Cognitive foundations of strategic behavior : from games to revolutions

Cognitive Foundations of Strategic Behavior:

From Games to Revolutions

by

David Jimenez-Gomez

Licenciado, Universidad de Murcia (2008)

M.A., Universitat Autonoma de Barcelona (2010)

Submitted to the Department of Economics

in partial fulfillment of the requirements for the degree of

PhD in Economics

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 2015

@

David Jimenez-Gomez, MMXV. All rights reserved.

The author hereby grants to MIT permission to reproduce and to

distribute publicly paper and electronic copies of this thesis document

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May 8, 2015

Daron Acemoglu

Elizabeth and James Killian Professor of Economics

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Abhijit Banerjee

Ford Foundation International Professor of Economics

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Cognitive Foundations of Strategic Behavior:

From Games to Revolutions

by

David Jimenez-Gomez

Submitted to the Department of Economics on May 8, 2015, in partial fulfillment of the

requirements for the degree of PhD in Economics

Abstract

Game theory is one of the main tools currently used in Economics to model strate-gic behavior. However, game theory has come under attack because of the strong assumptions it makes on people's behavior. Because of that, alternative models of bounded rationality, with more realistic assumptions, have been proposed.

In the first chapter, I develop a game theoretic model where players use two different reasoning processes: cooperative and competitive. The model generalizes Level-k and team reasoning, and provides a unified explanation for several im-portant phenomena. In Rubinstein's Email game, players coordinate successfully upon receiving enough messages. In 2 x 2 games of complete information, the solu-tion concept lies between Pareto dominance and risk-dominance. In coordinasolu-tion games, the model explains several experimental facts that cannot be accounted for by global games, especially the fact people coordinate more with public rather than private information. I show the importance of public events in revolutions: a self-interested government prevents the generation of common knowledge among the citizenry when times are bad.

In the second chapter, I develop a model of cognitive type spaces which incor-porates Level-k and Cognitive Hierarchy (CH) models into games of incomplete information. CH models make two assumptions: agents of higher level have richer

beliefs and can perform more computations. In my model, like in Level-k and CH models, an agent's level determines how complex her beliefs are. However, given their beliefs, agents are fully rational and behave according to Interim Correlated Rationalizability. My main result is that, restricted to cognitive type spaces, the product topology and the uniform strategic topology coincide, what implies that two players with similar beliefs behave similarly. This means that, unlike for general type spaces, predictions will be robust to small specification errors and suggests that incorporating cognitively plausible assumptions into game theory can increase robustness. As an application, I show that in the Email game, when players receive few messages they never attack; however, when they receive enough messages, they behave as if there was complete information, and both actions are rationalizable.

In the third chapter, I develop a dynamic model of forward-looking agents in the presence of social pressure. I show that social pressure is effective in generat-ing public good provision: after an agent starts contributgenerat-ing to the public good, other agents decide to contribute as well because of fear of being punished, what generates contagion in the network. In contrast with the previous literature, con-tagion happens fast as part of the best response of fully rational individuals. The network topology has implications for whether the contagion starts and the extent to which it spreads. I find conditions under which an agent decides to be the first to contribute in order to generate contagion in the network, as well as conditions for contribution due to a self-fulfilling fear of social pressure.

Thesis Supervisor: Daron Acemoglu

Title: Elizabeth and James Killian Professor of Economics Thesis Supervisor: Abhijit Banerjee

Acknowledgments

I want to express my gratitude to my advisors, Daron Acemoglu, Abhijit Baner-jee and Muhamet Yildiz, for their invaluable guidance throughout the PhD. They provided me with advice at all levels of research, from the big ideas to the small details. Their breath of knowledge and commitment to research was an example for me to follow and derive inspiration from. As I explored the frontiers of the knowledge in Economics, they gave me honest and encouraging feedback, and de-voted many hours of their time to help me in my endeavor.

I have built on the work of countless academics, too many to be mentioned here; they are cited where it corresponds in the respective chapters. Some of them kindly devoted time to give me feedback on my work. For the first chapter, I thank Gabriel Carroll, Alp Simsek, Tomasz Strzalecki, Nils Wernerfelt, and the participants of seminars at MIT and Chicago. For the second chapter, I thank Nemanja Antic, Dan Barron, Ben Brooks, Gabriel Carroll, Elliot Lipnowski, Georgy A. Lukyanov, Stephen Morris, Jawwad Noor, Alp Simsek, Tomasz Strzalecki, Nils Wernerfelt, and the participants of seminars at MIT, Universitat Autnoma de Barcelona and EconCon. For the third chapter, I thank Glenn Ellison, Ben Golub, Giovanni Reggiani, John Tsitsiklis, Xiao Yu Wang, and the participants of seminars at MIT.

Prior to my studies at MIT I had several teachers and mentors who inspired me through their example, and supported my lifelong desire of learning. Among

them, I am especially indebted to Joss Orihuela Calatayud at Universidad de Mur-cia, and Salvador Barbera and Miguel Angel Ballestcr at Universitat Aut6noma de Barcelona, for transmitting to me their passion for research and encouraging me in pursuing such a fascinating path.

I am thankful to my friends for many moments of inspiration and joy. I am especially indebted to Dana Chandler, Caitlin Lee Cohen, David Colino, Dong Jae Eun, Sara Herndndez, Elena Manresa, Giovani Reggiani, Alejandro Rivera, Xiao Yu Wang, Nils Wernerfelt and Leonardo Zepeda-Nninez in Cambridge, and Pedro Garcia-Ares, Tugce Cuhadaroglu, Ezgi Kaya and Pau PujolAs Fons in Barcelona, for their support and affection; and to my many friends, in Murcia and elsewhere, who were far but always felt so close.

Finally, I am extremely grateful to my family. My grandmother Soledad taught me how to read at an early age, thus setting me on a path of curiosity and discov-ery. My grandfather Prudencio showed me the value of effort through his living example. My parents, Prudencio and Marisol, transmitted to me their love for learning and teaching, and have always done everything in their hand to help me advancing in my path, with unwavering love and encouragement. My sister Alicia and I have rejoiced together in the good moments, and supported each other in the less good ones. It is because of all of you that I am where I am today.

