with γ when they report a profile of types that is not in A. Myerson (1983) shows that there is no loss of generality in assuming that all types of players agree to play this same grand mechanism µ without any communication during the negotiating process. 14 This idea of rolling in all the information leakage that could occur in any process of mediator selection into the grand mechanism is the inscrutability principle. Therefore, whatever the equilibrium outcome of any bargaining procedure is, without loss of generality, I can reduce that equilibrium into a grand mechanism that accomplishes the same thing and is incentive feasible. I call this an inscrutable mechanism, or an inscrutable mediator. Under the inscrutability principle, the set of all equilibria under a noncooperative game once defined is attainable as the set of all inscrutable mechanisms. That is, if all inscrutable mechanisms have some property, then every equilibrium of a noncooperative game has this property. In light of Myerson (1983), I assume that players bargain inscrutably in the negotiating process. 15 As we have seen in the previous section, there are multiple interim incentive efficient mediators including the unique ex ante incentive efficient one in my model. I would next like to know how small a solution set I can refine for the bargaining problem in this paper to get the strongest prediction. Then, focusing on working in the domain of all incentive feasible mediators that player would inscrutably choose, I can ask further within the set of interim incentive efficient mediators what properties must be satisfied for the mediators which players inscrutably agree upon. 14 That is, for any outcome achievable via any equilibrium under some noncooperative game, I can think about that as an incentive feasible grand mechanism that basically everyone selects. Moreover, for any incentive feasible grand mechanism for some bargaining problem, there exists some noncooperative game that generates it as an equilibrium. 15 In other words, neither player will ever deliberately reveal any information about his true type until the mediator is agreed upon in the negotiating process. If two players agree, they simultaneously and confidentially report their types to the mediator, and then the decision is adopted according to µ. In effect, any information that the players could have revealed during the mechanism selection process can be revealed instead to the mediator after it has been selected, and the mediator can use this information in the same way that it would have been used in the mechanism selection process. (Myerson, 1991, 504) Also see Myerson (1984, 463). 15

16 The notion of interim incentive efficiency implicitly uses U(µ) = ((U i (µ t i ) ti T i ) i {1,2} given any mechanism µ as the relevant utility allocation vector. Since each player already knows his true type at the time of bargaining, it would not be appropriate to average each player s expected utility over his various types. It might seem that just the components corresponding to the two actual types are significant in determining whether a mediator with such µ is chosen. However, all of the components of U(µ) is relevant in determining whether it is chosen, because each player must express a compromise among the preferences of all of his possible types in bargaining, in order to conceal his true type during the bargaining process. That is, even though each player already knows his true type at the time of bargaining, in order to bargain inscrutably, each player must use a bargaining strategy which maintains a balance between the conflicting goals that he would have if he were of different types. Therefore, a bargaining solution must specify criteria for equitable compromise between different possible types of each player, as well as between two players. In this sense, I put a further refinement in choosing a subset within the set of interim incentive efficient mediators with a very simple set of axioms using Myerson (1984b). Myerson (1984b) develops a generalization of the Nash bargaining solution for two-person bargaining games with incomplete information from some basic properties that a fair bargaining solution should satisfy. The neutral bargaining solutions (Myerson, 1984b) form the smallest set satisfying two axioms: an extension axiom and a random-dictatorship axiom. 16 The first axiom is a stronger version of Nash s independence of irrelevant alternatives axiom. If there are ways to extend the Bayesian bargaining problem Γ in a way that the set of decision options is increased, without changing the disagreement outcome such that both players would be willing to settle for expected utility payoffs that are arbitrarily close to what they can get from the feasible mechanism µ, then the players would be willing to settle for the mechanism µ even when these extra decision options are not available. The second axiom is related to Nash s symmetry axiom. In cases where we can predict what mechanism each player would select if he were the principal who selects the mechanism dictatorially, if giving each player a probability.5 of acting as the principal (the random dictatorship) is interim incentive efficient, then the random dictatorship mechanism should be a fair bargaining solution. While the hypotheses of this axiom are quite restrictive, it just means that only in a bargaining game where the random dictatorship is both well understood and interim incentive efficient, it should be a fair claim to being a bargaining solution. A neutral bargaining solution of a bargaining problem Γ is defined to be any mediator associated with a mechanism with the property that it is a solution for every bargaining solution concept which satisfies these two axioms. As the set of all interim incentive efficient mediators would satisfy these two axioms, I can think about picking a neutral bargaining solution over the set of all interim incentive efficient mediators S(Γ). Let me first reproduce the characterization theorem in Myerson (1984b) generated by the two axioms, slightly modified for my case which gives the most tractable 16 See Myerson (1984b) for detailed exposition of these axioms. 16

17 conditions for computing neutral bargaining solutions. Theorem (Myerson, 1984b). A mechanism µ is a neutral bargaining solution for Γ if and only if, for each positive number ε, there exist vectors λ, α, β, and ϖ (which may depend on ε) such that λ i (t i ) + α i (s i t i ) + β i (t i ) ϖ i (t i ) α i (t i s i )ϖ i (s i ) / p(t i ) s i T i s i T i = v i (d, t, λ, α, β) p(t i ) max, i, t i T i ; d {d 0, d 1 } 2 t i T i j {1,2} λ i (t i ) > 0, α i (s i t i ) 0, β i (t i ) 0, i, s i T i, t i T i ; and U i (µ t i ) ϖ i (t i ) ε, i, t i T i, where v i ( ) is the virtual utility payoff defined by v i (d, t, λ, α, β) = ((λ i (t i ) + s i T i α i (s i t i ) + β i (t i ))u i (d, t) s i T i α i (t i s i )u i (d, (t i, s i )))/ p i (t i ); and α i (s i t i ) denotes the Lagrange multiplier for the informational incentive constraints; and β i (t i ) denotes the Lagrange multiplier for the participation constraints. Moreover, (λ, α, β) satisfy the interim incentive efficiency conditions for the mechanism µ. The above theorem asserts that a neutral bargaining solution can be characterized as an incentive feasible mediator who is not only efficient in terms of interim incentive efficiency but also equitable in terms of balancing out intertype compromise, maximizing some virtual utility allocations for the two players. 17 A player might bargain as if he wanted to maximize some virtual utility scale which exaggerates the difference from the other type in response to the pressure that he feels from the other player s distrust of any statements that he could make about his type. The first equation in the theorem defines intertype equity conditions such that the weighted sums of players possible conditionally expected utilities should be equal. The third condition says that a neutral bargaining solution should generate type-contingent expected utilities that are equal to or interim Pareto superior to a limit of virtually equitable utility allocations. The neutral bargaining solution can be interpreted as if some principal maximizes ex ante social welfare with virtual utility weights that are different than the probability distribution of types, where the distribution of virtual utility weights are subject to interim incentive constraints. If a mediator is a neutral bargaining solution, then necessarily there exists some distorted type- 17 The players can negotiate over the set of virtual utility maximizing allocations without revealing information about their types. 17

