The Value of a ∗ Preliminary and Incomplete

Willemien Kets† Alvaro Sandroni‡

February 27, 2017

Abstract

How much are you willing to pay to play a given coordination game? Schelling (1960) noted that traditional 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. †Kellogg School of Management, Northwestern University. E-mail: [email protected] ‡Kellogg 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 (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 (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 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 ε ∈ (0, 1], so that we obtain the following game: a b a 2,2 0,0 b 0,0 1 + ε,1 + ε

HLε 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 HLε? We show that as ε increases to 1, the value of game HLε decreases, despite the fact that the payoffs in game HLε are weakly higher than in game HL. Intuitively, as ε 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 HLε 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). Game PC in the introduction is of course the prototype for this class. A matching game is a Hi-Lo game if coordinating on a is better than coordinating on b (i.e., uaa > ubb).

2.2 Introspection

As coordination games have multiple Nash equilibria, the payoff structure gives little guidance and players face considerable strategic uncertainty, that is, uncertainty about the other player’s action. To form a belief about others’ behavior, individuals can use what is called theory of mind in psychology, that is, a cognitive ability to attribute mental states (e.g., beliefs) to others (Premack and Woodruff, 1978). A central part of theory of mind is introspection: players use their observation of their own mental state to form an initial belief about others’ mental state. That is, they put themselves into others’ shoes using their own subjective experience as a guide.4 This is a rapid and instinctive process referred to as first-person simulation (Goldman, 2006). It is followed by a slower, more deliberative process whereby individuals reason about others’ mental states and the implications for behavior using a naive understanding of psychology. This may lead them to adjust their initial belief (Gopnik and Wellman, 1994). Under this view, individuals use the observation of their own mental state to form an initial belief and then use “folk psychology” to adjust their belief based on their understanding of the other person’s goals and attributes (Epley and Waytz, 2010). In strategic environments, players have to take into account that the other player will likewise adjust his beliefs. Player can thus be viewed as “folk game theorists” who, rather than engaging in a full-fledged equilibrium analysis, use introspection to form an initial belief and then adjust their belief after reasoning about others’ beliefs, taking into account that the other player also adjusts her beliefs.

We model this as follows. Each player j receives an impulse ij to play a certain action

(i.e., ij ∈ S). Impulses are privately observed, payoff-irrelevant signals. The distribution

4These ideas also have a long history in philosophy. 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.” Russell(1948) observes that “[t]he behavior of other people is in many ways analogous to our own, and we suppose that it must have analogous causes.”

5 of impulses and players’ beliefs about impulses are described by a common-prior type space

T = (T1,T2, I1, I2, µ), where Ti is the set of types for player j = 1, 2, Ij(tj) = a, b is the impulse associated with type tj ∈ Tj, and µ is the common prior on the set T1 × T2 of type profiles.5 A player’s instinctive reaction is to follow his impulse. This defines defines a level-0 0 0 σj (i.e., σj (tj) = sj if Ij(tj) = sj). Through introspection, a player realizes that the other player will also have an instinctive reaction. By observing his own mental state (i.e., impulse), the player can form a belief µ(I−j = s−j | tj) about the other player’s impulse using Bayes’ rule. 0 The process of first-person simulation thus yields a belief µ(σ−j(t−j) = s−j | tj) = µ(I−j =

