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The Value of a Coordination Game Preliminary and Incomplete

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The Value of a Coordination Game Preliminary and Incomplete The Value of a Coordination Game∗ Preliminary and Incomplete Willemien Ketsy Alvaro Sandroniz February 27, 2017 Abstract How much are you willing to pay to play a given coordination game? Schelling (1960) noted that traditional game theory does not assign a value to coordination games, and suggested that this is because people's ability to coordinate cannot be understood from the economic primitives alone (e.g., pp. 97{98). We thus incorporate insights from psychology into a standard game-theoretic model to model how people take the perspective of others, and use this to assign a unique value to every symmetric (2 × 2) coordination game. In making their choice, players take into account both the payoffs and the strategic uncertainty that they face. As a result, the value may be nonmonotonic in payoffs. ∗We thank Larry Blume, Peter Hammond, and David Schmeidler for helpful comments and stimulating discussions. yKellogg School of Management, Northwestern University. E-mail: [email protected] zKellogg School of Management, Northwestern University. E-mail: [email protected] 1 1 Introduction How much are you willing to pay to play a given coordination game? The answer to this question defines the value of the game. Defining the value of a game thus requires a theory of coordination. We want the theory to match some key stylized facts. Consider the following games: a b a b a 1,1 0,0 a V ,V 0,0 b 0,0 1,1 b 0,0 1,1 , V > 1 PC HL We follow game-theoretic convention by referring to the actions in these games by mean- ingless labels (viz., a; b). But, in real life, action labels have meanings and thus various asso- ciations and connotations. Such associations and connotations can make one of the actions more salient than another. In that case, it is reasonable to expect that in the pure coordina- tion game PC, players will coordinate at a higher rate than in the mixed Nash equilibrium (where each player randomizes uniformly and the coordination rate is 25%). At the same time, unless one of the actions is highly salient (e.g., it is labeled \pick me!"), we would expect some miscoordination. Thus, the coordination rate can be expected to lie strictly between that of the mixed Nash equilibrium and the two pure Nash equilibria. This is indeed what is found in experiments. For example, Schelling(1960, p. 55ff.) conducted a small experiment where the actions where labeled \heads" and \tails." The vast majority of subjects (86%) chose heads, giving a coordination rate (under uniform random matching) of 73%.1 So, in game PC, we expect non-Nash behavior, and the value of a game should be strictly between the expected 1 payoff (viz., 4 ) under the mixed Nash equilibrium and the expected payoff (viz., 1) under one of the pure Nash equilibria. On the other hand, if the payoff V to coordinating on a is sufficiently high in the Hi-Lo coordination game HL, then we can expect players to select a even if b's label is highly salient. So, in that case, the value will be V or very close to it.2 The fundamental issue is that players who aim to coordinate their actions face, in fact, two distinct types of coordination problems. The first is to avoid miscoordination. The second 1Schelling's informal experiment has been replicated by Mehta, Starmer, and Sugden(1994) who found similar results; see Section3. 2One might think that players will select a whenever V > 1, i.e., that payoff salience always dominates label salience, at least in the absence of risk. This may be true in laboratory settings where differences in payoffs tend to be highly salient. However, in more natural environments, other factors may be more salient and choices can be shaped by payoff-irrelevant factors (Bordalo, Gennaioli, and Shleifer, 2013). 2 is to coordinate on the \right" Nash equilibrium. In a Hi-Lo game such as HL, we expect the second consideration to dominate if V is sufficiently high. Any theory of coordination ought to select the efficient Nash equilibrium (viz., (a; a)) in such games. On the other hand, in pure coordination games such as PC, no Nash equilibrium gives higher payoffs than another, and the principal concern is to minimize the risk of miscoordination. By using the salience of action labels, players can do better than pure chance (i.e., 25%). However, they are unlikely to coordinate perfectly, except perhaps in unusual circumstances. Thus, any theory of coordination must predict non-Nash behavior in pure coordination games. Moreover, as the payoffs of the two pure Nash equilibria come closer to each other, turning a Hi-Lo game into a pure coordination game, we expect play in a