information processing, learning, and analogy-based expectations

Information Processing, Learning, and Analogy-BasedExpectations: An Experiment�

Ste¤en Hucky Philippe Jehielz Tom Rutterx

October 19, 2006

Abstract

Mental content comes to mind more or less easily depending on its �accessibility�(Higgins 1996).We consider how learning is a¤ected by the framing of feedback information about opponents�play when subjects are engaged in several interactions. Speci�cally, we examine an experimentalsetting where the feedback�s content is identical across treatments, but the presentation di¤ers,inducing di¤erent degrees of accessibility. Behavior di¤ers strongly between treatments and weshow that the observed di¤erences can be organized by applying the concept of analogy-basedexpectation equilibrium (Jehiel 2005). Additional treatments were designed to examine behaviorwhen feedback mirrors the theoretically assumed mental representations, and behavior in thesetreatments con�rms the analogy-based expectations hypothesis. More generally, our results sug-gest the need to incorporate framing considerations into equilibrium analysis.

Keywords: Analogy-based expectation; information processing; experiments; accessibility;interactive learning.

JEL Classification: C72; D82.

1 Introduction

Information comes to mind more or less easily depending on its accessibility (Higgins 1996). For the

sake of illustration, consider Figures 1a and 1b.1 As we look at the object in Figure 1a, we have

an immediate impression of tower�s height, but not of the total area that its building blocks would

cover if the tower were dismantled (the total area can be estimated by a deliberate procedure, such

as multiplying the base area of a block by the number of blocks, but clearly it is less immediately

accessible). The situation is reversed in Figure 1b in which the total area is immediately accessible,

but the height of the tower that could be constructed with these building blocks is not.

The accessibility dimension of information may have important implications for the understanding

of how information is being processed in a variety of applications. In this paper, we consider the

question of how information is being processed in a setting where players play di¤erent games and

try to learn the behaviors of others so as to make the best choice of strategy in each game. Our

�Acknowledgements to be added.yUCL & ELSE.zPSE, UCL & ELSE. Corresponding author at address: PSE, 48 Bd Jourdan, 75014 Paris. email: [email protected] & ELSE.1This example is taken from Kahneman (2003).

1

Figure 1a Figure 1b

main insight concerns the comparison of play in treatments in which the same objective feedback

about opponents�past play is given to subjects� but with various degrees of accessibility. Speci�cally,

subjects play two di¤erent games about which they only know the structure of their own payo¤s. In

each round and prior to the action choice, it is announced whether subjects play game A or game

B. In addition, subjects receive some information about the past play of their opponents in both

games. In principle, the information is precise in that it details past play for both games separately.

However, the precise information is relatively accessible in the �rst treatment but not in the second.

As a consequence, subjects in the second treatment do not make full use of the precise information.

Rather they behave as if they were using much coarser information. We observe that the di¤erence

in accessibility yields very di¤erent patterns of play even after 60 rounds. In addition, we run two

control treatments, one where accessibility is not an issue because the information is precisely spelled

out for subjects and one where the information is exogenously coarse, and observe a marked similarity

between behaviors in the control and test treatments.

We will explain the di¤erence of behaviors in our various treatments by relying on the recent

approach of analogy-based expectation equilibrium (Jehiel 2005). This approach relaxes the traditional

Nash equilibrium concept by assuming that players may have a partial rather than total understanding

of their opponents� strategies, and it parameterizes players by how precisely they understand the

strategy of their opponents. The basic motivation for the analogy-based expectation approach is

that in complex environments it may be too hard to learn the reaction of other players for every

situation. Instead, players are viewed as relying on coarse statistics that summarize partial aspects of

other players�choices. Speci�cally, players bundle situations or games into analogy classes, and form

expectations only about the aggregate behavior of opponents in each such class. Players are assumed

to best-respond against these beliefs and, in equilibrium, the beliefs coincide with the aggregate play

in each class. In general, the equilibrium that is so obtained is sensitive to the analogy partitions used

by the players.2 It is worth mentioning that the analogy-based expectation equilibrium (or ABEE)

2See Jehiel (2005), Jehiel and Koessler (2006) and Ettinger and Jehiel (2006) for various applications of the approachto the so called �nite horizon paradoxes, bargaining/ultimatum games, global games, strategic information transmissiongames, betting games and reputation games.

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is viewed as the limiting outcome of a learning process in which players would only keep track of the

aggregate statistics of the opponents�average play in each analogy class; it is not viewed as the result

of introspective reasoning (see Jehiel 2005).

In our experiment, there were two underlying normal form games A and B, both involving a

Column Player and a Row player. In both A and B, the Row player had to choose between three

rows �; �; or , and the Column player had to choose between �ve columns a; b; c; d; or e. A priori

there are two possible analogy groupings for each player: the �ne analogy grouping (the two games

are treated separately) and the coarse analogy grouping (the two games are bundled into one class).

The payo¤s in games A and B were chosen so that the prediction of the analogy-based expectation

equilibrium approach is markedly di¤erent depending on whether the Row player uses the �ne or the

coarse analogy grouping (assuming the Column player uses the �ne analogy grouping)� with a unique

equilibrium employing pure strategies in each case (see Section 2).

We have conducted several experiments in which subjects were assigned either to the role of the

Row player or to the role of the Column player for the entire experiment. Each subject played an

equal number of games A and B over 60 rounds. Subjects were informed of the action space structure

of the two players and of their own payo¤ structure, but not of the payo¤ structure of their opponent.3

In each round, subjects in each role were randomly matched to subjects in the other role. The various

treatments di¤ered in the feedback given to subjects between each round of the experiment. In all

cases, the feedback was about the past play of opponents in the last �ve rounds. Note that subjects

received no feedback about their own performance until the end of the experiment at which time they

were informed about how much they had earned over the 60 rounds.4

Intuitively, when subjects have access to the behavior of their opponents game by game this

corresponds to the �ne analogy grouping and when they only have access to the aggregate behavior

this corresponds to the coarse analogy grouping. In all treatments, Column players had access to past

choices of Row players game by game. Speci�cally, as a Column player was to play game ! = A or B

he or she was being informed of the distribution of play of Row players (in the entire population) over

the last �ve rounds in game !. So the corresponding analogy grouping of Column players is always

the �ne grouping.