18 dependent λ i (t i ) weights constrained by interim incentive feasibility, maximizing the virtual utilities; given the weights, I can interpret this as though the probability distributions have changed. The theorem gives a set of conditions which must hold for every mechanism in the set of neutral bargaining solutions. Myerson (1984b) proved that the set of neutral bargaining solutions is nonempty for any finite two-player Bayesian bargaining problem. Let N S(Γ) denote the set of neutral bargaining solutions identified for the class of two-person Bayesian bargaining game Γ that satisfies assumptions (A1) through (A3). Then, it can be shown that for the bargaining problem considered in this paper, there is only one mediator among S(Γ) that satisfies the conditions in the above theorem. Theorem 1. For any two-person Bayesian bargaining problem Γ that satisfies (A1) (A3) with independent and symmetric priors, the neutral bargaining solution is unique. Proposition 3. The neutral bargaining solution in S(Γ) is If p(s) p : NS(Γ) = {µ 1,0 }. If p(s) (p, p ): NS(Γ) = {µ 1, z( p(s)) }. If p(s) p : NS(Γ) = {µ 0,0 }. Theorem 1 asserts that, for the class of environments in my framework, it identifies the smallest solution set, and surprisingly, gives a unique prediction. Proposition 3 implies that when the probability of the strong type is sufficiently small, the unique ex ante incentive efficient mediator who puts the lowest probability on war would not be contained in the solution set refined by requiring the axioms of the neutral bargaining solution. In fact, the notion of a neutral bargaining solution picks out a unique mediator in S(Γ) who is associated with the highest probability of war and is the farthest away from ex ante incentive efficiency. The reason behind this is precisely the intuition that the weak types will want to mimic the strong types, since expressing preferences for the ex ante incentive efficient mediator will immediately convince the other player to go to war. Since higher weight will be placed on the strong types, any negotiating process should lead to the neutral bargaining solution and it is the one we should reasonably expect to be chosen in any negotiating process. I elaborate more on the intuitions and implications behind the distortion within the set of neutral bargaining solutions in the next subsection. 3.3 Failure of Ex Ante Efficiency The resulting features of the neutral bargaining solution characterized in Proposition 3 are that it is the worst mediator, who has the highest probability of failure to make a peaceful settlement, and 18

19 Figure 3.1: Structure of Solution Set Refinements in the Cooperative Approach with whom players most of the time get involved in war. Why is this mediator the unique neutral bargaining solution, and how do these properties relate to the distortion in the neutral bargaining solution? Ultimately, I argue that the cooperative notion of the neutral bargaining solution leads to inefficiency of bargaining. I assumed that the players in the bargaining process with private information bargain over a mediator in an inscrutable way, without any communication during the mediator selection stage. If a player did anything different, the other player might learn something about him, and so, both types do not want to reveal their true types and nothing is learned in the negotiating process. The inscrutability principle basically states that whatever theory we have about bargaining, we can focus on, without loss of generality, an inscrutable theory of bargaining when we talk about the choice of a mediator. However, the inscrutability principle does not imply that the possibility of information leakage during a mediator selection process is irrelevant. 18 There may be some interim incentive efficient mediators that we should expect not to be selected by the players in such a process, precisely because some player would choose to reveal or conceal information about his type rather than let some mediators be selected. In this sense, the notion of the neutral bargaining solution provides the answer to which mediator, or communication system, is chosen inscrutably. So then, in the class of bargaining games that I described, what is it that a player does not want the other to learn about him? Among all the interim incentive efficient mediators, players would have preferred the ex ante incentive efficient solution before they learned their own types from the ex ante point of view. But since players already know their own types, what each player 18 See Myerson (1991, 505). 19

20 does not want the other to learn via his choice of mediator is whether he is a weak type, because a player expressing preference for the ex ante incentive efficient mediator, who is known to be better at implementing a peaceful settlement, may convey information that he is in a weaker position. Whether a player is strong or weak, he would not want the other to infer that he is weak, even if the probability of the strong type is small. Thus even when both players are weak, they will tilt toward what they would do when they are strong, choosing the mediator who would be better for the strong type, to be avoid being revealed as weak. Therefore, in that process, we expect the players to behave in such a way as if the strong type is more important. That is, one notable distortion in the neutral bargaining solution is that it is as if both types pool on pretending to be strong in order to conceal their types. In particular, both players have to act like they know their types and are strong, and they pick the mediator accordingly. Weak types, in order to maintain inscrutability, have to imitate the strong type at least in bargaining. Therefore, the players in a bargaining process inscrutably agree on a neutral bargaining solution: players are going to set up a bargaining process in a way that would be better for when they are strong because they already know their types, and along the way will never reveal their types. The feature of this distortion in the neutral bargaining solution concept has a signaling component. In a cooperative sense, players pick a mediator such that no information is revealed, but this is effectively asking the players to pool in a way such that no information is revealed. That nature of pooling is in the spirit of a pooling equilibrium of a signaling outcome in a noncooperative game. Although formally I do not have a signaling game where one player is choosing a strategy that signals his intention or strength, it is as if we built the signaling distortion in a noncooperative game directly into the cooperative mathematics. That is, the flavor of signaling implicitly comes into the argument of Myerson (1984b) s neutral bargaining solution, in which all the distorted welfare weights together with the inscrutability principle would lead to the result that the strong types would get more weight in the final mechanism. This result implies that the inscrutable intertype compromise between the strong type and the weak type tends to get resolved in favor of the strong type. Even if the probability of strong type is fairly small, the strong type is going to end up being more influential on players behavior in the bargaining process. I can interpret the neutral bargaining solution as putting extra welfare weights on the strong types in virtual utility characterizations. 19 What is best for the strong type is typically very far from ex ante efficiency for the fundamental reason that the strong type wants to separate itself from the weak type. As a result, in a model where ex ante efficient outcomes are technically feasible, there exists an equilibrium of any mediator selection game in which players systematically do not choose the 19 Otherwise, the mediator would not have been an equilibrium in any communication game defined. This can be seen as an analog to renegotiation-proof. Putting more weight on the strong type s utility allows the equilibrium to be interim incentive efficient and inscrutable. 20

