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 different
epistemic conditions that form the foundations of several game theoretic solution
concepts. It employs strategic choice data from a carefully chosen set of ringnetwork games to obtain individual-level estimates of the following epistemic
conditions: rationality; beliefs about the rationality of others; and consistent
beliefs about strategies. We find that 94 percent of subjects are rational, 72
percent are rational and believe others are rational, and 44 percent are rational
and hold at least second-order beliefs about the rationality of others. Of the
72 percent of subjects that satisfy the sufficient rationality conditions needed
to observe consistent beliefs, none of them satisfy consistent beliefs. Not a
single subject satisfies all three sufficient epistemic conditions required for Nash
equilibrium. The unique design allows us to weigh the relative plausibility of
alternatives 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. Email: [email protected]. I am extremely grateful to my supervisor Yoram Halevy for support
and guidance throughout the process of developing and writing this paper. I am also grateful to
Mike 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 thank
Doug 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 helpful
conversations and comments.
⇤
1
1
Introduction
Over the past 20 years there has been an accumulation of experimental evidence that
challenges the empirical validity of Nash equilibrium predictions when subjects interact in novel environments. This evidence not only prompts the need for alternative
solution concepts to explain behavior in one-shot games, but an understanding of
why Nash equilibrium fails in the laboratory can provide valuable insight into the
plausibility of alternative solution concepts.
Solution concepts do not simply provide a description of behavioral predictions
but embody assumptions about players’ rationality and beliefs that underpin these
predictions. Three epistemic assumptions form the foundations for several common
solution concepts. The first is rationality: whether a player plays a best response to
her beliefs. The second is higher-order rationality: whether a player believes others
are 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 hold
correct beliefs about the action she is playing.
The Nash equilibrium solution concept requires players to be 2nd-order rational (rational and believe others are rational) and have consistent beliefs (Aumann
and Brandenburger 1995). Rationalizability relaxes the assumption of consistent beliefs but strengthens the rationality assumptions to common knowledge of rationality
(Bernheim 1984; Pearce 1984). Behavioral solution concepts also impose a combination of the three assumptions. The level-k solution concept assumes rationality,
relaxes consistent beliefs, and allows for heterogeneity in beliefs about the rationality of others (Nagel 1995; Stahl and Wilson 1995; Costa-Gomes and Crawford 2006;
Camerer et al. 2004). While quantal response equilibrium (QRE) relaxes the assumption 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 epistemic
assumptions hold at the individual level. The methodology here differs substantially
from previous experimental work that evaluates solution concepts. Existing game
theoretic work compares different solution concepts by jointly testing the epistemic
assumptions in combination with additional structural assumptions. For example, in
1
See Kneeland (2012) “Rationality and consistent beliefs: theory and experimental evidence”, an
earlier version of the current paper, for epistemic characterizations of the level-k and QRE solution
concepts.
2
the case of Nash equilibrium, rationalizability, and level-k, an error distribution is
typically 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 specification then anchors the beliefs of all other levels by having players play an action that
is a finitely iterated best response to the L0 behavior. In the case of QRE, it is typically assumed that player’s mistakes when best responding are determined according
to a logistic error structure. Players then noisily best respond to other players’ noisy
best responses. Models are then judged in terms of goodness-of-fit under an ex post
model-fitting approach. Under this approach, the structural assumptions cannot be
separated from the epistemic assumptions themselves and it is difficult to assess how
much power they allow when fitting the different models to data.2 The approach in
this paper is to directly test the three main epistemic assumptions underlying our
solution concepts. Untangling the role that rationality and consistent beliefs play in
strategic reasoning will allow direct comparison of solution concepts without relying
on structural assumptions that risk introducing bias.
In order to directly test the epistemic assumptions, we use choice data to make
inferences about player’s beliefs. However, this approach has an identification problem that stems from the fact that there are generally a number of different beliefs
that lead to the same behavior. The main problem comes in identifying a players
order of rationality: the behavioral implications of higher orders of rationality are
contained in the behavioral implications of lower orders of rationality. To deal with
this identification problem, we focus on a special class of games, ring games. Ring
games allow a natural exclusion restriction to be incorporated into the experimental
design because there are features of the payoff environment that affect higher-order
rational types but not lower-order rational types. The payoff features are due to the
relaxed opponent structure of the ring game. A ring game is essentially a series of
two-player normal form games where the opponent structure is relaxed relative to
standard game forms. For example, in the 3-player ring game in Figure 1, player 1’s
payoff depends on the action of player 2. Player 2’s payoff depends on the action of
player 3. And, player 3’s payoff depends on the action of player 1.
This payoff structure allows one to define games which induce differences only
2
Wright and Leyton-Brown (2010) provide the most exhaustive test to date between Nash equilibrium and a set level-k and QRE models by performing a meta-analysis on a data set generated
from existing normal-form game experiments using standard structural methodology. Level-k and
QRE perform equally well in these tests.
3
in higher-order payoffs without affecting lower-order payoffs. This is not possible in
standard game forms (e.g. bimatrix games) because there is a tight link between
higher- and lower-order payoffs. A player’s own payoffs affect her 2nd, 4th, 6th-order
payoffs and so on, while her opponent’s payoffs affect her 1st, 3rd, 5th-order payoffs
and so on. Thus, it is not possible to change higher-order payoffs without affecting
lower-order payoffs. But, adding a new player into the game creates an additional
degree of freedom. In the 3-player ring game, we can change player 3’s payoffs without
affecting player 1’s own payoffs or her 1st-order payoffs. Thus, ring games can be used
to isolate the behavioral implications of different orders of rationality under a natural
exclusion restriction: lower-order rational players do not respond to changes in higherorder payoffs and by focusing on a carefully chosen set of ring games that exploits this
identifying assumption. To the best of my knowledge, this is the first paper to use
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of rationality and consistent beliefs to derive testable behavioral implications.
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consistent beliefs. Given that a player satisfies the sufficient rationality conditions,
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whether
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this paper
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a review of the literature on network games, or the analogous concept in computer science
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non-interactive belief conditions, which allows us to characterize behavior in terms
of one player’s beliefs (rather than conditions on multiple players simultaneously as
in Aumann and Brandenburger (1995)) which is naturally suited to the application
of individual decision data. In addition, we provide a tighter characterization than
Perea, analogous to Aumann and Brandenburger, allowing us to identify consistent
beliefs for a larger set of subjects.
This paper uses strategic choice data to obtain individual-level estimates of the
three epistemic conditions. We find that 94 percent of subjects behave consistently
with rationality. 72 percent behave consistently with 2nd-order rationality (subjects
are rational and believe others are rational). 44 percent behave consistently with
3rd-order rationality (subjects are 2nd-order rational and hold 2nd-order beliefs that
others 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 2ndorder rationality and consistent beliefs about strategies. This means that 28 percent
of subjects do not satisfy the sufficient rationality conditions for Nash equilibrium.
However, of the 72 percent of subjects that satisfy the sufficient rationality conditions
required to identify consistent beliefs, none of them satisfy consistent beliefs. Not a
single subject satisfies all three sufficient epistemic conditions for Nash equilibrium.
The results provide support for the key features of the level-k and cognitive hierarchy
models; heterogeneous orders of rationality and inconsistent beliefs are important
features 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 consistent beliefs in the lab. The main differences between the papers is the methodology.
