The Value of a Coordination Game∗ Preliminary and Incomplete Willemien Ketsy Alvaro Sandroniz 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. yDepartment of Economics, University of Oxford. E-mail: [email protected]. zKellogg 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) coordi- nation 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., un- derstanding 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 stan- dard 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.