I gratefully acknowledge financial support from "La Caixa" Foundation, the Bank of Spain, Rafael del Pino Foundation, the MIT Economics department and the George and Obie Shultz Fund. The opinions in this thesis are exclusively my own and do not represent those of the aforementioned institutions.

Contents

1 Cooperative and Competitive Reasoning: from Games to

Revo-lutions 11 1.1 Introduction ... ... ... ... 11 1.1.1 Literature review . . . 16 1.2 The m odel . . . . 20 1.2.1 Bounded representation . . . . 20 1.2.2 Cooperative reasoning . . . . 25

1.2.3 Competitive reasoning: Level-k . . . . 30

1.2.4 Putting it all together: Solving the Email game . . . 31

1.2.5 Games of complete information . . . . 34

1.3 Global games . . . 36

1.3.1 Theoretical predictions . . . . 38

1.3.2 Revolutions and Collective Action . . . . 42

1.4 Generalized Model and Applications . . . . 43

1.4.2 Signal jamming by the government . . . . 1.5 Conclusion . . . . 2 You Are Just Like Me: Bounded Reasoning and Recursive Beliefs 73

2.1 Introduction ... 2.2 The Model .... 2.3 Application: The 2.3.1 Analysis 2.4 Main Result ... 2.5 Conclusion . . . 3 Social Pressure 3.1 Introduction 3.2 The model 3.3 Contagion 3.3.1 Simpl 3.3.2 Gener 3.4 Leadership . 3.4.1 Spear 3.4.2 Social 3..5 Bounded rati 3.6 Conclusion . . . . . 73 Email Game

in Networks Induces Public Good Provision

. .. .. . . ... ...

al ...

.. ad..qulri...

e-: Case: An Illustration . . . . al Case . . . .

headed equilibria . . . .

pressure equilibria: Supermodularity and MPE . onality and visibility . . . . . . . . 50 54 80 85 88 90 93 111 111 119 122 122 125 132 135 137 142 145

Chapter 1

Cooperative and Competitive

Reasoning: from Games to

Revolutions

1.1

Introduction

"For nearly thirty years, the price of a loaf of bread in Egypt was held constant; Anwar el-Sadat's attempt in 1977 to raise the price was met with major riots. Since then, one government tactic has been to make the loaves smaller gradually, another has been to replace a fraction of the wheat flour with cheaper corn flour." - Chwe

(2001)

example from Crawford et at. (2008): two people must choose between X and Y; if their choices coincide they both receive $5. Around 80% of people choose X in this situation. What happens if we tell one person she will receive 10 extra cents for coordinating on X, and the same to the other person about coordinating on Y? Now around 80% of people choose the action for which they are not paid extra; and this results in massive miscoordination, and therefore lower payoffs. If X was salient, why do players stop coordinating on it once the small extra incentive is added? Another example comes from Chwe's quote above: people could not coor-dinate their protests when the subsidized bread became smaller or of lower quality; but when the price rose, protests erupted. Why these differences in coordination?

I propose a model which provides a unified explanation for these apparently disconnected phenomena. The main idea is that people use two different kinds of reasoning: cooperative and competitive. Seemingly small differences in the structure of a game, or of a political scenario, lead to a switch from one reasoning modality to the other, and can therefore generate a large change in behavior. The model has three main components.

1. Players have a bounded understanding of the game, indexed by a player's level k: the higher k is, the better the agent is able to understand the game. Intuitively, a player is unable to consider the other player's beliefs of very high order, and k roughly measures the order at which a player stops thinking. 2. Cooperative reasoning: players attempt to play a Pareto dominant

equi-librium a* whenever there is one. The likelihood of this happening depends on a parameter 0, which measures the probability that others reason coop-eratively, as well as the riskiness of a*.

3. Competitive reasoning when cooperative reasoning fails, players behave as level-k thinkers: level-0 players randomize uniformly, and level-k players best-respond to players of level k - 1.

The solution concept is dual reasoning: players first attempt to engage in cooperative reasoning, and use competitive reasoning otherwise. The model is psychological, in the sense that it attempts to capture, as much as possible, the reasoning process that people follow in strategic interactions. Consider the well-known Email game, where players exchange emails of confirmation back and forth until a message is lost. Because no player is ever sure whether she was the last to receive a message, there is no common knowledge in this game, Rubinstein (1989); however, most people would agree that after receiving a large quantity of confir-mation emails (millions for example), then both players have common knowledge, for all practical purposes, that the message is known. This intuition is captured by my model because, after having received enough messages, players represent the game as if there was complete information, and are able to coordinate on the Pareto dominant equilibrium (Theorem 1). This is consistent with experimental evidence for this game, Camerer (2003). Note that we need all three components of the model to obtain this result. Firstly, the boundedly rational representation is necessary so that players who have received enough emails consider the message is

commonly known. Secondly, cooperative reasoning is needed to have players coor-dinate on Attacking when they receive enough messages - without it both actions would become rationalizable.1 _{Finally, we need competitive reasoning in order to}

have a default behavior that happens when players cannot engage in cooperative reasoning, and Level-k reasoning is a particularly appropriate because it has been widely applied and it coincides with risk-dominance in several important scenarios.

Global games (which are games of incomplete information where payoffs are observed with a small amount of noise) usually have a unique equilibrium, Morris and Shin (2003).2 However, this relies on assuming that players are extremely rational, and able to perform a long chain of reasoning. Heinemann et al. (2004)

showed that when tested experimentally, subjects follow some of the predictions of global games but violate several others. In particular, players are able to coor-dinate, and to use public signals, better than global games predict; facts which are captures by dual reasoning (Theorems 4 and 5). This is because dual reasoning captures the cognitive limitations of players, as well as their ability to engage in cooperative reasoning.

In the limit when the noise is zero, global games become games of complete information. I show that in 2 x 2 complete information games with two pure Nash

'In Jimenez-Gomez (2013) both actions are rationalizable after players observe enough mes-sages. In that paper this is something desirable, as we want to use the Interim Correlated Rationalizability as a solution concept.

2

equilibria, the Pareto dominant equilibrium or the risk-dominant will be chosen, as a function of risk-dominance and parameter o1 (Proposition 2). This is remark-able because game theorists have long debated whether the Pareto dominant or the risk dominant equilibrium should be chosen in such situations, Harsanyi and Selten (1988). I provide a simple condition under which the Pareto dominant equi-librium will be selected over the risk-dominant: this imposes a clear structure to the discussion on when we will expect players to coordinate on either equilibrium.

I model revolutions as coordination global games, where a mass of citizens chooses whether to participate or not and payoffs increase in the level of coordina-tion and the weakness of the fundamentals of the state. In that context I show that the government will choose to make public announcements when the fundamentals are strong and players assign low probability to others using cooperative reasoning (Proposition 6). When the government can jam the signal to make individuals be-lieve fundamentals are better than they truly are, the government jams the signal for intermediate values of the fundamentals: when they are very good, there is no risk of revolution; when they are very bad, revolution cannot be averted, Propo-sition 10. This is in contrast to the fully rational global games result, in which the government is always able to prevent a revolution, as the signals become more precise.

mind of the players, when facing a strategic situation. The formal way to do so is by merging three assumptions which have been successful in their respective domains: players have a bounded understanding of the game, they attempt to cooperate when doing so is not too risky (team reasoning), and they otherwise attempt to predict what the other player will choose, and best-respond to it (Level-k). There are two justifications for doing so. The first is the evidence: Decety et at. (2004)

showed that different neural circuits are involved in cooperative vs. competitive reasoning (cooperative reasoning engages orbitofrontal cortex differentially, and competitive reasoning engages inferior parietal and medial prefrontal cortices); Crawford et al. (2008) and Bardsley et al. (2010) showed that their experimental data could be explained by a combination of team reasoning and Level-k. The second, and perhaps more convincing reason, is that the model works: it offers a series of novel predictions and is better able to explain evidence from experiments (Heinemann et al., 2004; Cornand, 2006) than global games and Level-k models, what opens the possibility of applying the model to several interesting applications; I discuss this in more detail in the Conclusion.

1.1.1

Literature review

It has been long recognized by economists that game theory models assume too much rationality on the part of players. While rationality has a formal game-theoretic definition (that of maximizing utility given beliefs), what economists usually refer to is not a failure of this narrowly defined notion of rationality, but

rather the notion that people exhibit biases in their beliefs and limits in their computational capacity.3 The paper is connected to the literature on bounded ra-tionality on game theory, and especially to Level-k (Nagel (1995), Stahl and Wilson (1995, 1994)) and Cognitive Hierarchy Camerer et al. (2002) models. Two recent papers in this literature are especially relevant: Strzalecki (2014), who applies a Cognitive Hierarchy model to the Email game, and Kneeland (2014), who applies it to global games. In Kets (2014) agents can have finite and infinite hierarchies of beliefs, and players can have common certainty of events, even if they have finite hierarchies of beliefs. Her model is very general and encompasses standard game theory; in contrast I offer a model which is very simple and attempts to improve the accuracy of predictions, following the tradition of Level-k models.

In Economics, there is a literature, pioneered by Michael Bacharach and Robert Sugden on team reasoning: the game-theoretic idea that players are able, under certain circumstances, to coordinate on Pareto optimal outcomes, and therefore "reason as a team"; Sugden (1993), Bacharach (1999), Bacharach and Stahl (2000), Bacharach (2006). Outside of Economics, the literature on collective intentional-ity has been dominated by philosophers, Sellars (1968, 1980), Bratman (1999), Gilbert (2009). This literature has studied fundamental issues such as how do people represent a shared intention in their minds but, unlike the literature on team reasoning, little attention is paid to whether the actions taken by the players

3

Yildiz (2007) is an excellent example of a model where players are fully rational in this narrow sense, and yet they are wishful thinkers because they hold delusional beliefs.

are incentive compatible. Because of that, this literature is less relevant for the model in this paper, although it provides an intellectual basis for future research into the cognitive foundations of cooperative reasoning.

The false consensus effect refers to a cognitive bias by which a person over-estimates the extent to which her beliefs are representative of those of others, Ross et al. (1977). There is a long tradition in the psychology, and more recently game theory literatures of eliciting beliefs of players in games, which shows that the false consensus effects is also relevant to strategic interactions, Dawes et al. (1977), Mess6 and Sivacek (1979), Rubinstein and Salant (2014). Rubinstein and Salant (2014) is especially relevant, as they elicit both beliefs about other player's actions as well as profiles in the Chicken game. They find that people have a disproportionate tendency to assign their own beliefs to others.4 In the model, the assumption about how players represent information about the game implies that individuals exhibit the false consensus effect.

This paper is also deeply connected to the literature on equilibrium selection, 4_Other

papers related to the false consensus effect includes Kelley and Stahelski (1970), who found that in a repeated prisoner's dilemma, players who tended to defect often believed that others were like themselves, whereas those who were more cooperative where aware of the true distribution of defectors. Kuhlrnan and Wimuberley (1976) classified players as "individualistic", "competitive" and "cooperative" in a series of one-shot games and showed that each type assumed most others to be of their own type. A modern study is Iriberri and Rey-Biel (2013), which studies beliefs of players in a Dictator game, and find that selfish players exhibit more false consensus than other categories of players. These observations fit nicely within my framework, where players who engage in cooperative reasoning are one level of awareness above those who engage in competitive reasoning.

and global games in particular. Carlsson and van Damme (1993) show that in 2 x 2 games, by adding infinitesimally small noise to the original games, players conform to risk dominance. In "A General Theory of Equilibrium Selection in Games", Harsanyi and Selten endeavor to find a selection criterion that is univer-sal. They developed the risk dominance criterion, but concluded that whenever in conflict, payoff dominance should have precedence has a selection criterion. My model reconciles both criteria: risk-dominance affects the behavior in competitive reasoning, and Pareto dominance in cooperative reasoning.

The paper is also connected to notions of correlated equilibrium, Aumann (1987), Forges (1993), Dekel et al. (2007). While in correlated equilibrium there is some external correlation device which enlarges the set of equilibria, in this paper the correlation arises endogenously from the reasoning processes of the players: when players engage in cooperative reasoning, they are de facto coordinating their actions. This means that, in contrast to this literature, correlation reduces the set of outcomes when using dual reasoning.

The remainder of the paper is organized as follows. Section 3.2 presents the model, using the Email game as a running example. At the end of the section the model is applied to games of complete information, to discuss the connection between Pareto dominance and risk dominance. Section 1.3 discusses the empirical evidence on global games, presents the main theoretical results (consistent with the

evidence), and applies the model to revolutions. Section 1.4 ofers a generalization of model, which is applied to normally distributed signals and the case when the government can jam the signals. Section 3.6 concludes. The Appendix contains

proofs of all the theoretical results.

1.2

The model

We start the description of the model with some preliminary notation and con-cepts. There are only two players, I =

{1,

2}. Player i represents an arbitrary player, and -i represents the Player who is not i. Let

E

be the set of states of the world; each Player i has utility ui(O, ai, a-i). Given a set X, we represent by A(X) the set of probability measures on X. Let T be the type space for Player

i, such that there is a belief function

fl

: T -+ A(E x Ti), so that /

#(O,

3 Elti) is

the probability that type ti assigns to the state being 0 E

E

and Player -i being of type t-i E E C T-i. Finally, if Y is a random variable, we will denote by E[Y]

the expectation of Y.

1.2.1

Bounded representation

We turn now to describe how players use a boundedly rational representation of the game. In particular, Player i with level k can only reason about the first k-order beliefs: beyond that, Player i will "approximate" her higher order beliefs

using her lower order beliefs. Before expressing this idea formally, we need to introduce some standard concepts from the literature on epistemic game theory. For each E C

E

x T, x T₂ and each type tj E T, we define Et, as5

Eti

= {(0, t-i) : (9, t1, t2) E E}.

We define the belief of Player i as,

Bi(E) = {ti C : ,3(Et~Itj) = 1}.

We can define mutual belief B1(E) as:'

B'(E) =

E

x B1(E) x B2(E),

and k-mutual belief as

k-i

Bk (E) =

f

Bm (E) n B1(Bk-1 (E)). m=1

We define common certainty as 00

C(E) = k

B(E).

k=1

That is, C(E) is the set of states of the world at which players believe E, believe others believe E, believe others believe others believe E, etc. Note that common

5I follow Chen et al. (2014) in their exposition.

certainty requires that players be able to perform an arbitrarily long chain: "I believe you believe I believe..." However, most people are unable to perform such types of reasonings for more than 4 iterations, Kinderrnan et al. (1998). Because of that, we will assume that when there is mutual k-belief of an event, a player of

level-k considers that as evidence that there is common certainty of the event.

Assumption 1. If ti of level k believes E is k-mutually believed, then she believes

E is common certainty:

,i (Bk(E)Iti) = 1 ==-> #i(C(E)Iti) =

1.

Let T be the type space of the game, which we will call the objective type space. Assumption 1 implies that each type ti has a bounded representation of the game, which we will denote by Tk(ti) (where k is the level of the type), and call the representation. In this bounded representation, whenever there is k-mutual belief of an event, ti believes the event is common certainty, even if in T the event E might not be common certainty.8 In order to illustrate this, we analyze the email game, popularized by Rubinstein (1989).

Example 1 (Email game, Part I). There are two generals, and each of them in

charge of a division. Their objective is to conquer an enemy city, and in order to

7

An exception is Kets (2014), where players can have common certainty of an event, even when their beliefs are of finite order.

8

accomplish so, each of them must attack from a different location. Because of that, they can only communicate through email. Each general has two actions: to attack

(A), or not to attack (N). The enemy can be of two types: strong or weak. Both

generals share a common prior: the enemy is either weak or strong with probability

1/2 each.

Because the generals cannot communicate directly, they have implemented a communication system through email. General 1, who is closer to the enemy, learns whether the enemy is strong or weak. If the enemy is weak, an email is sent automatically to general 2, who sends an email back with the confirmation that 2 received the email from 1, etc. Each time an email is sent, it is lost with probability e. This process of emailing back and forth is repeated until an email is lost, in which case no more emails are sent.

Let's denote by tT the type for Player i who sent m messages. Note that to is the type who is certain that the enemy is strong and of Player 2 having sent 0 messages. For other types, their beliefs are as follows:

1. Player 2 who sends 0 messages (i.e. type t|) assigns probability 1/2 to the enemy being strong, and e/2 to the enemy being weak and the message lost. Therefore type tO assigns probability 1/(1 +,e) to the enemy being strong and type to, and e/(1 + c) to the enemy being weak and t' respectively.

probability E on the message being lost before Player -i received it, and (1- 6)E

on the message being lost after Player -i received. Therefore type t' assigns probability -I- and 1- to t'" 1 and t' respectively; and t"j assigns probability

- and to tm and tm+1 respectively.

Therefore, the objective type space T can be represented as follows:

)3(to) W 01(tI) W S 01(tM) W S to 1 0 -.. t1 0 to 0 1 2-E 2i 2-E 0 2 2-c 0 02(t ) W S 2(t),) W S 02(t2) W S tl 0 1 l ... t"' 1 0 0 0 1+ _2-c l E t2 1-E t+1 1-C

S T+E 1 2-E 1 2-E

Let's consider now how different types would represent the game. For example, consider Player 1 who sent 2 messages and has k = 1. Note that it is mutual belief that the enemy is weak, because both players know that the other received at

least 1 message. Assumption 1 implies that in T1 (t2), t2 believes that it is common

certainty that the enemy is weak.

there is no mutual belief that the enemy is weak, because Player 1 is not certain whether her email was received by Player 2. Because of that, the condition in Assumption 1 is not satisfied, and in T2_{(t') there is no common certainty that the}

enemy is weak. This hints to a very important role for k: the lower the k, the easier it is for a player to (maybe mistakenly) believe there is common certainty of an event.