21 mediator who would maximize ex ante efficient gains because they want to signal in a cooperative sense that they are the strong type. Hence, they systematically do worse than the ex ante efficiency because they already know their types at the time the mediator is chosen. This explains why interim bargaining theory is not going to look like ex ante bargaining theory, as well as how one might get exactly the wrong answer if we do ex ante bargaining. Therefore, I argue that the very process of selecting a mediator itself exhibits an inherent inefficiency of bargaining in the following senses. The mediator who is good at achieving peace gets defeated, and the players end up choosing a mediator who over-implements the war payments, implicitly putting more weight on the payoffs for the strong type. This results in too much war, and players expected gains from trade are less than the optimal (the second best). This result evokes Myerson and Satterthwaite (1983), in which the status quo disagreement payments are likewise over-implemented. It also gives us a new way to think about distortions that go on in political conflicts in the context of international relations, as if we are taking the countries who are most difficult to negotiate with and making them more likely to arise in equilibrium. 4 Noncooperative Approach The cooperative approach, while illuminating which mediator we should reasonably expect to arise endogenously, by itself is not satisfactory without also considering a noncooperative game. So, let us now back out from the cooperative approach, and take a more explicit noncooperative approach. Consider a ratification game, where some uninformed third party can announce an alternative mechanism and players can switch from some status quo mechanism if both players ratify (or, unanimously vote in favor of) the alternative. Importantly, I must allow for the fact that if one of them votes in favor and one of them votes against, then they are going to learn from disagreement. Considering such a pairwise ratification game, I ask: Given some status quo mechanism, can we find another mechanism that can beat it in some sequential equilibrium in a ratification game, where beliefs following disagreement are required to satisfy credibility conditions? 20 I introduce a notion of threat-security, based on the extended concept of ratifiability in Cramton and Palfrey (1995). The characterization of threat-security is adapted and revised from their formulation. Before going into formal construction, I briefly lay out a comparison between Holmström and Myerson (1983), Cramton and Palfrey (1995), and my setting. 4.1 Durability, Ratifiability, and Assumptions Another possibility is to think about durability in Holmström and Myerson (1983) as durability also formalizes the idea of a mechanism being invulnerable to proposals of alternative mechanisms 20 While a cooperative approach is to think about a reasonable solution set compared to the status quo of unmediated Bayesian game, this approach looks at whether one mediator survives in comparison to another. Nonetheless, we can also think about a mediator versus an unmediated game using this approach. 21

22 in a pairwise comparison. Holmström and Myerson (1983) show that the set of durable interim incentive efficient mechanisms is non-empty for any finite Bayesian bargaining problem and that with one-sided incomplete information, all interim incentive efficient mechanisms are durable. I argue that the notion of durability, however, would not give a strong prediction and is limiting in my framework. For the games that I study in this paper, not only is the neutral bargaining solution durable, but in fact the entire interim incentive efficient set is durable. It would be a weak concept in the sense that it has no bite for the fundamental reasons I state below. The distinction can be made due to different assumptions about passive beliefs, voting, and the mechanisms. First, durability relies on the assumption that passive beliefs hold whenever an alternative is rejected, allowing only for learning from agreement to the alternative. 21 So when the players go back to play the status quo mechanism, they play as they would have in the first place. What I focus on though in the games I consider is the fact that information does get leaked when there is disagreement in the negotiations over selecting a mediator. This distinction reflects somewhat different implicit assumptions in that durability tries to capture the idea of one type of one player proposing an alternative at the interim stage (although it is not literally talking about the player s proposal-making stage) while I consider the proposed alternative as selected by an uninformed third party, and so no information is revealed by a proposal itself. Therefore, I assume that if an alternative is ratified it is played with the original beliefs, considering only learning from disagreement. Second, players are not told the details of the vote using a secret ballot in Holmström and Myerson (1983), while I assume a non-anonymous voting procedure. It is realistic to assume that the vote is public since I am particularly concerned with international relations settings, where one type of country may have an incentive to publicly announce displeasure with the alternative when the countries are negotiating over possible change to another mediator, because of the information the announcement conveys. Third, Holmström and Myerson (1983) assume a fixed direct revelation status quo mechanism and arbitrary mechanisms as proposals. Therefore, if players commit themselves to a status quo mechanism, then they cannot alter the outcome selected by the status quo mechanism afterward. However, my status quo is not necessarily a direct mechanism and I only consider direct mechanisms as alternatives. These reasons are what make durability not quite the right noncooperative idea for the class of games I am considering and so, I do not pursue it here, but rather proceed with the notion of ratifiability in Cramton and Palfrey (1995), which I expound on in the next section. It is a mirror image of durability, but their setting is much more compelling to relate to my game since it also 21 A mechanism is durable if there exists a sequential equilibrium such that if an alternative is offered, at least one player rejects the alternative for every equilibrium path. If a player strictly prefers the alternative, he votes for it even if he thinks the other is not, and if a player is indifferent (the trembling point), he is allowed to vote against the alternative. 22