s−j | tj) about other players’ instinctive responses. Players are rational, so if the player’s instinctive reaction is not optimal in the light of this belief, then he adjusts his response. The to his belief about the other player’s level-0 strategy defines his level-1 strategy 1 6 σj . The reasoning does not stop here. As a folk game theorist, the player realizes that other 1 0 0 player will likewise formulate a best response σ−j(t−j) to her belief µ(σj (tj) = sj | t−j) about 1 his level-0 strategy. This leads him to formulate a best response to σ−j; this, in turn, defines 2 k his level-2 strategy σj . In general, for k > 1, the level-k strategy σj is a best response to k−1 the level-(k − 1) strategy σ−j of the other player. Players go through this infinite reasoning process in their mind before making a decision. Accordingly, player j’s behavior is given by the k 7 limit σj := limk→∞ σj of the introspective process. The profile of limiting strategies (σj)j=1,2 defines an introspective equilibrium (Kets and Sandroni, 2015). In our model, a player starts by forming an initial belief about others by observing his own impulse. He then adjust this instinctive belief if it is not consistent with what he knows about other players’ preferences, beliefs, and rationality. In interactive situations, players need to take into account that the other player may adjust her belief as well. This necessitates higher- order reasoning, modeled here by the different levels. Thus, unlike a “folk psychologist” who faces a single-person decision situation, a “folk game-theorist” must continue to reason after adjusting his initial belief. As in Bayesian games, the type space is part of the primitives for the model. Unlike in Bayesian games, the state of the world (i.e., the impulses and players’ types) does not have a direct impact on payoffs. However, it can influence payoffs indirectly if types with different beliefs or impulses choose different actions. While impulses cannot be observed directly, the

5This definition is more general than in our earlier work (Kets and Sandroni, 2015) where there is a one- to-one relation between a player’s type and his impulse. 6 1 0 That is, if the player’s type is tj, then σj (tj) is a player’s best response against σ−j given his belief µ(· | tj). If there are multiple best responses, a player chooses an action according to a fixed tie-breaking rule. The choice of tie-breaking rule does not affect our results. 7Section3 shows that such limiting strategies exist under weak conditions.

6 associated reasoning process can be tested experimentally. Agranov, Caplin, and Tergiman (2015) introduce a mechanism that incentivizes subjects to make an initial choice and to revise their choice immediately when further reflection causes their choice to change. This mechanism can be used to elicit the distribution of instinctive responses and to test whether individual choices over time are consistent with an introspective process.

2.3 Impulses

An unreasoned impulse to take a certain action is often guided by what is salient: if an action stands out from the others, a player’s instinctive reaction is to choose it.8 Salience may be the result of emotional and cultural factors and is thus inherently subjective and context- dependent (Fiske and Taylor, 2013). As such, it is perhaps more a question for psychology than for economics which alternative stands out in any given situation.9 However, even if our understanding of the precise determinants of salience in any particular situation is incomplete at best, we can still develop a theory of aggregate behavior if the qualitative properties of the predictions do not depend on the particulars of the type space. This is the approach we will take: we impose some basic conditions on the type space that reflect the nature of the introspection process, and prove our results for any type space that satisfies these conditions. To state the assumptions on the type space, fix a type space

T = (T1,T2, I1, I2, µ) and write µ(I1,I2) for the probability that players 1 and 2 have impulses

I1 and I2, respectively. We also write µ(Ij = k | ti) for the conditional probability that player j 6= i has an impulse to play action k given that player i has type ti, and µ(Ij = k | Ii = m) is the conditional probability that player j 6= i has impulse k given that player i has an impulse to play action m. We want our model to capture that individuals typically (but not always) agree on what is salient. For example, when asked to pick “heads” or “tails,” the vast majority of subjects select heads (Schelling, 1960, pp. 54–58) and expect others to choose heads (Mehta, Starmer, and Sugden, 1994), but not everyone does so. This suggests that impulses are (imperfectly) correlated. This is reflected in the following assumption:

8For example, Mehta, Starmer, and Sugden(1994, p. 659) write: “Some [action] labels are more promi- nent or conspicuous or salient than others; they ‘stick out’ or ‘suggest themselves’ . . . Players tend to choose . . . strategies [that] are salient.” Bardsley, Mehta, Starmer, and Sugden(2009, pp. 76–77) discuss the relation between saliency and “pre-reflective inclinations.” 9 Schelling(1960, p. 57) emphasizes that which actions are salient depends “on imagination more than on logic; it may depend on analogy, precedent, accidental arrangement, symmetry, aesthetic or geometric config- uration, casuistic reasoning” or even on “whimsy” and rejects any suggestion that salience can be captured by a single formal theory.