1.2.2 Cooperative reasoning

Now that we have defined what a bounded representation of the game means, we need to properly address what is it that cooperative reasoning accomplishes. In doing so, I build on a vast literature from Philosophy and Economics, which have delved on the intricacies of cooperative reasoning and intention.9 A central concept in all of the traditions is the idea of a collective intention (also known as a "we-intention"): an intention that is shared by a group of people, and which encompasses the individuals in that group. For example, if Alice and Bob want to go for a walk together, this is a collective intention: it is not sufficient that each of them attempts to walk next to other person, but that is is a goal shared by both. I will avoid a philosophical analysis and will instead use a game theoretic approach, following the Bacharach-Sugden tradition in Economics.

9_{See Bacharach (2006) for a review of the idea in Economics, and Gold and Sugden (2007)}

Collective intention

In game theoretic terms, we will consider that a collective intention involves agents who attempt to play a Pareto dominant equilibrium. We say action profile a* is a Pareto dominant equilibrium at a given E

c

E

x T, x T2, if a* is a

Pareto dominant equilibrium of the game of complete information defined by 0, for each 0 E proj9E. We next introduce a slight generalization of the notion of q-dominance from Kajii and Morris (1997). We say a* is q-dominant at E C

E

x T, x T2 if for any type tj in E, it is a best response to play a* whenever other

types in E play a*i with probability at least q.

Definition 1. An action profile a* is q-dominant at E C 8 x T x T2 if for all tj E projr E and all i E I, it holds that:

/A-

XT-[ui(0, a*, a-j) - ui(0, a', a-i)] drq(a-i, tj)do(0, ti Iti)

0,

for any a' =L a* and for any rj E A(Ai x T-i) such that ri(a*i, tj) q for all

t_i E projT_.E.

We are now ready to define the key concept of p-collective intention, which is an action profile a* such that, for some types, it is common certainty that a* is a Pareto optimal equilibrium which is p-dominant.'(

10_{The definition of collective intention is based on the similar concept by Shoham and}

Definition 2. Given action profile a* and given E C

E x T x T

2, we say that a*

is a p-collective intention at E if

1. E c C(E)

2. a* is a Pareto optimal equilibrium at E

3. a* is p-dominant at E.

The intuition behind a collective intention a* is that all players believe that a* is a Pareto dominant equilibrium, all players believe that all players believe this, etc. Therefore, there is common certainty that it is in the best interest of everybody to coordinate on a*. However, playing a* might be risky, depending on the game payoffs. Because a* is p-dominant, the higher p is, the more risky is to play a*. When p = 0 playing a* is dominant for Player i, and when p = 1, Player i would play a* only if she is certain that everybody else is playing a*_ with probability 1.

To simplify the notation, given F C T, if there is S C

E

such that a* is a p-collective intention at E = S x F, we will also say a* is a p-collective intention

at F.

Example 2 (Email game, part II). We continue our example of the Email game.

The payoffs are as follows."

When the enemy is strong:

When the enemy is weak:

A N A -2,-2 -2,0 N 0, -2 0,0 A N A 1,1 -2,0 N 0,-2 0,0

In Part I of the example, we saw that in T1 (t2) there is common certainty that

0 = W. Moreover, when 0 = W, a* = (A, A) is a Pareto optimal equilibrium, so the first two conditions in Definition 2 are satisfied. Note that a* is 2/3-dominant because it the other player chooses A with probability at least 2/3, then the payoff from choosing A is at least 2/3 - (-2)/3 = 0. Therefore, a* is a 2/3-collective

intention in E.

We saw in Part I of the example, in T 2_{(tI ) there is no common certainty either}

that 0 = W or that 0 = S. The only event which is common certainty at T 2(ti) is

the event E =

E

x T 2(t'), but there is no action profile which is a Pareto dominant equilibrium at E. Therefore, there is no collective intention for t' of level k = 2.

From intention to action

Let

'

E [0, 1] be a parameter that measures the probability that others will engage in cooperative reasoning. In other words, if there is a collective intention a*, V) is the probability each player assigns to other playing according to a*. The following assumption determines when do players engage in cooperative reasoning.

Assumption 2 (Cooperative reasoning). For each Player i E I and each type ti E T, if there is a p-collective intention a* at Tk(ti) with > p, then ti plays ai.

Otherwise, ti engages in competitive reasoning.

The intuition behind Assumption 2 is as follows. Player ti conjectures that

t-i will engage in cooperative reasoning and therefore play a* i with probability

at least 0 and, because a* is p-dominant, a* is a best response to such conjecture whenever 0 > p. When this condition is not met, players cannot use cooperative reasoning, and will engage in competitive reasoning, which will be described in the Section below.

Example 3 (Email game, Part II continued). Because (A, A) is a 2/3-collective

intention at T'(t'), Assumption 2 implies that type t2 plays A whenever

4

> 2/3.

Intuitively, t2 of level 1 has a low ability to represent the game, and upon sending 2 messages she believes that "enough messages have been sent" so that it has become common certainty that 0 = W. While this is not the case in the objective type

play Attack when she assigns enough probability

4'

on Player 2 also Attacking. On the other hand, type t' of level 2 cannot engage in cooperative reasoning, because, as we saw, in her representation T 2(tl) there exists no collective intention.

1.2.3 Competitive reasoning: Level-k

In standard Level-k models of complete information, a Level-0 player chooses ac-cording to a given exogenous distribution, usually taken to be uniform, and a player of level k > 1 best responds to the belief that she is playing against a player of level k -1. We will preserve that basic intuition, and will apply it to games of in-complete information. Given a type ti, and a conjecture q (a-iItUi) E A(A-i x Ti), we define the best response as

BRt, (,q) = arg max ui(0, di, a-i)d (a-ilti)di (0, ti ti)

Given a type ti with level k, we will define the behavior of ti recursively. Let

Lk(tj) denote the behavior of type tj of level k. The level-0 behavior is defined,

as is usual in the literature, to be a uniform distribution over the set of actions. Then, we can define the behavior of types of level k > 1 recursively.

When Lk-1(t-i) _{is a singleton, Equation 1.1 is well-defined. When Lkt-1} )

is not a singleton, Equation 1.1 should be interpreted as the set that results from all possible conjectures for Lk-1 (t_. Lk(-) is the intuitive extension of Level-k to games of incomplete information. Type ti believes that the other players are of level k - 1; because it is a game of incomplete information, type ti also has beliefs about the state of the world and types of the other players. The combination of the beliefs about the level of others and the beliefs about the state of the world jointly determine behavior. We make this explicit in the following assumption.

Assumption 3 (Competitive reasoning). Type ti of level k who engages in

com-petitive reasoning plays according to Lk(t,).

Example 4 (Email game, Part III). Let's consider the behavior of a type of

Level-k. First, level-O players randomize uniformly. Because of that, it is a best-response

for a player of level 1 to play N, independently on her beliefs about 9 (when 0 = S, N is dominant, and when 0 = W, N is a best response to uniform randomization).

Therefore, all level 1 players choose N. This means that, for all k > 1, it is a best response to choose N, and therefore all players with k > 1 choose N

1.2.4

Putting it all together: Solving the Email game

Assumptions 2 and 3 imply that a type ti E T with level-k engages in a dual reasoning process: first she attempts to engage in cooperative reasoning, and if she cannot, she engages in cooperative reasoning. We can define this process formally as the solution concept we will use in the remainder of the paper.

Definition 3. Given k and

4',

we define the solution concept of dual reasoning Di(ti) as

a* _{if V' > p, where a* is a collective p-intention,}

Di (ti) -{i

Lk(t,) otherwise.

Going back to the Email game, let m be the number of messages received by

tv', and k her level. Rubinstein (1989) showed that the only rationalizable action, irrespectively of how many messages a player receives, is to never attack. The

intuition for his result is as follows. Players who sent 0 messages never attack, because they put enough probability on 0 = S (so that A is dominated by N).

Player 1 who sent 1 message puts high probability on Player 2 having sent 0 mes-sages, and therefore best respond by playing N. Player 2 who sent 1 message puts high probability on Player 1 having sent only 1 message, and also best re-sponds by playing N. Proceeding inductively, all players best respond by playing

N, irrespectively of how many messages they receive. We see here that there is a

contagion of behavior, by which the actions chosen by those who sent 0 messages affect the actions of everybody else. Note, however, that this argument requires that players be extremely rational, and able to follow such a long chain of reasoning.

dif-ferent result: tT attacks when m and 0 are large enough.12

Theorem 1. Player tT Attacks if and only if m > k+i=1 and ) > 2/3.

The intuition for this result is as follows. Because tT uses a boundedly rational k-representation, she believes that there is common certainty of 0 = W upon

re-ceiving enough messages. In that case, (A, A) is a 2/3-collective intention, and she attacks whenever 0 > 2/3. When she receives too few messages, or ' is not high enough, tn engages in competitive reasoning and, as we saw in Example 4, that means she plays N. Dual reasoning formalizes our intuition that after observing enough messages there is "common certainty" of 0 = W for all practical purposes,

and players Attack when they believe others engage in cooperative reasoning with high enough probability.

Compare Theorem 1 with the following result by Strzalecki (2014). He assumes that players of Level-0 always Attack (instead of randomizing), and shows that if

m > k+i=, Player i Attacks. Note that Theorem 1 and Strzalecki's result are

remarkably similar in the prediction of what will happen when enough messages are received. Their explanation, however, relies on mechanisms which are almost opposite. Strzalecki's result relies on a contagion process starting from Level-0 players, and showing that when enough messages have been received, a Player of

1 2

Unless otherwise stated, all the results in this paper are for k > 1. I follow the literature of Level-k models, which assumes that level-O agents do not exist in the population, but only on the minds of the players.

level k

+

1 is certain that Players of level k will Attack. The proof of Theorem 1 relies on the fact that when m is large enough, players cannot include the type who receives no messages in their representation, and therefore the contagion process from Rubinstein (1989) is prevented to happen. The game is thus transformed into a complete information game where players coordinate on the Pareto optimal equilibrium. '3

1.2.5 Games of complete information

Note that in a game of complete information, the first step of representing the objective type space is redundant, because

e

is a singleton. However, the model still has interesting predictions to make in this context. In order to obtain the cleanest possible results, I focus on the simplest possible games: 2 x 2 symmetric 2-player games of complete information, which can be represented by 4 parameters as follows:

L R

L mw xy

R y, xz, z

Table 1.1: Symmetric game

Let's consider the case where the games has two pure-strategy Nash equilibria:

13_{Theorem 1 is also connected to a result in Jimnenez-Gomez (2013). In that model, agents} also represent the game as if there was complete information when they receive enough messages; unlike here, agents cannot engage in competitive reasoning, but behave according to interim correlated rationalizability, Dekel et at. (2007).

(L, L) and (R, R). Moreover, consider that (L, L) is the Pareto optimal equilibrium

(hence w > z). Note that (L, L) is p-dominant, for p defined as:

z - x

pw+(1-p)x=py+(1-p)z == p=

If p < 1/2, then (L, L) is risk-dominant, otherwise (R, R) is risk-dominant. We have the following result.

Proposition 2. Player i chooses L if and only if 1/2 > p or

4

> p."

The intuition is as follows: because this is a game of complete information, the first two conditions of Definition 2 of collective action (i.e. that it is common certainty that (L, L) is Pareto dominant) are immediately satisfied. Therefore, players choose action L, corresponding to the Pareto dominant equilibrium, when this is not too risky, i.e. when

4

> p (by Assumption 2). Otherwise, Player i

engages in competitive thinking, which will lead to playing according the risk-dominant equilibrium (and hence choosing L when p<1/2).