Treatments di¤ered in the accessibility dimension of the feedback given to Row players. In the

�rst benchmark treatment, Fine, Row players received feedback about Column players�past behavior

in the same way as Column players. In the second benchmark treatment, Coarse, they only received

aggregate feedback, that is, they were told how often Column players had chosen each of the �ve

actions in both games without being told how the behavior was distributed between game A and B.

In addition, we have two treatments, our main treatments, where, in principle, the (relevant)

3We adopted this framework so as to make introspective reasoning about opponents play impossible. This alsoavoided the di¢ culty pointed out by Ehrblatt et al. (2005) that belief formation may be a¤ected by the knowledge ofopponents�payo¤s.

4Not providing feedback about past performance allowed us to focus on our central issue of interest, e.g. whethersubjects process information about opponents�past play game by game or on aggregate over the two games, rather thanon the issue whether subjects rely on belief-based like learning or reinforcement learning-like procedures (see Erev andRoth, 1998, and Camerer and Ho, 1999, for an investigation of that alternative issue).

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information provided is identical as in treatment Fine but not quite as immediately accessible.5 The

�rst treatment is designed to make �ne analogy grouping feasible (without making it obvious). We

shall refer to this treatment as Feasible. In this treatment feedback is given in a matrix form as

depicted in Figure 2 (where the letter inside a square indicates the game and the color of the square

indicates the action played by the Column player in that speci�c interaction� actions appeared as

colors also when playing the game; grey squares indicate that not enough rounds had yet been played

to �ll the grid).

Figure 2

In treatment Hard, the feedback is given in two consecutive screens, as shown in Figures 3a and

3b where the �rst screen informs subjects about the distribution of actions (in color form) that have

been played in the last �ve rounds and the second screen shows a string of letters indicating the games

in which these actions have been taken. It is worth emphasizing that, even though the two screens

appeared in di¤erent formats, subjects were informed about the correspondence between the two.6

Thus, feedback in the two treatments is objectively the same (and the same as in treatment Fine),

even though accessibility is not: the distribution of play in each game A and B is obviously made

more accessible in Feasible than in Hard.7 ;8 So, what can we expect subjects to do in treatment

5Admittedly, in treatment Fine, row players received only information about the past play in the game they wereplaying, whereas in treatments Feasible and Hard feedback was given for both, but as this is the only relevantinformation we don�t consider the information to be practically di¤erent. Note also that the games under study aredominance solvable, thereby ensuring that, in each game, the Nash Equilibrium is also the only correlated equilibrium.

6The order is as follows: the �rst element of the string is the top left box of the colored grid and the sequencecontinues down the �rst column, then the second etc, until the last element which corresponds to the bottom right boxin the colored grid. The instruction sheet handed out to explain this is included in the appendix. Note also that�gures 2 and 3 show the same information.

7 In the same fashion as it is easier to estimate the height of the tower in Figure 1a than in Figure 1b, it is easier toestimate the distribution of colors game by game in Figure 2 than in Figure 3.

8Some readers may worry that the frame in which feedback was provided (colors for actions and letters for gamesand separate screens for the two pieces of information in Hard) does not echoe real life scenarios. Yet, one has to

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Hard? One possibility would be that subjects might simply ignore the information (say, best respond

to uniform beliefs). However, an alternative to discarding all the information is to construct a simpler

(coarser) statistic and, in fact, the treatment is designed in a way that allows subjects far more easily

to distill the equivalent of coarse information (about the aggregate distribution of play over the two

games) from what is, in fact, �ne information. If subjects distill such coarse statistics in Hard then,

of course, we might expect that behavior in Hard is similar to the behavior in Coarse where coarse

statistics are given exogenously.

Figure 3a

Figure 3b

We focus on the following research questions:

Question 1 Do the observed distributions of play in treatments Fine, Coarse, Feasible and Hard

stabilize?

Question 2 Do the long run behaviors in Feasible and Hard correspond to the behaviors in Fine

and Coarse, respectively?

remember that the ABEE approach is primarily designed to model interactions in complex environments and somehowour environment consisting of just two games A and B is less complex than most real life interactions of interest toeconomists. In treatment Hard, we voluntarily framed the feedback in a hard to access way so as to compensate forthe relative simplicity of the underlying environment. Note that despite this apparent di¢ culty, some (few) subjectswere able to disaggregate the feedback between the two games (see Table 4 in Section 4).

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Question 3 Do the long run behaviors in Fine and Feasible on the one hand, and Coarse and

Hard on the other correspond to the prediction of the analogy-based expectation equilibrium

approach?

With the required caveats, we obtain positive answers to all three questions. The behaviors do

stabilize; treatments Fine and Feasible are similar to each other and so are Coarse and Hard;9

and all observations are remarkably well organized by the theory.

Our positive answers to Questions 1-2-3 illustrate that accessibility matters even in the long run,

and our positive answer to Question 3 suggests that accessibility considerations should be added to

the description of the environment so as to inform theorists about which pro�le(s) of behaviors can

be expected to be stable in a given application (i.e., which equilibrium notion to consider).10

Some readers may wonder why no alternative approach (i.e. other than the ABEE approach) is

being considered to explain the experimental data. One reason is that the data are very well organized

by the ABEE approach. Another� more fundamental� reason is that as subjects were not informed of

their opponent�s payo¤ structure nor of their own past performances during a session, most alternative

approaches are not meaningful in the present context. We already mentioned the approaches based on

reinforcement learning models which would require that subjects be informed of past performances.