Healy estimates epistemic conditions by eliciting subjects’ beliefs about actions, payoffs 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 (actions are not always a best response to stated 1st-order beliefs), but the results for
2nd-order rationality (a direct elicitation of whether subjects believe their opponents
5
Of note, the estimated order of rationality distribution in this paper is interpretable as an
estimate of the level-k level distribution that is independent of the L0 specification. The only other
paper to provide a L0 independent distribution is Burchardi and Penczynski (2010). They incentivize
communication between group members and analyze the communication in order to determine a
subject’s depth of reasoning.
5
are rational)6 are roughly consistent. As well, he finds little evidence of consistent
beliefs though our definitions of consistent beliefs differ. This paper defines consistent beliefs as a condition on one player’s belief hierarchy (does a subject believe her
opponent believes she is playing the action she is actually playing) whereas Healy
defines consistent beliefs based on beliefs and actions (are a subject’s elicited beliefs
about her opponent’s action consistent with the action her opponent actually plays).
Costa-Gomes and Weizsäcker elicit beliefs and choices in normal form games. They
find that players tend not to best respond to their own stated beliefs (suggesting a
large proportion of subjects are not rational), however they also find that players behave differently when asked to state beliefs. Whether or not belief elicitation affects
strategic behavior remains an open question. See Healy (2011) for a more complete
discussion of these issues. Thus, this paper takes a different approach and uses a
design specifically chosen to allow identification of epistemic conditions from choice
data.
This paper is also related to the literature on iterated dominance (Beard and Beil
1994; 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 dominance
solvable games. The existing literature tends to focus on violations of iterated dominance, with the exception of Ho et al. (1998) which measures a subject’s capability to
perform levels of iterated dominance as her level-k level. The current paper measures
subjects order of rationality by classifying subjects into levels of iterated dominance
using a more general identifying assumption than the level-k structure.
This paper proceeds as follows. Section 2 introduces the ring game and motivates the experimental design through examples. Section 3 formalizes the epistemic
concepts of rationality and consistent beliefs and the identifying assumptions and
characterization used to identify the epistemic conditions. Section 4 discusses the experimental design. Section 5 gives the experimental results. And, Section 6 provides
a discussion of the solution concepts of Nash equilibrium, rationalizability, quantal
response equilibrium, and level-k models in relation to the results. Section 7 con6
While it is possible to test rationality by testing whether a subject’s choices are a best response
to her stated 1st-order beliefs about actions, it is not possible to test higher-order rationality in this
matter (i.e. failure of a subject’s stated 1st-order beliefs about actions to be a best response to her
stated 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 the
rationality of their opponents. This has the added difficulty of requiring players to understand the
concept of rationality directly.
6
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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 to
strategic reasoning. Are people rational? To what order do people believe others are
rational? And, do people have consistent beliefs? In this section, we illustrate the
challenge in estimating these three features of reasoning from strategic choice data
and 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 can
be played by a subject who satisfies kth-order rationality7 : a subject who satisfies
kth-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 rationalizable
if and only if it survives k rounds of iterated deletion of strictly dominated strategies.
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7
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any payoffs besides her own.
that analyze the information
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Observing the action profile of a subject in both games B1 and B2 would then allow us to separate the behavioral implications of 2nd-order rationality from 1st-order
rationality.
Following this logic, the behavioral implications of kth-order rationality can be
separated
from lower-orders of rationality by looking at games that differ only in (k!
1)th-order
but have different
rationalizable implications. However,
!
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Player!2!
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15! 5!payoffs,
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b! 5! 10! !
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this problem
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changes
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R1 is 1st-order
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Player!3's!actions
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rationalizable for player 2, and 3rd-order rationalizable for player 1. Player 3 must
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! In addition, R1 and R2 are related to each other in a structured way. Player 1
!
has
! the same payoffs and the same 1st-order payoffs in games R1 and R2 but the
games
have different 3rd-order rationalizable implications. This is possible because
!
! relaxed opponent structure of the ring game breaks the tight link between higherthe
and lower-order payoffs that bimatrix games impose.
R1 and R2 differ only in 2nd-order payoffs for player 1. Thus, these games separate
the behavioral implications of 3rd-order rationality and lower-order rationality under
the assumption that a subject who satisfies 2nd-order rationality but not 3rd-order
rationality will not respond to changes in 2nd-order payoffs. Under this assumption
such a subject will play either (a, a) or (b, b) as player 1 in games R1 and R2 respectively, 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 the
implications of 3rd-order rationality from lower-orders of rationality. Each additional
player in the ring game allows an additional degree of independence between orders
of payoffs. For example, a 4-player ring game could be used to separate 4th-order
rationality from lower orders of rationality.
Once a subject’s order of rationality is identified, a separate set of games can
then be used to estimate whether or not a player has consistent beliefs (a subject
believes 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 characterization of Nash equilibrium based on Aumann and Brandenburger (1995) and
Perea (2007), that combines assumptions about consistent beliefs and rationality to
give testable behavioral implications. In any 2-player complete information game, if
a player satisfies 2nd-order rationality and consistent beliefs, then she has to play
an action consistent with a Nash equilibrium. Thus to estimate whether consistent
beliefs hold, we consider a bimatrix game that has a unique Nash equilibrium but
where 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 indication
of a failure of consistent beliefs.
3
The model
In this section, we apply the tools of epistemic game theory to formally define rationality, higher-order rationality, and consistent beliefs. We formalize the exclusion
restriction that is applied to identify a subject’s order of rationality and present the
characterization of Nash equilibrium that is applied to identify consistent beliefs.
Our experimental design involves both 2-player bimatrix games and n-player ring
games. Thus, we formally consider only n-player ring games which contain the 2player 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 Si are a finite sets, ⇡i : Si ⇥ So(i) ! R, and o : I ! I with
o(i) = 1 + imodn.
The set I represents the set of players, Si represents the set of actions for player
i, ⇡i represents the payoffs for player i which depend upon player i’s action and the
action of her opponent o(i), and the function o represents the opponent mapping
function where o(i) is the opponent of player i. The restriction o(i) = 1 + imodn
restricts 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 ; ŝ1 , . . . , sˆn i where Ti is
a finite set, bi : Ti ! 4(To(i) ), and ŝi : Ti ! 4(Si ).
11
Every player has a set of types Ti . The function ŝi defines a strategy for each
type of player i. The function bi represents each type’s beliefs about the types of her
opponent. Thus, the function bi (ti ) together with the function ŝo(i) defines type ti ’s
beliefs about the strategies of her opponent.
3.1
Epistemic conditions
The expected utility of a type ti can be defined9
ui (si , ti ) ⌘
X
bi (ti )(to(i) )⇡(si , ŝ(to(i) )).
to(i) 2To(i)
Given this specification for utility, a type is rational if the strategy ŝi (ti ) is a best
response for player ti given her beliefs.
Definition. A type ti is rational if ŝi (ti ) maximizes player i’s expected payoff under
the measure bi (ti ). That is, if for any s 2 supp(ŝi (ti ))
ui (s, ti )
ui (s0 , ti ) 8s0 2 Si .
We are interested not only in whether types are rational, but also if they believe
others are rational, if they believe others believe others are rational, and so on. To
define higher-order rationality, we first define what we mean by belief. We say a type
believes an event if she places probability 1 on that event happening.
