Testing behavioral game theory: higher-order

rationality and consistent beliefs

Terri Kneeland⇤

October 17, 2013

Abstract

This paper investigates the behavioral and experimental support for differentepistemic conditions that form the foundations of several game theoretic solutionconcepts. It employs strategic choice data from a carefully chosen set of ring-network games to obtain individual-level estimates of the following epistemicconditions: rationality; beliefs about the rationality of others; and consistentbeliefs about strategies. We find that 94 percent of subjects are rational, 72percent are rational and believe others are rational, and 44 percent are rationaland hold at least second-order beliefs about the rationality of others. Of the72 percent of subjects that satisfy the sufficient rationality conditions neededto observe consistent beliefs, none of them satisfy consistent beliefs. Not asingle subject satisfies all three sufficient epistemic conditions required for Nashequilibrium. The unique design allows us to weigh the relative plausibility ofalternatives to Nash equilibrium; the data tend to support the level-k model.

Keywords: epistemic game theory, behavioral game theory, Nash equilibrium,

level-k, qre

⇤Department of Economics, University College London, Gower Street, London WC1E 6BT. E-mail: t.kneeland@ucl.ac.uk. I am extremely grateful to my supervisor Yoram Halevy for supportand guidance throughout the process of developing and writing this paper. I am also grateful toMike Peters for numerous conversations, without which this paper would not have been possible,and Ryan Oprea for invaluable feedback and encouragement. In addition, I would like to thankDoug Bernheim, Colin Camerer, Vince Crawford, David Freeman, Amanda Friedenberg, Ed Green,Li Hao, Andrew Hill, Kevin Leyton-Brown, Wei Li, Ran Spiegler, and Charles Sprenger for helpfulconversations and comments.

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1 Introduction

Over the past 20 years there has been an accumulation of experimental evidence thatchallenges the empirical validity of Nash equilibrium predictions when subjects inter-act in novel environments. This evidence not only prompts the need for alternativesolution concepts to explain behavior in one-shot games, but an understanding ofwhy Nash equilibrium fails in the laboratory can provide valuable insight into theplausibility of alternative solution concepts.

Solution concepts do not simply provide a description of behavioral predictionsbut embody assumptions about players’ rationality and beliefs that underpin thesepredictions. Three epistemic assumptions form the foundations for several commonsolution concepts. The first is rationality: whether a player plays a best response toher beliefs. The second is higher-order rationality: whether a player believes othersare rational, whether she believes others believe others are rational, and so on. And,the third is consistent beliefs about strategies: whether a player believes others holdcorrect beliefs about the action she is playing.

The Nash equilibrium solution concept requires players to be 2nd-order ratio-nal (rational and believe others are rational) and have consistent beliefs (Aumannand Brandenburger 1995). Rationalizability relaxes the assumption of consistent be-liefs but strengthens the rationality assumptions to common knowledge of rationality(Bernheim 1984; Pearce 1984). Behavioral solution concepts also impose a combi-nation of the three assumptions. The level-k solution concept assumes rationality,relaxes consistent beliefs, and allows for heterogeneity in beliefs about the rational-ity of others (Nagel 1995; Stahl and Wilson 1995; Costa-Gomes and Crawford 2006;Camerer et al. 2004). While quantal response equilibrium (QRE) relaxes the as-sumption of 2nd-order rationality but maintains the assumption of consistent beliefs(McKelvey and Palfrey 1995).1

This paper uses theory and an experiment to identify whether the three epistemicassumptions hold at the individual level. The methodology here differs substantiallyfrom previous experimental work that evaluates solution concepts. Existing gametheoretic work compares different solution concepts by jointly testing the epistemicassumptions in combination with additional structural assumptions. For example, in

1See Kneeland (2012) “Rationality and consistent beliefs: theory and experimental evidence”, anearlier version of the current paper, for epistemic characterizations of the level-k and QRE solutionconcepts.

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the case of Nash equilibrium, rationalizability, and level-k, an error distribution istypically imposed ex post to allow the solution concepts to fit the data. In addition,level-k models always impose a specification for the behavior of L0 types. This specifi-cation then anchors the beliefs of all other levels by having players play an action thatis a finitely iterated best response to the L0 behavior. In the case of QRE, it is typi-cally assumed that player’s mistakes when best responding are determined accordingto a logistic error structure. Players then noisily best respond to other players’ noisybest responses. Models are then judged in terms of goodness-of-fit under an ex postmodel-fitting approach. Under this approach, the structural assumptions cannot beseparated from the epistemic assumptions themselves and it is difficult to assess howmuch power they allow when fitting the different models to data.2 The approach inthis paper is to directly test the three main epistemic assumptions underlying oursolution concepts. Untangling the role that rationality and consistent beliefs play instrategic reasoning will allow direct comparison of solution concepts without relyingon structural assumptions that risk introducing bias.

In order to directly test the epistemic assumptions, we use choice data to makeinferences about player’s beliefs. However, this approach has an identification prob-lem that stems from the fact that there are generally a number of different beliefsthat lead to the same behavior. The main problem comes in identifying a playersorder of rationality: the behavioral implications of higher orders of rationality arecontained in the behavioral implications of lower orders of rationality. To deal withthis identification problem, we focus on a special class of games, ring games. Ringgames allow a natural exclusion restriction to be incorporated into the experimentaldesign because there are features of the payoff environment that affect higher-orderrational types but not lower-order rational types. The payoff features are due to therelaxed opponent structure of the ring game. A ring game is essentially a series oftwo-player normal form games where the opponent structure is relaxed relative tostandard game forms. For example, in the 3-player ring game in Figure 1, player 1’spayoff depends on the action of player 2. Player 2’s payoff depends on the action ofplayer 3. And, player 3’s payoff depends on the action of player 1.

This payoff structure allows one to define games which induce differences only2Wright and Leyton-Brown (2010) provide the most exhaustive test to date between Nash equi-

librium and a set level-k and QRE models by performing a meta-analysis on a data set generatedfrom existing normal-form game experiments using standard structural methodology. Level-k andQRE perform equally well in these tests.

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in higher-order payoffs without affecting lower-order payoffs. This is not possible instandard game forms (e.g. bimatrix games) because there is a tight link betweenhigher- and lower-order payoffs. A player’s own payoffs affect her 2nd, 4th, 6th-orderpayoffs and so on, while her opponent’s payoffs affect her 1st, 3rd, 5th-order payoffsand so on. Thus, it is not possible to change higher-order payoffs without affectinglower-order payoffs. But, adding a new player into the game creates an additionaldegree of freedom. In the 3-player ring game, we can change player 3’s payoffs withoutaffecting player 1’s own payoffs or her 1st-order payoffs. Thus, ring games can be usedto isolate the behavioral implications of different orders of rationality under a naturalexclusion restriction: lower-order rational players do not respond to changes in higher-order payoffs and by focusing on a carefully chosen set of ring games that exploits thisidentifying assumption. To the best of my knowledge, this is the first paper to usering games (or network games in general) to study strategic reasoning empirically.3,4

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Figure 1: 3-player ring game

Identification of consistent beliefs is also problematic, as on its own, consistentbeliefs has no behavioral implications. To solve this problem, we combine assumptionsof rationality and consistent beliefs to derive testable behavioral implications. Wedevelop a characterization of Nash equilibrium that requires both rationality andconsistent beliefs. Given that a player satisfies the sufficient rationality conditions,identified from behavior in ring games, playing a Nash equilibrium is indication ofwhether or not a player has consistent beliefs. The characterization developed inthis paper follows Aumann and Brandenburger (1995) and Perea (2007). We usean epistemic belief structure similar to Perea, characterizing behavior in terms of

3Cubitt and Sugden (1994) use a ring game to illustrate a paradox related to iterated deletion ofweakly dominated strategies.

4For a review of the literature on network games, or the analogous concept in computer science(graphical games) see Jackson (2005) and Kearns (2007).

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non-interactive belief conditions, which allows us to characterize behavior in termsof one player’s beliefs (rather than conditions on multiple players simultaneously asin Aumann and Brandenburger (1995)) which is naturally suited to the applicationof individual decision data. In addition, we provide a tighter characterization thanPerea, analogous to Aumann and Brandenburger, allowing us to identify consistentbeliefs for a larger set of subjects.

This paper uses strategic choice data to obtain individual-level estimates of thethree epistemic conditions. We find that 94 percent of subjects behave consistentlywith rationality. 72 percent behave consistently with 2nd-order rationality (subjectsare rational and believe others are rational). 44 percent behave consistently with3rd-order rationality (subjects are 2nd-order rational and hold 2nd-order beliefs thatothers are rational). And, 20 percent behave consistently with 4th-order rationality(subjects are 3rd-order rational and hold 3rd-order beliefs that others are rational).5

The epistemic conditions sufficient for Nash equilibrium in two-player games are 2nd-order rationality and consistent beliefs about strategies. This means that 28 percentof subjects do not satisfy the sufficient rationality conditions for Nash equilibrium.However, of the 72 percent of subjects that satisfy the sufficient rationality conditionsrequired to identify consistent beliefs, none of them satisfy consistent beliefs. Not asingle subject satisfies all three sufficient epistemic conditions for Nash equilibrium.The results provide support for the key features of the level-k and cognitive hierarchymodels; heterogeneous orders of rationality and inconsistent beliefs are importantfeatures of strategic reasoning at the individual level.

This paper is related to Healy (2011) and Costa-Gomes and Weizsäcker (2008).Both Healy’s work and this paper test the epistemic conditions of rationality and con-sistent beliefs in the lab. The main differences between the papers is the methodology.Healy estimates epistemic conditions by eliciting subjects’ beliefs about actions, pay-offs and rationality, and about their opponents’ beliefs about actions and payoffs.Healy finds that subjects are less rational than we find in the current paper (ac-tions are not always a best response to stated 1st-order beliefs), but the results for2nd-order rationality (a direct elicitation of whether subjects believe their opponents

5Of note, the estimated order of rationality distribution in this paper is interpretable as anestimate of the level-k level distribution that is independent of the L0 specification. The only otherpaper to provide a L0 independent distribution is Burchardi and Penczynski (2010). They incentivizecommunication between group members and analyze the communication in order to determine asubject’s depth of reasoning.

