STUDYING LEARNING IN GAMES
USING EYE-TRACKING
Daniel T. Knoepfle
Stanford University
Joseph Tao-yi Wang
National Taiwan University
Colin F. Camerer
California Institute of Technology
Abstract
We report results from an exploratory study using eye-tracking recording
of information acquisition by players in a game theoretic learning
paradigm. Eye-tracking is used to observe what information subjects look
at in 4x4 normal-form games; the eye-tracking results favor sophisticated
learning over adaptive learning and lend support to anticipatory or
sophisticated models of learning in which subjects look at payoffs of other
players to anticipate what those players might do. The decision data,
however, are poorly fit by the simple anticipatory models we examine. We
discuss how eye-tracking studies of information acquisition can fit into
research agenda seeking to understand complex strategic behavior and
consider methodological issues that must be addressed in order to
maximize their potential.
10/13/08. This paper was prepared for an invited session at the European
Economics Association, 2008. We thank Andrew Caplin, Vince Crawford,
Rosemarie Nagel, and an anonymous referee for helpful comments. This
research was supported by NSF-HSD, HFSP, National Science Council of
Taiwan (NSC 96-2415-H-002-034-MY2), and the Gordon and Betty Moore
Foundation (grants to CFC).
1. Introduction
Many theories have been proposed about how players learn in games,
leading to a large experimental literature comparing theories econometrically
(see Camerer, 2003, chapter 6). Here, we explore using eye-tracking recording
of information acquisition to inform this empirical debate. The maintained
hypothesis underlying this approach is that theories can be taken as algorithms
that use specific information about historical actions and payoffs in a precise
way to guide choices. Furthermore, the extent to which people look at the
relevant information is likely to correlate with how that information influences
choices. Eye-tracking data can be useful because it is difficult to identify
learning rules precisely from decision data alone, especially once we consider
heterogeneity across subjects (or even across time, within subjects, as in “rule
learning” models). In an observational study of information acquisition, the
goal is to gain insights about behavior that would be difficult to establish using
choices alone (and which is inefficient to test only by exogenously varying
what information is available).
Eye-tracking and mouse-tracking (MouseLab) methods for measuring
information acquisition have been used previously to study various topics such
as backward and forward induction (Johnson et al., 2002) and to estimate
decision rules used in normal-form games (Costa-Gomes et al, 2001) and two
person p-beauty contest games (Costa-Gomes and Crawford, 2006).
The core of our analysis consists of (a) measures of how well models
predict the actual choices of players and (b) measures of how often players look
at information relevant to executing a particular learning rule.
An ideal outcome would be that one theory is supported by both
decision and information lookup choices. Unfortunately, that is not the case.
Instead, we found that in predicting choices, adaptive models in which people
learn by generalized reinforcement (EWA and self-tuning EWA) predict more
accurately than simple “sophisticated” (or “anticipatory learning”) models in
which agents anticipate that others are learning. However, the lookup data
suggest that players often look at payoffs as required by sophisticated models,
about equally as often as they look at adaptive-model payoffs. The fact that
behavioral fits favor adaptive models but lookups favor sophisticated ones
suggest there are some models of sophisticated learning (in which lookups
overlap with those in the models that do not fit well here) which could fit both
lookup and choice data well. We leave development of such models to future
research.
Details and further analyses are reported in our online working paper,
Knoepfle, Wang, and Camerer (2008), denoted KWC.
2. Design
2.1 Experiment Structure
1
Subjects play four asymmetric non-zero-sum two-player 4x4 normal
form games (see KWC A1 for the game matrices and A2 for design details).
Each of the four games was played 10 times in a random-matching protocol
with feedback. The order of games was fixed. Experiments were conducted in
groups of six; in each group, two subjects were eye-tracked. Periods ended
when all subjects had entered their strategy choices.
2.2 Models
We consider five models of learning: reinforcement (Re), experienceweighted attraction learning (EWA), self-tuning EWA (stEWA), and
anticipatory response level one (C1) and two (C2).
EWA is a general adaptive model that nests types of reinforcement and
belief-based learning as special cases. In EWA, strategies have numerical
attractions which begin as subjective initial attractions Aij(0), with initial weight
N(0), and are updated each period after receiving choice feedback, according to
(1)
Aij(t+1)= ϕN(t)Aij(t)/N(t+1) + [δ+(1-δ)I(sij,si(t))]πi(sij,s-i(t))]/N(t+1)
with N(t +1)=ϕ(1-κ)N(t)+1. Attractions are then mapped into strategy choice
probabilities using a logit form Pij(t+1)=exp(λAij(t))/(∑k exp(λAik(t))). The
EWA parameters δ, ϕ, and κ correspond, respectively, to the weight placed on
the payoffs one would have received had they chosen a different strategy, the
weight of old attractions, and the degree to which attractions cumulate rather
than average. Of these parameters, only δ has implications for information
search.
For the adaptive models, we assume that subjects must look up the
information required to update their attractions each period but have perfect
recall of their attractions in the previous period. If we constrain δ=0, we get a
reinforcement-type model (Re) in which the only relevant information is the
payoff received; reinforcement models have the own payoff matrix cell πi(si(t1),s-i(t-1)) as their lookup area. If δ>0, as in unconstrained EWA, the payoffs
that would have been received are also relevant information; thus, EWA has as
its lookup area the own payoff matrix column πi(·,s-i(t-1)) corresponding to s-i(t1), the opponent’s strategy choice in the previous period.
We also consider a “self-tuning” form of EWA (stEWA) proposed by Ho,
Camerer, and Chong (2007), in which two psychologically uninteresting
parameters are removed from EWA and the remaining free parameters are
replaced by functions of experience. The reinforcement weight δ on un-chosen
strategies is replaced by 1 for strategies which are weakly better responses than
the chosen strategy (i.e., πi(sij,s-i(t-1)) > πi(si(t-1),s-i(t-1))) and 0 for worse
responses. In addition, the weight on old experience ϕ is specified as a function
2
of observed choices. The lookup area for stEWA is the set of cells in the own
payoff matrix column corresponding to s-i(t) with positive weight.
The two “sophisticated” models C1 and C2 are myopic anticipatory models
that predict opponents’ play. Let BRi(s) denote subject i’s best-response to his
opponent’s strategy s.1 Consider the familiar Cournot rule in which subjects
simply best-respond to their opponent’s previous choice; that is, they choose
BRi(s-i(t-1)). In C1, a player assumes his opponent is using a Cournot rule and
best-responds accordingly, choosing BRi(BR-i(si(t-1))). The relevant
information for determining this choice is π-i(si(t-1),·) and πi(·,BR-i(si(t-1))).
The C2 model is one level above the C1 model: in C2, a player assumes his
opponent is using the C1 rule and best-responds accordingly. A C2 player
chooses BRi(BR-i(BRi(s-i(t-1)))). The relevant information for determining this
choice is πi(·,s-i(t-1)), π-i(BRi(s-i(t-1)),·), and πi(·,BR-i(BRi(s-i(t-1)))).
2.3. Screen Display
[FIGURE 1 ABOUT HERE]
A mock-up of an experimental task screen is shown in Figure 1
(specifically, for a row player in period 5 of game 3). Some key design features
can be seen. First, following Costa-Gomes et al. (2001), we separate the
subject’s payoffs and his opponent’s payoffs into two matrices: the left matrix
contains the subject’s own payoffs and the right matrix contains his opponent’s
payoffs. Rows in the matrices correspond to the subject’s own strategies, and
columns correspond to his opponent’s strategies. Second, we display the
history of the subject’s choices, his opponents’ choices, and his own received
payoffs at the bottom of the screen. Another screen (not shown here) appears at
the conclusion of each period, displaying the subject’s choice, his opponent’s
choice, and his received payoff for the period; we discuss a possible bias related
to this screen in KWC A6.2.2.
2.3. Model-Relevant Information and Lookup Areas
At this point, it is worthwhile to discuss and give examples of the model
lookup areas. The model lookup areas are areas of the payoff matrices
corresponding to the information relevant to determining the model choice,
defined with respect to the choices of a subject and his opponent in the previous
period. Consider the game and its on-screen display in Figure 1, and suppose
that in the previous period the player chose strategy 1 and his opponent chose
strategy 1. The reinforcement cell (the player’s experienced payoff) is row 1,
column 1 in his own payoff matrix; this is Re’s lookup area. His corresponding
payoffs for un-chosen strategies are in the other cells of column 1 in his own
payoff matrix; EWA’s lookup area is this entire column. His strategies 1 and 2
1
For simplicity, assume best responses are always unique.
3
are weakly better responses to his opponent’s previous choice than his own
previous choice; thus, stEWA’s lookup area are the cells in row 1 and 2 of
column 1 in his own payoff matrix. Observe that Re and stEWA’s lookup areas
are nested within EWA’s; this is always the case.
Recall that C1 plays BRi(BR-i(si(t-1))). Thus, C1 must first find his
opponent’s best response to his own previous strategy choice. This entails
finding the maximum payoff in the row of his opponent’s payoff matrix
corresponding to his own previous strategy choice, strategy 1. Thus, C1’s
lookup area includes row 1 of his opponent’s payoff matrix. Here, his
opponent’s best response is strategy 4. A C1 player must then find his best
response to his opponent’s strategy 4; to do so, he looks at the corresponding
column of his own payoff matrix (column 4) and finds the maximum payoff;
the strategy yielding this maximum is his strategy choice. Thus, C1’s lookup
area also includes column 4 of his own payoff matrix.
The C2 lookup area comes from a similar process. Recall that C2 plays
BRi(BR-i(BRi(s-i(t-1)))); thus, a C2 player must first find his best response to his
opponent’s previous strategy choice by looking for the maximum payoff in his
corresponding own payoff matrix column, column 1. Therefore, C2’s lookup
area includes column 1 of his own payoff matrix. The maximum payoff in this
column corresponds to strategy 2; that is, BRi(s-i(t-1)) = BRi(1) = 2. A C2
player must then find his opponent’s best response to this strategy, BR-i(BRi(si(t-1))) = BR-i(2); this is accomplished by looking for his opponent’s maximum
payoff in row 2 of his opponent’s payoff matrix. Thus, C2’s lookup area also
includes row 2 of his opponent’s payoff matrix. Here, the maximum payoff
corresponds to his opponent’s strategy 4; that is, BR-i(2) = 4. Finally, a C2
player finds his best response to this strategy; that is, BRi(BR-i(BRi(s-i(t-1)))) =
BRi(4). He does so by looking for his maximum payoff in the corresponding
column of his own payoff matrix, column 4; this maximum, strategy 1, is his
choice. Thus, C2’s lookup area includes column 4 of his own payoff matrix.
3. Results
Due to practical and technical reasons, some subjects were excluded
prior to analysis; we use decision data from 44 subjects and lookup data from
12 subjects. In some cases, players choose the same strategy across all ten
periods of a game. As learning is the topic of interest, all of the results we
report herein exclude these games and use only the “learning games” in which
players do not make the same choice every time (see KWC A5.3 and A6.3 for
results computed using all games).
3.1 Behavior
4
[FIGURE 2 ABOUT HERE]
Choices in the games appear to converge to Nash equilibrium
relatively rapidly; the frequency of Nash play tends to increase over the ten
periods of each game and averages 79.7% in the final period (see KWC 5.1).
We fit the learning models to observed behavior using maximum
likelihood. As a non-learning benchmark, we also estimate a model in which
players are predicted to choose their Nash strategy. The adaptive models have
likelihoods specified by the logit-form probability map; for the Ck and Nashtype models, we estimate the likelihood maximizing p such that players choose
the predicted strategy with probability p and randomize uniformly over the
other strategies. To control for different numbers of parameters, model
parameters are fit on periods 1-7 in each game and geometric mean likelihoods
of observed choices in the hold-out periods 8-10 are reported. Parameters are
estimated at the subject level (see KWC A5.2 for details on the behavioral
estimation).
Out of sample goodness-of-fit measures are reported in Figure 2
separately for eye-tracked subjects and for all subjects. The best models put
predicted probability of about 0.5 on the actual choices, well above a random
guessing benchmark of 0.25. The results are similar across the four games (see
KWC A5.3). Eye-tracked subject fits are worse for some models but not
significantly so (and we have no sensible explanation for the differences).
Pooling tracked and untracked subjects, within subject signed rank tests
between models find that EWA and stEWA significantly outperform C1, C2
and Nash at p<0.01 (all p-values two-tailed, uncorrected for multiple
comparisons). Re significantly outperforms C1 (p=0.024) but is not
significantly better than C2 or the Nash benchmark (p=0.067 and p=0.196,
respectively). EWA and stEWA are better than Re at p=0.055 and p=0.065,
respectively. The C1 and C2 models are clearly the worst-fitting overall, and
are strongly so within-subjects.
3.2 Eye-tracking
Eye-tracking produces a huge amount of data. With this increase in
data volume comes an increase in noise and uncertainty in interpretation. We
emphasize that our results are conditional on a set of assumptions and that the
numbers we report here derive from data which has been preprocessed,
filtered, and transformed in a number of ways. We find that our qualitative
results are generally robust to different preprocessing and analysis choices;
discussion of these preprocessing methods and analysis choices can be found
in the online appendix (KWC A6).
The eye-trackers recorded gaze location and pupil dilation samples at
250 Hz. These gaze location samples are processed into fixations and saccades
(which are rapid movements between fixations). Fixation locations are
compared with screen areas corresponding to cells in the payoff matrices and
5
other on-screen items; we consider a fixation in a payoff matrix cell’s area to be
a “lookup” of that cell’s information. Since we have limited prior knowledge
regarding the relationship between gaze and information acquisition, we take all
fixations within the payoff matrices’ cells as representing information lookups.
A notable analysis choice is our use of fixation counts rather than durations as
the atoms of analysis; see KWC A6.4.5 for discussion of this issue. In addition,
as our on-screen display did not visually separate different history components,
data on history lookups cannot help identify learning models without
undesirably strong assumptions. Furthermore, history lookups represent a
small fraction of the observed lookups. We consider only payoff matrix
lookups here, and discuss observed history lookup behavior in KWC A6.5.
The median (mean) number of payoff matrix lookups (henceforth,
“lookups”) per trial was 32 (42.24). There is a sharp decline in the average
number of lookups from the first period to the second and a more gradual
decline across the following nine periods of each game (see KWC A6.3).
Our first result is very simple but important: the mean share of payoff
matrix lookups corresponding to the opponent’s payoff matrix is 46.1%; while
there is subject-wise heterogeneity, eight of the twelve subjects have shares
above 40% and all have shares above 25%. This observation is in stark contrast
to the class of adaptive learning models, in which information about the
opponent’s payoffs is never relevant. Unless we assume an unrealistic level of
noise, we must conclude that many subjects are deliberately acquiring
information about their opponents’ strategic situation. This result demonstrates
the power of observing information acquisition: with minimal assumptions, we
establish subjects’ interest in opponents’ payoff structures, suggesting that
adaptive learning models ignore important determinants of choice.
To move from statements about classes of models to statements about
specific models, we make additional assumptions about memory and
information relevance. We measure the degree to which lookups conform to
theories by examining lookup rates in screen areas corresponding to the
information which is relevant in updating underlying attractions (or beliefs) in
each particular period. That is, we are implicitly assuming subjects recall
previous-period attractions and only look up the information needed to update
these attractions. (Note that we did display choice and own-payoff history at the
bottom of the screen, but subjects did not look at it frequently.)
Under this memory assumption, the theories predict that subjects will
look at a screen area containing certain target cells. We use a linear measure to
score how well lookups correspond to model predictions. Let x equal the “hit
rate”, the proportion of lookups in a period that fall in the target cells, and let a
equal the proportional area of the target cells. The linear measure (LM) equals
x-a, the proportional hit rate minus the proportional area. Under uniform
6
random looking, the LM has expectation zero. The LM has a number of
desirable properties (cf. Selten, 1991).2 Areas with positive LM scores are
looked at more than average. Thus, the skeptical can take our LM scores for
the various models as mere indications of what information subjects look at
more than would be predicted by uniform looking.
[FIGURE 3 ABOUT HERE]
Figure 3 shows the mean linear measure scores for the set of learning
models. The highest score is for the sophisticated learning rule C2, followed by
its sibling C1, then by EWA and stEWA, and finally by Re. All of the scores
are well above a uniform-looking benchmark (zero LM score).
Signed rank tests using subject-wise mean linear measure scores
(learning games only) find that EWA and stEWA have higher scores than Re at
p=0.09 and p=0.001, respectively; there is no significant difference between
stEWA and EWA (p=0.91). C1 is insignificantly superior to Re, EWA, and
stEWA, with p=0.07, p=0.13, and p=0.13, respectively. In contrast, C2 is
significantly superior to these adaptive models, with p=0.001, p=0.016, and
p=0.002, respectively. Furthermore, C2 is superior to C1 at the marginally
significant level p=0.052.
Taking the learning models as cognitive algorithms, the adaptive
models do not predict any particular order in which their relevant payoffs are
looked up. In contrast, the Ck models do suggest ordered stages of lookups. In
the Ck models, the player must find certain best responses in order to know
what other information to look at. For example, in C1, the player must first
find BR-i(si(t-1)) in “stage 1” in order to determine BRi(BR-i(si(t-1))) in “stage
2”. We impose a simple order restriction requiring at least one lookup in a
stage’s lookup area before lookups in the next stage’s area count as hits, and
adjust the model areas accordingly to maintain zero expected LM score under
uniform looking (see KWC 6.2.1 for additional details). Figure 3 shows the
resulting LM scores for the restricted Ck models (labeled C1r and C2r). Adding
the order restriction, the lookup scores drop sharply for C1r and slightly for C2r,
so that C1r now scores much worse on lookups but C2r still scores better than
the adaptive models. Signed rank tests find that applying this order restriction
significantly decreases the corresponding scores for C1 (p=0.007) and C2
(p=0.012). Restricted C2 remains superior to the adaptive models Re, EWA,
and stEWA with p=0.003, p=0.077, and p=0.034, respectively; restricted C1 is
2
An oft-suggested alternative measure is the ratio measure x/a. This ratio
averages around 2 for Re and stEWA and 1.4 for all other theories. Selten
(1991) notes that the ratio measure favors point theories (given a set of equalsized areas, the ratio measure is maximized by a model including only the
area(s) with the highest hit-rate). We consider it a poor summary of lookups
(see KWC A6.2 for discussion).
7
insignificantly superior to these models with p=0.110, p=0.266, and p=0.301. If
C1 and C2 are accurate models of subjects’ cognitive processes, the simple
lookup order restriction discussed above should not decrease their scores. That
the scores significantly decrease suggests that C1 and C2 include relevant
information lookups but are inaccurate descriptions of true cognitive processes.
4. Discussion
We think our qualitative results imply that learning has a large
component of sophistication (as the lookups suggest) but the behavioral fits
make it clear we have not yet found the appropriate model. Our fundamental
conclusion is that researchers should work towards specifying sophisticated
learning rules that can fit both choices and lookups; a model that does so is
likely to be a good approximation of true cognitive processes.
It is important to note that different learning processes can have
identical lookup areas. Given that C1 and C2 perform worse than the adaptive
models in predicting behavior, one wonders why they have superior lookup
scores. The true learning process may have a similar or identical set of relevant
information but map this information into choices differently. We explore this
possibility in KWC A5, varying the behavioral specifications corresponding to
the Ck models.
Future studies could improve on our design in myriad ways.
Convergence to equilibrium is rapid here, so games with slower convergence
might provide richer data. Introducing variation into the location and
orientation of on-screen elements could help us gain an understanding of biases
in looking patterns and permit more sophisticated analyses of eye-tracking data.
Fundamental changes in the observational paradigm could help us to
differentiate genuine information lookups from noise fixations with confidence.
Eye-tracking is very noisy. While mouse-tracking designs have much less noise,
they introduce exogenous costs for information acquisition that may bias
subjects’ lookups and behavior. As a complement to eye-tracking designs,
however, they offer a range of alternative balances between noise and bias.
Future studies could use a hybrid design combining mouse-tracking features
with eye-tracking recording. For instance, the payoff matrices’ structure could
be freely visible but subjects might be required to press and hold a key in order
to view the numerical payoffs. If this were combined with a strategically
irrelevant cost rate, we could infer that subjects’ fixations while the key is
depressed are genuine information lookups. With our current design, we get
fixations, which we take to be lookups, so long as subjects have their eyes open.
This does not invalidate our results, but does add noise and constrain what we
might hope to accomplish with the data. In eye-tracking studies, we accept an
increase in noise and uncertainty in interpretation in our pursuit of the ideal of
naturalistic observation; hybrid designs could lessen these tradeoffs.
8
References
Camerer, Colin (2003). Behavioral game theory: Experiments on strategic
interaction. Princeton, Princeton University Press.
Camerer, Colin and Teck Ho (1999). Experience-weighted attraction learning
in normal-form games. Econometrica, 67, 827-74.
Costa-Gomes, Miguel A. and Vincent P. Crawford (2006). Cognition and
Behavior in Two-Person Guessing Games: An Experimental Study.
American Economic Review, 96, 1737-1768.
Costa-Gomes, Miguel A., Vincent P. Crawford and Bruno Broseta (2001).
Cognition and Behavior in Normal-Form Games: An Experimental Study.
Econometrica, 2001, 69, 1193-1235.
Ho, Teck, Colin Camerer and Juin-Kuan Chong (2007). Self-tuning experience
weighted attraction learning in games. Journal of Economic Theory, 133,
177-198.
Johnson, Eric J., Colin Camerer, Sankar Sen, and Talia Rymon (2002).
Detecting Failures of Backward Induction: Monitoring Information Search
in Sequential Bargaining. Journal of Economic Theory, 104, 16-47.
Knoepfle, Daniel, Joseph Tao-yi Wang, