The Value of a Coordination Game∗ Preliminary and Incomplete

Willemien Kets† Alvaro Sandroni‡

February 19, 2018

Abstract

The value of a game is what a player can expect to receive a priori from playing the game. An important intuition is that the value of a game need not increase with (ex post) payoffs when the game has multiple Nash equilibrium. This is the case if an improvement in payoffs increases strategic uncertainty by reducing the payoff saliency of one of the equilibria. To model this, we build on insights from psychology on how players resolve strategic uncertainty. We assign a unique value to symmetric two-player, two-action coordination games and show that the value changes non-monotonically with payoffs.

∗We thank Larry Blume, Peter Hammond, Stephen Hansen, Wouter Kager, and David Schmeidler for helpful comments and stimulating discussions. †Department of Economics, University of Oxford. E-mail: [email protected]. ‡Kellogg School of Management, Northwestern University. E-mail: [email protected]

1 1 Introduction

Say that the value to an agent of an uncertain prospect is what the agent can expect a priori. That is, the value of a game to a player is the ex ante expected payoff from playing the game. The question of how the value depends on economic primitives is of obvious economic importance. For example, when designing institutions, it is important to know what payoffs players can expect to receive when they interact under different institutional constraints. Likewise, when advising a player whether or not to participate in a game (possibly at a cost), it is important to know the expected payoff to the player when he enters. In settings where economic primitives uniquely pin down behavior, defining the value of a game is simple. To give an example, in a standard principal-agent problem where the agent’s wage is made contingent on the profit to incentivize effort, it is straightforward to determine what the contract is worth to both parties. But it is less clear how to define the value for games with multiple equilibria. For two-player zero-sum games, von Neumann and Morgenstern(1944) have shown that the value is uniquely determined even if the game has multiple equilibria, as all equilibria yield the same payoff. However, their classic result does not extend to general games. This leaves open the question how to evaluate the value of a game with multiple equilibria beyond two-player zero-sum games. In this paper, we study the value of (simultaneous-move, complete-information) coordination games. Coordination games are simple in that they abstract away from incentive problems, yet they are nontrivial in that payoffs do not uniquely determine behavior. And while coordination games are admittedly stylized, they form the basis for understanding the general problem of how agents can coordinate their actions. This is important for, e.g., understanding how bargainers decide on a split of a pie or how long-term cooperation can be sustained by mutually understood punishment strategies. Despite the simple nature of coordination games, understanding how their value depends on economic primitives is a vexing problem. In his seminal work, Thomas Schelling(1960, pp. 97–98) already observed that “It is noteworthy that traditional game theory does not assign a “value” to [coordination games]: how well people [coordinate] is something that, though hopefully amenable to systematic analysis, cannot be discovered by reasoning a priori. This corner of game theory is inherently dependent on empirical evidence.” Since the publication of Schelling’s work, a large experimental literature on coordination games has produced valuable insights on which a theory of the value of coordination games can be based. To illustrate, consider the following game:

2 IN I 10, 10 2, 6 N 6, 2 6, 6

Each player has two actions, Invest (denoted I) and Not Invest (N). The game has two strict Nash equilibria: one in which both players invest, and one in which both players do not invest. At first sight, it might seem obvious that both players will invest – after all, if both players invest, then they receive 10; if neither invests, they receive 6 –, so that the value of the game is 10 to both players. However, in the one shot game, only 60% of the players invests (Schmidt, Shupp, Walker, and Ostrom, 2003).1 A theory of the value of coordination games must therefore account for the possibility of coordination failure: players may fail to coordinate on the Nash equilibrium that would give them the highest payoff. A second striking feature is that players may not just fail to coordinate on the “best” Nash equilibrium; they may fail to coordinate at all. Indeed, while a significant proportion of subjects do not invest, some do, and many end up choosing a different action from their opponent’s. A theory of the value of coordination games must therefore also account for the possibility of miscoordination, that is, non-Nash behavior associated with the failure to coordinate on a Nash equilibrium. A third striking feature is that increasing the (ex-post) payoffs in a game may not make players better off. Consider the following game:

IN I 10, 10 2, 8 N 8, 2 8, 8

In this game, the ex-post payoffs are (weakly) higher than in the original game: the payoffs to investing are the same as before while the payoffs to not investing have increased. Yet, players are not necessary better off in this game: the proportion of subjects who invest falls to 40% (Schmidt, Shupp, Walker, and Ostrom, 2003). Intuitively, the increase in the payoffs makes it less attractive to invest. This increases strategic uncertainty by reducing the payoff saliency of the Pareto-dominant Nash equilibrium (i.e. (I, I)). An increase in the payoffs to not investing thus has an indirect strategic effect on the value by reducing the probability that

1Also see Van Huyck, Battalio, and Beil(1990), Cooper, DeJong, Forsythe, and Ross(1990, 1992), and Straub(1995). In general, there is signiﬁcant coordination failure both when the game is played only once and when it is played repeatedly.

3 players coordinate on the Pareto-dominant Nash equilibrium. There is also a direct payoff effect on the value: an increase in the ex-post payoffs to not investing makes players better off if they do not invest. When the payoffs to not investing are small, then the indirect strategic effect dominates because players choose to invest with high probability. As the payoffs to not investing increase, the direct payoff effect becomes more important as the probability that players do not invest increases. Hence, any theory of the value of coordination game must allow for the value to be non-monotonic in ex-post payoffs. The value of the game thus critically depends on the strategic uncertainty that players face. A standard approach in game theory is to assume away strategic uncertainty by selecting a Nash equilibrium. But, as we argue below, Nash equilibrium or other standard approaches cannot account for the empirical regularities described above. We therefore depart from standard game theory in that we explicitly model the process by which players reason about others to resolve any strategic uncertainty. As observed by Schelling(1960), when a player is uncertain about another player’s action, “[the] objective is to make contact with the other player through some imaginative process of introspection” (p. 96). By modeling the process by which players reach such a “meeting of the minds,” we obtain a unique prediction for any coordination game, despite the fact that these games have multiple Nash equilibria and that every action is rationalizable (Kets and Sandroni, 2015). Importantly, the outcome that is selected by the introspective process depends crucially on the interplay between strategic uncertainty and payoffs. This allows us to capture the empirical regularities described above. When the payoffs to investing are sufficiently high (relative to the payoffs to not investing), investing is uniquely salient. In that case, strategic uncertainty is negligible and the introspective process selects the pure Nash equilibrium in which both players invest. But, as the relative payoffs to not investing increase, players begin to face significant strategic uncertainty and the rate of miscoordination increases. This indirect strategic effect leads to a fall in the value. As the payoffs to not investing increase further, players begin to coordinate on the pure Nash equilibrium in which no player invests. This reduces the rate of miscoordination. But while the direct payoff effect of an increase in ex-post payoffs is obviously positive (and can be significant), coordination failure makes that the value of the game is lower than in the case where investing is uniquely payoff salient. While these predictions are intuitive, they are difficult to obtain using existing methods as we argue in Section 5.2. In short, existing models, including Nash equilibrium, equilibrium refinements, and behavioral models, either do not capture miscoordination or coordination failure, or fail to assign a unique value to the game. As a result, existing methods cannot account for the empirical regularities described above. Coordination problems are a central feature of economic environments with nontrivial

4 strategic uncertainty and have been studied in a range of applications, including economies with technological complementarities (Bryant, 1983), technology adoption (Katz and Shapiro, 1986), search and matching (Diamond, 1982), currency crises (Obstfeld, 1996), and bank runs (Diamond and Dybvig, 1983). A central theme in the literature is that even when there is an equilibrium that all players prefer, they may fail to coordinate on it. This is especially the case when the payoff difference between the Pareto-dominant Nash equilibrium and other equilibria is limited. To give an example, cartels may become unstable when leniency programs are introduced, with at least some members deviating from the cooperative Nash equilibrium of the infinitely repeated game (Motta, 2004, Ch. 4).2 This is an instance of how an increase in ex-post payoffs (to defection) leads to a decrease in the value. While miscoordination has received less theoretical attention, it is commonplace in a range of settings. For example, in the absence of explicit communication, it can be challenging to settle on a Nash equilibrium, especially if there is no clear focal point. This can explain why firms choose explicit collusion over tacit collusion even though this potentially exposes them to significant fines or worse (Knittel and Stango, 2004).3 Perverse effects of policy changes can often be traced back to strategic effects that lead to nonmonotonicities in the value. To give an example, if workers and firms have to exert costly effort to search for a match, an increase in payoffs that increases the opportunity costs of searching may reduce search effort and lead a society to coordinate on a low-search equilibrium. While intuitive, this is difficult to capture using existing methods. Search models with multiple equilibria cannot explain why a high-search equilibrium is likely to be selected when the opportunity cost of search is low while the low-search equilibrium is selected when the opportunity cost is high. Equilibrium refinements, including those based on payoff per- turbations such as global games (Carlsson and van Damme, 1993), can capture how the risk of coordination failure depends on economic primitives. However, as we argue more exten- sively in Section 5.2, these methods cannot fully account for nonmonotonicities in the value. In particular, they cannot fully account for the effect of policies that target miscoordination payoffs. The remainder of this paper is organized as follows. Section2 introduces the introspective process and derives some basic properties. Section3 defines coordination games and the general strategic environment. Section4 characterizes strategic behavior when players are introspective. Section5 presents the main result on the comparative statics of the value (Section 5.1) and argues that existing approaches are unable to capture the key insights

2An infinitely repeated Prisoner’s Dilemma game can be viewed as a risky coordination game; see Dal Bó and Fréchette(2011). 3Also see Farrell and Klemperer(2007, Sec. 3.4) for a discussion of coordination failure and miscoordination in the context of network externalities.

5 (Section 5.2). Section6 discusses the related literature and Section7 concludes.

2 Introspection

Consider a ﬁnite (simultaneous-move) game Γ, that is, a game with a ﬁnite set N of players in

which each player j ∈ N has a finite set Sj of actions. There is no payoff uncertainty: payoffs are commonly known. Even when the payoffs are fixed and known, players may face considerable strategic uncertainty, that is, uncertainty about the actions of other players. As noted above, players can resolve strategic uncertainty by putting themselves into others’ shoes. Perspective-taking in- volves introspection: to form a belief about another person’s impulse, people first observe their own impulse.4 This is a rapid and instinctive process referred to as first-person simulation (Goldman, 2006). The introspective stage is followed by a slower, more deliberative process whereby individuals reason about others’ impulses using a naive understanding of psychology, which may lead them to adjust their initial belief (Gopnik and Wellman, 1994). Thus, individuals use the observation of their own mental state to form an initial belief and then use “folk psychology” to adjust this belief. This introspective process is formalized as follows. Each player has an impulse – i.e., a pre-reflective inclination – to take a certain action. Impulses do not affect payoffs, and a player’s impulse is not observable by other players. Players can use their own impulse to form a belief about the impulses of other players. Players’ beliefs are modeled in the standard way, using a type space. Each player j has a set Tj of types, taken to be a closed subset of the

real line. Each type tj ∈ Tj is associated with an impulse Ij(tj) ∈ Sj. Players know their own impulse (i.e., the impulse functions are measurable). Players’ beliefs are described by a Q common prior on the set j Tj of type proﬁles. We denote the type space by T and refer to a pair (Γ, T ) consisting of a game Γ and a type space T as a model. 0 A player’s instinctive reaction is to follow his impulse. The level-0 strategy σj for player

j ∈ N thus maps each type tj ∈ Tj into its impulse Ij(tj) ∈ Sj. Through introspection, players realize that other players also have an impulse. By updating their prior beliefs, players can form a belief about other players’ instinctive responses (i.e., their level-0 strategies). Of course, a player’s instinctive reaction need not be optimal in the light of others’ instinctive response. If a player’s level-0 strategy is not optimal given the belief that other players follow 1 their impulse, then the player can adjust his response. The level-1 strategy σj for a player

4These ideas also have a long history. Locke (1690/1975) suggests that people have a faculty of “Perception of the Operation of our own Mind,” and called introspection the “sixth sense.” Mill (1872/1974) writes that understanding others’ mental states ﬁrst requires understanding “my own case.”

6 j ∈ N is thus a best response to the belief that the other players follow their impulse: for 1 0 5 tj ∈ Tj, σj (t) is a best response against σ−j. The reasoning will generally not stop here. As a “folk game theorist,” each player realizes that other players go through a similar reasoning process and will likewise formulate a level-1 strategy. Again, if a player’s level-1 strategy is not optimal against the level-1 strategy of the other players, then he adjust his response. The 2 1 level-2 strategy σj for a player j ∈ N is thus a best response against the level-1 strategy σ−j. In k general, for k > 1, the level-k strategy σj for each player j is a best response to the level-(k−1) k−1 strategy σ−j of the other players. These levels do not represent actual decisions: players go through the entire reasoning process in their mind before taking a decision. Accordingly, k player j’s behavior is given by the limit σj := limk→∞ σj of the introspective process. If these limiting strategies exist, then σ = (σj)j∈N is an introspective equilibrium of the model (Γ, T ) (Kets and Sandroni, 2015). To summarize, to resolve the strategic uncertainty that they face, players try to put themselves into others’ shoes. Players introspect: they observe their own impulse and project it onto others. This is captured by the level-0 actions and beliefs. This fast, instinctive process is followed by a slower, deliberate reasoning process where players reason about others’ instinctive responses. As players reason about others, they may adjust their response. As others may likewise adjust their response, players’ contemplated course of action may not remain optimal. If that is the case, players engage in further reasoning. Thus, unlike a “folk psychologist” who faces a single-person decision situation, a “folk game-theorist” reasons to higher orders, which are represented here by the different levels. A first observation is that while introspective equilibrium models the behavior of “folk game-theorists” who do not engage in a full-fledged equilibrium analysis, its predictions are consistent with equilibrium.

Proposition 2.1. [Common Knowledge of Rationality]6 Fix a ﬁnite game and a common- prior type space. Every introspective equilibrium is a correlated equilibrium.

As correlated equilibrium is characterized by common knowledge of rationality (Aumann, 1987), Proposition 2.1 implies that, in introspective equilibrium, every player acts as if he is rational, believes that the other players are rational and that they believe that others are rational,. . . , ad inﬁnitum. This result holds very generally: it holds for any ﬁnite game and arbitrary type spaces. The intuition behind Proposition 2.1 is straightforward. As any introspective equilibrium is the limit of a best-response process, every player plays a best

5If there are multiple best responses, an action is chosen using a fixed tie-breaking rule. The choice of tie-breaking rule does not affect our results. 6Kets and Sandroni(2015) prove a similar result for binary action games with a continuum of players and finite type spaces.

7 response against other players’ strategies. So, the strategy proﬁle forms an equilibrium. Since beliefs can be correlated, the strategies of players can be correlated even if players do not follow their impulse in equilibrium. Hence, an introspective equilibrium need not be a Nash equilibrium. This will be critical for our results.

3 Coordination

We henceforth focus on symmetric two-player, two-action coordination games. The set of players is N = {1, 2}, and each player j ∈ N can choose whether to invest or not (so,

Sj = {I, N}). The payoﬀs are given by the following payoﬀ matrix:

I uII,uII uIN,uNI

N uNI,uIN uNN,uNN

where uII > uNI, and uNN > uIN. The game has two strict Nash equilibria: one in which both players invest, and one in which both players do not invest. We assume that uII ≥ uNN, so that the Nash equilibrium in which both players invest is payoﬀ dominant. It will be convenient to deﬁne

u − u ρ := NN IN . uII + uNN − uNI − uIN

The parameter ρ captures the (relative) risk associated with investing: investing is a best response for a type if and only if it assigns probability at least ρ to the other player investing. If ρ is close to 0, then investing is a best response even for types that assign a low probability to the other player investing; if ρ is close to 1, then investing is a best response only for types that assign a high probability to the other player investing. We refer to ρ as the risk parameter. The risk parameter is closely related to risk dominance: investing is risk dominant 1 1 if ρ < 2 , and not investing is risk dominant for ρ > 2 . We make the following assumptions on players’ beliefs:

(SYM) Players are ex ante identical. That is, players have the same type set (i.e., T1 = T2 =

T := [0, 1]) and impulse function (i.e., I1 = I2) and the cumulative distribution function F induced by the common prior is symmetric (i.e., F (x, y) = F (y, x)).

(MON-I) Impulses are monotone in type. That is, there is a threshold τ 0 ∈ (0, 1) such that 0 for each player j, a type t has an impulse to invest (i.e., Ij(t) = I) if and only if t ≥ τ .

8 (MON-B) Beliefs are monotone in type in the sense that higher types think it is likely that other players have a high type. That is, the conditional probability F (τ | t) that the other player has type at most τ (given the player’s type t) is decreasing in t.

(DENS) The common prior F has a continuous density f with full support on T × T .

The central assumptions are Assumptions (MON-I) and (MON-B), which ensure that the game is a monotone supermodular game (Van Zandt and Vives, 2007). Assumption (MON- B) implies that the conditional distributions over the types over the other players can be ordered by first-order stochastic dominance. This assumption is satisfied, for example, if types are affiliated. Assumptions (SYM) and (DENS) are for analytical convenience and can be relaxed. We refer to τ 0 as the level-0 threshold. A simple example that satisfies the above assumptions is a standard symmetric binary signals model. Suppose there is an underlying state θ ∈ {I, N}. The probability that θ = I is p ∈ (0, 1). Conditional on θ = s, player j ∈ N has an impulse to take action s with probability 1 qj ≥ 2 , independently across players. The idea is that if the underlying state is θ = s, then, 7 on average, most people’s impulse is to choose s. The normalized parameterq ˜j := 2qj − 1 is drawn from a distribution G on [0, 1] independently across players. The distribution G is

assumed to have a continuous density. If we map each pair (Ij, qj) into its posterior belief that θ = I, then we obtain a type space that satisﬁes the above assumptions. We use this example (with G the beta distributionwith parameter α = 2, β = 1) throughout the paper to illustrate our results. In the remainder of the paper, we will assume (SYM), (MON-I), (MON-B), and (DENS) without explicitly stating them.

4 Introspective Equilibrium

Before analyzing the value of a game, we ﬁrst study players’ strategic behavior. The next result shows that for coordination games, the introspective equilibrium is (essentially) unique and has a monotone structure.

Proposition 4.1. [Existence, uniqueness, monotononicity]8 Fix a coordination game and a type space. An introspective equilibrium exists and it is essentially unique. Moreover,

7It could be, for instance, that action s is more likely to be salient than the other action in certain contexts (i.e., when θ = s). For experimental evidence that (primary) salience of actions influences players’ pre-reflective inclinations, see Mehta, Starmer, and Sugden(1994) and Bardsley, Mehta, Starmer, and Sugden(2009) 8Kets and Sandroni(2015) show the existence of introspective equilibrium in coordination games in which each player has finitely many types. In that case, introspective equilibrium is unique for generic games.

9 it is in monotone strategies: there is a threshold τ ∈ T such that, in introspective equilibrium, any type t < τ does not invest and any type t > τ invests.

By “essentially unique,” we mean that introspective equilibrium uniquely determines behavior for all but a measure-0 set of types (viz., the threshold type t = τ). Notice that uniqueness is with respect to the payoff structure and the type space: different type spaces may yield different introspective equilibria even if the payoff structure is kept fixed. To obtain testable predictions, we therefore focus on comparative statics below. The proof of Proposition 4.1 is straightforward. Recall that F (˜τ | t˜) is the conditional probability that the other player has type at mostτ ˜ given that a player has type t˜. It will be convenient to define Fe(˜τ | t˜) := 1 − F (˜τ | t˜) to be the conditional probability that the other player has type greater thanτ ˜ given that the player has type t˜. At level 1, then, investing is a strict best response for the level-0 threshold type τ 0 if and only if

0 0 0 0 0 0 0 0 Fe(τ | τ ) · uII + (1 − Fe(τ | τ )) · uIN > Fe(τ | τ ) · uNI + (1 − Fe(τ | τ )) · uNN,

or, equivalently, Fe(τ 0 | τ 0) > ρ.

If this holds (i.e., investing is a strict best response for τ 0), then the level-1 threshold is the largest type τ 1 ≤ τ 0 that satisﬁes Fe(τ 0 | τ 1) = ρ

if such a type exists, and τ 1 = 0 otherwise (i.e., all types invest). By induction, for k > 1, if τ k−2 ≤ τ k−1, then the level-k threshold is the largest type τ k ≤ τ k−1 that satisﬁes

Fe(τ k−1 | τ k) = ρ

k k if such a type exists, and τ = 0 otherwise. The sequence {τ }k=0,1,..., being a monotone sequence in a compact space, converges to a threshold τ ∈ T . The equilibrium threshold τ is either identical 0 (all types invest) or it is the largest type less than τ 0 that satisﬁes Fe(τ | τ) = ρ, or, equivalently, F (τ | τ) = 1 − ρ. (1)

In this case, some types invest even if they lack an incentive to do so. By a similar argument, if not investing is a best response at level 1 for the level-0 threshold τ 0, then the level-k thresholds τ k converge to τ ∈ T such that τ ≥ τ 0. The equilibrium threshold is either identical 0 (no type invests) or is the smallest type greater than τ 0 that satisﬁes (1). In this case, some types do not invest even though they have an incentive to do so.

10 t 0

Figure 1: The best-response curve.

Figure1 illustrates the result for our ongoing example. It plots the best response curve that gives for any threshold t the threshold b(t) that is a best response if the other player uses a monotone strategy with threshold t. Starting from the level-0 threshold τ 0, the sequence τ 1 = b(τ 0), τ 2 = b(b(τ 0)),... that converges to a threshold τ such that τ = b(τ). Since investing is a strict best response for the level-0 threshold type (i.e., b(τ 0) < τ 0), in introspective equilibrium, some types invest even though they have an impulse to not invest (i.e., τ < τ 0). To derive comparative statics on the value, we must understand how equilibrium behavior varies with ex-post payoﬀs. The next result shows that the probability that players invest increases when the risk parameter falls.

Proposition 4.2. [Investment Decreases with Risk] Fix a type space. The probability that a player invests in introspective equilibrium decreases with the risk parameter. That is, the equilibrium threshold τ increases with ρ.

These comparative statics are valid for any type space. Hence, Proposition 4.2 has the testable and intuitive implication that the probability that players invest increases when the risk parameter decreases. Moreover, the proof shows that only the risk parameter matters for equilibrium behavior: for a given type space T , two games Γ, Γe that have the same risk parameter ρ have the same introspective equilibrium (i.e., τ). The result is illustrated in Figure2 for our ongoing example.

11 τ

0 ρ ρ¯ ρ Figure 2: The equilibrium threshold τ as a function of the risk parameter ρ with tipping points ρ andρ ¯.

In deriving these comparative statics, we have kept players’ impulses and beliefs fixed. A natural conjecture is that when the payoff to investing increases, players are more likely to have an impulse to invest. This can easily be accommodated in our model. In Appendix A, we show that if players are more likely to have an impulse to invest when the payoffs to investing increase (so that the risk parameter falls) , then we obtain the same comparative static prediction. In fact, a decrease in the risk parameter leads to even more investment in introspective equilibrium. This intuitive result illustrates the robustness of our approach. Figure2 shows that, for our ongoing example, there are “tipping points” ρ, ρ¯ for the risk parameter ρ: For intermediate values of ρ, both actions are chosen with strictly positive probability (i.e., τ is bounded away from 0 and 1). However, when ρ falls below the tipping point ρ, introspective equilibrium selects the pure Nash equilibrium in which both players invest. Similarly, when ρ increases beyond the tipping pointρ ¯, introspective equilibrium selects the pure Nash equilibrium in which both players invest (i.e., τ = 0). Hence, for extreme values of the risk parameter (i.e., ρ close to 0 or close to 1), introspective equilibrium selects a pure Nash equilibrium. The next result shows that this holds generally:

Proposition 4.3. [Tipping Points and Equilibrium Selection] For extreme values of the risk parameter, introspective equilibrium selects a unique Nash equilibrium. That is, there exist tipping points ρ, ρ¯, with 0 < ρ ≤ ρ¯ < 1, for the risk parameter such that all types invest if ρ < ρ (i.e., τ = 0) and no type invests if ρ > ρ¯ (i.e., τ = 1). Moreover, there is a discontinuous change in behavior at the tipping points ρ, ρ¯.