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are rational)6 are roughly consistent. As well, he finds little evidence of consistentbeliefs though our definitions of consistent beliefs differ. This paper defines consis-tent beliefs as a condition on one player’s belief hierarchy (does a subject believe heropponent believes she is playing the action she is actually playing) whereas Healydefines consistent beliefs based on beliefs and actions (are a subject’s elicited beliefsabout her opponent’s action consistent with the action her opponent actually plays).Costa-Gomes and Weizsäcker elicit beliefs and choices in normal form games. Theyfind that players tend not to best respond to their own stated beliefs (suggesting alarge proportion of subjects are not rational), however they also find that players be-have differently when asked to state beliefs. Whether or not belief elicitation affectsstrategic behavior remains an open question. See Healy (2011) for a more completediscussion of these issues. Thus, this paper takes a different approach and uses adesign specifically chosen to allow identification of epistemic conditions from choicedata.

This paper is also related to the literature on iterated dominance (Beard and Beil1994; Andrew et al. 1994; Huyck et al. 2002; Ho et al. 1998; and Costa-Gomes et al.2001), as we estimate a player’s order of rationality based on her choices in dominancesolvable games. The existing literature tends to focus on violations of iterated domi-nance, with the exception of Ho et al. (1998) which measures a subject’s capability toperform levels of iterated dominance as her level-k level. The current paper measuressubjects order of rationality by classifying subjects into levels of iterated dominanceusing a more general identifying assumption than the level-k structure.

This paper proceeds as follows. Section 2 introduces the ring game and moti-vates the experimental design through examples. Section 3 formalizes the epistemicconcepts of rationality and consistent beliefs and the identifying assumptions andcharacterization used to identify the epistemic conditions. Section 4 discusses the ex-perimental design. Section 5 gives the experimental results. And, Section 6 providesa discussion of the solution concepts of Nash equilibrium, rationalizability, quantalresponse equilibrium, and level-k models in relation to the results. Section 7 con-

6While it is possible to test rationality by testing whether a subject’s choices are a best responseto her stated 1st-order beliefs about actions, it is not possible to test higher-order rationality in thismatter (i.e. failure of a subject’s stated 1st-order beliefs about actions to be a best response to herstated 2nd-order beliefs about actions does not necessarily indicate a failure of 2nd-order rationality).To avoid this problem, Healy estimates 2nd-order rationality by eliciting subjects’ beliefs about therationality of their opponents. This has the added difficulty of requiring players to understand theconcept of rationality directly.

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cludes.

2 Example

The main goal of this paper is to separate the extent to which the three assumptions:(1) rationality, (2) beliefs about rationality, and (3) consistent beliefs contribute tostrategic reasoning. Are people rational? To what order do people believe others arerational? And, do people have consistent beliefs? In this section, we illustrate thechallenge in estimating these three features of reasoning from strategic choice dataand motivate the ring game as a solution to this problem.

Consider the problem of estimating a subject’s order of rationality. Bernheim(1984), Pearce (1984), and Tan and Werlang (1988) establish the strategies that canbe played by a subject who satisfies kth-order rationality7: a subject who satisfieskth-order rationality must play a kth-order rationalizable action. In many games(and in all the games considered in this paper) a strategy is kth-order rationalizableif and only if it survives k rounds of iterated deletion of strictly dominated strategies.

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Figure 2: B1

Consider the game B1 in Figure 2 (the first matrix represents player 1’s payoffs andthe second matrix represents player 2’s payoffs). This game is 2nd-order rationalizablefor player 1 and 1st-order rationalizable for player 2. If player 2 is rational she mustplay action a. If player 1 is rational she can play either a or b. If she satisfies 2nd-orderrationality (is rational and believes player 2 is rational) then she must play action a.

7kth-order rationality corresponds to holding finite-orders of beliefs about the rationality of oth-ers. For example, 1st-order rationality means you are rational, 2nd-order rationality means you arerational and you believe your opponent is rational. 3rd-order rationality means you are rational,believe that your opponent is rational, and believe that your opponent believes her opponent isrational. This is formalized in Section 3.

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Suppose we observe a subject play the action a as player 1. She may have playeda because she satisfies 2nd-order rationality. However, we cannot rule out the possi-bility that the subject only satisfies lower orders of rationality. This is because therationalizable sets necessarily contain one another. The 1st-order rationalizable setcontains the 2nd-order rationalizable set which contains the 3rd-order rationalizableset and so on. This leads to an identification problem.

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Figure 3: B2

This problem can be resolved by observing behavior in related sets of games undera natural exclusion restriction. Consider the game B2 in Figure 3. This game is 2nd-order rationalizable for player 1 and 1st-order rationalizable for player 2. If player 2is rational she must play action b. If player 1 satisfies 2nd-order rationality she mustplay action b. In addition, B2 is related to B1 in a structured way: player 1 has thesame payoffs in both B1 and B2.

This structure along with an exclusion restriction allows us to solve our identi-fication problem. We assume that subjects satisfying lower-order rationality but not

higher-order rationality do not respond to changes in higher-order payoffs.

8 For ex-ample, we assume a subject that satisfies rationality but not 2nd-order rationalitywill not respond to changes in 1st-order payoffs. Under this assumption, such a sub-

8This exclusion restriction seems empirically valid. In our experimental data, subjects follow therestrictions of this assumption 83 percent of the time. In addition, the assumption is supportedby the limited depth of reasoning literature. If a subject has a finite depth of reasoning k, thenshe satisfies kth-order rationality, but not (k+1)th-order rationality and will ignore any informationthat has to be processed at k+1 depths of reasoning. For example, if a subject is rational (and nothigher-order rational) because she has a depth of reasoning of 1 then she will not base her decision onany payoffs besides her own. There is empirical support for this type of behavior from experimentsthat analyze the information search patterns of subjects. Costa-Gomes et al. (2001), Costa-Gomesand Crawford (2006), Wang et al. (2009), Brocas et al. (forthcoming), Camerer et al. (2002), andJohnson et al. (1993) all analyze strategic behavior by investigating the information search patternof subjects. They find a correlation between the play of kth-order rationalizable strategies andpatterns of search that are associated with k depths of reasoning.

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ject would play either (a, a) or (b, b) as player 1 in games B1 and B2 respectively,however a subject who satisfies 2nd-order rationality would play action profile (a, b).Observing the action profile of a subject in both games B1 and B2 would then al-low us to separate the behavioral implications of 2nd-order rationality from 1st-orderrationality.

Following this logic, the behavioral implications of kth-order rationality can beseparated from lower-orders of rationality by looking at games that differ only in (k-1)th-order payoffs but have different kth-order rationalizable implications. However,no such bimatrix games exist. This is because there is a tight link between higher-and lower-order payoffs in bimatrix games. A player’s opponent’s payoffs determineher 1st-, 3rd-, 5th-order payoffs, and so on. While her own payoffs determine her2nd, 4th-, 6th-order payoffs, and so on. Higher-order payoffs are not independent oflower-order payoffs. Therefore, any two bimatrix games with the same payoffs up toat the 2nd-order must be the same game (and hence have the same rationalizableimplications).

This paper solves this problem by making use of a novel class of games: ringgames. A ring game is essentially a series of two-player games that have a uniqueopponent structure. In a standard 2-player game, player 1 and player 2 are eachother’s mutual opponent. But, in a ring game, player 1’s opponent is player 2 butplayer 2 has an entirely different opponent, player 3. The opponent structure of ringgames allows us to solve the identification problem because it allows us to inducechanges in higher-order payoffs independently of lower-order payoffs.

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a! 5! 0! !! ! b! 5! 10! ! b! 5! 10! ! b! 10! 5! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

!!!!!!!

! ! ! ! Player!1! ! ! ! Player!2!! ! ! ! Player!2's!actions! ! ! ! Player!1's!actions!! ! ! ! a! b! ! ! ! a! b!B2! !

Player!1's!actions!

a! 15! 5! !

Player!2's!actions!

a! 5! 0!! ! b! 5! 10! ! b! 10! 5!! ! ! ! ! ! ! ! ! ! !

Figure 4: R1

Consider the game R1 in Figure 4. R1 is 1st-order rationalizable for player 3,2nd-order rationalizable for player 2, and 3rd-order rationalizable for player 1. Thus,player 3 must play a if she is rational, player 2 must play a if she satisfies 2nd-order

9

rationality and player 1 must play a if she satisfies 3rd-order rationality (rational,believes her opponent is rational and believes her opponent believes her opponent isrational). Game R2, in Figure 5, is 1st-order rationalizable for player 3, 2nd-orderrationalizable for player 2, and 3rd-order rationalizable for player 1. Player 3 mustplay b if she is rational, player 2 must play b if she satisfies 2nd-order rationality andplayer 1 must play b is she satisfies 3rd-order rationality.

!! ! ! ! Player!1! ! ! ! Player!2!! ! ! ! Player!2's!actions! ! ! ! Player!1's!actions!! ! ! ! a! b! ! ! ! a! b!B1! !

Player!1's!actions!

a! 15! 5! !

Player!2's!actions!

a! 10! 5!! ! b! 5! 10! ! b! 5! 0!! ! ! ! ! ! ! ! ! ! !

!!

!!! ! ! ! Player!1! ! ! ! Player!2! ! ! ! Player!3! !! ! ! ! Player!2's!actions! ! ! ! Player!3's!actions! ! ! ! Player!1's!actions! !! ! ! ! a! b! ! ! ! a! b! ! ! ! a! b! !R1! !

Player!1's!actions!

a! 15! 5! !

Player!2's!actions!

a! 15! 5! !

Player!3's!actions!

a! 10! 5! !! ! b! 5! 10! ! b! 5! 10! ! b! 5! 0! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

!! ! ! ! Player!1! ! ! ! Player!2! ! ! ! Player!3! !! ! ! ! Player!2's!actions! ! ! ! Player!3's!actions! ! ! ! Player!1's!actions! !! ! ! ! a! b! ! ! ! a! b! ! ! ! a! b! !R2! !

Player!1's!actions!

a! 15! 5! !Pla

yer!2's!actions!

a! 15! 5! !

Player!3's!actions!

a! 5! 0! !! ! b! 5! 10! ! b! 5! 10! ! b! 10! 5! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

!!!!!!!

! ! ! ! Player!1! ! ! ! Player!2!! ! ! ! Player!2's!actions! ! ! ! Player!1's!actions!! ! ! ! a! b! ! ! ! a! b!B2! !

Player!1's!actions!

a! 15! 5! !

Player!2's!actions!

a! 5! 0!! ! b! 5! 10! ! b! 10! 5!! ! ! ! ! ! ! ! ! ! !

Figure 5: R2

In addition, R1 and R2 are related to each other in a structured way. Player 1has the same payoffs and the same 1st-order payoffs in games R1 and R2 but thegames have different 3rd-order rationalizable implications. This is possible becausethe relaxed opponent structure of the ring game breaks the tight link between higher-and lower-order payoffs that bimatrix games impose.

R1 and R2 differ only in 2nd-order payoffs for player 1. Thus, these games separatethe behavioral implications of 3rd-order rationality and lower-order rationality underthe assumption that a subject who satisfies 2nd-order rationality but not 3rd-orderrationality will not respond to changes in 2nd-order payoffs. Under this assumptionsuch a subject will play either (a, a) or (b, b) as player 1 in games R1 and R2 respec-tively, however a subject who satisfies 3rd-order rationality would play action profile(a, b). Observing the behavior of player 1 in R1 and R2 will allow us to separate theimplications of 3rd-order rationality from lower-orders of rationality. Each additionalplayer in the ring game allows an additional degree of independence between ordersof payoffs. For example, a 4-player ring game could be used to separate 4th-orderrationality from lower orders of rationality.

Once a subject’s order of rationality is identified, a separate set of games canthen be used to estimate whether or not a player has consistent beliefs (a subjectbelieves her opponent believes she is playing the strategy that she actually plays).

10

On its own, consistent beliefs has no behavioral implications. Thus, we use a char-acterization of Nash equilibrium based on Aumann and Brandenburger (1995) andPerea (2007), that combines assumptions about consistent beliefs and rationality togive testable behavioral implications. In any 2-player complete information game, ifa player satisfies 2nd-order rationality and consistent beliefs, then she has to playan action consistent with a Nash equilibrium. Thus to estimate whether consistentbeliefs hold, we consider a bimatrix game that has a unique Nash equilibrium butwhere all actions are rationalizable. For all subjects that satisfy 2nd-order rationality(estimated from the ring game), failure to play the Nash equilibrium is an indicationof a failure of consistent beliefs.

3 The model

In this section, we apply the tools of epistemic game theory to formally define ra-tionality, higher-order rationality, and consistent beliefs. We formalize the exclusionrestriction that is applied to identify a subject’s order of rationality and present thecharacterization of Nash equilibrium that is applied to identify consistent beliefs.

Our experimental design involves both 2-player bimatrix games and n-player ringgames. Thus, we formally consider only n-player ring games which contain the 2-player bimatrix game as a special case when n=2.

Definition. A n-player ring game � is a tuple � = hI = {1, . . . , n};S1, . . . , Sn

;

⇡1, . . . , ⇡n

; oi where I and S

i

are a finite sets, ⇡i

: Si

⇥ S

o(i) ! R, and o : I ! I witho(i) = 1 + imodn.

The set I represents the set of players, Si

represents the set of actions for playeri, ⇡

i

represents the payoffs for player i which depend upon player i’s action and theaction of her opponent o(i), and the function o represents the opponent mappingfunction where o(i) is the opponent of player i. The restriction o(i) = 1 + imodnrestricts the opponent relationship to that of a ring: player 1’s opponent is player 2,player 2’s opponent is player 3, and so on, with player n’s opponent being player 1.

Given a game, an epistemic type space describes players’ beliefs about strategies.

Definition. Let � = hI;S1, . . . , Sn

; ⇡1, . . . , ⇡n

; oi be an n-player ring game. A finite�-based epistemic type space is a set hT1, . . . , Tn

; b1, . . . , bn; s1, . . . , sni where T

i

isa finite set, bi : T

i

! 4(To(i)), and s

i

: Ti

! 4(Si

).

11

Every player has a set of types T

i

. The function s

i

defines a strategy for eachtype of player i. The function b

i represents each type’s beliefs about the types of heropponent. Thus, the function b

i(ti

) together with the function s

o(i) defines type t

i

’sbeliefs about the strategies of her opponent.

3.1 Epistemic conditions

The expected utility of a type t

i

can be defined9

u

i

(si

, t

i

) ⌘X

t

o(i)2To(i)

b

i(ti

)(to(i))⇡(si, s(to(i))).

Given this specification for utility, a type is rational if the strategy s

i

(ti

) is a bestresponse for player t

i

given her beliefs.

Definition. A type t

i

is rational if si

(ti

) maximizes player i’s expected payoff underthe measure b

i(ti

). That is, if for any s 2 supp(si

(ti

))

u

i

(s, ti

) � u

i

(s0, ti

) 8s0 2 S

i

.

We are interested not only in whether types are rational, but also if they believeothers are rational, if they believe others believe others are rational, and so on. Todefine higher-order rationality, we first define what we mean by belief. We say a typebelieves an event if she places probability 1 on that event happening.

Definition. A type t

i

believes an event E ✓ T

o(i) if b

i(ti

)(E) = 1. Let the setB

i(E) = {ti

2 T

i

|bi(ti

)(E) = 1} be the set of types for player i that believe event E.

Higher-order beliefs about rationality is then defined recursively in the followingway. Define

R

1i

= {ti

2 T

i

|ti

is rational}

R

m+1i

= R

m

i

\B

i(Rm

o(i))

Definition. If ti

2 R

m

i

then we say that t

i

satisfies mth-order rationality.9Let ⇡(s

i

, s(to(i))) be redefined in the standard way whenever s is a mixed-strategy.

12

The assumption of consistent beliefs places restrictions on a player’s beliefs aboutstrategies. Consistent beliefs ensure that player i believes her opponent believes she isplaying the action she is actually playing.10 For simplicity, we define consistent beliefsonly in the case when n=2 as the assumption is only applied to 2-player games.

Definition. A type ti

has consistent beliefs if for any t�i

2 T�i

such that bi(ti

)(t�i

)

> 0, thenPt2T

i

s

i

(t)(s) · b�i(t�i

)(t) = s

i

(ti

)(s) for all s 2 supp(si

(ti

)).

3.2 Identifying orders of rationality

In this section, the exclusion restriction used to identify a subject’s order of rationalityis discussed. The exclusion restriction essentially assumes that a player with a loworder of rationality will not respond to changes in higher-order payoffs. To state thisassumption, we first formalize what is meant by higher-order payoffs.

Every ring game can be characterized by its payoff hierarchy for each player. Fora given ring game � = hI = {1, . . . , n};S1, . . . , Sn

; ⇡1, . . . , ⇡n

; oi, the payoff hierarchyh

i(�) of player i is defined by

h

i(�) = {⇡o

k(i)}1k=0.

The payoffs of player i are given by h

i

0(�), her 1st-order payoffs are given byh

i

1(�) (i.e. the payoffs of player o(i)), her 2nd-order payoffs are given by h

i

2(�) (i.e.the payoffs of player o

2(i)), and so on. Payoff hierarchies for 2-player games arerelatively redundant. For a 2-player ring game, h1(�) = {⇡1, ⇡2, ⇡1, ⇡2, . . .}. However,increasing the number of players in the ring permits us to provide a richer set of payoffhierarchies. For example, in a 3-player ring game, h1(�) = {⇡1, ⇡2, ⇡3, ⇡1, ⇡2, ⇡3, . . .}.Thus, each additional player in the ring game adds an additional degree of freedombetween higher- and lower-order payoffs.

Our identification strategy exploits the flexibility of the payoff structure ringgames. The assumption used to identify a player’s order of rationality requires aplayer that satisfies kth-order rationality but not (k + 1)th-order rationality to notrespond to changes in her mth-order payoffs whenever m is greater than k. This

10The definition of consistent beliefs places restrictions on a single player’s belief hierarchy and is atype of internal consistency. This is in contrast to an assumption of external consistency which wouldrequire a player to hold correct beliefs about the action her opponent is actually playing. Whetherexternal consistency holds depends not just on a subject’s own beliefs but the actual action of heropponent, and hence cannot be attributed to an individual subject but a situation.

13

assumption cannot be applied effectively in standard game forms (bimatrix games)because any two bimatrix games must differ in either a player’s own payoffs or her1st-order payoffs (or they are the same game). However, there exists different n-playerring games that differ only in a player’s kth-order payoffs or higher (possible whenevern > k).

ER

⇤: A player i who satisfies kth-order rationality but not (k+1)th-order rationality(k � 1) will play the same action in any two games, � and �0, where h

i

j

(�) =

h

i

j

(�0) for all j <k

ER⇤ acts as an exclusion restriction that separates the behavioral implications ofdifferent orders of rationality. Without an exclusion restriction, the behavioral pre-dictions of the different orders of rationality necessarily nest one another. A playerthat satisfies only lower-order rationality could always play an action that a playerthat satisfies higher orders of rationality could play. Under ER⇤, it is possible toseparate the predictions of lower and higher order rationality by considering behaviorin sets of games that differ only in higher-order payoffs. Thus, by observing strategicchoices in specifically chosen sets of ring games it is possible to separately identifydifferent orders of rationality. This was demonstrated with examples in Section 2.

3.3 Identifying consistent beliefs

To identify consistent beliefs, we apply a characterization of Nash equilibrium thatcombines both rationality and consistent belief assumptions to form behavioral im-plications. The definition of Nash equilibrium is standard.

Definition. Consider an n-player ring game � = hI = {1, . . . , n};S1, . . . , Sn

; ⇡1,

. . . , ⇡

n

; oi. The strategy � 2 4(S1)⇥ · · ·⇥4(Sn

) is a Nash equilibrium of � if forall i 2 I and for all a 2 supp(�

i

), then

u

i

(a, �o(i)) � u

i

(a0, �o(i)) 8a0 2 S

i

The characterization of Nash equilibrium we provide below is a characterization fora single player. If a subject satisfies 2nd-order rationality and consistent beliefs, thenshe has to play an action consistent with a Nash equilibrium. Thus, for any subjectthat satisfies 2nd-order rationality, the characterization can be applied to identifywhether or not a subject has consistent beliefs (failure to play a Nash equilibrium

14

action in a 2-player game is taken as evidence that a subject does not have consistentbeliefs). Proposition 1 characterizes Nash equilibrium. The proof is given in AppendixA.11

Proposition 1. Consider an 2-player ring game � = h{1, 2};S1, S2; ⇡1, ⇡2i and a �-

based epistemic type space hT1, T2; b1, b2; s1, s2i. Suppose type t

i

satisfies the following

conditions: (i) 2nd-order rationality and (ii) consistent beliefs. Then it must be that

� defined by �

i

= s

i

(ti

) and ��i

(s) =P

t�i

2T�i

s(t�i

)(s) · bi(ti

)(t�i

) for all s 2 S�i

constitutes a Nash equilibrium.

This characterization of Nash equilibrium is similar to existing characterizations.The closest is the characterization by Aumann and Brandenburger (1995) which re-quires mutual knowledge of conjectures as the consistent belief assumption (if player1 plays a and player 2 plays b, player 1 must believe player 2 plays b and player 2 mustbelieve player 1 plays a) and for both players to be rational. We rewrite these as non-interactive conditions (restrictions on only one player’s belief hierarchy): a player isrational, believes her opponent is rational and has consistent beliefs (i.e. if she playsa, she believes her opponent believes she plays a). The non-interactive framework isbetter suited to testing the epistemic assumptions using individual decision data.

An alternative characterization due to Perea (2007) is also in terms of non-interactive conditions, though the current characterization tightens the sufficient con-ditions to be inline with Aumann and Brandenburger. Relative to Perea, both therationality and consistent belief assumptions are relaxed. This allows us to broadenthe set of players for which we can identify consistent beliefs. In addition, Perea hastwo aspects of consistent beliefs that operate (believing you are correct about youropponent’s beliefs (you put probability 1 on your opponents belief type) and believingthat your opponent is correct about your beliefs (your opponent puts probability 1on your belief type)) which is a considerably stronger statement than our concept ofconsistent beliefs which is only one directional and on actions rather than belief types

11Throughout the body of this paper we define ’belief’ to be a belief with probability 1 (i.e. if youbelieve your opponent is rational, that means you believe your opponent is rational with probability1). However, in Appendix B, we relax the notion of belief from that of belief with probability 1 tobelief with probability p. We show that with a sufficiently high p, Proposition 1 still characterizesNash equilibrium behavior in 2-player games. This allows the identification of consistent beliefs evenif we estimate each player to only believe her opponent is rational with probability p. The robustnessof our identification to a generalization of belief with probability 1 to belief with probability p isdiscussed in Appendix B.

15

(believe your opponent puts probability 1 on the action you play).

4 Experimental design

This section discusses the details of the experimental design that allows us to sepa-rately identify the three key epistemic conditions: rationality, higher-order rationality,and consistent beliefs. The design is motivated by the results of the previous sections.

4.1 Rationality

Each subject’s order of rationality is estimated from strategic choice data in 8 4-playerring games. This separates subjects into five rationality categories: irrational (R0),rational but not 2nd-order rational (R1), 2nd-order rational but not 3rd-order rational(R2), 3rd-order rational but not 4th-order rational (R3), and 4th-order rational (R4).!"#$%&!'(!)&

!! ! "#$%&'!(! ! ! ! "#$%&'!)! ! ! ! "#$%&'!*! ! ! ! "#$%&'!+!! ! "#$%&'!),-!$./012-! ! ! ! "#$%&'!*,-!$./012-! ! ! ! "#$%&'!+,-!$./012-! ! ! ! "#$%&'!(,-!$./012-!! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .!$! 4! )5! ()! ! $! (+! (4! +! ! $! )5! (+! 4! ! $! ()! (6! (+!3! 5! 4! (6! ! 3! )5! 4! (+! ! 3! (6! )! (4! ! 3! 4! ()! (5!

"#$%&'!(,-!$./012-!

.! (4! ()! 6! ! "#$%&'!),-!$./012-!

.! 5! (6! (4! ! "#$%&'!*,-!$./012-!

.! 5! (6! (6! ! "#$%&'!+,-!$./012-!

.! 6! (5! 4!!!! ! "#$%&'!(! ! ! ! "#$%&'!)! ! ! ! "#$%&'!*! ! ! ! "#$%&'!+!! ! "#$%&'!),-!$./012-! ! ! ! "#$%&'!*,-!$./012-! ! ! ! "#$%&'!+,-!$./012-! ! ! ! "#$%&'!(,-!$./012-!! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .!$! 4! )5! ()! ! $! (+! (4! +! ! $! )5! (+! 4! ! $! 4! ()! (5!3! 5! 4! (6! ! 3! )5! 4! (+! ! 3! (6! )! (4! ! 3! 6! (5! 4!

"#$%&'!(,-!$./012-!

.! (4! ()! 6! ! "#$%&'!),-!$./012-!

.! 5! (6! (4! ! "#$%&'!*,-!$./012-!

.! 5! (6! (6! ! "#$%&'!+,-!$./012-!

.! ()! (6! (+!!!! ! "#$%&'!(! ! ! ! "#$%&'!)! ! ! ! "#$%&'!*! ! ! ! "#$%&'!+!! ! "#$%&'!),-!$./012-! ! ! ! "#$%&'!*,-!$./012-! ! ! ! "#$%&'!+,-!$./012-! ! ! ! "#$%&'!(,-!$./012-!! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .!$! 6! (5! 7! ! $! +! 5! 4! ! $! 6! (5! 7! ! $! +! 5! 4!3! )! ()! )! ! 3! )! (5! (4! ! 3! )! ()! )! ! 3! )! (5! (4!

"#$%&'!(,-!$./012-!

.! )! 6! 4! ! "#$%&'!),-!$./012-!

.! )! ()! (6! ! "#$%&'!*,-!$./012-!

.! )! 6! 4! ! "#$%&'!+,-!$./012-!

.! )! ()! (6!!!!! ! "#$%&'!(! ! ! ! "#$%&'!)!! ! "#$%&'!),-!$./012-! ! ! ! "#$%&'!*,-!$./012-!! ! $! 3! .! ! ! ! $! 3! .!$! 6! (5! 7! ! $! +! 5! 4!3! )! ()! )! ! 3! )! (5! (4!

"#$%&'!(,-!$./012-!

.! )! 6! 4! ! "#$%&'!),-!$./012-!

.! )! ()! (6!!!!!

Figure 6: G1

!"#$%&!'(!)&!! ! "#$%&'!(! ! ! ! "#$%&'!)! ! ! ! "#$%&'!*! ! ! ! "#$%&'!+!! ! "#$%&'!),-!$./012-! ! ! ! "#$%&'!*,-!$./012-! ! ! ! "#$%&'!+,-!$./012-! ! ! ! "#$%&'!(,-!$./012-!! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .!$! 4! )5! ()! ! $! (+! (4! +! ! $! )5! (+! 4! ! $! ()! (6! (+!3! 5! 4! (6! ! 3! )5! 4! (+! ! 3! (6! )! (4! ! 3! 4! ()! (5!

"#$%&'!(,-!$./012-!

.! (4! ()! 6! ! "#$%&'!),-!$./012-!

.! 5! (6! (4! ! "#$%&'!*,-!$./012-!

.! 5! (6! (6! ! "#$%&'!+,-!$./012-!

.! 6! (5! 4!!!! ! "#$%&'!(! ! ! ! "#$%&'!)! ! ! ! "#$%&'!*! ! ! ! "#$%&'!+!! ! "#$%&'!),-!$./012-! ! ! ! "#$%&'!*,-!$./012-! ! ! ! "#$%&'!+,-!$./012-! ! ! ! "#$%&'!(,-!$./012-!! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .!$! 4! )5! ()! ! $! (+! (4! +! ! $! )5! (+! 4! ! $! 4! ()! (5!3! 5! 4! (6! ! 3! )5! 4! (+! ! 3! (6! )! (4! ! 3! 6! (5! 4!

"#$%&'!(,-!$./012-!

.! (4! ()! 6! ! "#$%&'!),-!$./012-!

.! 5! (6! (4! ! "#$%&'!*,-!$./012-!

.! 5! (6! (6! ! "#$%&'!+,-!$./012-!

.! ()! (6! (+!!!! ! "#$%&'!(! ! ! ! "#$%&'!)! ! ! ! "#$%&'!*! ! ! ! "#$%&'!+!! ! "#$%&'!),-!$./012-! ! ! ! "#$%&'!*,-!$./012-! ! ! ! "#$%&'!+,-!$./012-! ! ! ! "#$%&'!(,-!$./012-!! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .! ! ! ! $! 3! .!$! 6! (5! 7! ! $! +! 5! 4! ! $! 6! (5! 7! ! $! +! 5! 4!3! )! ()! )! ! 3! )! (5! (4! ! 3! )! ()! )! ! 3! )! (5! (4!

"#$%&'!(,-!$./012-!

.! )! 6! 4! ! "#$%&'!),-!$./012-!

.! )! ()! (6! ! "#$%&'!*,-!$./012-!

.! )! 6! 4! ! "#$%&'!+,-!$./012-!

.! )! ()! (6!!!!! ! "#$%&'!(! ! ! ! "#$%&'!)!! ! "#$%&'!),-!$./012-! ! ! ! "#$%&'!*,-!$./012-!! ! $! 3! .! ! ! ! $! 3! .!$! 6! (5! 7! ! $! +! 5! 4!3! )! ()! )! ! 3! )! (5! (4!

"#$%&'!(,-!$./012-!

.! )! 6! 4! ! "#$%&'!),-!$./012-!

.! )! ()! (6!!!!!

Figure 7: G2

Consider the games G1 and G2 in Figures 6 and 7. The games are similar to thegames R1 and R2 from Section 2 except that G1 and G2 are 4-player, 3 action games.Adding an additional player permits identification of a player’s order of rationality up

16

to the 4th-order. Each subject’s order of rationality is estimated by her choice patternin the 8 games (2 games ⇥ 4 player positions). Only the payoffs of Player 4 changeacross these two games. Thus, player 1 has the same payoffs up to the 2nd-order,player 2 has the same payoffs up to the 1st-order, and player 3 has the same payoffs,in both G1 and G2. The predicted actions for a kth-order rational player (satisfyingER⇤) are given in Table 1. A R1 subject plays the same action in G1 and G2 whenshe plays as player 1, 2, or 3. A R2 subject plays the same action in G1 and G2 whenshe plays as player 2 or player 1. And, a R3 subject plays the same action in G1 andG2 when she plays as player 1.12

G1 G2 Type P4 P3 P2 P1 P4 P3 P2 P1

Rational (R1) a ! ! ! b ! ! !

2nd-order Rational (R2) a a ! ! b b ! !

3rd-order Rational (R3) a a a " b b b " 4th-order Rational (R4) a a a a b b b b

Action G1 G2

Type P4 P3 P2 P1 P4 P3 P2 P1

Rational (R1) a ! " " c ! " "

2nd-order Rational (R2) a a ! # c b ! #

3rd-order Rational (R3) a a b ⌘ c b a ⌘

4th-order Rational (R4) a a b a c b a c P1 P2 P3 P4 Type G1 G2 G1 G2 G1 G2 G1 G2

Rational (R1) (a,a)(b,b)(c,c) (a,a)(b,b)(c,c) (a,a)(b,b)(c,c) (a,c)

2nd-order Rational (R2) (a,a)(b,b)(c,c) (a,a)(b,b)(c,c) (a,b) (a,c)

3rd-order Rational (R3) (a,a)(b,b)(c,c) (b,a) (a,b) (a,c)

4th-order Rational (R4) (a,c) (b,a) (a,b) (a,c) Rational + PE + Consistent

Type R1 R2 R1 R2 R1 R2

< 3rd-order Rational a,b,c a,b,c (a,a)(b,b)(c,c) (a,a)(b,b)(c,c)

3rd-order Rational a b (a,b) (a,b)

Table 1: Predicted actions under rationality and assumptions ER⇤

A player’s order of rationality is estimated by matching her action profile in the 8games to one of the profiles in Table 1. Consider the 4 types: R1, R2, R3 and R4. Theaction of each type is determined by the exclusion restriction ER⇤ and her order ofrationality over the set of 8 games as given in Table 1. Let a

i

2 {a, b, c}8 be an 8-tuplerepresenting the action of subject i in each of the 8 games. Each of the types, R1-R4,has a predicted action (or set of actions) for these 8 games, Sk ⇢ {a, b, c}8. Forexample, type R4 has a unique prediction. She must play the 4th-order rationalizablestrategy in each of the 8 games, S4 = {(a, b, a, a, c, a, b, c)}. Type R3 has a uniqueprediction for 6 of the 8 games, but can play a number of different strategies forthe other 2 games, S3 = {(a, b, a, a, a, a, b, c), (c, b, a, a, c, a, b, c), (b, b, a, a, b, a, b, c)}.Notice that ER⇤ generates an exclusion restriction and ensures that S4 \ S3 = ;.

12We estimate a player’s order of rationality from a set of finite games. In these games, thebehavioral implications of a belief with probability 1 will be the same for a belief with probabilityp, for high enough p. Relaxations of a 1-belief in rationality to a p-belief in rationality is discussedin Appendix B. We show that allowing for a relaxation to belief with probability p does not changethe identification of orders of rationality or consistent beliefs.

17

We can define similar sets for types R1 and R2 such that Sn \ Sm = ; for anyn 6= m 2 {1, 2, 3, 4}.

If a subject’s action profile matches one of the predicted action profiles of typeRk, then we assign them as that type. However, a subject’s action profile does notalways coincide precisely with a predicted action profile, in this case we would liketo allow for the possibility that a player might make an error. We allow subjects tomake at most one error. If they are within one error of an exact type match of typeRk (i.e. the action played in one of the 8 games can be changed so that the actionprofile matches a predicted action profile of type Rk) then we assign the subject astype Rk. If a subject makes two or more errors, then we assign them as an irrational(R0) type.13 If a subject’s action profile is within one error of two types, we assignthem to the lower type (assignment to the lower type implicitly assumes that errorsdue to assumption ER⇤ are more likely than errors due to rationality).14

Order of Rationality Fail Percent

R0 3 60% R1 6 35% R2 4 17% R3 3 16% R4 0 0%

*Uniform L0

L1 L2 L3 L4 n = 63 14 choices 0 0 0 0 0 13 choices 2 0 0 3 5 12 choices 10 2 0 10 22 11 choices 14 6 5 17 42

*G3/G4 Stability

n=63 Player 1 Player 2 Both 3 of 3 actions 28 21 8 2 of 3 actions 61 62 60 5 of 6 actions 41

Assignment # of subjects assigned # of action profiles Exact type match 51 40 Type match with 1 error 24 255 R0 (2+ errors) 5 6266 Total 80 6561

Table 2: Subjects assigned by category

The number of subjects assigned by exact type match and by error is given inTable 2. Each subject could play 3 possible actions in each of the 8 games for a totalof 6561 possible action profiles. The rate of exact type match is extremely high, 51of the 80 subjects were assigned to types because their action profiles matched one ofthe 40 action profiles of types R1-R4 exactly. An additional 24 subjects were assignedto types R1-R4 because their action profiles matched one of R1-R4’s action profilesby one error (255 possible action profiles match type R1-R4 with one error). And, theremaining 5 subjects were assigned as R0 types as they deviated from the predictedprofiles of R1-R4 types by more than one error. Thus, 95 percent of the possible

13The assignment mechanism can be generated by a finite mixture model where each subjecthas some probability of being each of the five types and the probability is chosen to maximize thelikelihood that the subject’s action profile is generated by the given type. Such an approach closelyfollows Costa-Gomes and Crawford (2006).

14We also assume that a subject has to play the rational action as P4 to be matched to R1.

18

action profiles (6266 of 6561 action profiles) that could be played are assigned to theR0 type.

4.2 Consistent beliefs

Next, we test whether a player has consistent beliefs about strategies based on strate-gic choices in game Game G3 in Figure 8. G3 is a 2-player ring game. There is aunique Nash equilibrium (a, a) in game G3.

& ! Player!1! ! ! ! Player!2!! ! Player!2's!actions! ! ! ! Player!1's!actions!! ! a! b! c! ! ! ! a! b! c!a! 4! 10! 9! ! a! 6! 0! 8!b! 2! 12! 2! ! b! 2! 10! 18!

Player!1's!actions!

c! 2! 6! 18! ! Player!2's!actions!

c! 2! 12! 16!!!!!!!Rationalizable!Sets!!G1& ! Player!1! ! ! ! Player!2! ! ! ! Player!3! ! ! ! Player!4!! ! Player!2's!actions! ! ! ! Player!3's!actions! ! ! ! Player!4's!actions! ! ! ! Player!1's!actions!! ! a! b! c! ! ! ! a! b! c! ! ! ! a! b! c! ! ! ! a! b! c!

a! 8! 20! 12! ! a! 14! 18! 4! ! a! 20! 14! 8! ! a! 12! 16! 14!b! 0! 8! 16! ! b! 20! 8! 14! ! b! 16! 2! 18! ! b! 8! 12! 10!

Player!1's!actions!

c! 18! 12! 6! ! Player!2's!actions!

c! 0! 16! 18! ! Player!3's!actions!

c! 0! 16! 16! ! Player!4's!actions!

c! 6! 10! 8!!G2& ! Player!1! ! ! ! Player!2! ! ! ! Player!3! ! ! ! Player!4!! ! Player!2's!actions! ! ! ! Player!3's!actions! ! ! ! Player!4's!actions! ! ! ! Player!1's!actions!! ! a! b! c! ! ! ! a! b! c! ! ! ! a! b! c! ! ! ! a! b! c!

a! 8! 20! 12! ! a! 14! 18! 4! ! a! 20! 14! 8! ! a! 8! 12! 10!b! 0! 8! 16! ! b! 20! 8! 14! ! b! 16! 2! 18! ! b! 6! 10! 8!

Player!1's!actions!

c! 18! 12! 6! ! Player!2's!actions!

c! 0! 16! 18! ! Player!3's!actions!

c! 0! 16! 16! ! Player!4's!actions!

c! 12! 16! 14!!!!G1& ! Player!1! ! ! ! Player!2! ! ! ! Player!3! ! ! ! Player!4!! ! Player!2's!actions! ! ! ! Player!3's!actions! ! ! ! Player!4's!actions! ! ! ! Player!1's!actions!! ! a! b! c! ! ! ! a! b! c! ! ! ! a! b! c! ! ! ! a! b! c!

a! 8! 20! 12! ! a! 14! 18! 4! ! a! 20! 14! 8! ! a! 12! 16! 14!b! 0! 8! 16! ! b! 20! 8! 14! ! b! 16! 2! 18! ! b! 8! 12! 10!

Player!1's!actions!

c! 18! 12! 6! ! Player!2's!actions!

c! 0! 16! 18! ! Player!3's!actions!

c! 0! 16! 16! ! Player!4's!actions!

c! 6! 10! 8!!

Figure 8: G3: 2-player ring game

We apply the characterization of Nash equilibrium to estimate consistent beliefs.If a subject satisfies 2nd-order rationality and consistent beliefs then she must playthe Nash equilibrium strategy a in game G3 (as both player 1 and 2). If a subjectsatisfies 2nd-order rationality as determined from games G1 and G2 (all R2, R3 andR4 subjects), and fails to play a in G3, then it must be because the assumption ofconsistent beliefs fails.

The action a is also a rationalizable action. A player who satisfies any levelof rationality can play a. Because of this, our estimate of consistent beliefs willnecessarily be an upper bound on the proportion of players who satisfy consistentbeliefs.

Strategic choices from games G1, G2 and G3 permit us to categorize subjectsinto 3 epistemic types: boundedly rational types that do not satisfy the sufficientrationality conditions required to play Nash equilibrium in 2-player games (R0 andR1 types), 2nd-order rational types that satisfy the sufficient rationality conditionsrequired to play Nash equilibrium in 2-player games but do not satisfy consistentbeliefs (R2, R3 and R4 types who do not play (a, a) in G3), and those that satisfyboth the rationality and consistent belief requirements (R2, R3 and R4 types whoplay (a, a) in G3).

19

4.3 Laboratory implementation

Sessions were conducted in Arts ISIT computer labs with undergraduate studentsat the University of British Columbia. No subject participated in more than onetreatment. Subjects made all decisions through an online interface. In order to ensureindependence across subjects, subjects did not interact with one another during theexperiment and were not informed of one another’s decisions.

We used a within-subjects design in which subjects played all games in each ses-sion. Each subject played the games G1, G2 and G3 in each of the player positions,for a total of 10 games.15 The within subject design allows us to estimate the orderof rationality and consistent beliefs for each subject.

Subjects played the games in a random order without feedback. Subjects wererequired to spend at least 90 seconds on each of the games. Once subjects madechoices in all games they were given the opportunity to revise their choices (withoutany feedback). If subjects chose to revise their choices they could review each of thegames (in the same original order) and make changes to their choices.

Subjects were paid a $5 showup fee and received payment from their choices inone of the games. One of the games was randomly selected for payment at the endof the experiment. Subjects were randomly and anonymously assigned to 2 or 4player groups (depending on whether the game to be paid was a 2- or 4-player game).Subjects were paid based on their choice and the choices of their group members in theselected game and received the dollar value of their payoff in the game. Subjects werepaid in cash at the end of the experiment. The average session lasted 45 minutes andthe average subject earned approximately $17 dollars (maximum payment was $25and minimum payment was $7, including showup fee). Payments were in Canadiandollars.