Definition. A type ti believes an event E ✓ To(i) if bi (ti )(E) = 1. Let the set
B i (E) = {ti 2 Ti |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 following
way. Define
Ri1 = {ti 2 Ti |ti is rational}
m
Rim+1 = Rim \ B i (Ro(i)
)
Definition. If ti 2 Rim then we say that ti satisfies mth-order rationality.
9
Let ⇡(si , ŝ(to(i) )) be redefined in the standard way whenever ŝ is a mixed-strategy.
12
The assumption of consistent beliefs places restrictions on a player’s beliefs about
strategies. Consistent beliefs ensure that player i believes her opponent believes she is
playing the action she is actually playing.10 For simplicity, we define consistent beliefs
only 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 )
P
> 0, then
ŝi (t)(s) · b i (t i )(t) = ŝi (ti )(s) for all s 2 supp(ŝi (ti )).
t2Ti
3.2
Identifying orders of rationality
In this section, the exclusion restriction used to identify a subject’s order of rationality
is discussed. The exclusion restriction essentially assumes that a player with a low
order of rationality will not respond to changes in higher-order payoffs. To state this
assumption, we first formalize what is meant by higher-order payoffs.
Every ring game can be characterized by its payoff hierarchy for each player. For
a given ring game = hI = {1, . . . , n}; S1 , . . . , Sn ; ⇡1 , . . . , ⇡n ; oi, the payoff hierarchy
h̄i ( ) of player i is defined by
h̄i ( ) = {⇡ok (i) }1
k=0 .
The payoffs of player i are given by h̄i0 ( ), her 1st-order payoffs are given by
h̄i1 ( ) (i.e. the payoffs of player o(i)), her 2nd-order payoffs are given by h̄i2 ( ) (i.e.
the payoffs of player o2 (i)), and so on. Payoff hierarchies for 2-player games are
relatively redundant. For a 2-player ring game, h̄1 ( ) = {⇡1 , ⇡2 , ⇡1 , ⇡2 , . . .}. However,
increasing the number of players in the ring permits us to provide a richer set of payoff
hierarchies. For example, in a 3-player ring game, h̄1 ( ) = {⇡1 , ⇡2 , ⇡3 , ⇡1 , ⇡2 , ⇡3 , . . .}.
Thus, each additional player in the ring game adds an additional degree of freedom
between higher- and lower-order payoffs.
Our identification strategy exploits the flexibility of the payoff structure ring
games. The assumption used to identify a player’s order of rationality requires a
player that satisfies kth-order rationality but not (k + 1)th-order rationality to not
respond to changes in her mth-order payoffs whenever m is greater than k. This
10
The definition of consistent beliefs places restrictions on a single player’s belief hierarchy and is a
type of internal consistency. This is in contrast to an assumption of external consistency which would
require a player to hold correct beliefs about the action her opponent is actually playing. Whether
external consistency holds depends not just on a subject’s own beliefs but the actual action of her
opponent, 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 her
1st-order payoffs (or they are the same game). However, there exists different n-player
ring games that differ only in a player’s kth-order payoffs or higher (possible whenever
n > 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 h̄ij ( ) =
h̄ij ( 0 ) for all j <k
ER⇤ acts as an exclusion restriction that separates the behavioral implications of
different orders of rationality. Without an exclusion restriction, the behavioral predictions of the different orders of rationality necessarily nest one another. A player
that satisfies only lower-order rationality could always play an action that a player
that satisfies higher orders of rationality could play. Under ER⇤ , it is possible to
separate the predictions of lower and higher order rationality by considering behavior
in sets of games that differ only in higher-order payoffs. Thus, by observing strategic
choices in specifically chosen sets of ring games it is possible to separately identify
different 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 that
combines both rationality and consistent belief assumptions to form behavioral implications. 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 for
all i 2 I and for all a 2 supp( i ), then
ui (a,
o(i) )
ui (a0 ,
o(i) )
8a0 2 Si
The characterization of Nash equilibrium we provide below is a characterization for
a single player. If a subject satisfies 2nd-order rationality and consistent beliefs, then
she has to play an action consistent with a Nash equilibrium. Thus, for any subject
that satisfies 2nd-order rationality, the characterization can be applied to identify
whether 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 consistent
beliefs). Proposition 1 characterizes Nash equilibrium. The proof is given in Appendix
A.11
Proposition 1. Consider an 2-player ring game = h{1, 2}; S1 , S2 ; ⇡1 , ⇡2 i and a based epistemic type space hT1 , T2 ; b1 , b2 ; ŝ1 , ŝ2 i. Suppose type ti satisfies the following
conditions: (i) 2nd-order rationality and (ii) consistent beliefs. Then it must be that
P
defined by i = ŝi (ti ) and
ŝ(t i )(s) · bi (ti )(t i ) for all s 2 S i
i (s) =
constitutes a Nash equilibrium.
t
i 2T i
This characterization of Nash equilibrium is similar to existing characterizations.
The closest is the characterization by Aumann and Brandenburger (1995) which requires mutual knowledge of conjectures as the consistent belief assumption (if player
1 plays a and player 2 plays b, player 1 must believe player 2 plays b and player 2 must
believe player 1 plays a) and for both players to be rational. We rewrite these as noninteractive conditions (restrictions on only one player’s belief hierarchy): a player is
rational, believes her opponent is rational and has consistent beliefs (i.e. if she plays
a, she believes her opponent believes she plays a). The non-interactive framework is
better suited to testing the epistemic assumptions using individual decision data.
An alternative characterization due to Perea (2007) is also in terms of noninteractive conditions, though the current characterization tightens the sufficient conditions to be inline with Aumann and Brandenburger. Relative to Perea, both the
rationality and consistent belief assumptions are relaxed. This allows us to broaden
the set of players for which we can identify consistent beliefs. In addition, Perea has
two aspects of consistent beliefs that operate (believing you are correct about your
opponent’s beliefs (you put probability 1 on your opponents belief type) and believing
that your opponent is correct about your beliefs (your opponent puts probability 1
on your belief type)) which is a considerably stronger statement than our concept of
consistent beliefs which is only one directional and on actions rather than belief types
11
Throughout the body of this paper we define ’belief’ to be a belief with probability 1 (i.e. if you
believe your opponent is rational, that means you believe your opponent is rational with probability
1). However, in Appendix B, we relax the notion of belief from that of belief with probability 1 to
belief with probability p. We show that with a sufficiently high p, Proposition 1 still characterizes
Nash equilibrium behavior in 2-player games. This allows the identification of consistent beliefs even
if we estimate each player to only believe her opponent is rational with probability p. The robustness
of our identification to a generalization of belief with probability 1 to belief with probability p is
discussed 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 separately 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-player
ring 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
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G2
! !
"#$%&'!(!
!! ! !
"#$%&'!)!
!! ! !
"#$%&'!*!
!! ! !
"#$%&'!+!
()! )!
(5! (4!
()! )!
(5! (4!
!
!
!
!
! 3!! )!"#$%&'!),-!$./012! ! 3!! )!"#$%&'!*,-!$./012! ! 3!! )!"#$%&'!+,-!$./012! ! 3!! )!"#$%&'!(,-!$./012!
!
!
!
!
! .!! )!
$! 6!
3! 4!
.!
! .!! )!
3! (6!
$! 6!
.!
$! ()!
3! (6!
.!
()!
)!
4!
Consider
the
games
G1
and$! G2
in.!Figures 6.!! and
7.3! The
games.!!are)!
similar
to the
!
!
!
$! +! 5! 4!
$! 6! (5! 7!
$! +! 5! 4!
! $! 6! (5! 7!
games
R1 and R2 from Section 2 except that G1 and G2 are 4-player, 3 action games.
!
3! )! ()! )! !
3! )! (5! (4! !
3! )! ()! )! !
3! )! (5! (4!
!
Adding
an additional player permits identification of a player’s order of rationality up
! .!! )!"#$%&'!(!
6! 4! !! ! .!! )!"#$%&'!)!
()! (6! !
.! )! 6! 4! !
.! )! ()! (6!
"#$%&'!),-!$./012-!
"#$%&'!*,-!$./012-!
! !
! ! !
!
! ! ! $! 3! .! 16
! ! ! $! 3! .!
! $! 6! (5! 7! !
$! +! 5! 4!
! !
"#$%&'!(!
! ! !
"#$%&'!)!
!
()! )!
(5! (4!
!
!
! 3!! )!"#$%&'!),-!$./012! ! 3!! )!"#$%&'!*,-!$./012! .!! )!
$! 6!
3! 4!
.! ! ! .!! )!
$! ()!
3! (6!
.!
Type
G1
G2
P4 P3 P2 P1 P4 P3 P2 P1
Rational (R1)
a
!
!
!
b
!
!
!
2nd-order Rational (R2)
a
a
!
!
b
b
!
!
3rd-order Rational (R3)
4th-order Rational (R4)
a
a
a
a
a
a
"
b
b
b
b
b
b
"
a
b
to the 4th-order. Each subject’s order of rationality is estimated by her choice pattern
in the 8 games (2 games ⇥ 4 player positions). Only the payoffs of Player 4 change
across these two games. Thus, player 1 has
the same payoffs up to the 2nd-order,
Action
G2player 3 has the same payoffs,
player 2 has the same payoffs up to G1
the 1st-order, and
P4 P3actions
P2 P1forP4
P3 P2 P1
in bothType
G1 and G2. The predicted
a kth-order
rational player (satisfying
⇤
"
"
" c the!same action
"
Rational
!
ER ) are
given(R1)
in Table 1. A aR1 subject
plays
in G1 and G2 when
!
#plays
! action
#
2nd-order
Rational
c the
b same
she plays
as player
1, 2, (R2)
or 3. Aa R2asubject
in G1 and G2 when
⌘
⌘
3rd-order
Rational
a ba R3 subject
c b plays
a the same action in G1 and
she plays
as player
2 or (R3)
player a1. And,
4th-order
Rational
(R4) 1.a12 a b a c b a c
G2 when
she plays
as player
P1
Type
Rational (R1)
G1
P2
G2
G1
P3
G2
G1
P4
G2
(a,a)(b,b)(c,c) (a,a)(b,b)(c,c) (a,a)(b,b)(c,c)
G1 G2
(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)
Table 1: Predicted actions under rationality and assumptions ER⇤
A player’s order of rationality
is estimated
her action profile in the 8
Rational
+ PEby matching
+ Consistent
games to one of the profiles in Table 1. Consider the 4 types: R1, R2, R3 and R4. The
Type
R1
R2
R1
R2
R1
R2
action of each type is determined by the exclusion restriction ER⇤ and her order of
< 3rd-order Rational a,b,c a,b,c (a,a)(b,b)(c,c) (a,a)(b,b)(c,c)
rationality
over the set of 8 games as given in Table 1. Let ai 2 {a, b, c}8 be an 8-tuple
3rd-order
Rational
a
(a,b)
representing
the action
of subject
ib in each (a,b)
of the 8 games.
Each of the types, R1-R4,
has a predicted action (or set of actions) for these 8 games, S̄k ⇢ {a, b, c}8 . For
example, type R4 has a unique prediction. She must play the 4th-order rationalizable
strategy in each of the 8 games, S̄4 = {(a, b, a, a, c, a, b, c)}. Type R3 has a unique
prediction for 6 of the 8 games, but can play a number of different strategies for
the other 2 games, S̄3 = {(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 S̄4 \ S̄3 = ;.
12
We estimate a player’s order of rationality from a set of finite games. In these games, the
behavioral implications of a belief with probability 1 will be the same for a belief with probability
p, for high enough p. Relaxations of a 1-belief in rationality to a p-belief in rationality is discussed
in Appendix B. We show that allowing for a relaxation to belief with probability p does not change
the identification of orders of rationality or consistent beliefs.
17
Order of
Rationality
Fail
Percent
R0
R1
R2
R3
R4
3
6
4
3
0
60%
35%
17%
16%
0%
We can define similar sets for types R1 and R2 such that S̄n \ S̄m = ; for any
n 6= m 2 {1, 2, 3, 4}.
If a subject’s*Uniform
actionL0profile matches one of the predicted action profiles of type
Rk, then we assign them as that
type. However,
aL3subject’s
actionn profile
does not
L1
L2
L4
= 63
14 choices with a0 predicted
0 action profile,
0
always coincide precisely
in 0this case0we would like
13 choices
2
0
0
3
5
to allow for the possibility
a player might
make
12 choices that10
2
0 an error.
10 We allow
22 subjects to
choicesIf they14are within
6 one error
5 of an exact
17
make at most one11error.
type42match of type
Rk (i.e. the action played in one of the 8 games can be changed so that the action
*G3/G4 Stability
profile matches a predicted action profile of type Rk) then we assign the subject as
1 errors,
Player
type Rk. If a subjectn=63
makes two Player
or more
then2 we assignBoth
them as an irrational
3
of
3
actions
28
21
8
(R0) type.13 If a2 subject’s
is within
of 3 actions action profile
61
62 one error of60two types, we assign
5
of
6
actions
41 assumes that errors
them to the lower type (assignment to the lower type implicitly
due to assumption ER⇤ are more likely than errors due to rationality).14
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 in
Table 2. Each subject could play 3 possible actions in each of the 8 games for a total
of 6561 possible action profiles. The rate of exact type match is extremely high, 51
of the 80 subjects were assigned to types because their action profiles matched one of
the 40 action profiles of types R1-R4 exactly. An additional 24 subjects were assigned
to types R1-R4 because their action profiles matched one of R1-R4’s action profiles
by one error (255 possible action profiles match type R1-R4 with one error). And, the
remaining 5 subjects were assigned as R0 types as they deviated from the predicted
profiles of R1-R4 types by more than one error. Thus, 95 percent of the possible
13
The assignment mechanism can be generated by a finite mixture model where each subject
has some probability of being each of the five types and the probability is chosen to maximize the
likelihood that the subject’s action profile is generated by the given type. Such an approach closely
follows Costa-Gomes and Crawford (2006).
14
We 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 the
R0 type.
4.2
Consistent beliefs
Next, we test whether a player has consistent beliefs about strategies based on strategic choices in game Game G3 in Figure 8. G3 is a 2-player ring game. There is a
unique Nash equilibrium (a, a) in game G3.
0!