This also applies to the approach of cognitive hierarchies of types developed by Stahl (1993), Nagel

(1995), Costa-Gomes et al. (2001) and Camerer et al. (2004) (only levels 0 and 1 are meaningful in

the present context, since subjects were not informed of their opponent�s payo¤ structure).11

The remainder of the paper is organized as follows. In Section 2 we describe more formally the

analogy-based expectation approach and we apply it to the environment considered in our experiment.

In Section 3 we present more precisely the experimental design. In Section 4 we present the results

of the experiments. Section 5 puts this work in a broader perspective stressing the importance of

incorporating framing considerations into equilibrium analysis.

2 Background and Theory

We �rst describe the analogy-based expectation approach (in a simple setup) and then apply it to the

environment considered in our experiment. We next describe the corresponding learning model that

will be used as a theoretical reference point when we analyse the data from our experiment.

Consider a family of normal form games denoted by ! 2 . Each game has two players i and j.For each !, the action space of player i is Ai and the action space of player j is Aj . Action spaces

9Speci�cally, we compared Fine and Feasible in terms of the frequency with which Row players played best-responses to the distribution of actions game by game as resulting from the feedback they received. Similarly, wecompared Coarse and Hard in terms of the frequency with which Row players played best-responses to the aggregatedistribution of actions over A and B as resulting from the feedback they received. We did the same comparisons interms of the distribution of the payo¤ losses vis a vis these two theories.10Traditional game theory would describe in the same way the environments in Feasible and Hard, even though

di¤erent pro�les of behaviors should be considered as stable, as we illustrate.11Considering treatments in which subjects receive extra information (say on opponent�s payo¤ structure) and/or

feedback (say on their past performance) so as to combine the present investigation with earlier investigations shouldbe the subject of future work.

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Ai and Aj are �nite. The payo¤ obtained by player i in game ! when (ai; aj) 2 Ai � Aj is played isdenoted by ui(ai; aj ;!). The probability of game ! is denoted by p(!). We assume that each player

i knows which game ! 2 he is playing.A strategy of player i is a mapping �i : ! �Ai where �i(ai j !) denotes the probability with

which action ai 2 Ai is chosen by player i in game !.Each player i is endowed with an analogy partition Ani over . The element of Ani containing !

is denoted by �i(!) and called the analogy class of player i at !. Player i is assumed to understand

only the aggregate behavior of player j in every analogy class in Ani. Formally, given the strategy

�j of player j, the strategy of player j perceived by player i (given Ani) is de�ned by the function

�j : ! �Aj such that for all ! 2 and aj 2 Aj

�j(aj j !) =P

!02�i(!) p(!0)�j(!

0)P!02�i(!) p(!

0)=X!02

p(!0 j �i(!))�j(aj j !0) (1)

That is, given the strategy �j of player j, player i perceives only the average behavior of player j in

each analogy class where the weight assigned to a speci�c game !0 of an analogy class is proportional

to p(!0).

De�nition A strategy pro�le � = (�1; �2) is an analogy-based expectation equilibrium (ABEE) given

the analogy partitions An1, An2 if for all i, ! 2 and a�i in the support of �i(!):

a�i 2 arg maxai2Ai

X!02

�j(aj j !)ui(ai; aj ;!);

where �j(aj j !) is given by (1).

In other words, in equilibrium each player i plays a best-response to the belief that player j behaves

in each game ! according to the aggregate behavior in �i(!), i.e. �j .

In our experiment, we considered two games ! = A and B whose payo¤ matrices for the Row

(i = 1) and Column player (i = 2) are depicted below:

A a b c d e B a b c d e� 25,10 0,10 10,20 0,0 0,0 � 15,0 20,10 10,0 5,5 0,0� 20,15 15,0 5,0 10,0 0,0 � 0,10 25,10 0,0 10,20 0,0 15,0 10,0 0,0 5,25 25,0 10,0 15,0 5,25 0,0 25,0

Table 1: Normal form games used in the experiment

The two games A and B were played with the same frequency so that p(A) = p(B) = 12 . Given

the feedback given to Column player (to be described more precisely in the next Section), the relevant

analogy partition for Column player was the �ne partition Anf2 = ffAg ; fBgg. For the Row player,two analogy partitions can be considered: either the �ne partition, Anf1 = ffAg ; fBgg, or the coarsepartition, Anc1 = ffA;Bgg.

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When the Row player�s analogy partition is �ne:

When both players use the �ne analogy partition, ABEE coincides with Nash equilibrium. In game

A the only Nash equilibrium requires that the Row player plays �, �1(A) = �, and the Column player

plays c, �2(A) = c. In game B the only Nash equilibrium requires that the Row player plays �,

�1(B) = �, and the Column player plays d, �2(B) = d. This is easily seen as the iterated elimination

of strictly dominated strategies single out a unique pair of actions in each game.12

When the Row player�analogy partition is coarse:

When An1 = Anc1 and An2 = Anf2 , the only (analogy-based expectation) equilibrium is (�1; �2)

where �1(A) = �, �1(B) = � and �2(A) = a, �2(B) = b. That is, the Row player plays � in game A

and � in game B; the Column player plays a in game A and b in game B. Thus, the strategies are

markedly di¤erent from the Nash equilibrium strategies.

It is easy to understand why this strategy pro�le de�nes an analogy-based expectation equilibrium.

The Column player plays a in A because this is the best-response to �; she plays b in B because this

is the best-response to �. The aggregate behavior of the Column player is a balanced mix of a

and b (remember that p(A) = p(B) = 12 ). Thus, �2(a j !) = �2(b j !) = 1

2 for ! = A and B.

Given the expectation that the Column player plays a and b with an equal frequency, the Row player

�nds it optimal to play � in game A (because 20+152 > max(20+0;15+10)

2 ) and � in game B (because15+202 > max(0+25;10+15)

2 ). It is a routine exercise to check that there is no other equilibrium in this

case.13

Learning models:

Jehiel (2005) (see also Jehiel and Koessler, 2006) motivates the analogy-based expectation approach

by a learning story. More precisely, Jehiel (2005) interprets the ABEE as the limiting outcome of a

learning process in which each player i would base his strategy on the sole feedback about the aggregate

play of player j in the various analogy classes of Ani .

In our experiment, each session of each treatment consisted of several rounds t = 1; 2::: In each

round, four Column players and four Row players were randomly matched to play one of the games

! = A or B, and each game was played exactly twice, e.g. in two of the four matches m = 1; 2; 3; 4.

We call M t! the set of matches in which game ! = A or B was played in round t, and we refer to

!t(m) as the game being played in match m of round t.

The feedback given to subjects concerned the behaviors of opponents (i.e., subjects assigned to the

12 In both games A and B action e is strictly dominated (by a mixture over a and d in A and a mixture of b and cin B) for the Column player. After eliminating action e action is strictly dominated for the Row player (by action� in A and action � in B). Following these eliminations, in game A, actions d and b are strictly dominated by a anda mixture of a and c, respectively. In game B, actions c and a are strictly dominated by b and a mixture of b and d,respectively. Finally, with the remaining actions, � in A and � in B strictly dominate � and � respectively, and we canconclude.13A similar argument to that for the �ne equilibrium su¢ ces. Action e is strictly dominated for the Column player

in both games. After eliminating e action is strictly dominated in both games. Column players have a �ne analogypartition so we can again eliminate actions d and b in A and a and c in B. Finally, we use the analogy partitions tonote that if the column players choose any mixture of a and c in A and b and d in B (such that the row player perceivesan equal mixture of both strategies in both games), then action � in A and action � in B dominates actions � and �respectively. The equilibrium follows from the column player�s best response to this.

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role of the other player) in the last �ve rounds, and two options were considered according to whether

the feedback is about the aggregate behaviors in the two games or the behaviors game by game. For

each game ! = A or B, we refer to

BR!i (xj) = arg maxai2Ai

X!02

xj(aj)ui(ai; aj ;!)

as player i�s best response in game ! to the distribution xj 2 �Aj of player j�s actions where xj(aj)denotes the weight assigned to action aj in xj .

For each player i, we let at;mi denote the action played by the player assigned to the role of player

i in match m of round t. Given the belief xj(t + 1) about player j�s behavior considered by player i

in round t+ 1, our learning dynamics requires that for t > 5,

at+1;mi 2 BR!i (xj(t+ 1))

where

! = !t+1(m)

The two feedback cases di¤er only in how xj(t) is determined. In the �ne feedback case, we have:

xFj (t+ 1) =1

10

tXk=t�4

Xm02Mk

!

ak;m0

j

That is, xFj (t + 1) is the empirical distribution of actions of subjects assigned to the role of player

j over those matches in which game ! = !t+1(m) (the one to be played) was played in the last �ve

rounds.

In the coarse feedback case, we have:14

xCj (t) =1

20

tXk=t�4

Xm0=1;2;3;4

ak;m0

j

That is, xCj (t) is the empirical distribution of actions of subjects assigned to the role of player j in all

matches over the last �ve rounds.

The two (benchmark) learning models that we consider have the Column players following the �ne

feedback belief dynamics. In one learning model, Row players follow the �ne feedback belief dynamics,

and in the other they follow the coarse feedback belief dynamics.

When Row players follow the �ne feedback belief dynamics, this corresponds to a �ctitious play

dynamics with a �ve period window and a population of players. Our games ! = A and B are

dominance solvable. Thus, applying Milgrom and Roberts (1991)�s result (see also Nachbar, 1991), we

can conclude that whatever the starting point (the actions in the �rst �ve rounds) one must converge

14An essential di¤erence with the (stochastic) �ctitious play learning model with cross over considered in Cooper andKagel (2004) is that the behaviors and beliefs in the two games in�uence each other at the same time: there is no priorbelief attached to one game that can be exported to the other.

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to the unique Nash Equilibrium in the two games.

No general convergence result is available for the learning dynamics in which Row players follow

the coarse feedback belief dynamics.15 Yet, it can be checked that whatever the initial actions (in the

�rst �ve rounds and in the four matches) the system converges to the unique ABEE, which has been

described above.

3 Experimental Design

The computerized experiments16 were conducted at the UCL-ELSE Economics Laboratory between

February and December 2005. Upon arrival at the lab, subjects sat down at a computer terminal to

start the experiment. Instructions were presented on the computer screen and a written summary of

the instructions was also handed out. Subjects were invited to raise their hands at any time to ask

questions which would be answered privately.

The experiment consisted of four treatments which varied in the accessibility of the information

available to subjects about the actions of others. Each session involved eight subjects and four sessions

were run for each treatment. In total 128 subjects participated in the experiment, drawn from the

student population at UCL. Their subjects of study included a cross section of arts, humanities, science

and medical subjects. Subjects were paid a turn-up fee of £ 5 and in addition to this were given £ 0.05

per point won during the experiment. The average payment was around £ 13 per subject, including

the turn-up fee. All of the sessions lasted between 45 minutes and 1 hour, with the Hard treatments

taking the longest. And subjects took longer to consider their choices at the start of the experiment:

generally, over our sessions of 60 rounds, the �rst 20 rounds took a similar length of time as the last

40 rounds.

In all treatments, subjects were split up equally into two roles, Row and Column. Each session

consisted of sixty rounds where Row and Column subjects were randomly matched into four pairs to

make a choice in one of two normal form games, the Row subject choosing the row in the game matrix

and the Column subject choosing the column. The two normal form games chosen were detailed

in table 1. In each round, two pairs were allocated to �game A� and two to �game B�, and this

information was common knowledge to both subjects in each pair. Subjects could only see their

own payments in each game, and were given information about the choices made by the subjects in

the other role in the previous �ve rounds, and the games in which these choices were made. The

treatments di¤ered only in how this feedback was presented to subjects with role Row.

In all treatments the Column subjects were presented in every round with the number of times

each row had been chosen, in the current game, over the last �ve rounds. The number was shown

against the row on their payo¤ matrix, and the experiment instructions explained the meaning of the

numbers and that they were being provided �to help you make your decision.�Column subjects were

never given any feedback about play in the game not currently seen. For example, if a Column subject

15 It is readily veri�ed though that in general if there is convergence it must be to an ABEE.16The experiments were programmed using Tomlinson�s (2005) Expecon software.

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was in game A in round 25, she only saw the distribution of choices for game A on the screen. In

a later round, she may have been in game B, and only feedback for game B would have been seen.

Appendix A shows the instruction sheets handed out at the start of the experiment, which also show

screenshots from the experiment software.

The two main treatments, Feasible and Hard were designed to examine how the accessibility of

feedback information a¤ects how the information is used. In both treatments Row subjects were given

full information about the distribution of columns chosen in the last �ve rounds, and the games in

which the choices were made. For the Row subjects, each column was given a color, and these colors

were used to indicate the choices. In the Feasible treatment, before seeing the current game and

being able to make a choice, each row player was shown a screen containing a grid of squares. Each

square was given the color of a previous choice, and contained the letter of the game it was made

in. The ordering of this grid showing the actions chosen in each of the four matches of each of the

last �ve rounds was randomized independently each round, and subjects were informed of this. Row

subjects were allowed to consider this information for as long as they wanted, and then continue to

the next screen to see the game to which they had been allocated and to make a choice. The choice

screen contained no information about previous choices� it was necessary for subjects to choose how

to interpret the feedback grid and remember what they thought relevant before going on to make a

choice.

The Hard treatment made it harder to connect the game to the choices. In this treatment, Row

subjects were shown a similar grid of colored squares, but the game letters in each square were removed.

They were presented on a following screen as a string of As and Bs. The �rst letter corresponded to

the top left in the grid, and it worked down each column in turn, to the last letter giving the game

for the choice indicated in the bottom right of the grid. The instruction sheet handed out at the

start of the experiment gave a clear indication of this ordering. Again, screenshots are included in

Appendix A. Subjects were free to consider each screen for as long as desired before continuing to the

next, but could not go back once they had moved on. Clearly now, although the same information is

being presented, it is cognitively much more di¢ cult to create two separate distributions of play for

each game. E¤ectively, the whole grid of colored squares needs to be remembered when connecting

the distribution of moves to the games. The game information is thus less accessible in the Hard

treatment, even though the subjects are presented with screens giving identical information content.17

The two control treatments Coarse and Fine were simpler. Here the Row subjects were given

similar information to that of the Column subjects. Every subject was told the number of times each

choice had been made by subjects in the other role, over the last �ve rounds. In the Fine treatment

the Row subjects saw this information just for the current game, identically to the information given

to Column subjects. However, in the Coarse treatment the row players saw the total frequency of

each column choice aggregated over the two games, and were not given any information about the

game in which the choices were made.

17This isn�t necessarily as di¢ cult as it sounds as, near equilibrium, only two colors would mainly be seen, and wouldbe partitioned identically to the situations, making it much easier to match the two parts of the feedback.

11

These two treatments establish an obvious benchmark as accessibility is trivial in both cases. From

the viewpoint of ABEE these treatments correspond to di¤erent exogenous analogy classes that are

directly induced. Thus, the prediction for the Fine treatment is given by ABEE with �ne analogy

classes (i.e. Nash equilibrium), while the prediction for Coarse corresponds to the ABEE with coarse

analogy classes (see section 2).

4 Results

4.1 Aggregate data

A �rst set of summary statistics is given in Table 2. They show for all four treatments the frequencies

of each choice in each of the two games for row and column players. A �rst observation is that

there is a signi�cant di¤erence of behaviors across treatments. Second the modal behaviors of Row

players and Column players in Fine and Coarse coincide with the behaviors arising in the Nash

equilibrium and the ABEE with coarse grouping, respectively. Third, the distributions of behaviors

in Fine and Feasible on the one hand and Coarse and Hard on the other are similar, con�rming

our intuition that the feedback about opponents�play is accessible game by game in Feasible but

only in aggregate over the two games in Hard.

Notice though that there are still some systematic deviations from equilibrium play, such as column

players choosing a in game A in the Fine and Feasible treatments. However, in all cases these

deviations are the same in the Fine and Feasible treatments on the one hand and the Coarse and

Hard treatments on the other.

The results, in particular the comparative statics, appear to be very encouraging for the theoretical

model. Speci�cally, the table suggests that the answer to Question 2 in Introduction is yes. Behavior

in Feasible appears to be very similar to behavior in Fine and behavior in Hard very similar

to behavior in Coarse. There is also some support for an a¢ rmative answer to Question 3. The

theoretical predictions are modal (though not met all the time).

The key question is, of course, whether the aggregate frequencies are coincidental or can be traced

to individual behavior. In order to investigate this question we examine whether individual decisions

are best responses to the information provided. Of course, in doing so we shall distinguish between

�ne and coarse information, taking into account that in some instances best replies to both types of

information may coincide. We consider three di¤erent beliefs that the row players might hold about

the strategy of the column players. The �rst two are constructed from the empirical distributions of

past play over the previous 5 rounds. The Fine beliefs use the �ne feedback, thereby considering only

previous play of the Column players for the relevant game. The Coarse beliefs use the coarse feedback,

i.e., the total frequencies of past play of the Column players over both games. To these beliefs we add

a third, the Uniform belief, which assumes a uniform strategy of the column players. This could also

be interpreted as the row players ignoring the feedback and taking the row with the highest average

12

Row Player game A Fine Feasible Hard Coarse� (Nash Equilibrium) .73 .80 .24 .24� (Coarse ABEE) .18 .19 .64 .66

.9 .2 .12 .10

Row Player game B Fine Feasible Hard Coarse� (Coarse ABEE) .16 .23 .62 .73� (Nash equilibrium) .76 .73 .21 .17

.8 .4 .17 .10

Col Player game A Fine Feasible Hard Coarsea (Coarse ABEE) .43 .33 .66 .75

b .0 .1 .0 .0c (Nash equilibrium) .47 .60 .8 .12

d .10 .5 .25 .13e .0 .0 .0 .0

Col Player game B Fine Feasible Hard Coarsea .0 .0 .0 .0

b (Coarse ABEE) .15 .29 .42 .51c .8 .7 .30 .16

d (Nash equilibrium) .77 .64 .28 .33e .0 .0 .0 .0

Table 2: Summary of choices made

Row None Uniform Coarse Fine Coarse & Fine19

Fine .5 .8 .17 .76 .7Feasible .2 .3 .21 .80 .6Hard .8 .15 .60 .43 .26Coarse .5 .10 .68 .28 .11

Table 3: Best response frequencies

payo¤.18 The Uniform belief corresponds to the �Level 1�reasoning of Camerer, Ho and Chong (2004),

with Level 0 employing a uniform strategy. Higher level beliefs in this model would require knowledge

of the payo¤s of the opponent, which are not available to subjects. Hence, we cannot consider them

here. Having constructed these beliefs from the information made available to row players we are

able to calculate the expected payo¤ for each row. Table 3 details the frequencies of best responses,

i.e., instances of row choices that maximize expected payo¤s given some method of forming beliefs.

For matters of comparison we also show the best response frequencies for column players to Fine and

Uniform beliefs which are constructed similarly to those for the row players. (Column players cannot

hold coarse beliefs because they always get game-speci�c feedback only.)

The table is very suggestive. In fact, the table lends even stronger support to our theoretical

reasoning than the previous table that summarized outcomes. Both, in Fine and Feasible, row

subjects best respond to �ne information around 80% of the time while in Coarse and Hard row

subjects best respond to coarse information around 65% of the time. It is maybe not surprising that

18Conveniently, the best response to the uniform beliefs is always in both games, meaning that near our suggestedequilibria, the best responses to our three candidates for beliefs span the available action space.

13

the numbers for the coarse treatments are lower than those for the �ne treatments, since after all �ne

information is more reliable than coarse information. Moreover, the number is lowest in Hard, which

may be attributed to the fact that in this treatment even accessing the coarse information requires

some substantial cognitive e¤ort and is much harder than in the case where the coarse information is

simply given exogenously.

In a further test of the similarity or di¤erence between the treatments we directly consider the

distribution of expected losses with respect to the �ne and coarse models of opponents�play. We

calculated the theoretical expected losses for the choices taken, if the row players were to treat the

two kinds of feedback as the true distribution of column play. Hence, for a subject who best replies

against a given belief the expected loss of taking this action is zero. For all non-best replies the

expected loss is simply the expected payo¤ from the chosen action (given some beliefs) minus the

expected payo¤ of the best reply (given the same beliefs). Figure 4 shows a histogram with the

empirical distribution of expected losses implied by the choices made by the row players, when using

the Fine beliefs, i.e the past distribution of Column players�actions in the current game. It shows a

clear similarity between the Fine and Feasible treatments on the one hand, and the Coarse and

Hard treatments on the other. The next graph, Figure 5, shows the expected loss of the choice taken

when using the Coarse beliefs, i.e. the past distribution of Column players�actions over both games

A and B. Again, this clearly shows the similarity between the Fine and Feasible treatments on the

one hand, and the Coarse and Hard treatments on the other.

Figure 4: Expected loss using the Fine beliefs

The results strongly support the hypothesis that the accessibility of information determines how

the information is used when making decisions. The Feasible treatment shows that when the game

information is feasibly accessible, it is used (i.e, separate distribution of opponents�actions are con-

14

Figure 5: Expected loss using the Coarse beliefs

sidered for each game, similar to that shown in the Fine treatment). The Hard treatment makes

the speci�c game information considerably less accessible. But crucially subjects do not throw the

information away. Rather they distill a statistic from that information that coincides with coarse in-

formation. Consequently, subjects behave in Hard very similarly to subjects in the Coarse control

treatment.

4.2 Individual data

Next we consider individual behavior of the row subjects. Our objective is to see whether we can

identify di¤erent types of subjects, where types are de�ned by their �typical�best response behavior.

Subjects who �typically� (to be de�ned later) best respond to a belief of type � will be classi�ed

as a � -type. Since we have three relevant types of beliefs (�ne, coarse, uniform) this gives us three

di¤erent types of subjects. To this we add a fourth type for whom we cannot say that they follow any

particular type of beliefs. This type we call �other.�20 The di¤erence with Subsection 4.1 is that we

follow each individual row player across all 60 rounds, and we ask ourselves whether the subject�s use

of feedback over time is consistent with either the Fine, the Coarse or the Uniform belief throughout

the duration of the experiment.

The use of the word �typical�in the above paragraph might sound ambiguous but in what follows

we shall precisely explain our procedures. The key idea is that for each type of belief we calculate a

variable that indicates the realized gains or losses from the actual choice compared with the respective

20This is justi�ed as in almost all rounds the choices made by the �rst 3 types span the strategy set for row players.The �Other� type could therefore be interpreted as playing a strategy that mixes between the strategies of the othertypes or has another reason for stochastic choices. In this sense our candidate strategies span all possible observedbehaviors.

15

best response. More speci�cally, when a player does not best respond to a particular belief this

variable is equal to the di¤erence between the expected pro�t from taking the optimal choice, minus

that of the choice taken. This is the expected loss that we have also used above in the analysis of the

aggregate data.