Instructions were read aloud by the experimenter at the beginning of the session.The instructions to all subjects were the same. Subjects then completed a shortunderstanding quiz to make sure they understood the instructions. The completeinstructions and quiz can be found in Appendix C.

The dataset includes observations from 6 sessions with either 12 or 16 subjects ineach session. In total, 80 subjects are represented in the dataset.

154 additional, related games were played by each subject. The data from those games is notanalyzed in this paper

20

5 Experimental results

Figure 9 gives the proportion of subjects classified as each type R0-R4. 6 percent ofsubjects are classified as R0 types, 22 percent are classified as R1 types, 27 percentare classified as R2 types, 25 percent are classified as R3 types, and 20 percent areclassified as R4 types. Overall, 94 percent of subjects are rational, 72 percent ofsubjects are 2nd-order rational, 45 percent are 3rd-order rational, and 20 percent are4th-order rational.

R0#

R1#

R2#

R3#

R4#

0%#

10%#

20%#

30%#

40%#

50%#

60%#

70%#

80%#

90%#

100%#

Prop

or%o

n'of'Sub

jects'

Orders'of'Ra%onality'

Figure 9: Subjects classified by order of rationality

The estimated orders of rationality suggest that 28 percent of subjects fail to playNash equilibrium because of a failure of bounded rationality. This includes subjectswho are irrational and those who are rational but fail to account for the rationality ofothers. The sufficient epistemic conditions for Nash equilibrium (in 2-player games)require a subject to be 2nd-order rational. 72 percent of the population satisfies thesufficient rationality conditions for Nash equilibrium.

Next, we estimate whether each subject satisfies consistent beliefs based on theirchoices in game G3. Subjects are categorized into 3 epistemic types: (1) bound-edly rational types: types that do not satisfy the sufficient rationality conditionsrequired for Nash equilibrium (and for whom we can not observe consistent beliefs)(2) 2nd-order rational/inconsistent types: types that satisfy the sufficient rationalityconditions of Nash equilibrium but do not satisfy consistent beliefs, and (3) 2nd-orderrational/consistent types: types that satisfy all three sufficient conditions for Nash

21

equilibrium. Not a single subject in our sample satisfies all 3 sufficient con-

ditions for Nash equilibrium. It is also important to notice that while failure ofNash equilibrium is due in part to bounded rationality, failure to play Nash equilib-rium rarely occurs because subjects are not rational. In addition, the failure to playNash equilibrium is largely due to a failure of having consistent beliefs. 28 percentof the subject pool are boundedly rational and 72 percent of the subject pool are2nd-order rational with inconsistent beliefs.

0%#

10%#

20%#

30%#

40%#

50%#

60%#

70%#

80%#

Boundedly#Ra6onal####### 2nd7order#Ra6onal#+#inconsistent#beliefs#

2nd7order#Ra6onal#+#consistent#beliefs#

Prop

or%o

n'of'Sub

jects'

Epistemic'Types'

Figure 10: Subjects classified by epistemic type

5.1 Secondary results

During the experiment, each subject answered a short 5-question quiz after the in-structions were read aloud. The quiz was designed to test the subject’s understandingof the game structure and to make sure they understood the instructions. The quizcan be found in Appendix C. 16 subjects failed the quiz. Table 3 breaks down thenumber of subjects who failed the quiz by order of rationality. 60 percent of theirrational subjects (R0) failed the quiz. 35 percent of R1 subjects failed the quiz.Approximately, 16 percent of both R2 and R3 types failed. None of the R4 subjectsfailed. This suggests an explanation for the boundedly rational subjects - they didnot have a clear understanding of the games they were asked to play.16

16Arguably, we may want to limit our analysis only to subjects who pass the quiz if we areinterested in investigating the strategic reasoning of subjects who understand the game structure.This changes the results in the predictable way: 20 percent of subjects are characterized as boundedlyrational and 80 percent are sufficiently rational but have inconsistent beliefs.

22

Order of Rationality Fail Percent

R0 3 60% R1 6 35% R2 4 17% R3 3 16% R4 0 0%

*Uniform L0

L1 L2 L3 L4 n = 63 14 choices 0 0 0 0 0 13 choices 2 0 0 3 5 12 choices 10 2 0 10 22 11 choices 14 6 5 17 42

*G3/G4 Stability

n=63 Player 1 Player 2 Both 3 of 3 actions 28 21 8 2 of 3 actions 61 62 60 5 of 6 actions 41

Assignment # of subjects assigned # of action profiles Exact type match 51 40 Type match with 1 error 24 290 R0 (2+ errors) 5 6231 Total 80 6561

Table 3: Subjects who failed quiz, classified by order of rationality

Subjects were also given the opportunity to revise their choices after completingall of the games for the first time. The subjects could review each of the games (inthe original order) and make changes if they desired. 39 subjects chose to reviewtheir choices. Table 4 gives the estimated orders of rationality based off of initialchoices versus final choices. The rows give the estimated orders of rationality basedoff initial choices and the columns give the estimated orders of rationality based offfinal choices. The estimated distribution is largely unaffected by the opportunity toreview. The estimated order of rationality changes for only 11 subjects from initialto final choices. 7 subjects increase their order of rationality and 4 subjects decreasetheir order of rationality.

*G1 and G2: Only subject who pass quiz

R0 R1 R2 R3 R4 n

Rationality + PE + errors 2 15 14 18 16 65

Rationality + PE 5 9 13 16 13 56

Rationality 5 18 25 19 13 80

Players who play same action in G1 and G2, by type P1 P2 P3 P4 total

L0 1 1 0 0 1 L1 11 9 7 0 11 L2 12 17 0 0 20 L3 8 0 0 0 11 L4 0 0 0 0 11

Final Choices R0 R1 R2 R3 R4 #Review

R0 4 0 0 1 0 2 R1 1 16 0 0 0 7 R2 0 2 22 2 4 17 R3 0 0 1 16 0 6

Inital Choices

R4 0 0 0 0 12 7

Table 4: Order of rationality determined by initial versus final choices

The estimated distribution of orders depends upon the assumption ER⇤. If ER⇤

holds then subjects are not misidentified. However, if ER⇤ does not hold, a subject’sorder of rationality may be mispecified. However, it is possible to bound our ob-served rationality distribution under different assumptions on subjects’ behavior. Forexample, suppose a player satisfied kth-order of rationality, did not satisfy ER⇤, butplayed all actions in the kth-order rationalizable set with equal probability. Then theprobability that a Rk type is really a R(k-1) type is 1/9. Additionally, the behavior ofirrational types (R0) have not been specified. The design limits misidentification of

23

irrational types in two ways. First, the design is weighted against the over identifica-tion of R1-R4 types as 95 percent of the action profiles, if played, would get a subjectassigned as an R0 type. Second, if an irrational type plays randomly, as is oftenassumed, then there is only a very small chance of such a type getting misspecifiedas an R1-R4 type. Specifically, there is a 1/81 chance that an irrational type wouldget assigned as a R1 type, a 1/243 chance an irrational type would get assigned asan R2 type, and so on. We might expect then that only 1 of our 80 subjects wasmisidentified as a R1 type rather than a R0 type.

6 Discussion

The experimental results show that no subject satisfies all of the sufficient conditionsrequired to play Nash equilibrium. However, there are a number of alternative solu-tion concepts to Nash equilibrium. Popular alternatives are rationalizability, quantalresponse equilibrium (QRE), and level-k models.17 Each of these solution conceptsrelaxes at least one of the assumptions underlying Nash equilibrium. Rationalizabilityrelaxes the consistent belief assumption of Nash equilibrium but requires players tosatisfy common knowledge of rationality (i.e satisfy kth-order rationality as k tendsto infinity). QRE relaxes the rationality requirements of Nash equilibrium but main-tains the consistent belief assumption. Level-k models maintain the assumption thatsubjects are rational but relax the consistent belief assumption by allowing playersto hold heterogenous beliefs about the rationality of others.

In addition to imposing different assumptions on players’ rationality and beliefstructures, these alternative solution concepts impose additional structural assump-tions. Rationalizability, in the most general terms, imposes no additional assump-tions, but only makes precise predictions in rationalizable games. When testing themodel it is often assumed that players play all possible rationalizable actions withequal probability. The QRE model imposes a logistic error structure to describe theprobability of error in best responding. Level-k models impose a particular behaviorfor L0 types and particular assumptions about the beliefs of each level about the lev-els of other players. The success of these alternative models is typically judged by thelikelihood of a model to explain a given data set evaluated at likelihood-maximizing

17This paper does not separate between level-k and cognitive hierarchy, hence we limit discussionto level-k models for simplicity.

24

parameter estimates. Thus, each solution concept is judged based on the completepackage of assumptions about rationality, beliefs, and additional structural assump-tions.

The standard methodology makes it difficult to separately identify the role thateach of these assumptions plays in explaining a given data set. In addition, it is unclearto what extent the structural assumptions improve the models fit or to which extentthe different models make different testable predictions. For example, when fittingthe models to data it is assumed that both QRE and level-k types make mistakesaccording to a logistic error distribution. However, this is explicitly modeled underQRE and players take into account the fact that others are making mistakes whenthey make their choices. Whereas, under level-k, mistakes happen ex post and playersdo not take into account the fact that others are making mistakes when they maketheir decisions.

The approach in this paper allows us to abstract from the additional structuralassumptions and directly assess the assumptions of rationality and consistent beliefsthat underlie these solution concepts. Our results are consistent with certain classes ofexisting solution concepts and not others. We find 92 percent of subjects are rational,but that 80 percent of subjects do not satisfy 4th-order rationality. This is inconsistentwith the assumptions imposed under rationalizability which would require all subjectsto satisfy 4th-order rationality. Quantal response equilibrium imposes the assumptionthat players have consistent beliefs but may not be rational. However, our resultssuggest that the opposite holds, most subjects are rational but consistent beliefs doesnot hold. The level-k model is the most promising as it incorporates the assumptionthat players are rational, relaxes the assumption of consistent beliefs, and allows forheterogeneity in subjects’ beliefs about the rationality of their opponents. We take acloser look at the QRE and level-k solution concepts below.

QRE is modeled by assuming players make mistakes according to a logistic errorfunction. Each player’s strategy must be a best response to her opponent’s strategy,taking into account that her opponent makes errors. Errors are determined by thelogistic error distribution and depend upon the relative payoff of each action. If thepayoff to a given action is high, then it is more likely to be played. The QRE modelis typically tested by fitting the solution concept to data by choosing the precisionparameter � according to maximum likelihood analysis.

Definition. Consider a n-player ring game. The strategy � 2 4(S1)⇥· · ·⇥4(Sn

) is a

25

quantal response equilibrium (QRE) of � if for all i 2 I and for all a 2 supp(�i

),then

�

i

(a) =exp(� · u

i

(a, �o(i)))P

s

i

2SI

exp(� · ui

(si

, �

o(i)))

Under the level-k model, a Lk type plays an action determined by �(k) in thelevel-k equilibrium. Every Lk type must play a best response to what the Lk-1 typeis doing, with the exception of the L0 type. L0 types do not need to play a bestresponse (and hence can play any action). Typically, the level-k model is tested byspecifying a behavior for L0 types ex ante and then fitting the model to the data bychoosing the proportion of level-k types (and the error with which each type makesdecisions) according to maximum likelihood analysis.18

Definition. Consider a n-player ring game. The strategy � = �1 ⇥ · · · ⇥ �

n

, where�

i

: N ! 4(Si

) is a level-k equilibrium of � if for all i 2 I, for all k � 1 2 N andfor all a 2 supp(�

i

(k))

u

i

(a, �o(i)(k � 1)) � u

i

(a0, �o(i)(k � 1)) 8a0 2 S

i

The majority of our subject pool satisfies rationality, which does not fit QREassumptions unless the precision parameter, �, is high (people do not make mistakeswhen best responding often). But, even if a high � could explain the high proportionof rational subjects, QRE is not able to predict the pattern of behavior across the 8games described by G1 and G2 for any value of the precision parameter. Figure 11gives the experimental proportion of subjects who play the rationalizable actions ineach of the player positions in games G1 and G2. The empirical frequency of playingthe rationalizable strategy increases across the player positions (position 1 to 4) inboth games G1 and G2. The QRE predictions are given in Figure 12. The QREpredictions are not consistent with aggregate data for any �. For any value of �, theQRE equilibrium does not predict that the rationalizable action will be played mostoften by player 4, then player 3, player 2 and player 1 in descending order. Thisis because QRE imposes the assumptions that player’s make mistakes when bestresponding and that consistent beliefs hold. This effectively means that QRE cannot

18This paper does not differentiate between level-k and cognitive hierarchy models (hence weconsider only the former for simplicity).

26

allow a player to be both rational (high �) and believe that her opponent is not rational(low �), or that her opponent does not believe her opponent is rational, or so on. Thismeans, more or less, that a high probability of playing the rationalizable action forplayer 4 will impose a high probability of playing the rationalizable action for therest of the players or vice versa). However, the logit error distribution (probabilityyou play particular actions depends upon relative payoffs) does interact somewhatnon-intuitively with the assumption of consistent beliefs. From Figure 12 note thatthe relative likelihoods of playing the rationalizable action for the players vary as� changes. However, the model never predicts the pattern that we observe in thedata, that the likelihood of playing the rationalizable action increases from player 1to player 4.

P4## P4#P3#

P3#

P2#

P2#

P1#

P1#

0%#10%#20%#30%#40%#50%#60%#70%#80%#90%#

100%#

G1# G2#

Prop

or%o

n'of'Sub

jects'P

laying'

Ra%o

nalizab

le'Strategy'

Game'

Figure 11: Proportion of subjects who play rationalizable action in G1 and G2

0.3$

0.4$

0.5$

0.6$

0.7$

0.8$

0.9$

1$

0$ 0.15$ 0.3$ 0.45$ 0.6$ 0.75$

Prob

ability*of*P

laying*Ra/

onalizab

le*

Strategy*

Lambda*

Player$1$

Player$2$

Player$3$

Player$4$

0.3$

0.4$

0.5$

0.6$

0.7$

0.8$

0.9$

1$

0$ 0.15$0.3$0.45$0.6$0.75$0.9$1.05$1.2$1.35$1.5$

Prob

ability*of*P

laying*Ra/

onalizab

le*

Strategy*

Lambda*

Player$1$

Player$2$

Player$3$

Player$4$

Figure 12: QRE predictions for games G1 and G2

However, level-k models predict the aggregate behavior in Figure 11. Consider anydistribution over levels L1, L2, L3 and L4 (Lk type plays the strategy determined by

27

�(k) in the level-k equilibrium). L4 types play the rationalizable strategy as player1. L3, and L4 types play the rationalizable strategy as player 2. L2, L3 and L4types play the rationalizable strategy as player 3. L1, L2, L3 and L4 types play therationalizable strategy as player 4. No matter what the distribution over levels orthe specification for L0, the proportion of subjects playing the rationalizable strategywill weakly increase over the player positions 1-4.

Further, the level-k model (without allowing for errors) is only consistent with 1percent of the action profiles that could be played in the 8 games defined by G1 andG2. This is under any specification for L0 and any distribution of levels (for k � 1).However, 64 percent of the subject pool plays actions that are consistent with thelevel-k model (i.e. play those 1 percent of actions).19 This provides a strong test ofsome of the key assumptions underlying the level-k model.

In addition, kth-order rationality and the ER⇤ assumption capture the main as-sumptions defining the behavior of an Lk type. L1 types are rational and do notrespond to changes in 1st-order payoffs as they best respond to a fixed L0 behavior.L2 types are 2nd-order rational and do not respond to changes in 2nd-order payoffs.L3 types are 3rd-order rational and do not respond to changes in 3rd-order payoffs,and so on. Thus, the distribution of orders of rationality estimated in Figure 9 givesus a ‘L0’ independent estimate of the level-k distribution. The distribution of levelsin Figure 9 puts more weight on higher levels than is generally estimated in level-kexperiments. This could suggest that the usual assumption of uniform L0 or thestructural estimation approach, biases the estimation of the level distribution.

7 Conclusion

This paper applies epistemic game theory to experimental design in order to estimatekey assumptions underlying strategic reasoning. This approach differs from the typ-ical approach in behavioral game theory that implicitly tests epistemic assumptionsby jointly testing all epistemic and structural assumptions embodied in a solutionconcept. Instead, we directly test epistemic assumptions which gives direct insightinto the plausibility of different solution concepts.

To draw direct inferences about epistemic assumptions from strategic choice data,this paper develops a novel experimental design based around ring games. Ring games

19This increases to 94 percent if we allow for an error as modeled in Section 4.

28

allow for the application of a natural exclusion restriction due to the flexibility of theirpayoff structure relative to standard game forms. Using such a design allows us tomake weaker identification assumptions than those that are typically imposed in theliterature.

The main findings are that players are rational, have finite and heterogenousorders of beliefs about the rationality of others, and do not satisfy consistent beliefs.These features have implications for how we should model strategic behavior. Theclass of level-k and cognitive hierarchy models are consistent with these experimentalresults. However, the results in this paper do not speak to other features of this classof models such as the nature of the L0 specification, how salient the L0 specificationis for different players, nor the differences between the level-k and cognitive hierarchymodels. Further research is needed to address these assumptions.

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Healy, P. J. (2011). Epistemic Foundations for the Failure of Nash Equilibrium.working paper.

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Theory, chapter Cognition and Framing in Sequential Bargaining for Gains andLosses, pages 27–47. MIT Press.

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31

A Proof of Proposition 1

Let t

i

be a type that satisfies conditions (i) and (ii).Let T = {t�i

2 T�i

|bi(ti

)(t�i

) > 0}.Since t

i

is rational it must be true that for any s 2 supp(�i

)

u

i

(s, ��i

) � u

i

(a0, ��i

) 8a0 2 S

i

.

In other words, �i

is a best response to ��i

Further, since t

i

2 R

2i

) t�i

2 R

1�i

for all t�i

2 T

And, since t

i

has consistent beliefs it must be thatPt2T

i

s

i

(t)(s) · b�i(t�i

)(t) = �

i

(s)

8s 2 S

i

, for all t�i

2 T .Therefore, for any t�i

2 T and any action s 2 supp(s�i

(t�i

))

u�i

(s, �i

) � u�i

(s0, �i

) 8s0 2 S�i

.

In other words, s�i

(t�i

) is a best response to �

i

for all t�i

2 T .But, since supp(��i

) =S

t�i

2Tsupp (s�i

(t�i

)). It must be that for any s 2 supp(��i

)

u�i

(s, �i

) � u�i

(s0, �i

) 8s0 2 S�i

.

And, hence ��i

is a best response to �

i

.Therefore, � is a Nash Equilibrium.⇤

32

B Relaxing belief to p-belief

In the experiment ran in this paper, we technically do not estimate whether a sub-ject’s beliefs hold with probability 1. This is because for finite games, the behavioralimplications of a belief with probability 1 will be the same for a belief with probabilityp, for high enough p. In this appendix, we show that allowing for a relaxation of beliefto that of p-belief (belief with probability 1 to belief with probability p) is consistentwith our experimental results.

We say that a type p-believes an event if he believes that event with probabilityat least p.

Definition. A type t

i

p-believes an event E ✓ T

o(i) if bi(ti

)(E) � p. Let the setB

i

p

(E) = {ti 2 T

i|bi(ti

)(E) � p} be the set of types for player i that p-believe eventE.

We define higher-order p-rationality by recursively defining the following sets:

R

1i,p

= {ti

2 T

i|ti

is rational}

R

m+1i,p

= R

m

i,p

\ B

i

p

(Rm

o(i),p)

Definition. If ti

2 R

m

i,p

then we say that t

i

satisfies mth-order p-rationality

We can also define a relaxed version of consistent beliefs for 2-player games. Thisdefinition ensures that a player believes her opponent beliefs she is playing the actionshe is actually playing with at least probability p.

Definition. A type t

i

has consistent p-beliefs if for all t�i

2 T�i

such thatb

i(ti

)(t�i

) � 0 thenPt2T

i

s

i

(t)(s) · b�i(t�i

)(t) � s

i

(ti

)(s) · p 8s 2 S

i

R

m

i,p

= R

m

i

for each of the players in Games G1 and G2 as long as p � 78 (with the

bound tight for all players except player 1). In other words, the behavioral predictionsunder kth-order p�rationality are the same as the behavioral predictions under kth-order 1-rationality (when p is sufficiently high). Thus, we do not identify whethera subject believes his opponent is rational with probability 1, but rather a lowerbound p such that she believes her opponent is rational with probability p. Given

33

this the best we observe is whether a subject believes others are rational with at leastprobability 7

8 .Our identification of consistent beliefs, relies along the assumption that subject’s

satisfied the sufficient rationality conditions for Nash Equilibria in G3. Thus, it isnecessary to relax the sufficient epistemic characterization of Nash equilibrium toallow for a p-belief in rationality.