Proposition 4.3 shows that introspective equilibrium selects a pure Nash equilibrium for extreme values of the risk parameter. The next result shows that if introspective equilibrium does not select a pure Nash equilibrium, then behavior is not consistent with Nash equilibrium except in knife edge cases:

12 Proposition 4.4. [Non-Nash Behavior] Fix a type space. There is at most one value ρ of the risk parameter such that the introspective equilibrium for ρ is a (strictly) mixed Nash equilibrium.

Proposition 4.4 shows that if the introspective process does not select a pure Nash equilibrium, then introspective equilibrium is generically inconsistent with Nash equilibrium, in the strong sense that there is at most one value of the risk parameter for which the introspective equilibrium coincides with mixed Nash equilibrium. In particular, if an introspective equilibrium is a mixed Nash equilibrium, then the introspective equilibrium of a game with slightly perturbed payoﬀ parameters is not a Nash equilibrium. That means that while for intermediate values of the risk parameter, there is scope for miscoordination (i.e., τ 6= 0, 1), equilibrium behavior is (generically) correlated so that the problem of miscoordination is less severe than in the mixed Nash equilibrium where players mix independently. Together, Propositions 4.3 and 4.4 show that introspective equilibrium either selects a pure Nash equilibrium (for extreme values of the risk parameter) or yields behavior that is (generically) inconsistent with Nash equilibrium. In the former case there is scope for coordination failure, with players being stuck at a Pareto-dominated Nash equilibrium. In the latter case, there is miscoordination. We next explore the conditions under which we obtain non-Nash behavior. Say that an introspective equilibrium is interior if the equilibrium threshold lies strictly between 0 and 1 (i.e. τ 6= 0, 1). We provide a suﬃcient condition under which the introspective equilibrium is interior for intermediate values of the risk parameter. Say that the rank belief of a type t is the conditional probability F (t | t) that the other player has a lower type.9 The rank belief function F (· | ·) maps each type t into its rank belief F (t | t). We have the following condition:

(NMRB) The type space has nonmonotone rank beliefs. That is, there is an open interval (t, t¯) ⊂ T such that the rank belief function F (· | ·) is decreasing on (t, t¯) and F (τ 0 | τ 0) ∈ (F (t¯| t¯),F (t | t)).

A type space satisﬁes condition (NMRB) if the rank belief function is decreasing for types that are bounded away from 0 and 1. We refer to this property as nonmonotone rank beliefs since the rank belief function is always increasing for t close to 0 and 1 (see the proof of Proposition 4.3). The rank belief function for our ongoing example has nonmonotone rank

9Rank beliefs play a prominent role in the global games literature. Morris, Shin, and Yildiz(2016) show that coordination games have a unique rationalizable outcome if beliefs about payoﬀs are perturbed in such a way that there is approximate common belief in approximately uniform rank beliefs, where a rank belief 1 F (t | t) is uniform if F (t | t) = 2 . Type spaces that satisfy this assumption generally do not have nonmonotone rank beliefs.

13 1 F (t|t)

0 t ¯ 0 0τ t t τ¯ 1 Figure 3: Nonmonotone rank beliefs. beliefs, as illustrated in Figure3. 10 Notice that F (τ 0 | τ 0) ∈ (F (t¯ | t¯),F (t | t)) for any τ 0 ∈ (τ 0, τ¯0) in Figure3. When the type space has nonmonotone rank beliefs, then the introspective equilibrium is interior for intermediate values of the rank parameter:

Proposition 4.5. [Interior Equilibrium] If the type space has nonmonotone rank beliefs, then there is an open interval (ρ, ρ¯), 0 < ρ < ρ¯ < 1, such that for ρ ∈ (ρ, ρ¯), the introspective equilibrium is interior (i.e., τ 6= 0, 1).

This is illustrated in Figure2 for our ongoing example: For intermediate values of the risk parameter (i.e., ρ ∈ (ρ, ρ¯)), the equilibrium threshold lies strictly between 0 and 1. By Proposition 4.4, behavior is not consistent with Nash equilibrium for these values of the risk parameter (except perhaps for a knife-edge case).

5 The value of a game

5.1 Main result

The value of a model is the ex ante expected payoﬀ for a player in introspective equilibrium.

That is, the value of a model (Γ, T ) (where Γ := (uII, uIN, uNI, uNN) and T = (T,F, I)) is Z

V (σ;Γ, T ) := uσ1(t1),σ2(t2)dF (t1, t2), T ×T

10Another example of nonmonotone rank beliefs is the case where there is a common shock with a fat-tailed distribution, as in Morris and Yildiz(2016).

14 where σ = (σ1, σ2) is the introspective equilibrium uσ1(t1),σ2(t2) ∈ {uII, uIN, uNI, uNN} is the (ex-

post) payoff that the player receives in introspective equilibrium when he has type t1 and the other player has type t2. By Proposition 4.1, coordination games have a unique introspective equilibrium so that the expected equilibrium payoff for each player is well-defined. Moreover, the introspective equilibrium is symmetric, so that the expected payoff is the same for both players. Hence, the value of a coordination game is well-defined. The value of a coordination game is now easy to characterize: By Proposition 4.1, the value of a coordination game is

pII · uII + pIN · uIN + pNI · uNI + pNN · uNN,

where τ = τ(ρ) is the equilibrium threshold and pss0 = pss0 (ρ) is the prior probability that players 1 and 2 choose actions s and s0, respectively. Our main result, Theorem 5.1, shows how the value of a coordination game changes with ex-post payoﬀs.

Theorem 5.1. [The Value of a Coordination Game] The value of a coordination game is a nonmonotonic function of ex post payoﬀs. That is, for any model (Γ, T ), there exist tipping points ρ, ρ¯ with 0 < ρ ≤ ρ¯ < 1 such that

(a) for ρ < ρ, the value is equal to the expected payoﬀ in the pure Nash equilibrium in which

both players invest (i.e., uII);

(b) for ρ > ρ¯, the value is equal to the expected payoﬀ in the pure Nash equilibrium in which

no player invests (i.e., uNN);

    ∂V ∂ρ  p  ∂uII ∂uII II  ∂V   ∂ρ    ∂τ    pIN  ∂uIN = f(τ) · (u − u ) · · ∂uIN +   , (2)  ∂V  IN NI  ∂ρ      ∂ρ   pNI  ∂uNI   ∂uNI    ∂V ∂ρ pNN ∂uNN ∂uNN

where τ = τ(ρ) is the equilibrium threshold and pss0 = pss0 (ρ) is the prior probability that players 1 and 2 choose actions s and s0, respectively.

Moreover, the value changes discontinuously at the tipping points ρ, ρ¯, and for any ρ > ρ, the value is strictly lower than the expected payoﬀ in the pure Nash equilibrium in which both players invest.

15 uII

0 uNN uNN

Figure 4: The value of a stag hunt game.

Figure4 illustrates the result for our ongoing example when uNN increases from uIN to uII, for uII = 5, uNI = 0, and uIN = −1. When the risk parameter is close to 1, the value of the game is equal to the expected payoff in the Pareto-dominant Nash equilibrium in which both players invest (i.e., uII). When ρ reaches the tipping point ρ, the value falls discontinuously when the probability that players coordinate on the Pareto-dominant Nash equilibrium falls below 1. The value then changes non-monotonically with uNN as an increase in the ex-post payoff uNN increases the scope for miscoordination but also gives higher payoffs conditional on players coordinating successfully. To see the intuition behind Theorem 5.1, consider the effect of an increase the (ex-post) payoff uNN that a player receives if both players do not invest. There are three effects, illustrated in Figure5. The first is a direct effect on payoffs: conditional on both players choosing to not invest, the player receives a higher payoff. This is indicated by the gray area in Figure 5. In addition, there are indirect strategic effects on the value, that is, effects that are not driven by changes in the payoffs per se, but rather by changes in equilibrium strategies driven by changes in payoffs. There are two types of indirect effects in this case, indicated by the blue and red arrows in Figure5. Indirect effects arise because players change their strategy when payoffs change (Proposition 4.2). The first indirect effect, indicated by the red arrows, is that the probability of miscoordination increases: As the probability that players invest decreases, some probability mass moves from the diagonal (i.e., (I, I)) to off-diagonal outcomes (i.e., (N, I) and (I, N)). The second indirect effect, indicated by the blue arrows, is that the probability of coordination failure increases: some probability mass moves from the outcome