23 allows public voting and takes into account that the beliefs are changed going back to the status quo after unsuccessful ratification of an alternative. Other assumptions that I make that conform with Cramton and Palfrey (1995) are that the players voluntarily decide to participate in the alternative mediation at the interim stage and if participation is unanimous, then the players are bound by the alternative and cannot renegotiate during implementation. However, they also consider the status quo mechanism to be fixed as in Holmström and Myerson (1983), although not necessarily a direct mechanism, as a reduced form game of Bayesian Cournot competition. That is, whatever beliefs from disagreement would be injected into the exogenously imposed status quo, they are ultimately committed to whatever the fixed status quo is with possibly altered beliefs only about the vetoer. My innovations over Holmström and Myerson (1983) and Cramton and Palfrey (1995) are, first, that the status quo is designed, and second, what is central in my approach is that by assuming public voting with two players, both players would learn about each other after disagreement. Therefore, beliefs following disagreement would be required to satisfy credibility conditions, where the conditions are appropriately modified since a direct application of Cramton and Palfrey (1995) s ratifiability concept is not suitable in my setting as I will show. 4.2 The Threat-Secure Mediator The essential idea is that an interim incentive efficient mediator δ should be considered threat-secure if and only if there does not exist an alternative mediator that can be always unanimously ratified in favor of the alternative. I want to establish whether there is at least one type in the unanimous ratification game shutting down the alternative offer, so the renegotiation does not happen. In order to develop this idea, I first formulate the concept of a ratifiable alternative mediator. Different from the standard mechanism design approach, 22 I focus on each player s decision to switch to an alternative mediator γ, where a failure to ratify may reveal information. What is learned from the unilateral veto with public voting may alter both players status quo payoffs after non-ratification; and so, in deciding whether to veto or not veto the mediator γ, a player considers how the other s belief about him and his belief about the other might change as a result of the unilateral veto. 23 I link together a player s decision to veto with rational expectations about the outcome in the continuation game, taking into account the appropriate individual rationality constraints. Again, I interchangeably use the terms mediator and mechanism in my setting. An (uninformed) alternative mediator proposes to implement an incentive compatible directrevelation mechanism γ : D T R. Participation constraints are handled by requiring that every type of every player prefers to participate in γ than to play the status quo mechanism G δ with 22 In many mechanism design settings, learning from disagreement is not an issue, since the status quo mediator is not affected by a change in the players beliefs. (Cramton and Palfrey, 1995, 257) 23 This is possible based on correspondence with Peter Cramton. 23

24 finite message space and outcome function. I assume that a status quo G δ depends on strategic actions and is identified with a mediator that players are initially involved with who implements an incentive compatible direct-revelation mechanism δ : D T R. For each i {1, 2}, let A i = {0} R i, where 0 denotes going to war and each set R i represents a nonempty finite set of possible reports r i as an informational input into G δ. I write R = R 1 R 2 and r = (r 1, r 2 ). Then, G δ is a function of the form G δ : D A 1 A 2 R such that G δ (d 0 a 1, a 2 ) = 1 if a 1 = 0 or a 2 = 0 G δ (d a 1, a 2 ) = δ(d r) otherwise. The strategies in the status quo mechanism G δ consist of: Each player reports (possibly deceptively) or chooses to go to war, given its type and its belief about the other player s type. With prior beliefs, the equilibrium outcome in the status quo mechanism G δ is simply hiring mediator δ and both players reporting honestly their types. 24 So participation constraints are considered by analyzing a two-stage ratification game. I want to establish whether there exists an equilibrium of vetoing strategies such that γ is unanimously ratified over G δ along every equilibrium path. An individual s optimal vetoing strategy in this ratification game should depend on what he would believe about the other s type and what the other would believe about his type, conditioned on the (unexpected) failure to ratify the alternative; and how he would expect the status quo to be played as a result of the veto. Thus, I shall need the following notations. I maintain the independence and symmetry assumptions on the probability of types. In the first stage, the players each vote simultaneously for or against the alternative mediator γ. A strategy for i in the first stage specifies the probability that each player i vetoes γ if his type were t i. Let v i (t i ) denote the vetoing strategy of player i of type t i. An outcome to the first stage indicates the players M {1, 2} that vetoed γ. In the second stage, γ is implemented if M = ; Otherwise, the status quo mechanism G δ is played with the knowledge that players M vetoed γ. Because ratification requires a unanimous vote, I must specify the beliefs of all players if i unilaterally deviates from this ratification equilibrium by vetoing. For each i vetoing, for all j {1, 2} and for all t j T j, let q j,i (t j ) denote player j s belief about player j s type, conditional on the knowledge that M = {i}. In particular, suppose that if i alone vetoes γ and the players believe that the other s type is given by the distribution q j,i, the status quo mechanism G δ is then played with the updated beliefs about the other player s type to be q j,i (t j ). For example, if player 1 vetoed (i = 1), then q 1,1 (t 1 ) is the probability that player 2 (ratifier) would believe about player 24 If one of the players rejects γ, the inference drawn from the vote outcome alters both vetoer and ratifier s equilibrium plays in such a way that they play G δ (mediator δ appended to the war option) differently than as with prior beliefs, since with updated beliefs mediator δ may not be individually rational anymore and players may instead prefer to go to war. If either side chooses war, players get the war outcome. Therefore, individual rationality must embody the altered equilibrium play of G δ following unsuccessful ratification. 24