In the proof below, we show that that for any p >

67 , as long as a subject satisfies

2nd-order p-rationality and consistent p-beliefs then she must play the Nash strategy a

in game G3 (as player 1 and 2). Taking subjects who are classified as R2 or higher forgames G1 and G2 ensures that they satisfy 2nd-order 7

8 -rationality. Thus, if they failto play the Nash equilibrium in G3 it must be because they fail to satisfy consistent78 -beliefs. The following proof establishes that p-belief with p >

67 is sufficient to

guarantee that (a, a) is the unique action profile in G3.

Proof

probability#player#2#plays#a#

prob

ability#player#2

#plays#b#

7/9#

9/13#

9/11#

b#

a#c#

probability#player#1#plays#a#

prob

ability#player#1

#plays#b#

1/7#

1/2#

5/7#

b#

c#

a#

probability#player#2#plays#a#

prob

ability#player#2

#plays#b#

7/9#

9/13#

9/11#

b#

a#c#

probability#player#1#plays#a#

prob

ability#player#1

#plays#b#

1/7#

1/2#

5/7#

b#

c#

a#

Figure 13: Best response correspondences for game G3 (player 1 and 2 respectively)

Player 1: Consider the behavior of a type t1 that satisfies 2nd-order p-rationalityand consistent p-belief with p >

67 .

Define �1 = s1(t1) and µ2(s) =Pt2T2

s2(t)(s) · b1(t1)(t) 8s 2 S2. And, let t2 2 T2 such

that b

1(t1)(t2) > 0 and define µ1(s) =Pt2T1

s1(t)(s) · b2(t2)(t) 8s 2 S1.

Consider 2 cases:(i) Suppose t1 plays b (s1(t1)(b) > 0 and hence s1(t1)(c) = 0 since player 1 will neverrationally mix between b and c )Since t1 is rational and believes opponent playing according to µ2, it must be that:

34

µ2(b) � 79 �

59µ2(a)

Therefore, µ2(b) � 12 .

Also, since t1 has consistent p-beliefs, it must be that µ1(a)+µ1(b) � p) µ1(c) 1�p.Now, consider the behavior of any rational type t2 such that b1(t1)(t2) > 0. We knowthat with µ1(c) 1 � p, as long as p >

67 , then any best response will be to mix

between only a and c ) µ2(a) + µ2(c) � p ) µ2(b) 1� p. Contradiction.

(ii) Suppose t1 plays c (s1(t1)(c) > 0 and hence s1(t1)(b) = 0 since player 1 will neverrationally mix between b and c)Since t1 is rational and believes opponent playing according to µ2, it must be that:u2(b) 9

13 �1113µ2(a).

Therefore,µ2(c) � 2

11 .This also means that µ1(a) + µ1(c) � p ) µ1(b) 1� p.Now, consider the behavior of any rational type t2 such that b1(t1)(t2) > 0. We knowthat with µ1(b) 1 � p, as long as p >

67 , then any best response will be to mix

between only a and b ) µ2(a) + µ2(b) � p ) µ2(c) 1� p. Contradiction.

Player 2: Now, consider the behavior of a type t2 that satisfies 2nd-order p-rationalityand consistent p-belief with p >

67 .

Define �2 = s2(t2) and µ1(s) =Pt2T1

s1(t)(s) · b2(t2)(t) 8s 2 S1. And, let t1 2 T1 such

that b

2(t2)(t1) > 0 and define µ2(s) =Pt2T2

s2(t)(s) · b1(t1)(t) 8s 2 S2.

Consider 2 cases:(i) Suppose t2 plays b (i.e. s2(t2)(b) > 0)Thus, µ1(c) � 1

7

Now, consider the behavior of any rational type t1 such that b2(tt

)(t1) > 0

3 cases to consider:(a) t1 plays bTherefore, µ2(b) � 1

2 .Thus all rational types t such that b

2(t2)(t) > 0 must mix between a and b, hence,µ1(a) + µ1(b) � p ) µ1(c) 1� p. This means that type t2 can only mix between aand c. Therefore µ2(a) + µ2(c) � p ) µ2(b) p . Contradiction.(b) t1 plays c

35

Therefore, µ2(c) � 211 .

Thus all rational types t such that b

2(t2)(t) > 0 must mix between a and c, hence,µ1(a) + µ1(c) � p ) µ1(b) 1� p. This means that type t2 will only mix between aand b. Therefore µ2(a) + µ2(b) � p ) µ2(c) p . Contradiction.(c) Type t1 plays a (mixing ruled out in case (a) and (b)), then any rational typeplays a. Thus µ1(a) � p and µ1(b) + µ1(c) 1� p. Contradiction.

(ii) Suppose t2 plays c (i.e. s2(t2)(c) > 0)Then, µ1(b) � 1

7 .3 cases to consider:(a) A rational type t1 (such that b

2(t2)(t1) > 0) is mixing between a and bTherefore, µ2(b) � 1

2 .Thus all rational types t such that b

2(t2)(t) > 0 must mix between a and b, hence,µ1(a) + µ1(b) � p ) µ1(c) 1� p. This means that type t2 can only mix between aand c. Therefore µ2(a) + µ2(c) � p ) µ2(b) p . Contradiction.(b) Any rational type t1 (such that b

2(t2)(t1) > 0) is mixing between a and cTherefore, µ2(c) � 2

11 .Thus all rational types t such that b

2(t2)(t) > 0 must mix between a and c, hence,µ1(a) + µ1(c) � p ) µ1(b) 1� p. This means that type t2 will only mix between aand b. Therefore µ2(a) + µ2(b) � p ) µ2(c) p . Contradiction.(c) Any rational type t1 (such that b

2(t2)(t1) > 0) plays a, they all play a (mixingruled out in case (a) and (b)) Thus µ1(a) � p and µ1(b)+µ1(c) 1�p. Contradiction.⇤

36

C Instructions and quiz

**Do not use the BACK or REFRESH Buttons**

Instructions

You are about to participate in an experiment in the economics of decision-making. If you follow these instructions carefully and makegood decisions, you can earn a considerable amount of money, which will be paid to you in cash at the end of the experiment.

To insure best results for yourself please DO NOT COMMUNICATE with the other participants at any point during the experiment. If youhave any questions, or need assistance of any kind, raise your hand and one of the experimenters will approach you.

The Basic Idea

You will play 14 games. In each of these games, you will be randomly matched with other participants currently in this room. For eachgame you will choose one of three actions. Each other participant in your game will also choose one of three actions.

Your Earnings

Player 2's actions

d e f

a 10 4 16

b 20 8 0

c 4 18 12

Player 2's Earnings

Player 3's actions

g h i

d 12 16 4

e 0 12 8

f 4 4 20

Player 3's Earnings

Your actions

a b c

g 20 12 8

h 6 8 18

i 0 16 4

Your earnings will depend on the combination of your action and player 2's action. These earnings possibilities will be represented in atable like the one above. Your action will determine the row of the table and player 2's action will determine the column of the table.You may choose action a, b, or c and player 2 will choose action d, e, or f. The cell corresponding to this combination of actions willdetermine your earnings.

For example, in the above 3-player game, if you chose a and player 2 chooses d, you would earn 10 dollars. If instead player 2 chose e,you would earn 4 dollars.

Player 2 and Player 3's earnings are listed in the other two tables. Player 2 may choose action d, e or f and Player 3 may choose action g,h, or i. Player 2's earning depends upon the action he chooses and the action player 3 chooses. Player 3's earnings depend upon theaction he chooses and the action you choose.

For example, if you choose c, player 2 chooses e, and player 3 chooses h then you would earn 18 dollars, player 2 would earn 12 dollarsand player 3 would earn 18 dollars.When you start each new game, you will be randomly matched with different participants. We do our best to ensure that you and yourcounterparts remain anonymous.

The earnings tables in a new game are not always the same as in the previous game, so you should always look at the earnings carefullyat the beginning of each game. You will be required to spend at least 90 seconds on each game. You may spend more time on each gameif you wish.

Earnings

You will earn a show-up payment of $5 for arriving to the experiment on time and participating.

In addition to the show-up payment, one game will be randomly selected for payment at the end of the experiment. Every participant inthis room, will be paid based on their actions and the actions of their randomly chosen group members in the selected game. Any of thegames could be the one selected. So you should treat each game like it will be the one determining your payment.

You will be informed of your payment, the game chosen for payment, what action you chose in that game and the action of yourrandomly matched counterpart only at the end of the experiment. You will not learn any other information about the actions of otherplayer's in the experiment. The identity of your randomly chosen counterparts will never be revealed.

Frequently Asked Questions

Q1. Is this some kind of psychology experiment with an agenda you haven't told us? Answer. No. It is an economics experiment. If we do anything deceptive or don't pay you cash as described then you can complain to thecampus Human Subjects Committee and we will be in serious trouble. These instructions are meant to clarify how you earn money, andour interest is in seeing how people make decisions.

Your

act

ions

Play

er 2

's ac

tion

s

Play

er 3

's ac

tion

s

37

**Do not use the BACK or REFRESH Buttons**

Quiz

Your Earnings

Player 2's actions

d e f

a 10 4 16

b 20 8 0

c 4 18 12

Player 2's Earnings

Player 3's actions

g h i

d 12 16 4

e 0 12 8

f 4 4 20

Player 3's Earnings

Your actions

a b c

g 20 12 8

h 6 8 18

i 0 16 4

Consider the above game. You are Player 1. Your earnings are given by the blue numbers. You may choose a or b or c.

1. Your earnings depend on your action and the action of which other player?

(a) Player 1

(b) Player 2

(c) Player 3

2. Suppose you choose a, Player 2 chooses f, and Player 3 chooses i. What will your earnings be?

(a) 10

(b) 0

(c) 16

(d) 6

3. Suppose Player 2 chooses d and Player 3 chooses h. Which action will give you the highest earning?

(a) a

(b) b

(c) c

4. Suppose you choose c. What is your highest possible earning?

(a) 20

(b) 18

(c) 4

5. Suppose you choose b. What is your lowest possible earning?

(a) 0

(b) 4

(c) 8

Continue

You

r ac

tion

s

Pla

yer

2's

acti

ons

Pla

yer

3's

acti

ons

38

39