7 Assumption CORR (Positive Correlation). Impulses are positively correlated: for each pair 1 of players i, j 6= i, and impulse k = a, b, µ(Ij = k | Ii = k) ≥ 2 .

In addition, a player who thinks that an alternative is highly salient is likely to think that the other player will think the same.10 Likewise, a player who thinks that none of the alternatives stand out is unlikely to think that the other player thinks one of the options is strongly salient. This suggests that beliefs about impulses are also correlated. Moreover, if a player thinks it is highly likely that the other player has the same impulse, then he is likely to think that she thinks it is likely that he has the same impulse. And so on. To capture this, we assume that types with an impulse to play a certain action think it is likely that the other player has the same impulse. Moreover, types that assign a higher likelihood to the other player having the same impulse assign a higher likelihood to the other player believing that her opponent has similar beliefs, etc. Formally, type sets are ordered, with higher types having an impulse to play action a and assigning a higher probability to higher types:

Assumption MON-B (Monotone Beliefs). For each player i = 1, 2 the type set Ti is endowed 0 0 with a complete order ≥i. For every type ti ∈ Ti, if Ii(ti) = a and ti ≥i ti, then Ii(ti) = a. 0 0 Moreover, if ti ≥i ti, then µ(· | ti) first-order stochastically dominates µ(· | ti).

To abstract away from any asymmetries in the environment that would allow players to coordinate, we assume that beliefs and impulses are symmetric across players:

Assumption SYM (Symmetry). We have T1 = T2, ≥1=≥2, I1 = I2, and the prior µ is symmetric in players’ types.

Thus, if players have the same type, then their impulses are the same and they have the same conditional beliefs. In the remainder of the paper, we will write ≥ for the (common) order ≥j on Tj.

Finally, we make some technical assumptions. First, we take the functions Ij : Tj → {a, b}, j = 1, 2, that associate an impulse Ij(tj) with each type tj to be measurable. This ensures that all types know their own impulses. Second, it will be convenient to assume that any impulse is possible a priori (even if the probability that a player receives a given impulse may be vanishingly small). That is, for each player i = 1, 2 and impulse k = a, b, the probability µ that player i has impulse k is strictly greater than 0. We will also assume that if FI (q) is the cumulative distribution function that a type with impulse I assigns conditional probability q µ to the other player having the same impulse (under µ), then FI (q) has a density with full 10Bacharach and Bernasconi(1997) present some evidence of “noticer bias”: a subject who has identified a certain alternative as salient has a tendency to think that other subjects also think it is salient.

8 1 support on [ 2 , 1]. This rules out atoms in players’ beliefs. This is stronger than necessary, but simplifies the statement of results. For the remainder of the paper, we restrict attention to type spaces that satisfy the above conditions. We start with some basic results on introspective equilibrium. First, 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. [Rationality of introspective equilibrium, Kets and Sandroni, 2015] Every introspective equilibrium is a .

Importantly, since correlated equilibrium is characterized by of ratio- nality (Aumann, 1987), the result implies that introspective equilibrium does not presume that players are boundedly rational. 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 response against other players’ strategies. Moreover, the correlation in impulses allows the strategies of players to be correlated even if players do not follow their impulse in equilibrium. As a result, an introspective equilibrium need not be a Nash equilibrium. While every introspective equilibrium is a correlated equilibrium, the two concepts do not coincide. First, unlike in correlated equilibrium, a player may not follow his impulse in an introspective equilibrium, as we will see. Second, while coordination games have multiple (Nash or correlated) equilibria, the introspective process selects an essentially unique one:11

Proposition 2.2. [Existence and Uniqueness, Kets and Sandroni, 2015] Every coor- dination game has an introspective equilibrium. The introspective equilibrium is essentially unique.