8!
b! 2! 10! 18!
c! 2! 12! 16!
!
Player!3's!actions
Player!2's!actions
Player!1's!actions!
!
!
! 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 play
Rationalizable!Sets!
the Nash equilibrium strategy
a in game G3 (as both player 1 and 2). If a subject
!
satisfies 2nd-order rationality
determined! from
G1 and G2
R3 and
G1& ! asPlayer!1!
! !games
Player!2!
! (all
! !R2, Player!3!
Player!2's!actions!
Player!3's!actions!
Player!4's!actions!
!
!
!
!
!
!
!
!
R4 subjects), and fails to play a in G3, then it must be because the assumption of
!
! a! b! c! ! ! ! a! b! c! ! ! ! a! b! c!
consistent beliefs fails.
a! 8! 20! 12! !
a! 14! 18! 4! !
a! 20! 14! 8!
b! 0!
8! 16!
!
!
Player!3's!actions
!
!
!
Player!2!
!
a! b! c!
a! 14! 18! 4!
!
!
!
!
b! 20! 8! 14!
!
Player!3's!actions
!
!
!
!
!
!
!
!
Player!1!
!
!
!
a! b! c! 19 !
a! 8! 20! 12! !
Player!2's!actions
Player!2's!actions
Player!1's!actions!
G1& !
!
!
!
!
Player!3's!actions
Player!2's!actions
Player!1's!actions!
!
The action a is also a rationalizable action. A player who satisfies any level
0! 8! 16! !
20! 8! 14! !
16! 2! 18!
of rationality can play a. b!Because
of this, our b!estimate
of consistentb!beliefs
will
!
!
c! 18!on12!
16! 18!
c! consistent
0! 16! 16!
necessarily be an upper bound
the 6!proportionc! of0!players
who satisfy
!
beliefs.
G2& !
Player!1!
! ! !
Player!2!
! ! !
Player!3!
Strategic choices from
games
G1,
G2
and
G3
permit
us
to
categorize
subjects
Player!2's!actions
!
Player!3's!actions
!
Player!4's!actions
!
!
!
! ! !
! ! !
! ! the
! a! b!rational
c! ! types
! ! that
a! do
b! not
c! satisfy
! sufficient
a! b! c!
into 3 epistemic types: ! boundedly
rationality conditions required
play12!
Nash! equilibrium
in 2-player
(R014!
and8!
a! 8!to 20!
a! 14! 18!
4! ! games
a! 20!
!
!
R1 types), 2nd-order rational
the
sufficient
b! types
0! 8! that
16! satisfy
b! 20!
8! 14!rationality
b! conditions
16! 2! 18!
required to play Nash equilibrium in 2-player
games but do not! satisfy consistent
c! 18! 12! 6! !
c! 0! 16! 18!
c! 0! 16! 16!
beliefs (R2, R3 and R4! types who do not play (a, a) in G3), and those that satisfy
! consistent belief requirements (R2, R3 and R4 types who
both the rationality and
!
play (a, a) in G3).
!
!
!
Player!3!
!
a! b! c!
!
!
!
!
!
!
!
!
!
!
!
!
a! 20! 14! 8!
!
!
!
!
b! 16! 2! 18!
!
Player!4's!actions
!
!
!
!
!
6! 18!
a! 6!
Player!4's!actions
c! 2!
Player!2!
!
a! b! c!
Player!1's!actions
!
!
!
!
!
!
!
!
Player!4's!actions
b! 2! 12! 2!
!
!
!
!
!
!
!
a! 4! 10! 9!
!
!
!
!
Player!4's!actions
Player!1!
!
a! b! c!
Player!2's!actions
!
!
!
!
Player!2's!actions
Player!1's!actions!
&
!
!
!
!
!
Play
Player!
a!
a! 12! 1
b! 8! 1
c!
6! 1
!
!
!
Player!
Play
a!
a! 8! 1
b! 6! 1
c! 12! 1
!
!
!
Play
Player!
a!
a! 12! 1
b! 8! 1
4.3
Laboratory implementation
Sessions were conducted in Arts ISIT computer labs with undergraduate students
at the University of British Columbia. No subject participated in more than one
treatment. Subjects made all decisions through an online interface. In order to ensure
independence across subjects, subjects did not interact with one another during the
experiment and were not informed of one another’s decisions.
We used a within-subjects design in which subjects played all games in each session. 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 order
of rationality and consistent beliefs for each subject.
Subjects played the games in a random order without feedback. Subjects were
required to spend at least 90 seconds on each of the games. Once subjects made
choices in all games they were given the opportunity to revise their choices (without
any feedback). If subjects chose to revise their choices they could review each of the
games (in the same original order) and make changes to their choices.
Subjects were paid a $5 showup fee and received payment from their choices in
one of the games. One of the games was randomly selected for payment at the end
of the experiment. Subjects were randomly and anonymously assigned to 2 or 4
player 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 the
selected game and received the dollar value of their payoff in the game. Subjects were
paid in cash at the end of the experiment. The average session lasted 45 minutes and
the average subject earned approximately $17 dollars (maximum payment was $25
and minimum payment was $7, including showup fee). Payments were in Canadian
dollars.
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 short
understanding quiz to make sure they understood the instructions. The complete
instructions and quiz can be found in Appendix C.
The dataset includes observations from 6 sessions with either 12 or 16 subjects in
each session. In total, 80 subjects are represented in the dataset.
15
4 additional, related games were played by each subject. The data from those games is not
analyzed in this paper
20
5
Experimental results
Figure 9 gives the proportion of subjects classified as each type R0-R4. 6 percent of
subjects are classified as R0 types, 22 percent are classified as R1 types, 27 percent
are classified as R2 types, 25 percent are classified as R3 types, and 20 percent are
classified as R4 types. Overall, 94 percent of subjects are rational, 72 percent of
subjects are 2nd-order rational, 45 percent are 3rd-order rational, and 20 percent are
4th-order rational.
100%#
80%#
70%#
R3#
60%#
50%#
R2#
40%#
30%#
Propor%on'of'Subjects'
90%#
R4#
20%#
R1#
10%#
R0#
Orders'of'Ra%onality'
0%#
Figure 9: Subjects classified by order of rationality
The estimated orders of rationality suggest that 28 percent of subjects fail to play
Nash equilibrium because of a failure of bounded rationality. This includes subjects
who are irrational and those who are rational but fail to account for the rationality of
others. 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 the
sufficient rationality conditions for Nash equilibrium.
Next, we estimate whether each subject satisfies consistent beliefs based on their
choices in game G3. Subjects are categorized into 3 epistemic types: (1) boundedly rational types: types that do not satisfy the sufficient rationality conditions
required for Nash equilibrium (and for whom we can not observe consistent beliefs)
(2) 2nd-order rational/inconsistent types: types that satisfy the sufficient rationality
conditions of Nash equilibrium but do not satisfy consistent beliefs, and (3) 2nd-order
rational/consistent types: types that satisfy all three sufficient conditions for Nash
21
equilibrium. Not a single subject in our sample satisfies all 3 sufficient conditions for Nash equilibrium. It is also important to notice that while failure of
Nash equilibrium is due in part to bounded rationality, failure to play Nash equilibrium rarely occurs because subjects are not rational. In addition, the failure to play
Nash equilibrium is largely due to a failure of having consistent beliefs. 28 percent
of the subject pool are boundedly rational and 72 percent of the subject pool are
2nd-order rational with inconsistent beliefs.