Proposition 2 has two conditions under which players choose L. The first con-dition of 1/2 > p corresponds to risk-dominance, Harsanyi and Selten (1988). The second condition of

4'

> p happens when players engage in cooperative

reason-ing and choose the Pareto dominant equilibrium. In their classical discussion, Harsanyi and Selten (1988) argued that Pareto dominance should be used as an

"At p = 1/2, dual reasoning predicts that either L or R are played, as individuals are indif-ferent. I will omit conditions for those knife-edge cases in the rest of the paper, as they would needlessly complicate the statements of the theorems.

equilibrium refinement and, in the absence of a Pareto dominant equilibrium, risk dominance should be used.-5 _{Proposition 2 gives a very clear prediction for this}

simple class of games: when 4 is high enough as compared to p, players

coordi-nate on the Pareto dominant outcome; when 4' is not large enough, they choose

the risk dominant outcome. This captures Harsayi and Selten's intuition that in games where the Pareto dominant equilibrium is more rewarding as compared to the risk dominant equilibrium, players are more likely to coordinate on the Pareto dominant equilibrium. Moreover, the model as applied to 2 x 2 symmetric games is a generalization of Pareto dominance (when 4 = 1) and risk dominance (when

4' = 0). Following Matthew Rabin's argument for extending economic models in a testable way, Rabin (2013b,a), the model allows us to perform a statistical test the hypothesis HO : 0 = 0 (which corresponds to a Level-k model) or HO : 4' 1 (which corresponds to a team-reasoning model).

1.3

Global games

The idea behind global games is to turn a game of complete information into a game of incomplete information, by adding a small amount of noise, Carlsson

"They wrote: "risk dominance is important only in those situations where the players would

be initially uncertain whether the other players would choose one equilibrium or the other. Yet,

if one equilibrium would give every player higher payoffs than the other would ( ... ) every player can be quite certain that the other players will opt for this equilibrium which will make risk dominance considerations irrelevant" p. 358); "a theory that considers both payoff and risk, dominance is more in agreement with the usual image of what constitutes rational behavior"

and van Damme (1993). Heinemann et al. (2004) took this idea to the lab, and run an experiment. The game is a coordination game with two actions: action

A is "safe" and gives a constant payoff T. Action B is "risky", yielding a payoff

of Y when the number of subjects who choose B is at least C1 - C2 - Y, (where C1, C2 are constants). Subjects had access to this formula and to a table in the instructions. Let U[a, b] denote a uniform distribution of the interval [a, b]. The state Y is randomly selected from U[10, 90]. In the Common Information (CI) treatment, Y is common certainty. In the Private Information (PI) treatment, each individual receives a signal Xi, i.i.d. U[Y - 10, Y + 10]. The following is a

summary of the evidence from Heinemann et al. (2004) :16

1. Undominated thresholds: strategies are consistent with undominated thresholds

2. PI thresholds: with PI, mean thresholds are close to or below the global games equilibrium

3. CI thresholds: with CI, there is more coordination on the Pareto dominant equilibrium than what global games predict

4. Comparative statics: mean thresholds follow the predicted comparative statics

5. Information effect: with PI, mean thresholds are higher than with CI "The original fact 6 in Heinemann et al. (2004)has been omitted because it is about how ordering affects play, and none of the theories is concerned with that.

6. Coordination: with PI, there is more variation in individual thresholds than with CI

7. Predictability: variation in mean thresholds across sessions is similar for CI and PI

The following table summarizes whether these experimental facts are predicted by three different theories: global games, Level-k model as in Kneeland (2014), and the dual reasoning prediction offered in this paper. As we can see, the theory of global games predicts too conservative behavior with PI, and fails to predict that there is more coordination with CI.

Global games Level-k model Dual reasoning

1. Undomin. thresholds Y Y Y

2. PI thresholds Y/N Y Y

3. CI thresholds N N Y

4. Comparative statics Y Y Y

5. Information effect N 'Y/N Y

6. Coordination N Y Y

7. Predictability N Y Y

1.3.1

Theoretical predictions

Consider the following game, which has been used extensively in the global games literature.

Attack Not attack

Attack 7, - 17,0

Not attack 0, 0 - 1 0,0

When 0 > 1, Attack is strictly dominant, whereas if 0 < 0, Not attack is strictly dominant. When 0 E [0, 1] both actions are rationalizable. With complete information, a direct consequence of Proposition 2 is the following corollary.

Corollary 3. Player i Attacks if and only if 0 > 1/2 or 0 > 1 - ).

Now suppose that each player observes a signal xi = 0 + ej, where ei ~ i.i.d

U[-u, o-]. The original game of complete information has two pure equilibria;

the powerful result from global games is that by adding a small amount of noise, the equilibrium is unique and it coincides with risk dominance, Carlsson and van Damme (1993). In particular, as the noise vanishes (i.e. a -+ 0), players attack if and only if they receive a signal greater than 1/2, Morris and Shin (1998). However, this result depends on players following an arbitrarily long chain of reasoning. The argument is as follows: players with xi + o < 0 have Not attack as a dominant action, and those with xi - o > 1 have Attack as dominant. Given this, players

receiving nearby signals will play the same action (i.e. those with xi + a close to 0 will not attack, those with xi - a close to 1 will attack). Proceeding in this way a contagion happens, where the action of types closer and closer to intermediate signals is determined by a chain of best-responses started by those with extreme signals. For a -+ 0, however, this contagion takes an unbounded number of steps,

and our intuition should warn us that players might not be using such reasoning process. This intuition is formally captured in the following result, in which we see that, for contagion to happen, the cognitive level k needs to increase as o- becomes small.

Theorem 4. Player i with signal xi Attacks if and only if xi > 1/2 or xi -o-- k >

I - b.