When a player does best respond, the variable is set equal to the di¤erence between the second

best choice and the best one that he did choose (which is thus negative). One might refer to this as the

avoided expected loss. Thus, the variable is positive when subjects do not best respond, and negative

when they do. Let�s call this variable the subject�s incentive with respect to a particular belief. IF ,

IC and IU give the incentives with respect to Fine, Coarse and Uniform beliefs, respectively. For each

subject we, thus, have a sequence of 59 values (one for each round other than round 1) for all three

incentives.

If a player typically best responds to a given belief, then we can expect two things. First, the

median of the observed incentives will be negative. This is straightforward� we are just saying that the

player best responds more often than not and thus that there are more negative values than positive.

Second, in line with standard models of noisy decision making we should expect that subjects who,

in principle, want to follow a particular best response mode, but are prone to mistakes, would more

often deviate if the incentives were small than if they were large. (This is the same as saying that

more costly errors are less likely to occur than less costly errors.) Hence, for a player who �typically�

best responds against a certain belief we should expect the observed distribution of incentives to be

skewed towards larger negative and smaller positive values.21

Following this line of reasoning we de�ne the following two-step hypothesis test:

1. First try to reject the null hypothesis that Median(I) � 0

2. If we cannot reject in part 1, then try to reject the second hypothesis that Skew(I) � 0 (assumingthat Median(I) = 0)

For each method of forming beliefs, we have di¤erent bounds for the incentives and di¤erent

distributions under both the null hypothesis, and the di¤erent alternative hypotheses. We therefore

require a test for our null hypothesis that is invariant to the scale and the particular shape of the

distribution of the incentives, apart from median and skewness. We can do this by employing the

following composite test. For the �rst part of the null hypothesis we employ a sign test, which simply

relies on a binomial test statistic based on the number of positive and negative values. For the second

part we follow Gibbons (2003) who suggests that the Wilcoxon matched-pairs rank-sum test can be

used as a test for distributional symmetry (assuming a known median).22

21We are slightly mis-using the concept of the skew of a sample in considering the relative sizes of samples on eachside of the median, rather than the mean, as is more conventional. However, this interpretation is consistent with ourtest procedure, and all mention of the �skew�of a distribution should be interpreted as using the median, rather thanthe mean as the location measure in its calculation.22 In fact, the Wilcoxon test can also be used as a test of location (when assuming a symmetric distribution). This

would open up the possibility of combining both parts of the test by using just the Wilcoxon test. However, we arereluctant to follow this path as it is not clear how the skew and location e¤ects are compensated in the test. Thiswould make the results ambiguous if the skew and median of a sequence of incentives had di¤erent signs. Also, wespeci�cally want to ignore the e¤ect of the skewness if the player best responds most of the time.

16

Type Fine Feasible Hard CoarseFine beliefs 12 14 3 3Coarse beliefs 1 1 7 12Uniform beliefs 1 0 0 1

Other 2 1 6 0

Table 4: Type allocation over all rounds

We proceed as follows. We take an individual subject and �x the type of belief. For belief � this

gives us, as explained above, 59 values of the incentive variable I� . We then proceed by using the

sign test to try to reject the �rst part of the null hypothesis. If we are able to reject the �rst part of

the null hypothesis, then we allocate the player to the corresponding belief type � . If we are unable

to reject it, then we employ the Wilcoxon test to see whether we can reject the second part of the

null hypothesis. If we can, we again allocate the player to the belief type � . If the subject is not

allocated to beliefe type � , we then proceed to the next belief type. And so on. If the subject cannot

be allocated to any of the three types of beliefs we classify him as �other.�23 For all (one-sided) tests

we rely on a critical signi�cance level of 5%. Notice that our procedure ensures that no subject can

be allocated to more than one type, unless the best responses coincides very often.

In essence, we are allocating players to a type only if we can reject that their incentive values

could be generated by a player who does not use them to make decisions. This may seem to be a test

of relatively low strength, and it may seem likely that there would be high numbers of �others.�Of

course, this implies that if we can classify subjects into Fine belief, Coarse belief or Uniform belief

types, we can be fairly con�dent in this classi�cation. Finally, we are presented with a problem if more

than one of our null hypotheses are rejected at each stage. Thankfully, this happens in only two cases

out of all 64 row players� in one particular session in the Hard treatment where the best responses

to the Fine and Coarse feedback coincided often. For these two cases, we allocated the subjects to the

type where the incentive gave the lowest p-value in our test (one was allocated to the �ne-belief and

one to the coarse-belief type). We also note that the second step in our test was used in only three

instances allowing us to allocate an additional three subjects to non-other types.

Table 4 shows our classi�cation results. In contrast to the fears uttered above, our conservative

procedure classi�es the vast majority of subjects as entertaining either �ne or coarse beliefs. The

only exception is treatment Hard in which one third of the subjects is classi�ed as �other.�Due to

the di¢ culties in using the information provided (and our conservative procedure) this is, of course,

hardly surprising.24

We �nd that in treatments Fine and Feasible the vast majority of subjects are indeed classi�ed

as entertaining �ne beliefs. Thus, what we have seen above in the aggregate is con�rmed on the

individual level. Subjects do use the �ne information systematically and systematically best reply

against it in these treatments. In Coarse and Hard the majorities are smaller but the most frequent

23Notice that this procedure ensures that no subject can be allocated to more than one type, unless the best responsescoincided very often (which they don�t).24 Incidentally, column players behave in treatment Hard in a way that lets the best responses to �ne and coarse

beliefs coincide quite often. Therefore, there are less rounds where the strategies can be distinguished which makes iteven more di¢ cult to reject the null.