As in Bayesian games, the equilibrium generally depends on the type space. Thus, the uniqueness of the introspective type space is with respect to a given type space. However, while the exact equilibrium strategies may depend on the details of the type space, our char- acterization of the value does not: it holds for any type space that satisfies our assumptions. Finally, every introspective equilibrium is in threshold strategies:

Proposition 2.3. [Characterization of introspective equilibrium] Let σ be the intro-

spective equilibrium of a game. Then there is a threshold τ such that each type tj chooses

action a if tj ≥ τ and plays b otherwise.

11That is, the set of types that have a best response at some level k that is not unique has measure 0. We pick their action using some fixed tie-breaking rule. The choice of tie-breaking rule does not affect the value of the game.

9 These basic results greatly simplify the analysis below. Proposition 2.2 ensures that the value of a coordination game is well defined for any (2 × 2) coordination game. Moreover, when we perform comparative statics, Proposition 2.3 implies that we need to keep track only of the threshold as a function of parameters.

3 The value of a coordination game

The value of the game is the (ex ante) expected payoff for a player in the introspective equilibrium: X Z V := σ1(s1 | t1) σ2(s2 | t2) us1,s2 dµ(t1, t2), T1×T2 s1,s2=a,b

where σi(ai | ti) is the probability that player i chooses ai in the introspective equilibrium given

that his type is ti, and us,s0 is the payoff to a player if he chooses action s and his opponent chooses s0. Given the symmetry of the of the strategic environment and the uniqueness of the introspective equilibrium, the value of the game is well-defined. While the value depends on the type space, we can derive testable predictions by focusing on comparative statics. We first present some results for pure coordination games. Fix a pure

coordination game, and let v := uaa = ubb be the prize, that is, the payoff that players receive if they pick the same action. Then:

Proposition 3.1. [Pure Coordination Games] Fix a pure coordination game with prize 1 v. Then, the value of the game is q · v for some q ∈ ( 4 , 1). Proposition 3.1 implies that the coordination rate in a pure coordination game such as game PC in the introduction lies strictly between the rate in the mixed Nash equilibrium (25%) and the rate for one of the pure Nash equilibria (100%). Thus, play cannot be described by Nash equilibrium or one of its refinements.

Next, consider a Hi-Lo game. A Hi-Lo game can be defined by the payoff V := uaa in the Pareto efficient equilibrium and the payoff v in the Pareto dominated equilibrium. The next result says that if the payoff in the Pareto efficient equilibrium is sufficiently high, then players will coordinate on the Pareto efficient equilibrium.

Proposition 3.2. [Hi-Lo Games] Fix v > 0. For every η > 0, there is V such that the value of the Hi-Lo game (V, v) is greater than (1 − η) · V .

Finally, we study the comparative statics for Hi-Lo games.

Proposition 3.3. [Comparative Statics] Fix V, v > 0 such that V > v. For any v0 ∈ (v, V ], the value of the Hi-Lo game (V, v0) is strictly lower than the value of the Hi-Lo game (V, v).

10 Proposition 3.3 says that the value of the game with v0 > v is strictly less than the value of the game with v0 = v. This may seem counterintuitive at first sight: the payoffs in the former game are higher than in the latter game. However, the result is natural once we take into account the process by which players choose their action. If there is a significant difference in payoffs between the two pure Nash equilibria, the efficient equilibrium is highly salient. This makes it easy for players to coordinate. Moreover, players who coordinate receive a high payoff. If the difference in payoffs is smaller (or even nonexistent), then the efficient Nash equilibrium is less salient. As a result, there is more miscoordination. The higher payoffs that players receive in the Pareto dominated Nash equilibrium can never compensate for this. For example, if v0 = V , then players receive V in both Nash equilibria. However, the chances that they will manage to coordinate on one of the pure Nash equilibria is lower than in a game where a single Nash equilibrium is payoff salient.

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