25 1 s type being t 1, and q 2,1 (t 2 ) is the probability that player 1 (vetoer) would believe about player 2 s type being t 2. Let q,i = ( q 2,i (t 2 ), q 1,i (t 1 )) be a vector of updated beliefs conditional on the event M = {i}, where M = {i} denotes the vote outcome that i vetoes the alternative while the other does not. Now, for any r j in R j, let σ j,i (r j t j ) denote the probability that j would use the report r j and ψ j,i (t j ) denote the probability that j would go to war, if t j were his type and G δ is played following a veto by i. For any j and any t j, a strategy (σ j,i (r j t j ), ψ j,i (t j )) in G δ represents a plan for player j if t j were his type and γ did not win as a result of i s veto. From these definitions the quantities (v, σ, ψ, q) must be nonnegative and must satisfy t j q j,i (t j ) = 1, r j R j σ j,i (r j t j ) = 1, ψ j,i (t j ) 1, v j (t j ) 1, i, j, t j T j. (T1) Given any types vector t = (t 1, t 2 ) in T and given any possible reports vector r = (r 1, r 2 ) in R, let σ i (r t) = 2 j=1 σ j,i(r j t j ). Thus, σ i (r t) is the probability that the individuals would give reports (r 1, r 2 ), if their types were (t 1, t 2 ) and if G δ is played by i s veto. I must impose some restrictions on the strategies σ j,i and ψ j,i, and the posteriors q j,i when G δ is played as a result of i s veto. For any i, let Σ( q,i ) denote a Nash equilibrium in the subgame when γ does not win by i s veto with respect to the revised beliefs q,i = ( q 2,i (t 2 ), q 1,i (t 1 )). Then, the strategy profile (σ,i, ψ,i ) = ((σ 1,i (r 1 t 1 ), σ 2,i (r 2 t 2 )), (ψ 1,i (t 1 ), ψ 2,i (t 2 ))) form an equilibrium Σ( q,i ) of G δ with respect to the posterior beliefs ( q 2,i (t 2 ), q 1,i (t 1 )) if and only if: U j (G δ t j, Σ( q,i )) = t i q j,i (t j ){ψ j,i (t j )u j (d 0, t) + (1 ψ j,i (t j )) (ψ j,i (t j )u j (d 0, t) + (1 ψ j,i (t j )) r R σ i (r t) d D δ(d r)u j (d, t)} U j (G δ, ˆr j, ψ j,i t j, Σ( q,i )) = t i q j,i (t j ){ψ j,i(t j )u j (d 0, t) + (1 ψ j,i(t j )) (ψ j,i (t j )u j (d 0, t) + (1 ψ j,i (t j )) r R σ i (r t) d D δ(d r j, ˆr j )u j (d, t)}, j, t j T j, ˆr j R j, ψ j,i [0, 1]. (T2) Condition (T 2) asserts that player j with type t j should not expect any other report ˆr j or any other war strategy ψ j,i to be better for him in Gδ than the strategies selected by his σ j,i and ψ j,i when i vetoes and G δ is played. Let q = { q,1, q,2 } be a collection of vote beliefs vectors following any veto by a single player when the other player votes for ratification. Let Σ( q) = {Σ( q,1 ), Σ( q,2 )} denote a corresponding collection of equilibria in the resulting play of G δ with those vote beliefs. 25

26 Unanimous ratification of γ followed by truthful revelation in γ is a sequential equilibrium in the two stage ratification game if and only if γ is incentive compatible and γ is individually rational relative to G δ, i.e., there exists q and Σ( q) such that for each i and each t i : p(t i ) γ(d t)u i (d, t) t i d D t i and p(t i ) γ(d t)u i (d, t) t i d D t i p(t i )γ(d t i, ˆt i )u i (d, t); p(t i ){ψ i,i(t i )u i (d 0, t) + (1 ψ i,i(t i )) (ψ i,i (t i )u i (d 0, t) (T3) + (1 ψ i,i (t i )) r R σ i (r t) d D δ(d r i, ˆr i )u i (d, t))}, ˆt i T i, ψ i,i [0, 1], ˆr i R i. That is, (T 3) asserts that player i cannot gain by vetoing the alternative γ (and then going to war with probability ψ i,i and reporting ˆr i if G δ is played) when t i is his true type and the other player is expected to use her equilibrium ψ i,i and σ i,i strategies (in G δ characterized by an equilibrium Σ( q,i ) given q,i ), together with honest reporting in γ. The left hand side is t i s interim payoff with the mediator γ which corresponds to the equilibrium path for the equilibrium in which all types of all players vote for ratification. I now impose restrictions on beliefs in the ratification game, requiring an equilibrium to be supported by all credible updating rules for rationalizing observations off the equilibrium path, based on Cramton and Palfrey (1995). If i vetoes, even if that has zero probability, i will try to rationalize the veto by identifying a veto belief that is consistent with i s incentive to veto. In addition, the vetoer i would have some posterior subjective probability distribution over T i upon observing he was the only one to veto and learning the other did not veto. Define the veto set V i T i as those types that veto with positive probability: V i = {t i T i v i (t i ) > 0} =, and the non-veto set W i = T i \V i as those types that never vetoes. Credibility Conditions. A probability distribution q i,i (t i ) is a credible veto belief about the vetoer i and a probability distribution q i,i (t i ) on T i is a credible non-veto belief about the ratifier i if there exists a continuation equilibrium Σ( q,i ) (= (σ,i, ψ,i )) in G δ with respect to the beliefs q,i = ( q i,i (t i ), q i,i (t i )) and veto probabilities v i ( ) such that q,i, Σ( q,i ), and v i ( ) together satisfy the conditions (T 4) through (T 9). v i (t i ) > 0 for some t i T i. (T4) 26

27 If i (vetoer) of type t i would benefit from a veto, then he must veto γ: if t i q i,i (t i ) d D γ(d t)u i (d, t) < t i q i,i (t i ){ψ i,i (t i )u i (d 0, t) + (1 ψ i,i (t i )) (ψ i,i (t i )u i (d 0, t) + (1 ψ i,i (t i )) r R σ i (r t) d D δ(d r)u i (d, t))}, (T5) then v i (t i ) = 1. Those types t i that lose from a veto must not veto γ: if t i q i,i (t i ) d D γ(d t)u i (d, t) > t i q i,i (t i ){ψ i,i (t i )u i (d 0, t) + (1 ψ i,i (t i )) (ψ i,i (t i )u i (d 0, t) + (1 ψ i,i (t i )) r R σ i (r t) d D δ(d r)u i (d, t))}, (T6) then v i (t i ) = 0. Also, i (ratifier) of type t i who benefit from a vote on γ when i vetoes must not veto γ: if t i q i,i (t i ){ψ i,i (t i )u i (d 0, t) + (1 ψ i,i (t i )) (ψ i,i (t i )u i (d 0, t) + (1 ψ i,i (t i )) r R > t i q i,i (t i ) d D δ(d t)u i (d, t) σ i (r t) d D δ(d r)u i (d, t))} (T7) then v i (t i ) = 0. Lastly, q,i = ( q i,i (t i ), q i,i (t i )) satisfies Bayes rule, given p and v: p(t i )v i (t i ) t q i,i (t i ) = i V p(t i )v i (t i ) for t i V i i 0 for t i / V i. p(t i )(1 v i (t i )) t q i,i (t i ) = i W p(t i )(1 v i (t i )) for t i W i i 0 for t i / W i. (T8) (T9) The set of types V i that veto with positive probability in q i,i (t i ) is called a credible veto set. The set of types W i that do not veto with positive probability in q i,i (t i ) is called a credible non-veto set. 27

28 (T 8) and (T 9) require that the ratifier i and the vetoer i will update their beliefs about the other, using Bayes theorem to compute their posteriors q i,i (t i ) and q i,i (t i ), respectively. Let q,i = ( q i,i (t i ), q i,i (t i )) be called a vector of credible vote beliefs if (T 4) through (T 9) are all satisfied. The credibility conditions have a flavor similar to the motivation behind Grossman and Perry (1986) and Farrell (1985). Roughly speaking, a set of types V i breaks an equilibrium with an off-the-equilibrium-path message if all types in V i improve their payoff by vetoing as long as the other (ratifier) believes that all and only the types in V i would always deviate and veto. Here, i is specifically assumed to update her priors over V i in accordance with Bayes Rule. Note that there is no restriction on v i for types that are indifferent between vetoing and not, reflecting the idea that such types may randomly choose whether to deviate. The credibility conditions are basically asking whether, upon deviating, the vetoer can induce the other to reason that it must have been sent by a type within V i. That is, whether a set of types can credibly distinguish themselves by explicitly sending an off the path equilibrium message we are the types in V i that identifies a set of potential deviating types. This is justified when V i is precisely the set of vetoer s types that would benefit from deviating, whenever, in response to deviating, the other plays any best response to i vetoing with her beliefs restricted to V i. Thus, the vetoer types that do not belong to V i are worse off (compared to their equilibrium payoff) under such best response. An equilibrium fails the credibility conditions if the types in V i are precisely the ones who gain by vetoing, when the ratifier s beliefs are a Bayesian update of her prior beliefs on V i. The credibility condition (T 7) assumes away the case of i not vetoing, in which case i s veto would switch from γ to G δ. But as long as i s beliefs are restricted to V i such that the other is expected to veto, i must expect that her vote cannot make any difference since G δ will be implemented in any case. (T 7) requires that if i with type t i would get higher expected utility from equilibrium Σ(q,i ) of G δ than the expected utility from her also vetoing, then i must reveal her desire for γ by voting for γ. Note that the left hand side of (T 7) is the interim payoff for player i of type t i when i ratifies but i vetoes and the right hand side of (T 7) is the interim payoff to player i of type t i when i also vetoes and G δ is played under the original beliefs (hiring δ and reporting honestly). This assumes that the likely vetoer i will have the prior belief about i if i does veto, which makes sense as all t i W i are expected to not veto (as i of t i would benefit from revealing her non-veto). This also justifies that in (T 5) and (T 6) any possible change of behavior of i in G δ when i also vetoes but learns that i vetoed is assumed away. Remark 2. I do not entertain the possibility of multiple ways to rationalize a veto or a non-veto. In my setting, there is at most one credible vote belief for any i and so, at most one credible veto set satisfying (T 4) through (T 9), and thus there is no issue of multiple credible vote beliefs. 28

29 Definition 2. (v, σ, ψ, q) is an equilibrium ratification of γ when the status quo is G δ, if and only if γ satisfies the conditions (T 1) (T 3) and for all i either (i) there does not exist a credible veto belief, (ii) for every credible vote belief q,i such that the conditions (T 4) (T 9) are all satisfied, and q i,i (t i ) γ(d t)u i (d, t) = q i,i (t i ){ψ i,i (t i )u i (d 0, t) t i d D t i + (1 ψ i,i (t i )) (ψ i,i (t i )u i (d 0, t) + (1 ψ i,i (t i )) σ i (r t) δ(d r)u i (d, t))}, t i V i. r R d D Definition 3. An alternative mediator γ is ratifiable against the mediator δ if and only if there exists an equilibrium ratification of γ when the status quo is G δ. An alternative mediator γ is ratifiable against the mediator δ if and only if there exists a sequential equilibrium of the two-stage game in which γ is unanimously voted for at every profile of types, where beliefs following disagreement are required to satisfy credibility conditions. If after a veto, the players beliefs are restricted to credible vote beliefs, then the alternative mediator is ratifiable. An alternative mediator γ is not ratifiable against δ if, for every equilibrium of the two stage ratification game, there exists a profile of types in which γ is not unanimously approved over δ. That is, there is no equilibrium to the two-stage game in which γ is unanimously approved over δ along every equilibrium path. The equation in (ii) of Definition 2 says that if for every credible vote belief there is some type t i that strictly gains by vetoing, then γ is not ratifiable relative to δ. This suggests that rejection of γ is not so severe. Given the definition of a ratifiable alternative mediator, I can now define a secure status quo mechanism and a threat-secure mediator. Definition 4. A status quo mechanism G δ is secure if and only if every interim incentive efficient mediator that is ratifiable against δ is an equilibrium of G δ under the prior beliefs. A secure mechanism cannot be overturned by a ratifiable alternative. Moreover, if δ is ratifiable against itself, then such a status quo mechanism G δ would not be vulnerable to allowing players to send preplay cheap talk messages that may communicate information about their types. Definition 5. An interim incentive efficient mediator δ is threat-secure if and only if there does not exist an alternative mediator that is ratifiable against δ. 29

30 I say that a threat-secure mediator is immune to the ratification of any alternatives and endures unilateral war threat between the players, given reasonable updating in the event of a veto. To establish that a mechanism is secure, we need to check every other incentive feasible direct mechanism in S(Γ) and verify that there is a credible vote belief in each case. This tedious search is simplified by establishing the following Propositions 4 and 5, which is also used to prove an existence result. Proposition 4. If γ is ex ante Pareto superior to δ, then γ is not ratifiable against δ. Proposition 5. If γ is ex ante Pareto inferior to δ, then unanimous ratification of γ, when the status quo is G δ, is a sequential equilibrium of the ratification game where beliefs following disagreement are required to satisfy credibility conditions. Proposition 4 implies that an alternative mediator is not ratifiable against any ex ante Pareto inferior mediator and Proposition 5 implies that a alternative mediator is ratifiable against any ex ante Pareto superior mediator. Now, I can establish existence of interim incentive efficient and threat-secure mediator in the class of environments discussed in this paper, and the following Proposition 6 characterizes the set of threat-secure mediators on the interim Pareto frontier, denoted as T S(Γ). 25 Theorem 2. For any two-person Bayesian bargaining problem Γ that satisfies (A1) (A3) with independent and symmetric priors, there exists a unique interim incentive efficient and secure status quo mechanism. Proposition 6. The threat-secure mediator in S(Γ) is If p(s) p : T S(Γ) = {µ 1,0 }. If p(s) (p, p ): T S(Γ) = {µ 1, z( p(s)) }. If p(s) p : T S(Γ) = {µ 0,0 }. Proposition 6 states that a threat-secure mediator is uniquely defined to be in the set T S(Γ): If p(s) p, the threat-secure mediator chooses war with probability 1 when two players are of different types; If p(s) (p, p ), the threat-secure mediator chooses war with probability 1 when two players are of different types and with positive probability when both players are strong types; 25 We can also think about the a ratification game between no mediation versus mediation, where now status quo G δ would be simply an unmediated Bayesian game. This will be considered in the online Appendix. It is easy to see that all interim incentive efficient mediators are ratifiable against the underlying disagreement game, and so, no mediation is not secure. 30

31 If p(s) p, the peaceful mediator who always chooses war with probability zero is trivially the threat-secure mediator since there are no other alternatives. In this ratification game, Proposition 6 together with Proposition 4 and 5 imply that, if the players start with the ex ante incentive efficient mediator, then there is a sequential equilibrium in which the players immediately ratify an alternative mediator to replace the ex ante incentive efficient one; on the other hand, if the players start with the threat-secure mediator, then there is no sequential equilibrium in which the players unanimously agree to an alternative ex ante Pareto superior one. That is, once players insist upon the threat-secure mediator, they are stuck with it and can never renegotiate for a more ex ante efficient mediator. This result implies that the unique threat-secure mediator is the one that is not ex ante incentive efficient, and indeed is the most ex ante inefficient. The suboptimal choice of such a mediator is in fact the most war-inducing mediator associated with the highest probability of failing to make peaceful settlement among all the other interim incentive efficient mediators. In the following section, I discuss some common features of the equilibrium choice of a mediator in the ratification game and compare that to the neutral bargaining solution which has a signaling flavor as discussed in Section Discussions: Equivalence of the Two Approaches In this section, I look at the relationship between the ex ante incentive efficient mediator, the neutral bargaining solution, and the threat-secure mediator and discuss the implications of the solution set refined by the cooperative approach and the equilibrium choice in the noncooperative approach. The following is the main result of this paper. Remarkably, it turns out that the neutral bargaining solution coincides with the threat-secure mediator in the stylized game in my framework. Theorem 3. For any two-person Bayesian bargaining problem Γ that satisfies assumptions (A1) (A3) with independent and symmetric priors, T S(Γ) = NS(Γ) S(Γ). Moreover, the interim incentive efficient and threat-secure mediator is not ex ante incentive efficient. Theorem 3 implies that there is a unique mediator that is interim incentive efficient, neutral bargaining solution, (durable,) and threat-secure; and is distinct from the ex ante incentive efficient solution. In general, the neutral bargaining solution and the threat-secure set are not nested, but in the class of bargaining games considered in this paper, they are the same. Then, we can ask the deeper game-theoretic question of why the threat-secure set generates the same thing as the neutral bargaining solution even though they stem from such different approaches, where one is from mostly noncooperative and the latter is from completely cooperative. Proposition 4 implies that, in the pairwise ratification game between mediators, there is no equilibrium where an alternative ex ante Pareto superior mediator (who puts lower probability on 31

32 war) is chosen in the first stage, when the probability of the strong type is sufficiently small. By voting (either vetoing or not vetoing), a player may send a signal that he is of a certain type. In any case, the strong type benefits from revealing his type when the possibility of other mediators is considered, and so, the strong type s signal is credible and satisfies credibility requirements. However, the weak type s veto does not give a credible signal, since if he gives away his identity there is disadvantage to vetoing. In equilibrium, every player strategically pools on the strong action. Thus, I can argue that an ex ante Pareto better mediator gets defeated and players end up choosing an ex ante inefficient mediator who puts more probability on war than the ex ante efficient mediator. If we think about iteratively playing this pairwise ratification game between all possible interim incentive efficient mediators, an ex ante Pareto superior mediator is not sustainable against ratification of any other ex ante Pareto inferior mediator, and thus, the threat-secure mediator defined in Proposition 6 is the only one that survives ratification of any other alternative mediator. The strategic intuition to why the ex ante incentive efficient mediator cannot survive in the noncooperative game underscores the intuition to information leakage problems behind the cooperative game. Proposition 3 implies that any negotiating process should lead to the neutral bargaining solution. Both types do not want to reveal their types in the negotiating process and the inscrutable intertype compromise between the two types leads to implicitly putting more weight on the strong type. Thus, both types pretend to be strong in order to conceal their types, and so, the worst mediator would turn out to be the reasonable thing we should expect to see endogenously arising in a mediator selection game. In a cooperative sense, players would pick a mediator, effectively pooling in a way that no information is revealed. That nature of pooling is exactly reminiscent of signaling in a noncooperative game. In the ratification game, every player pretends to be strong in order to signal their types, and so the worse mediator is chosen in equilibrium. It is as if we built this signaling distortion in a noncooperative game directly into the cooperative mathematics. The combination of the inscrutability principle and the requirements in the neutral bargaining solution is generating this kind of built-in signaling in a cooperative game. Likewise, I can argue that the notion we are going to observe players mimic other players in the noncooperative game comes out directly from a cooperative approach. Therefore, both approaches take into account, either directly or indirectly, the information leakage problem. By either approach, I can conclude that the selection of a mediator by privately informed players is endogenous and depends upon the information that the players reveal by expressing their preferences for mediation. Consequently, with both approaches, I can define the smallest solution set for the given bargaining problem: When two privately informed players bargain over the choice of a mediator, the reasonable solution of a mediator in a cooperative sense is the neutral bargaining solution; and the equilibrium solution of a mediator in a noncooperative sense is the threat-secure mediator. Remarkably, I obtain the same unique solution which is farthest 32