Hi-Lo game to converge smoothly to that in the limiting pure coordination game. To the best of our knowledge, no existing theory simultaneously selects the efficient Nash equilibrium in Hi-Lo games and predicts non-Nash behavior in pure coordination games. Equi- librium selection methods such as global games (Carlsson and van Damme, 1993), payoff and risk dominance (Harsanyi and Selten, 1988), team reasoning (Sugden, 1993), or those based on adaptive processes (e.g., Kandori, Mailath, and Rob, 1993; Young, 1993), all select the efficient Nash equilibrium in Hi-Lo games, and would thus give a value of V in game HL. But, none of these approaches has any bite in pure coordination games such as PC. The same is true for existing behavioral models, such as level-k models (Nagel, 1995) and quantal response equilibrium (McKelvey and Palfrey, 1995). Accounting for these basic stylized facts thus requires a novel theory of coordination. Our starting point is that successful coordination requires players to put themselves into the other player's shoes. To take the other's perspective, individuals use \theory of mind." Theory of mind is the cognitive capacity to attribute { beliefs, feelings, knowledge, etc. { to others. Rather than using an \off-the-shelf" theory about what others think { for example, one prescribed by an equilibrium refinement {, individuals form their beliefs based on what they know about the situation and the other person. Introspection is a critical part of this process: they form beliefs by considering their own mental state and projecting it onto the other player. As this may lead to false beliefs and biases, this fast, instinctive process is followed by a more deliberative process whereby players use \folk psychology" to adjust their belief based on their understanding of the other person's goals and attributes (Epley and Waytz, 2010). In a strategic environment, a player has to take into account that the other player will likewise adjust her initial belief. Thus, a \folk game theorist" follows an iterative reasoning process whereby he modifies his beliefs taking into account that his opponent reasons about his beliefs in turn. The actions that players choose following this reasoning process define an introspective equilibrium (Kets and Sandroni, 2015). 3 We use this solution concept to define the value of symmetric (2 × 2) coordination games. The value of a game is the (ex ante) expected payoff a player receives in an introspective equilibrium. The value is well-defined since coordination games have a unique introspective equilibrium. Moreover, the introspective equilibrium is symmetric in the environments we consider. Thus, the value is uniquely defined. We show that under some mild assumptions on 1 the reasoning process, the value of game PC is strictly between 4 and 1, while the value of game HL is close to V provided that V is sufficiently high. We then ask how the value changes with payoffs. Suppose we increase the payoffs in the (b; b) equilibrium in the Hi-Lo game HL by some small amount " 2 (0; 1], so that we obtain the following game: a b a 2,2 0,0 b 0,0 1 + ",1 + " HL" If a player is willing to pay 2 to play the Hi-Lo game HL, how much would he be willing to pay to play the game HL"? We show that as " increases to 1, the value of game HL" decreases, despite the fact that the payoffs in game HL" are weakly higher than in game HL. Intuitively, as " increases from 0 to 1, the payoffs in the two pure Nash equilibria move closer together, and the payoff salience of the (a; a) equilibrium loses its force. There is more miscoordination in HL" than in HL, due to the weakened payoff salience of the efficient Nash equilibrium, and its value is correspondingly lower. The outline of this paper is as follows. Section2 introduces the model. Section3 presents the main results. Section ?? discusses extensions of the theory. Section ?? discusses the related literature, and Section ?? concludes. 2 Model 2.1 Game There are two players, labeled j = 1; 2. Players play a symmetric (2 × 2) coordination game with actions s = a; b and payoffs given by3 a b a uaa,uaa uab,uba b uba,uab ubb,ubb , uaa ≥ ubb > uab, uaa > uba. 3The results can be extended to more general settings. See Section ??. 4 The game has two strict Nash equilibria. In one of these equilibria, both players choose a, and in the other, both players choose b. The former is potentially Pareto efficient (i.e., uaa ≥ ubb). Payoffs are commonly known. A matching game or diagonal coordination game is a coordination game where players receive a prize if and only if they choose the same action (i.e., uba = uab = 0). A matching game is a pure coordination game if the coordination payoffs do not depend on which action the players coordinate on (i.e., uaa = ubb).
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