17

Rounds 1 to 15Type Fine Feasible Hard Coarse

Fine beliefs 9 9 1 1Coarse beliefs 1 0 3 4Uniform beliefs 0 0 0 0

Other 6 7 12 11



Other 3 4 8 6



Other 3 3 5 3



Other 2 4 4 1

Table 5: Type allocation over di¤erent phases

type is the coarse-belief type. In treatment Coarse this is perhaps not very surprising. After all,

there is nothing much else a subject could do. But in treatment Hard subjects have to exert some

signi�cant cognitive e¤ort to derive even such coarse beliefs.

The picture drawn in Table 4 abstracts from the possibility of subjects becoming perhaps more

sophisticated over time. Perhaps some subjects start out ignoring the feedback but are able to interpret

it in some way or other later on in the session. Or perhaps some players are confused about how to

interpret the information at all, and require a few rounds to understand how to use the available

information to make decisions. Our previous method of distinguishing types takes no notice of such

dynamics. So, in our �nal test, we consider the possibility of such learning during the experiment,

and that each player�s method of forming beliefs (as well as the information contained in those beliefs)

might change during the experiment. For that purpose we split the experiment into four phases of 15

rounds each and we repeat the same tests, again with a 5% signi�cance level, for each phase of the

experiment and each row player. The results are shown in Table 5.

Initially, we �nd a large share of Other types but after the �rst quarter of the experiment there is

a huge jump in sophistication. From the second quarter onwards most subjects entertain either �ne

or coarse beliefs and then keep on doing so for the rest of the experiment. Interestingly, it seems that,

once subjects have settled on a way of accessing and interpreting the given information, they rarely

18

change this interpretation later in the experiment. Once they have found a method of forming beliefs,

their type is largely �xed. Thus, in our experiment, there is no evidence that subjects switch to more

sophisticated rules of forming beliefs with more experience.

5 Conclusion

In this paper we have provided evidence for the role of accessibility of information in decision making.

When information is presented in a complex manner it would, of course, be naive to assume that

subjects can process it with full accuracy. Our experiments con�rm this intuitive statement. But

our study goes one step further by showing that accessibility can be modelled� and in quite familiar

ways. Thus, on a more general level, acknowledging the role of framing does not imply giving up the

apparatus of game theoretic modeling and equilibrium analysis. We consider this as good news.

More speci�cally, we �nd that when we uncouple quantitative information about choice frequencies

from information about in which context these choices were made subjects do what appears to be very

natural: They bundle the frequency information, irrespective of context. This is very similar to the

analogy-based expectation equilibrium approach introduced by Jehiel (2005). The logic behind this

is probably that subjects are hesitant to throw away what is obviously useful information and decide,

in face of its complexity, to simplify its content. They may or may not know how this distorts their

actions from the optimal choice. Unconstrained optimization is not an issue for them� when the

presentation gets too di¢ cult they are simply not able to process information correctly. Of course,

some readers may not be surprised by this. The more important aspect of our result is, therefore,

that subjects do process the information nevertheless and in a systematic way. Insofar, our Hard

treatment does perhaps provide the most crucial results. Even under what are in fact very hard

circumstances subjects process the provided information and construct a statistic against which they

systemically best reply.

From the viewpoint of the analogy-based expectations approach (which nicely organizes the data

from all our treatments), our study suggests that the analogy partitions considered in that approach

should not be viewed as solely deriving from the underlying game(s) as such. Here, with identical

underlying games, we were able to generate di¤erent grouping simply by varying the accessibility of

the same objective feedback information. Thus, our approach suggests that the description of some

interaction should not only include the objective information and payo¤ structure but also information

on accessibilty or, more generally, framing. In the language of Jehiel (2005), it suggests including the

�analogy partitions�employed by the players as part of the description of the environment.

From a broader perspective, it seems legitimate to say that in the real world, we have, in principle,

access to myriads of bits of information, and one might also argue that information is more and more

symmetrically shared among agents (thanks to the internet). Yet, this view misses the point that

processing information is not free of cost contrary to what is usually assumed in economic theory.

Generally, it holds that in order to make use of information one has to simplify it. That is, one has to

19

throw away many of the dimensions of the real world problem in order to identify regularities25 and

eventually reach a decision.

Our experiment shows that information processing is likely to a¤ect the behavior of agents in

the long run. What kind of equilibrium is reached may depend on the framing of information. It is

thus of primary importance to understand the transmission mechanisms from framed information to

actual choice. Beyond the objective of better describing the real world, this may also be of primary

importance from a normative viewpoint, i.e. from a mechanism design perspective. A mechanism that

functions only if agents extract information they cannot be realistically expected to extract may fail

horribly. This stresses both, the need to consider accessibility of information when judging a mech-

anism and, for the mechanism designer, the necessity to take into account the framing/accessibility

of the information that is required for the mechanism to function well. Of course, such behavioral

mechanism design needs a profound understanding of the mapping from frames into outcomes for

which much further research is required.26 We view this experiment as a preliminary step toward this

end.27

25As recently shown by Aragones et al. (2005), this is a hard problem in a computer science sense.26Some interesting results on the role of strategies� labels and context can be found in Cooper and Kagel�s (2006)

study on cross-game learning. They �nd that subjects are better able to transfer what they have learned in one situationto a new situation if meaningful real-life (here: market) context is provided. A related �nding is reported in Huck,Normann, and Oechssler (2004) where a market frame helps subjects to understand and overcome the dilemma problemimposed in 2-player Cournot games.27The views that theories are simpli�cations of the world and that experiments may help design better theories agree

with the recent paper by Samuelson (2005).

20

Appendix

A Experiment Instructions

There follows the instruction sheets handed out at the start of the experiment. Step-by-step in-

structions were presented on the computer screen at the start of each session, and these sheets were

intended to be used as a reference during the experiment. They have all been amended to show the

name of the treatment they apply to, but are otherwise identical to those handed out.

21

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information processing, learning, and analogy-based expectations

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