33 away from ex ante incentive efficiency. Moreover, in the class of bargaining games considered in this paper, the intuition under a noncooperative game can be founded on the very intuition under a cooperative bargaining theory. 6 An Illustrative Example Let us consider an example to illustrate the core idea of interim incentive efficient set, the neutral bargaining solution, and the threat-secure set. More extensive illustration with detailed calculations is done in the online Appendix. Suppose that there are two players in the economy, and each player may be one of two possible types: s (strong) or w (weak). There are two possible decisions called d 0 (war) and d 1 (peace). By the revelation principle, players bargain over the selection of a mediator, and they have candidates for the job from the set of all incentive feasible mediators each associated with a mediation mechanism. Consider π i (t) and ω i (t) which satisfy the following: t (s, s) (s, w) (w, s) (w, w) (ω 1 (t), ω 2 (t)) -1, -1 5, -6-6, 5-2, -2 (π 1 (t), π 2 (t)) 3, 3 4, 1 1, 4 2, 2 The normalized utility payoffs (u 1, u 2 ) depending on the decisions and types are shown in the following table. The payoffs satisfy (A1) through (A3): Table 1: An Example t (s, s) (s, w) (w, s) (w, w) d = d 0 0, 0 0, 0 0, 0 0, 0 d = d 1 4, 4-1, 7 7, -1 4, The Interim Incentive Efficient Mediators With these payoffs, we have p = 0.2, p = 0.36, and p = When p(s) < 0.2 (Case 1 ), the mediator with µ y( p(s)),0, where y( p(s)) = 1 5 p(s) 1 p(s) is the unique ex ante incentive efficient mediator. Recall that y( p(s)) is such that the participation constraint binds for the strong type. When p(s) 0.2 (Case 2, 3, & 4 ), the mediator who promises to implement µ 0,0 such that µ 0,0 (d 1 t) = 1 t, uniquely maximizes the two players ex ante expected utilities among all incentive feasible mediators. As shown in Lemma 2 and Proposition 2, the unique ex ante incentive efficient mediator is the one associated with the lowest probability on war outcome for t {sw, ws} among all interim incentive efficient mediators. In fact, there are other interim (but not ex ante) incentive efficient mediators characterized by Proposition 1. When p(s) < 0.45 (Case 1, 2, & 3 ), the interim incentive efficient mediators µ y,z 33

34 put probability 0 on d 0 for t {ss, ww}, positive probability y on d 0 for t {sw, ws}, and positive probability z on d 0 for t {ss} with further restrictions on y and z depending on the different regions of p(s). t (s, s) (s, w) (w, s) (w, w) d = d 0 0, 0 [z] 0, 0 [y] 0, 0 [y] 0, 0 [0] d = d 1 4, 4 [1-z] -1, 7 [1-y] 7, -1 [1-y] 4, 4 [1] Figure 6.1 plots the probability y that an interim incentive efficient mediator can possibly put on d 0 when t = {sw, ws} for each case characterized in Proposition 1, except that the dotted line particularly plots z( p(s)). Figure 6.1: Interim Incentive Efficient Mediators S(Γ) = {µ y,z } When p(s) < 0.2 (Case 1 ), the dark orange line depicts y( p(s)) (decreasing and concave in p(s)) which is the lowest probability on war that an interim incentive efficient can have for t {sw, ws}. When p(s) (0.36, 0.45) (Case 3 ), µ y,z (d 0 ss) z is such that U 1 (µ y,z w) = U 1 (µ y,z, s w), where z can take values from [0, z( p(s))] depending on y and p(s). When µ y,z (d 0 sw) y = 1, then the corresponding probability on war for t {ss} would be z( p(s)) = 11 p(s) 4 7 p(s), which is increasing in p(s) depicted as an orange dotted line in Figure 6.1. Now, let us focus on the case where each player is known to believe that the probability of the strong type s is 0.15, i.e., p(s) = 0.15 and p(w) = In this case, S(Γ) = {µ y,z y [ 5 /17, 1], z = 0}. It can be shown that the set of interim incentive efficient utility allocations between the strong and 34

35 the weak type is a line in R 2 with end points (U 1 (µ y,z s), U 1 (µ y,z w)) as follows: (0, 4.14), (0.6, 3.4). 26 The first of these allocations is implemented by the unique ex ante incentive efficient mediator µ 5/17,0. The second of these allocations is implemented by µ 1,0. For an illustrative purpose, consider the utility allocation (0.26, 3.82) which can be implemented by µ 0.6,0. The type-contingent expected payoffs from hiring these mediators are depicted in Figure 6.2. The ex ante expected payoffs are also indicated for each of these three mediators shown with red dots that correspond to the probability that the mediator chooses d 0 for t = {sw, ws}. Figure 6.2: Conditional Expected Payoffs when p(s) = Because U 1 (µ 5/17,0 ) = 3.52 > U 1 (µ 0.6,0 ) = 3.29 > U 1 (µ 1,0 ) = 2.98, µ 5/17,0 is ex ante Pareto superior to µ 0.6,0 and µ 1,0. In fact, it is the unique ex ante interim incentive efficient mediator among all µ y,0 for y 5 /17. However, because U 1 (µ 5/17,0 s) < U 1 (µ 0.6,0 s) < U 1 (µ 1,0 s), U 1 (µ 5/17,0 w) > U 1 (µ 0.6,0 w) > U 1 (µ 1,0 w), µ 5/17,0 is worse than µ 0.6,0 which is worse than µ 1,0 for the strong type but better than µ 0.6,0 which is better than µ 1,0 for the weak type. So, no mediator is interim Pareto superior to the other. In fact, all mediators µ y,0 with y 5 /17 are interim incentive efficient and interim Pareto superior to µ y,0 for y < 5 /17 when p(s) = These confirm Proposition 1 (Case 1 ), Lemma 2, Proposition 26 Since players are symmetric, without loss of generality, I focus on player 1. 35