Propor%on'of'Subjects'
80%#
70%#
60%#
50%#
40%#
30%#
20%#
10%#
0%#
Boundedly#Ra6onal####### 2nd7order#Ra6onal#+#
inconsistent#beliefs#
2nd7order#Ra6onal#+#
consistent#beliefs#
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 instructions were read aloud. The quiz was designed to test the subject’s understanding
of the game structure and to make sure they understood the instructions. The quiz
can be found in Appendix C. 16 subjects failed the quiz. Table 3 breaks down the
number of subjects who failed the quiz by order of rationality. 60 percent of the
irrational 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 subjects
failed. This suggests an explanation for the boundedly rational subjects - they did
not have a clear understanding of the games they were asked to play.16
16
Arguably, we may want to limit our analysis only to subjects who pass the quiz if we are
interested 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 boundedly
rational and 80 percent are sufficiently rational but have inconsistent beliefs.
22
Order of
Rationality
*G1 and G2: Only subject who pass quiz
Rationality + PE +
errors
R0
R1
R0
R2
R3
2
R4
Rationality + PE
Rationality
Fail
Percent
15
3
6R2
4
3
0 14
5
9
13
16
13
56
5
18
25
19
13
80
R1
18
60%
35% R4
17%
16%
0% 16
65
R3
n
Table 3: Subjects who failed quiz, classified by order of rationality
Subjects were *Uniform
also givenL0the opportunity to revise their choices after completing
all of the games for the first time. The subjects could review each of the games (in
L1
L2
L3
L4
n = 63
the original order) 14
and
make changes
choices
0 if they 0desired. 39
0 subjects 0chose to review
0
Players
who play same
action in 0G1 and G2, by
type
13
choices
2
0
3
their choices. Table 4P1gives the estimated
orders
of rationality
basedtotal
off of 5initial
P2
12 choices
10
2P3
0 P4
10
22
choices versus final 11
choices.
rationality42
based
choices The rows
14 give the6estimated 5orders of 17
L0
1
1
0
0
1
off initial L1
choices and the
orders0 of rationality
11 columns 9give the estimated
7
11 based off
L2 The estimated
12
17
0 unaffected
0 by the opportunity
20
final choices.
distribution
is largely
to
*G3/G4
L3
8 Stability 0
0
0
11
review. The
changes
for only0 11 subjects11from initial
L4 estimated0order of rationality
0
0
n=63
Player 1
Player 2
Both
to final choices. 7 subjects increase their order of rationality and 4 subjects decrease
3 of 3 actions
28
21
8
their order of rationality.
2 of 3 actions
61
62
60
5 of 6 actions
Inital
Choices
R0
R1
R0
4
0
R1
1
16
Assignment
R2
0
2
Exact
R3 type match
0
0
R4
0
Type match with 1 error0
41
Final Choices
R2
R3
R4
#Review
0
1
0
2
0
0
0
7
# of subjects
assigned
# of4action profiles
22
2
17
1 51
16
0 40 6
0 24
0
12 290 7
R0 (2+ errors)
5
6231
Total
80
6561
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’s
order of rationality may be mispecified. However, it is possible to bound our observed rationality distribution under different assumptions on subjects’ behavior. For
example, suppose a player satisfied kth-order of rationality, did not satisfy ER⇤ , but
played all actions in the kth-order rationalizable set with equal probability. Then the
probability that a Rk type is really a R(k-1) type is 1/9. Additionally, the behavior of
irrational 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 identification of R1-R4 types as 95 percent of the action profiles, if played, would get a subject
assigned as an R0 type. Second, if an irrational type plays randomly, as is often
assumed, then there is only a very small chance of such a type getting misspecified
as an R1-R4 type. Specifically, there is a 1/81 chance that an irrational type would
get assigned as a R1 type, a 1/243 chance an irrational type would get assigned as
an R2 type, and so on. We might expect then that only 1 of our 80 subjects was
misidentified as a R1 type rather than a R0 type.
6
Discussion
The experimental results show that no subject satisfies all of the sufficient conditions
required to play Nash equilibrium. However, there are a number of alternative solution concepts to Nash equilibrium. Popular alternatives are rationalizability, quantal
response equilibrium (QRE), and level-k models.17 Each of these solution concepts
relaxes at least one of the assumptions underlying Nash equilibrium. Rationalizability
relaxes the consistent belief assumption of Nash equilibrium but requires players to
satisfy common knowledge of rationality (i.e satisfy kth-order rationality as k tends
to infinity). QRE relaxes the rationality requirements of Nash equilibrium but maintains the consistent belief assumption. Level-k models maintain the assumption that
subjects are rational but relax the consistent belief assumption by allowing players
to hold heterogenous beliefs about the rationality of others.
In addition to imposing different assumptions on players’ rationality and belief
structures, these alternative solution concepts impose additional structural assumptions. Rationalizability, in the most general terms, imposes no additional assumptions, but only makes precise predictions in rationalizable games. When testing the
model it is often assumed that players play all possible rationalizable actions with
equal probability. The QRE model imposes a logistic error structure to describe the
probability of error in best responding. Level-k models impose a particular behavior
for L0 types and particular assumptions about the beliefs of each level about the levels of other players. The success of these alternative models is typically judged by the
likelihood of a model to explain a given data set evaluated at likelihood-maximizing
17
This paper does not separate between level-k and cognitive hierarchy, hence we limit discussion
to level-k models for simplicity.
24
parameter estimates. Thus, each solution concept is judged based on the complete
package of assumptions about rationality, beliefs, and additional structural assumptions.
The standard methodology makes it difficult to separately identify the role that
each of these assumptions plays in explaining a given data set. In addition, it is unclear
to what extent the structural assumptions improve the models fit or to which extent
the different models make different testable predictions. For example, when fitting
the models to data it is assumed that both QRE and level-k types make mistakes
according to a logistic error distribution. However, this is explicitly modeled under
QRE and players take into account the fact that others are making mistakes when
they make their choices. Whereas, under level-k, mistakes happen ex post and players
do not take into account the fact that others are making mistakes when they make
their decisions.
The approach in this paper allows us to abstract from the additional structural
assumptions and directly assess the assumptions of rationality and consistent beliefs
that underlie these solution concepts. Our results are consistent with certain classes of
existing 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 inconsistent
with the assumptions imposed under rationalizability which would require all subjects
to satisfy 4th-order rationality. Quantal response equilibrium imposes the assumption
that players have consistent beliefs but may not be rational. However, our results
suggest that the opposite holds, most subjects are rational but consistent beliefs does
not hold. The level-k model is the most promising as it incorporates the assumption
that players are rational, relaxes the assumption of consistent beliefs, and allows for
heterogeneity in subjects’ beliefs about the rationality of their opponents. We take a
closer look at the QRE and level-k solution concepts below.
QRE is modeled by assuming players make mistakes according to a logistic error
function. 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 the
logistic error distribution and depend upon the relative payoff of each action. If the
payoff to a given action is high, then it is more likely to be played. The QRE model
is typically tested by fitting the solution concept to data by choosing the precision
parameter according to maximum likelihood analysis.
Definition. Consider a n-player ring game. The strategy
25
2 4(S1 )⇥· · ·⇥4(Sn ) is a
quantal response equilibrium (QRE) of
then
i (a)
if for all i 2 I and for all a 2 supp( i ),
exp( · ui (a, o(i) ))
= P
exp( · ui (si , o(i) ))
si 2SI
Under the level-k model, a Lk type plays an action determined by (k) in the
level-k equilibrium. Every Lk type must play a best response to what the Lk-1 type
is doing, with the exception of the L0 type. L0 types do not need to play a best
response (and hence can play any action). Typically, the level-k model is tested by
specifying a behavior for L0 types ex ante and then fitting the model to the data by
choosing the proportion of level-k types (and the error with which each type makes
decisions) according to maximum likelihood analysis.18
Definition. Consider a n-player ring game. The strategy = 1 ⇥ · · · ⇥ n , where
if for all i 2 I, for all k 1 2 N and
i : N ! 4(Si ) is a level-k equilibrium of
for all a 2 supp( i (k))
ui (a,
o(i) (k
1))
ui (a0 ,
o(i) (k
1)) 8a0 2 Si
The majority of our subject pool satisfies rationality, which does not fit QRE
assumptions unless the precision parameter, , is high (people do not make mistakes
when best responding often). But, even if a high could explain the high proportion
of rational subjects, QRE is not able to predict the pattern of behavior across the 8
games described by G1 and G2 for any value of the precision parameter. Figure 11
gives the experimental proportion of subjects who play the rationalizable actions in
each of the player positions in games G1 and G2. The empirical frequency of playing
the rationalizable strategy increases across the player positions (position 1 to 4) in
both games G1 and G2. The QRE predictions are given in Figure 12. The QRE
predictions are not consistent with aggregate data for any . For any value of , the
QRE equilibrium does not predict that the rationalizable action will be played most
often by player 4, then player 3, player 2 and player 1 in descending order. This
is because QRE imposes the assumptions that player’s make mistakes when best
responding and that consistent beliefs hold. This effectively means that QRE cannot
18
This paper does not differentiate between level-k and cognitive hierarchy models (hence we
consider 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. This
means, more or less, that a high probability of playing the rationalizable action for
player 4 will impose a high probability of playing the rationalizable action for the
rest of the players or vice versa). However, the logit error distribution (probability
you play particular actions depends upon relative payoffs) does interact somewhat
non-intuitively with the assumption of consistent beliefs. From Figure 12 note that
the relative likelihoods of playing the rationalizable action for the players vary as
changes. However, the model never predicts the pattern that we observe in the
data, that the likelihood of playing the rationalizable action increases from player 1
to player 4.
P4##
Propor%on'of'Subjects'Playing'
Ra%onalizable'Strategy'
100%#
90%#
P3#
P2#
P4#
P1#
P3#
80%#
70%#
60%#
P2#
50%#
P1#
40%#
30%#
20%#
10%#
0%#
G1#
G2#
Game'
1$
Probability*of*Playing*Ra/onalizable*
Strategy*
Probability*of*Playing*Ra/onalizable*
Strategy*
Figure 11: Proportion of subjects who play rationalizable action in G1 and G2
0.9$
0.8$
0.7$
Player$1$
0.6$
Player$2$
0.5$
Player$3$
Player$4$
0.4$
0.3$
0$
0.15$
0.3$
0.45$
0.6$
0.75$
1$
0.9$
0.8$
0.7$
Player$1$
0.6$
Player$2$
0.5$
Player$3$
Player$4$
0.4$
0.3$
0$ 0.15$0.3$0.45$0.6$0.75$0.9$1.05$1.2$1.35$1.5$
Lambda*
Lambda*
Figure 12: QRE predictions for games G1 and G2
However, level-k models predict the aggregate behavior in Figure 11. Consider any
distribution 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 player
1. L3, and L4 types play the rationalizable strategy as player 2. L2, L3 and L4
types play the rationalizable strategy as player 3. L1, L2, L3 and L4 types play the
rationalizable strategy as player 4. No matter what the distribution over levels or
the specification for L0, the proportion of subjects playing the rationalizable strategy
will weakly increase over the player positions 1-4.
Further, the level-k model (without allowing for errors) is only consistent with 1
percent of the action profiles that could be played in the 8 games defined by G1 and
G2. 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 the
level-k model (i.e. play those 1 percent of actions).19 This provides a strong test of
some of the key assumptions underlying the level-k model.
In addition, kth-order rationality and the ER⇤ assumption capture the main assumptions defining the behavior of an Lk type. L1 types are rational and do not
respond 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 gives
us a ‘L0’ independent estimate of the level-k distribution. The distribution of levels
in Figure 9 puts more weight on higher levels than is generally estimated in level-k
experiments. This could suggest that the usual assumption of uniform L0 or the
structural estimation approach, biases the estimation of the level distribution.
7
Conclusion
This paper applies epistemic game theory to experimental design in order to estimate
key assumptions underlying strategic reasoning. This approach differs from the typical approach in behavioral game theory that implicitly tests epistemic assumptions
by jointly testing all epistemic and structural assumptions embodied in a solution
concept. Instead, we directly test epistemic assumptions which gives direct insight
into 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
19
This 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 their
payoff structure relative to standard game forms. Using such a design allows us to
make weaker identification assumptions than those that are typically imposed in the
literature.
The main findings are that players are rational, have finite and heterogenous
orders of beliefs about the rationality of others, and do not satisfy consistent beliefs.
These features have implications for how we should model strategic behavior. The
class of level-k and cognitive hierarchy models are consistent with these experimental
results. However, the results in this paper do not speak to other features of this class
of models such as the nature of the L0 specification, how salient the L0 specification
is for different players, nor the differences between the level-k and cognitive hierarchy
models. Further research is needed to address these assumptions.
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31
A
Proof of Proposition 1
Let ti be a type that satisfies conditions (i) and (ii).
Let T̄ = {t i 2 T i |bi (ti )(t i ) > 0}.
Since ti is rational it must be true that for any s 2 supp( i )
ui (s, i ) ui (a0 , i ) 8a0 2 Si .
In other words, i is a best response to i
Further, since ti 2 Ri2 ) t i 2 R1 i for all t i 2 T̄
P
And, since ti has consistent beliefs it must be that
ŝi (t)(s) · b i (t i )(t) =
i (s)
t2Ti
8s 2 Si , for all t i 2 T̄ .
Therefore, for any t i 2 T̄ and any action s 2 supp(ŝ i (t i ))
u i (s, i ) u i (s0 , i ) 8s0 2 S i .
In other words, ŝ i (t i ) is a best response to i for all t i 2 T̄ .
S
But, since supp( i ) =
supp (ŝ i (t i )). It must be that for any s 2 supp(
t
i 2T̄
u i (s,
i)
u i (s0 ,
And, hence i is a best response to i .
Therefore, is a Nash Equilibrium.⇤
32
i)
8s0 2 S i .
i)
B
Relaxing belief to p-belief
In the experiment ran in this paper, we technically do not estimate whether a subject’s beliefs hold with probability 1. This is because for finite games, the behavioral
implications of a belief with probability 1 will be the same for a belief with probability
p, for high enough p. In this appendix, we show that allowing for a relaxation of belief
to that of p-belief (belief with probability 1 to belief with probability p) is consistent
with our experimental results.
We say that a type p-believes an event if he believes that event with probability
at least p.
Definition. A type ti p-believes an event E ✓ To(i) if bi (ti )(E)
p. Let the set
i
i
i i
Bp (E) = {t 2 T |b (ti )(E) p} be the set of types for player i that p-believe event
E.
We define higher-order p-rationality by recursively defining the following sets:
1
Ri,p
= {ti 2 T i |ti is rational}
m+1
m
m
Ri,p
= Ri,p
\ Bpi (Ro(i),p
)
m
Definition. If ti 2 Ri,p
then we say that ti satisfies mth-order p-rationality
We can also define a relaxed version of consistent beliefs for 2-player games. This
definition ensures that a player believes her opponent beliefs she is playing the action
she is actually playing with at least probability p.
Definition. A type ti has consistent p-beliefs if for all t
P
bi (ti )(t i ) 0 then
ŝi (t)(s) · b i (t i )(t) ŝi (ti )(s) · p 8s 2 Si
i
2 T
i
such that
t2Ti
m
Ri,p
= Rim 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 predictions
under kth-order p rationality are the same as the behavioral predictions under kthorder 1-rationality (when p is sufficiently high). Thus, we do not identify whether
a subject believes his opponent is rational with probability 1, but rather a lower
bound p̄ such that she believes her opponent is rational with probability p̄. Given
33
this the best we observe is whether a subject believes others are rational with at least
probability 78 .
Our identification of consistent beliefs, relies along the assumption that subject’s
satisfied the sufficient rationality conditions for Nash Equilibria in G3. Thus, it is
necessary to relax the sufficient epistemic characterization of Nash equilibrium to
allow 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 for
games G1 and G2 ensures that they satisfy 2nd-order 78 -rationality. Thus, if they fail
to play the Nash equilibrium in G3 it must be because they fail to satisfy consistent
7
-beliefs. The following proof establishes that p-belief with p > 67 is sufficient to
8
guarantee that (a, a) is the unique action profile in G3.
7/9#
b#
a#
7/9#
b#
9/13#
9/11#
probability#player#2#plays#a#
c#
a#
1/2#
1/7#
c#
b#
probability#player#1#plays#b#
c#
probability#player#1#plays#b#
9/13#
probability#player#2#plays#b#
probability#player#2#plays#b#
Proof
a#
1/2#
5/7#
9/11#
probability#player#2#plays#a#
c#
probability#player#1#plays#a#
1/7#
b#
a#
Figure 13: Best response correspondences for game G3 (player 1 and 25/7#respectively)
probability#player#1#plays#a#
Player 1: Consider the behavior of a type t1 that satisfies 2nd-order p-rationality
and consistent p-belief with p > 67 .
P
Define 1 = ŝ1 (t1 ) and µ2 (s) =
ŝ2 (t)(s) · b1 (t1 )(t) 8s 2 S2 . And, let t2 2 T2 such
t2T2
P
that b1 (t1 )(t2 ) > 0 and define µ1 (s) =
ŝ1 (t)(s) · b2 (t2 )(t) 8s 2 S1 .
t2T1
Consider 2 cases:
(i) Suppose t1 plays b (ŝ1 (t1 )(b) > 0 and hence ŝ1 (t1 )(c) = 0 since player 1 will never
rationally mix between b and c )
Since t1 is rational and believes opponent playing according to µ2 , it must be that:
34
µ2 (b) 79 59 µ2 (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 know
that 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 (ŝ1 (t1 )(c) > 0 and hence ŝ1 (t1 )(b) = 0 since player 1 will never
rationally mix between b and c)
Since t1 is rational and believes opponent playing according to µ2 , it must be that:
9
11
u2 (b)  13
µ (a).
13 2
Therefore,
2
µ2 (c) 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 know
that 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-rationality
and consistent p-belief with p > 67 .
P
Define 2 = ŝ2 (t2 ) and µ1 (s) =
ŝ1 (t)(s) · b2 (t2 )(t) 8s 2 S1 . And, let t1 2 T1 such
t2T1
P
that b2 (t2 )(t1 ) > 0 and define µ2 (s) =
ŝ2 (t)(s) · b1 (t1 )(t) 8s 2 S2 .
t2T2
Consider 2 cases:
(i) Suppose t2 plays b (i.e. ŝ2 (t2 )(b) > 0)
Thus, µ1 (c) 17
Now, consider the behavior of any rational type t1 such that b2 (tt )(t1 ) > 0
3 cases to consider:
(a) t1 plays b
Therefore, µ2 (b) 12 .
Thus all rational types t such that b2 (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 a
and c. Therefore µ2 (a) + µ2 (c) p ) µ2 (b)  p . Contradiction.
(b) t1 plays c
35
2
Therefore, µ2 (c) 11
.
Thus all rational types t such that b2 (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 a
and 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 type
plays a. Thus µ1 (a) p and µ1 (b) + µ1 (c)  1 p. Contradiction.
(ii) Suppose t2 plays c (i.e. ŝ2 (t2 )(c) > 0)
Then, µ1 (b) 17 .
3 cases to consider:
(a) A rational type t1 (such that b2 (t2 )(t1 ) > 0) is mixing between a and b
Therefore, µ2 (b) 12 .
Thus all rational types t such that b2 (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 a
and c. Therefore µ2 (a) + µ2 (c) p ) µ2 (b)  p . Contradiction.
(b) Any rational type t1 (such that b2 (t2 )(t1 ) > 0) is mixing between a and c
2
Therefore, µ2 (c) 11
.
Thus all rational types t such that b2 (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 a
and b. Therefore µ2 (a) + µ2 (b) p ) µ2 (c)  p . Contradiction.
(c) Any rational type t1 (such that b2 (t2 )(t1 ) > 0) plays a, they all play a (mixing
ruled out in case (a) and (b)) Thus µ1 (a) p and µ1 (b)+µ1 (c)  1 p. Contradiction.
⇤
36
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 make
good 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 you
have 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 each
game you will choose one of three actions. Each other participant in your game will also choose one of three actions.
Player 2's Earnings
Player 3's actions
d
e
f
g
h
i
a
10
4
16
d
12
16
4
b
20
8
0
e
0
12
8
c
4
18
12
f
4
4
20
Player 3's Earnings
Your actions
Player 3's actions
Your Earnings
Player 2's actions
Player 2's actions
Your actions
C
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 a
table 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 will
determine 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 the
action 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 dollars
and 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 your
counterparts 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 carefully
at 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 game
if 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 in
this room, will be paid based on their actions and the actions of their randomly chosen group members in the selected game. Any of the
games 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 your
randomly matched counterpart only at the end of the experiment. You will not learn any other information about the actions of other
player'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 the
campus Human Subjects Committee and we will be in serious trouble. These instructions are meant to clarify how you earn money, and
our interest is in seeing how people make decisions.
37
**Do not use the BACK or REFRESH Buttons**
Player 2's Earnings
Player 3's actions
d
e
f
g
h
i
a
10
4
16
d
12
16
4
b
20
8
0
e
0
12
8
c
4
18
12
f
4
4
20
Player 3's Earnings
Your actions
Player 3's actions
Your Earnings
Player 2's actions
Player 2's actions
Your actions
Quiz
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
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