Theorem 4 maintains xi > 1/2 as a sufficient condition to coordinate on Attack, but there is another condition under which Attack is possible: if the noise is sufficiently small (as compared to k), then players can approximate the type space as having common certainty that E[O] > 0, in which case players can coordinate on Attacking through cooperative reasoning. Therefore dual reasoning predicts successful coordination on Attack in strictly more cases than global games do, as the the experimental evidence suggests. For comparison, it is instructive to observe what happens when the players observe a public signal y ~ U[O - a, 0 + -].

Theorem 5. With a public signal y, players Attack if and only if y > 1/2 or y 1 -0.

Theorem 5 shows that, when players observe a public signal, the conditions are analogous to those we found in Corollary 3 for the game of complete information. It is instructive to compare this result to Izmalkov and Yildiz (2010). In their model, players believe that others received a higher signal than themselves with probability q: when q > 1/2, agents are overoptimistic. They find that Player i

attacks if and only if xi 1- q, a condition which is analogous to xi 1- 0. Dual reasoning, in addition, is able to explain why players are better able to coordinate with public than with private information.

Taken together, Theorems 4 and 5 prove that dual reasoning is consistent with the evidence from Heinemann et al. (2004). Players use undominated thresholds (Fact 1), since they play Attack when their signal is higher than a certain thresh-old in [0,11. The threshthresh-olds with private signals are lower than 1/2 and lower than the threshold for a public signal, and this implies that there is more coor-dination with the public signal (Facts 2, 5). Thresholds with private signals are more disperse than with a public signal, and the latter is between the threshold for risk-dominance and Pareto dominance (Facts 3, 6). The comparative statics hold: players attack less as 0 decreases (or if the cost increased from 1), Fact 4. Finally, the predictability of a successful attack is the same in both conditions: it depends on 0, 0 and k, Fact 7 (in particular, there is no multiplicity that could generate excess variability in public signal case).

Note that the fact that the model is able to accommodate the experimenta evidence so well is only possible because of the three components of the model: bounded rationality, cooperative reasoning and competitive reasoning. Without bounded rationality in the representation of the game, the conditions for cooper-ative reasoning would never be fulfilled, and the model would reduce to a Level-k

model. If we did not have cooperative reasoning, bounded rationality would imply that there would be multiple equilibria, as players with low enough k would have both actions rationalizable. Finally, competitive reasoning provides the "default" behavior when cooperative reasoning fails, which is why we maintain the sufficient condition for attacking when agents receive a signal greater than 1/2. It is all three components together that make the model fit the empirical results.

1.3.2

Revolutions and Collective Action

To model revolutions, I will use the simplest possible extension to the coordination model in Section 1.3. There is a continuum of mass 1 of citizens, and each of them chooses whether to Attack or Not attack. If an individual chooses not to attack, her payoff is constant and equal to 0. If the individual chooses to Attack, she obtains 0 if the revolution succeeds and 0 - 1 if it does not succeed. When fraction q of the citizens attack, the revolution succeeds with probability q. Therefore, conditional on a fraction q of individuals attacking, the expected payoffs are given by:

Fraction q attack

Attack 0-(1-q)

Not attack 0

Suppose that a self-interested government wants to avoid revolution. The gov-ernment observes 0 perfectly, and can choose one of two options:

. to disclose 9 as a public signal with perfect accuracy

* not to disclose any public signal, in which case each citizen receives a private signal xi - i.i.d U[9 - o-, 9 + o].

Proposition 6. The government discloses 9 if and only if 9 < 1/2 and '$ < 1 -0.

The intuition of this result is as follows. When 9 > 1/2, if players learn the true value of 9, they always attack because it is risk-dominant (and hence selected by competitive reasoning). Therefore, in that case, the government does not disclose 9. When 9 < 1/2, the government faces a tradeoff: disclosing 9 informs individuals that times are bad for them, but also aids them in coordination. The smaller 0 is, the larger the first effect; when

4

< 1 - 9, the informational effect is more important than the coordination effect.

1.4

Generalized Model and Applications

Up to now, we have required that players have common certainty of an event E in order for a* to be a collective intention at E. That requirement, however, seems intuitively too strict: what if agents are almost certain about E, but consider that things might be otherwise with a small probability? In this section we will extend the concept of collective intention for the case when E is common p-belief, Monderer and Samet (1989). Recall from Section 1.2.1, that for each E C

E

x

Eti = {(0, t-i) : (0, t1, t2) E E}.

We define the p-belief of Player i as,

B(E) = {tj E T : i (Eti Iti) > p}, and the mutual p-belief BP(E) as:

BP(E) =9 x B'(E) x B2(E).

We define B"'P(E) = BP(E), and the k-mutual p-belief for k > 1 by:

k-1

Bk'P(E) - n BM"'P(E) n BP(Bk-l'P(E)),

m=1 and we define common p-belief as

CP(E) = B p(E).

k=1

In order to extend the model to the case of common p-belief, we need to modify Assumption 1, so that it accommodates the concept of p-belief.

Assumption 4. If tj of level k p-believes E is k-mutually p-believed, then she

p-believes E is commonly p-believed. Formally:

Note that Assumption 4 generalizes Assumption 1, because the latter is a particular case when p = 1. We also need to re-define the concept of collective intention, to take into account the fact that the event E might believed with probability less than 1.

Definition 4. Given action profile a* and given E C E) x T x T2, we say that a*

is a p-collective intention at E if p = q/s, with

1. E c Cs(E)

2. a* is a Pareto optimal equilibrium at E

3. a* is q-dominant at E.

Definition 4 generalizes Definition 2, and Assumptions 2 and 3 remain un-changed: players engage in cooperative reasoning whenever a* is a p-collective intention, and 0 > p; otherwise they engage in competitive (Level-k) reasoning. Note that when s = 1, this boils down to the condition we had in Section 3.2; however when s < 1, the condition ?/ > q/s requires either that 0 is higher or q lower. This is intuitive: when event E is commonly s-believed for s < 1, there is probability 1 - s that the other player does not believe in E. Because of that, a* has to be less risky (hence q be lower), or that the other player engages in