16 IN

I uII uIN

N uNI uNN

(a)

Figure 5: The direct and indirect effects of an increase in uNN. where both players invest to outcomes where at most one player invests, or from outcomes where one player invests to the outcome where neither player invests. The total effect of a change in (ex-post) payoffs on the value can thus be decomposed into a direct payoff effect and indirect strategic effects. The direct payoff effect is easy to characterize. When the introspective process selects a unique Nash equilibrium (i.e., ρ < ρ or ρ > ρ¯), the direct payoff effect of a change in ex-post payoffs on the value is simply the change in the ex-post value in the pure Nash equilibrium selected by the introspective process. When introspective equilibrium is not consistent with Nash equilibrium (i.e., ρ ∈ (ρ, ρ¯)), the direct payoff effect of a change in ex post payoff associated with an outcome is proportional to the probability of that outcome. This is captured by the second term in (2). We next discuss the indirect effects. For ρ < ρ and ρ > ρ¯, the introspective process selects a pure Nash equilibrium, and a small change in ex post payoffs does not affect equilibrium behavior. Hence, there are no indirect effects. But when behavior is not consistent with Nash equilibrium (i.e., ρ ∈ (ρ, ρ¯) and τ 6= 0, 1), a small change in ex post payoffs changes equilibrium behavior (Figure2). Hence, the value is affected by indirect strategic effects. To characterize these indirect strategic effects, consider Figure6. The figure shows the payoffs to player 1 for different realizations of both players’ types. The the left vertical line and bottom horizontal line represent the equilibrium threshold τ for players 1 and 2, respectively. For example, in the upper right quadrant, both players’ type exceeds the threshold. Hence,

both players invest, and player 1’s payoﬀ is uII.

When the ex-post payoff uNN increases, the equilibrium threshold increases for both players. So, the line representing the equilibrium threshold for player 1 shifts to the right, and the line representing the threshold for player 2 shifts up. Since, in equilibrium, the marginal type of a player is indifferent between investing and not investing, a change in a player’s own threshold does not affect the value. Hence, we only need to consider the effect on the value of a change in the threshold for the other player. This indirect strategic effect can be decomposed into

17 two distinct eﬀects. First, a small increase in the threshold for the other player leads to more miscoordination: when types switch from investing to not investing, the probability that players both invest decreases. This is indicated by the red area in Figure6. The change

in value due to this strategic eﬀect is the probability pmc that players have types in the red

area times the payoﬀ loss, uIN − uII, associated with the increase in miscoordination. The second indirect strategic eﬀect represents the enhanced scope for coordination failure: there is a greater probability that players both decline to invest. This is indicated by the blue area

in Figure6. The change in value due to this strategic eﬀect is the probability pcf that players

have types in the blue area times the payoﬀ change, uNN − uIN, associated with the greater scope for coordination failure. The key insight is that, in introspective equilibrium, the ratio of

pcf relative to pmc +pcf is precisely the rank belief F (τ | τ). And, in introspective equilibrium, this is precisely the inverse 1 − ρ of the risk parameter. Rearranging the terms yields the ﬁrst term of (2).

uNI uII τ

t2 uNN uIN

t1 τ

Figure 6: The indirect strategic eﬀects when behavior is not consistent with Nash equilibrium.

5.2 Comparison with existing approaches

In this section, we compare our main results to predictions derived using existing approaches. A natural starting point is to consider Nash equilibrium, arguably the most commonly-used solution concept. One obvious difficulty with using Nash equilibrium for defining the game is that coordination games have multiple Nash equilibria, and, unlike in two-person zero- sum games considered by von Neumann and Morgenstern(1944), different equilibria generally have different payoffs. Hence, Nash equilibrium fails to assign a unique value to coordination games.11

11If one is interested solely in strategic behavior (and not the value), then equilibrium multiplicity need not be problematic. Binary-action coordination games are supermodular games and hence the extremal Nash equilibria change monotonically with ex-post payoﬀs (Milgrom and Roberts, 1990; Vives, 1990). However, comparative statics for equilibrium do not readily translate into comparative statics for expected payoﬀs, except in special cases (Milgrom and Roberts, 1990, Thm. 7).

18 A potential solution is to consider equilibrium refinements, such as risk dominance and payoff dominance (Harsanyi and Selten, 1988). Payoff dominance predicts that players coordinate on the Pareto-dominant Nash equilibrium (i.e., (I, I)). This may be the natural outcome in some settings, e.g., if players can communicate before play. However, in other settings, it is less clear that payoff dominance provides reasonable predictions. In particular, payoff dominance implies that the value of the game does not depend on the payoffs in the Pareto-dominated Nash equilibrium or the miscoordination payoffs. To illustrate, consider the following variant of the games in Section1:

IN I 10, 10 −100, 9 N 9, −100 9, 9

Payoff dominance predicts that the value of this game is 10, no matter how close the payoff in the Pareto-dominated Nash equilibrium (viz., 9) is to the payoff in the Pareto-dominant Nash equilibrium (viz., 9) and how risky the Pareto-dominant equilibrium is compared to the other equilibrium (viz., a potential loss of -100-10=-110 vs. 9-9=0). Such considerations have led economists to consider risk dominance. Risk dominance is motivated by the idea that strategic uncertainty and payoffs jointly determine behavior. It is often used to model coordination failure in applications. But, like payoff dominance, risk dominance completely rules out miscoordination: players always coordinate on one of the pure Nash equilibria (albeit not on the Pareto-dominant one if this equilibrium fails to be risk dominant). Risk dominance is therefore unable to model persistent miscoordination. As a result, risk dominance misses an important tradeoff between the strategic effect and the direct payoff effect. To illustrate, consider the following three (symmetric) games (where only the row player’s payoff are displayed for simplicity):

IN IN IN I 4 −1 I 4 −1 I 4 −1 N 0 2 N 0.9 2 N 1.1 2

Γ1 Γ2 Γ3

The games Γ1, Γ2, Γ3 diﬀer only in the miscoordination payoﬀ uNI. In Γ1, the Nash equilibrium in which both players invest is risk dominant, so, according to risk dominance, the value of

the game is uII = 4. As the miscoordination payoﬀ uNI is increased from 0 in Γ1 to 0.9 in

Γ2, the Nash equilibrium in which both players invest remains risk dominant, so the value

19 remains unchanged at uII = 4. When the miscoordination payoff is increased further to 1.1 in Γ3, the Nash equilibrium in which no player invests becomes risk dominant, and the value drops discontinuously to uNN = 2. Hence, an increase in the miscoordination payoff uNI can only have a negative impact on the value: it only has a strategic effect, which is negative, but it does not have a (positive) payoff effect.12 By contrast, under introspective equilibrium, changes in the miscoordination payoff uNI have both a negative strategic effect and a positive direct effect. The tradeoff between these two effects makes that the value is nonmonotonic in our setting. Miscoordination can of course be modeled using mixed Nash equilibrium; however, mixed Nash equilibrium makes the unintuitive prediction that the probability that players invest decreases when the payoffs to investing increase.13 As a result the value of a game increases monotonically with ex-post payoffs. Mixed Nash equilibrium is thus unable to capture the common intuition that miscoordination and coordination failure can reduce the value of a game.14 Hence, Nash equilibrium and its common refinements are unsuitable to capture the basic intuitions on how miscoordination and coordination failure affect the value of a game. Unfor- tunately, other existing concepts do not fare any better: either their predictions coincide with Nash equilibrium or its refinements, or they fail to yield a well-defined value. Global games (Carlsson and van Damme, 1993), level-k models (Crawford, Costa-Gomes, and Iriberri, 2013), stochastic best-response models (Young, 1993; Kandori, Mailath, and Rob, 1993), and quantal response equilibrium (McKelvey and Palfrey, 1995) all select the risk-dominant Nash equilibrium in the games considered here. Team reasoning (Sugden, 1993) and stochastic learning models with random matching (Robson and Vega-Redondo, 1995) select the payoff dominant Nash equilibrium. And deterministic learning and evolutionary dynamics fail to deliver a well-defined value as the predicted behavior depends on the initial state. By explicitly modeling the introspective process by which players reduce strategic uncertainty, we obtain predictions that do not reduce to Nash equilibrium or to a Nash refinement. Moreover, the introspective process always selects a unique outcome so that the value is well- defined. We obtain unambiguous comparative statics that account for key empirical regularities. When one of the actions is payoff salient (e.g., the payoffs to investing are much higher than the payoffs to not investing), then the introspective process selects the payoff-salient pure

12 A change in the coordination payoff uNN can have both a negative strategic effect and a positive payoff effect under risk dominance, and this can lead to nonmonotonicities in the value. See Morris and Shin(2003, Sec. 3.2) for a similar observation in the context of global games. 13This is because, in mixed Nash equilibrium, players mix to keep other players indifferent. 14In addition, mixed Nash equilibrium is unable to account for the experimental evidence on coordination games (e.g., Van Huyck, Battalio, and Beil, 1990; Cooper, DeJong, Forsythe, and Ross, 1990, 1992; Straub, 1995; Schmidt, Shupp, Walker, and Ostrom, 2003).

20 Nash equilibrium. But, when there is limited asymmetry between actions in terms of payoﬀs, behavior is not consistent with Nash equilibrium (pure or mixed). Instead, the introspective process selects a correlated equilibrium. This captures the idea that, while players may fail to coordinate on a pure Nash equilibrium when no Nash equilibrium clearly stands out, mixed Nash equilibrium may be too pessimistic in that it predicts the maximal degree of miscoordination. Importantly, the correlated equilibrium selected by the introspective process yields clear and intuitive comparative statics both on strategic behavior and on the value. Thus, the model helps account for empirical regularities that are diﬃcult to account for in standard models, and certainly in one model.

6 Related literature

Introspective equilibrium was introduced by Kets and Sandroni(2015) to model the costs and benefits of diversity. When players interact with people from a different cultural background, they may face more strategic uncertainty, and this influences behavior even when diversity is not directly payoff relevant. Akerlof and Holden(2017) and Akerlof, Holden, and Rayo(2017) use introspective equilibrium to study the rents associated with capital assembly and pricing and competition in markets with network externalities, respectively. Besides the difference in focus, these papers also do not consider how the value of a game depends on ex-post payoffs. The introspective process bears some resemblance to level-k models.; see Nagel(1995), Stahl and Wilson(1995), Costa-Gomes, Crawford, and Broseta(2001), and Costa-Gomes and Crawford(2006) for influential early papers, and see Crawford, Costa-Gomes, and Iriberri (2013) for a recent survey. Introspective equilibrium distinguishes itself from level-k models in two important ways. First, we introduce correlation in impulses to model payoff-irrelevant contextual factors. By contrast, the level-k literature generally assumes that level-0 players choose their actions independently and uniformly at random. Second, we focus on the limiting outcome of the reasoning process. By contrast, the level-k literature focuses on deviations from Nash equilibrium. As a result, level-k models yield different predictions (Section 5.2). Our paper also contributes to the extensive literature on coordination games, in particular on how payoff-irrelevant situational factors influence behavior (Crawford and Haller, 1990; Sugden, 1995; Bacharach and Stahl, 2000; Janssen, 2001). Rather than explicitly modeling the determinants of salience, we take a “detail-free” approach and focus on deriving testable comparative statics on strategic behavior and payoffs. One of the distinct predictions of our model is that, depending on a subtle interplay between economic primitives and strategic uncertainty, the introspective process either selects a pure Nash equilibrium or predicts behavior that is not consistent with Nash equilibrium.

21 Instead, behavior is consistent with correlated equilibrium. A small literature in experimental economics has studied correlation in coordination games. Mehta, Starmer, and Sugden(1994) show how payoff-irrelevant contextual factors can help players coordinate even when the payoff structure of the game provides little guidance. The resulting behavior is well described by correlated equilibrium: players coordinate at higher rates than in the mixed Nash equilibrium, but not perfectly as in pure Nash equilibrium. Duffy and Fisher(2005) and Fehr, Heinemann, and Llorente-Saguer(2017) show that uncertainty that is not directly payoff-relevant influences behavior in experimental markets with multiple equilibria and in coordination games, respectively. Costa-Gomes, Crawford, and Iriberri(2009) show that the experimental data in the classic coordination games of Van Huyck, Battalio, and Beil(1990, 1991) is best described by allowing for correlation in beliefs. Cason, Sharma, and Vadovic(2017) elicit subjects’ belief and show that beliefs are correlated. The present results may prove useful in understanding and unifying these experimental findings, and potentially inspiring and guiding future experimental work.

7 Conclusions

A contribution of our paper is that it provides a new way to deal with the problem of equilibrium multiplicity. Game theorists have traditionally followed one of two approaches in dealing with equilibrium multiplicity. The first approach says that the model is essentially incomplete. In this view, behavior is not determined by economic primitives alone: which equilibrium is selected depends, at least in part, on psychological or sociocultural factors.15 This is captured by payoff-irrelevant “sunspots” (Cass and Shell, 1983). The second approach uses refinements of Nash equilibrium to obtain sharp predictions. The idea is that the refinement completes the model in that it imposes restrictions not captured by the payoffs. Prominent examples of this approach include the tracing procedure of Harsanyi and Selten(1988) and global games (Carlsson and van Damme, 1993). Neither of these approaches is entirely satisfactory. The first approach is mostly silent on the question which, if any, equilibrium is selected in a particular situation. The second approach makes precise predictions that can account for coordination failure. However, it cannot capture the common intuition that strategic uncertainty not only leads to coordination failure but also to miscoordination. 15This view goes back all the way to the work of von Neumann and Morgenstern(1944), who wrote “[W]e shall in most cases observe a multiplicity of solutions. Considering what we have said about interpreting solutions as stable ‘standards of behavior’ this has a simple and not unreasonable meaning, namely that given the same physical background different ‘established orders of society’ or ‘accepted standards of behavior’ can be built” (4.6.3). It plays a prominent role in the work of Schelling(1960), who emphasized that behavior is not pinned down by economic primitives alone when there is strategic uncertainty.

22 Rather than choosing one approach over the other, our framework combines the strengths of both, thus avoiding their weaknesses. On the one hand, we take seriously the view that behavior is not determined by economic primitives alone and may depend on payoff-irrelevant situational factors. But that does not lead to indeterminate predictions in our model: We bring payoff-irrelevant factors into the model by assuming that the introspective process is anchored by payoff-irrelevant “impulses” that are presumably shaped by the context. But, unlike in the sunspot literature, we obtain unambiguous predictions on how the interplay between economic primitives and strategic uncertainty influences strategic behavior and payoffs. On the other hand, while the introspective process completes the model by putting restrictions on equilibrium behavior, the model allows for both miscoordination and coordination failure. What allows us to make progress on the difficult question of equilibrium multiplicity is that we do not make any a priori assumptions on how payoff-irrelevant contextual factors shape strategic behavior. Instead, we use a “detail-free” approach: We posit a distribution over payoff-irrelevant impulses and then derive testable comparative statics on the value and equilibrium behavior that do not depend on the specifics of the distribution. By reconciling existing approaches for dealing with equilibrium multiplicity, we are able to take a first step in answering the question how players’ expected payoffs vary with economic primitives in environments with nontrivial strategic uncertainty. A better understanding of this question is important for central questions in economics ranging from institutional design to the evaluation of policies that, through their strategic effects, may have nontrivial and potentially perverse effects on equilibrium behavior and payoffs.

23 Appendix A Impulse distribution

Appendix B Proofs

B.1 Proof of Proposition 2.1

Fix a finite game Γ = hN, {Sj}j∈N , {uj}j∈N i, where N is the (finite) player set and for each player j ∈ N, Sj is the (finite) set of actions and uj : Sj → R is the payoff function. Fix a type space, that is, a set Tj of types and an impulse function Ij for each player j ∈ N as Q well as a common prior on the set j Tj of type profiles. We impose no conditions on the type space beyond the conditions in Section ??: for each player j ∈ N, the type set Tj is a closed subset of the real line and that the impulse function Ij is measurable with respect to the σ-algebra B(Tj) on Tj induced by the Borel σ-algebra on R. For each player j ∈ N, let

Σj be the set of (pure) strategies, i.e., measurable functions σj : Tj → Sj. For simplicity, we write σ−j(t−j) for (σi(ti))i6=j. It will also be convenient to represent the common prior by its cumulative distribution function F . The ﬁrst step is to show that the level-k strategies are, in fact, strategies:

k Lemma B.1. ] Let j ∈ N. Then, for every k, σj is measurable.

Proof. For k = 0, the result follows from the assumption that the impulse functions are measurable. We prove the result for k > 0 by showing the following claim: for every player j ∈ N, tie-breaking rule ψj, and proﬁle σ−j ∈ Σ−j for the other player, the tie-breaking rule yields a strategy σj ∈ Σj such that for every tj ∈ Tj, σj(tj) is a best response to σ−j. Given that the level-0 strategies are measurable for all players, it then follows that for each player 1 k j ∈ N, σj is measurable. Iterating this argument gives that σj is measurable for all j ∈ N and k = 0, 1,....

It remains to prove the claim. Fix a player j ∈ N and a strategy profile σ−j ∈ Σ−j. Then, for sj ∈ Sj, the function mapping type tj ∈ Tj into its interim expected payoff Z Vj(sj, σ−j; tj) := uj(sj, σ−j(t−j))dF (t−j | tj) T−j is measurable (e.g., Aliprantis and Border, 2006, Thm. 15.13). Let ϕj(·, σ−j): Tj Sj be the best-response correspondence (given σ−j), i.e., ϕj(tj, σ−j) is the set of actions that maxi- mize the interim expected payoff Vj(·, σ−j; tj) for tj. By the Measurable Maximum Theorem

(e.g., Aliprantis and Border, 2006, Thm. 18.19), ϕj(·, σ−j) is measurable. That is, for every

24 collection Cj of subsets of Sj,

{tj ∈ Tj : ϕj(tj, σ−j) ∈ Cj} ∈ B(Tj).

Since Sj is ﬁnite, it now follows immediately that for every subset Bj ⊂ Sj of actions,

{tj ∈ Tj : ϕj(tj, σ−j) = Bj} ∈ B(Tj).

Fix a tie-breaking rule, i.e., a function ψj that maps each nonempty subset Bj ⊂ Sj into an

element sj of Bj. Then, ψj ◦ ϕj(·, σ−j): Tj → Sj is measurable. This proves the claim. k k Hence, for every player j ∈ N, tie-breaking rule ψj, and k > 0, σj , deﬁned by σj (tj) = k−1 ψj(ϕj(tj, σ−j ) for tj ∈ Tj, is measurable.

k It is now immediate that for each player j, the limit limk→∞ σj of the level-k strategies is measurable. (This follows because the (pointwise) limit of a sequence of measurable functions

is measurable.) Hence, if σ = (σj)j∈N is an introspective equilibrium, then for each player

j ∈ N, σj is a strategy.

It remains to show that if σ = (σj)j∈N is an introspective equilibrium, then for each player j ∈ N, Z Z uj(σj(tj), σ−j(t−j))dF (tj, t−j) ≥ uj(sj, σ−j(t−j))dF (tj, t−j) (3)

for sj ∈ Sj. By Lemma B.1, the integrals in (3) are well-deﬁned. By a standard integration to the limit result, for every player j ∈ N, Z Z k k−1 lim uj(σj (tj), σ−j (t−j))dF (tj, t−j) = uj(σj(tj), σ−j(t−j))dF (tj, t−j); k→∞

likewise, for every sj ∈ Sj, Z Z k−1 lim uj(sj, σ−j (t−j))dF (tj, t−j) = uj(sj, σ−j(t−j))dF (tj, t−j). k→∞

(Again, the integrals are well-deﬁned.) The result then follows from a standard continuity argument.

B.2 Proof of Proposition 4.1

Recall that F (·) is a cumulative distribution function. Deﬁne Fe(˜τ | t˜) := 1 − F (˜τ | t˜) to be the conditional probability that the other player has type greater thanτ ˜ given that a player

25 has type t˜. As a tie-breaking rule, we assume that a type that is indifferent between the two actions chooses to invest. We show below that the choice of tie-breaking rule is immaterial. ∗ Say that a strategy σj is a switching strategy with threshold t ∈ T if a type t ∈ T with ∗ ∗ t < t does not invest (i.e., σj(t) = N), and types t ∈ T with t > t invests (i.e., σ(t) = I). Notice that, if the other player follows a switching strategy, then, by (MON-B) the expected payoff of investing to a type t is increasing in t while the expected payoff of not investing is decreasing in t. By (MON-I), types t < τ 0 have an impulse to not invest and types t > τ 0 have an impulse to invest. (Type τ 0 can have either impulse.) By (SYM), the level-0 threshold type τ 0 is the same for both players. 0 At level 0, types follow their impulse. Hence, for each player j, the level-0 strategy σj is a switching strategy with threshold τ 0. As the threshold is the same for both players, the level-0 strategies for the two players coincide. First suppose that τ 0 has a strict best response to invest, i.e.,

Fe(τ 0 | τ 0) > ρ.

Then, let τ 1 be the largest type τ 1 ≤ τ 1 such that

Fe(τ 0 | τ 1) = ρ (4) if such a type exists; otherwise let τ 1 = 0 (i.e., all types invest). By (MON-B), investing is a strict best response for types t > τ 1 against the belief that the other player follows the level-0 strategy; and not investing is a strict best response for types t < τ 1. So, the level-1 strategy is a switching strategy with threshold τ 1. By (SYM), this threshold is the same for both players. A similar argument shows that if not investing is a best response for τ 0, then the level-1 strategy is a switching strategy whose threshold is the smallest type τ 1 ≥ τ 0 satisfies (4) if such a type exists, or τ 1 = 1 otherwise. For k > 1, suppose, inductively, that for each player j, the level-(k − 1) and level-(k − 2) strategies are switching strategy with thresholds τ k−1 and τ k−2, respectively. We need to consider two cases. First suppose that τ k−1 ≤ τ k−2. Then, by a similar argument as above, the level-k strategy is a switching strategy whose threshold is the largest type τ k ≤ τ k−1 that satisfies Fe(τ k−1 | τ k) = ρ if such a type exists, or τ k = 0 otherwise. Next suppose that τ k−1 ≥ τ k−2. Then, by a similar argument as above, the level-k strategy is a switching strategy whose threshold is the smallest type τ k ≥ τ k−1 that satisfies Fe(τ k−1 | τ k) = ρ if such a type exists, or τ k = 1 otherwise.

26 k It follows that for each level k, the level-k strategy σj is a switching strategy with threshold τ k ∈ T (the same across players). There are two cases: either τ k ≤ τ k−1 for all k, or τ k ≥ τ k−1 for all k. In either case, the sequence τ 0, τ 1,..., being a monotone sequence in a compact space, converges to some τ ∈ T . In the former case, the equilibrium threshold is the largest τ ≤ τ 0 such that Fe(τ | τ) = ρ if such a type exists, and τ = 0 otherwise. In the latter case, the equilibrium threshold is the smallest τ ≥ τ 0 such that Fe(τ | τ) = ρ if such a type exists, and τ = 1 otherwise. So, in introspective equilibrium, each player follows a switching strategy with threshold τ. The introspective equilibrium is essentially unique: it pins down the behavior for all types t 6= τ, and this set has probability 1. The choice of tie-breaking rule is immaterial. It is easy to see that for each level k, only type t = τ k is indiﬀerent at level k. But this type has probability 0. Hence, how we choose the best responses for type t = τ k does not aﬀect the best responses for types at level k + 1.

Remark B.1. The proof shows that when players introspect and formulate a best response to the lower-level strategies, then their behavior converges to unique limit. Similarly, the literature on supermodular games has shown that in supermodular games, a best-response process converges to the extremal (largest or smallest) (Bayesian-)Nash equilibria (Vives, 1990; Van Zandt and Vives, 2007). In the games considered here, the extremal Nash equilibria are the pure Nash equilibria. By contrast, the introspective process selects a correlated equilibrium in which behavior need not be consistent with Nash equilibrium (Propositions 4.3 and 4.4).

B.3 Proof of Proposition 4.2

Fix a type space T and risk parameters ρ, ρ˜ such thatρ ˜ ≥ ρ. Denote the games with risk parameters ρ andρ ˜ by Γ and Γ,e respectively. First suppose that investing is a best response for the level-0 threshold type in game Γ. If investing is not a best response for type τ 0 in game Γ,e then the result follows immediately from the proof of Proposition 4.1. So suppose that investing is a best response for τ 0 in Γ.e The level-1 threshold type for Γ is the largest τ 1 ≤ τ 0 such that Fe(τ 0 | τ 1) = ρ or τ 1 = 0. Likewise, the level-1 threshold type for Γe is the largestτ ˜1 ≤ τ 0 such that Fe(τ 0 | τ˜1) = ρ or τ 1 = 0. By (MON-B),τ ˜1 ≥ τ 1. For k > 0, a similar argument yields that the level-k thresholds τ k, τ˜k for Γ and Γ,e respectively, satisfyτ ˜k ≥ τ k for all k. Hence, the equilibrium thresholds τ, τ˜ for Γ and Γ,e respectively, satisfyτ ˜ ≥ τ. An analogous argument applies for the case where not investing is a best response for τ 0. Moreover, it is clear that if Γ and Γe have the same risk parameter (i.e.,ρ ˜ = ρ), then they have

27 the same introspective equilibrium (i.e.,τ ˜ = τ).

B.4 Proof of Proposition 4.3

We first show that the conditional probability F (t | t) that the other player has a lower type is increasing in t for t sufficiently close to 0 or 1. This implies that for ρ sufficiently close to 0 or 1, the equilibrium threshold τ = τ(ρ) is either identically 0 or 1. To see this, suppose that the equilibrium threshold τ lies strictly between 0 and 1 for some ρ ∈ [0, 1]. By the proof of Proposition 4.1, F (τ | τ) = 1 − ρ. By Proposition 4.2, the equilibrium threshold τ increases in ρ. But this can only hold if F (τ | τ) is (locally) decreasing. So it remains to show that F (t | t) is increasing in t for t sufficiently close to 0 or 1. This is straightforward. First note that F (1 | 1) = 1 and F (0 | 0) = 0. As the joint density f satisfies f(t, t0) > 0 for t, t0 ∈ T (by (DENS)), F (t | t) < 1 for t < 1 and F (t | t) > 0 for t > 0. The claim then follows by noting that F (t | t) ∈ [0, 1] for t ∈ T .

B.5 Proof of Proposition 4.4

We start with a preliminary result, which says that a proﬁle of switching strategies with threshold τ cannot be an introspective equilibrium if τ 6= 0, 1 and the rank belief function is increasing at τ:

Lemma B.2. Fix ρ. Consider a proﬁle σ of switching strategies with threshold τ 6= 0, 1. Then, if the rank belief function F (· | ·) is increasing at τ, then σ is not an introspective equilibrium.

Proof. Consider a proﬁle σ of switching strategies with threshold τ 6= 0, 1 and suppose that F (· | ·) is increasing at τ. By the proof of Proposition 4.1, if F (τ | τ) 6= 1 − ρ, then σ is not an introspective equilibrium. So suppose F (τ | τ) = 1−ρ. Observe that, by the proof of Proposi- tion 4.2, the best-response curve (cf. Figure1) shifts up when the risk parameter ρ increases: for every threshold t ∈ T , the best-response threshold b(t) increases with ρ. Recall that, if σ is an introspective equilibrium with threshold τ, then b(τ) = τ, that is, the best-response curve b(·) intersects the 45-degree line at τ. Since F (· | ·) is increasing at τ, the intersection of the best-response curve b(·) with the 45-degree line shifts down at τ (otherwise, (1) would be violated when ρ is increased). Hence, the best-response curve b(·) intersects the 45-degree line from below at τ and the introspective process cannot converge to τ.

We are now ready to prove Proposition 4.4. By (DENS), the probability that players invest in introspective equilibrium lies strictly between 0 and 1, so behavior is not consistent with a

28 pure Nash equilibrium. So it remains to consider mixed Nash equilibrium. Suppose that an introspective equilibrium with threshold τ is a (strictly) mixed Nash equilibrium (for a given risk parameter ρ). Then, τ 6= 0, 1 and F (τ | τ) = F (τ) = 1 − ρ, where F (τ) is the marginal distribution of a player’s type. Moreover, by Lemma B.2, F (· | ·) is decreasing at τ. Let R be the set of pairs (τ, ρ) such that for risk parameter ρ, the introspective equilibrium with threshold τ is a mixed Nash equilibrium. Notice that, for given ρ, there is at most one τ such that (τ, ρ) ∈ R. Fix (τ, ρ), (τ 0, ρ0) ∈ R such that ρ0 > ρ. Since F (·) is increasing and F (· | ·) is decreasing at τ and at τ 0, we have τ 0 < τ. But this contradicts Proposition 4.1. So, R is at most a singleton. Conclude that, for a given type space, there is at most one value of ρ such that the introspective equilibrium for ρ is a (strictly) mixed Nash equilibrium. .

Remark B.2. There exist introspective equilibria that are mixed Nash equilibria. For our ongoing example, F (τ 0 | τ 0) = F (τ 0). Hence, for ρ0 such that F (τ 0) = 1−ρ0, the introspective equilibrium (with threshold τ 0) is a mixed Nash equilibrium. Proposition 4.4 implies that there are no other introspective equilibria (for any value of ρ) that are mixed Nash equilibria.

B.6 Proof of Proposition 4.5

Recall (from the proof of Proposition 4.1) that the equilibrium threshold τ is strictly lower than τ 0 if and only if the level-0 threshold τ 0 has a strict best response to invest (i.e., F (τ 0 | τ 0) < 1 − ρ); similarly, the equilibrium threshold τ is strictly greater than τ 0 if and only if the level-0 threshold τ 0 has a strict best response to not invest (i.e., F (τ 0 | τ 0) > 1 − ρ). Suppose there is an open interval (t, t¯) ⊂ T such that the rank belief function F (· | ·) is decreasing on (t, t¯). Deﬁne ρ andρ ¯ by

F (t | t) = 1 − ρ; F (t¯| t¯) = 1 − ρ¯;

respectively (cf. (1)). Notice that, by the Proof of Proposition 4.3, 0 < t < t¯ < 1. Hence, by (DENS), 0 < ρ < ρ¯ < 1. Also suppose F (τ 0 | τ 0) ∈ (F (t¯ | t¯),F (t | t)). So, the type space has nonmonotone rank beliefs (i.e., (NMRB) holds). We show that the equilibrium threshold τ lies strictly between 0 and 1 for ρ ∈ (ρ, ρ¯). That is, the introspective equilibrium is interior. We need to consider three cases:

Case 1: τ 0 < t. For ρ > 1 − F (τ 0 | τ 0), the equilibrium threshold τ is strictly greater than the level-0 threshold τ 0. If ρ < ρ¯, there is t∗ ∈ [t, t¯] such that 1 − F (t∗ | t∗) = ρ. Hence, by Lemma B.2, for ρ ∈ (1 − F (τ 0 | τ 0), ρ¯), the equilibrium threshold τ is strictly greater

29 than τ 0 > 0 and at most t∗, so the introspective equilibrium is interior. (Notice that, since F (τ 0 | τ 0) > F (t¯| t¯) =ρ ¯, the interval (1 − F (τ 0 | τ 0), ρ¯) is nonempty.)

Case 2: τ 0 ∈ [t, t¯]. Again, for ρ ∈ (1 − F (τ 0 | τ 0), ρ¯], the equilibrium threshold τ satisﬁes τ > τ 0. Since τ 0 ∈ [t, t¯], we have in fact τ ∈ [t, t¯]. Hence, by Lemma B.2, the introspective equilibrium is interior.

Case 3: τ 0 > t¯. If ρ < 1 − F (τ 0 | τ 0) then the equilibrium threshold τ satisﬁes τ < τ 0. If ρ > ρ, then there is t∗ ∈ [t, τ 0) such that 1 − F (t∗ | t∗) = ρ. Hence, for ρ ∈ (ρ, 1 − F (τ 0 | τ 0)), the equilibrium threshold τ is strictly lower than τ 0 < 1 and at least t∗ > 0, so the introspective equilibrium is interior.

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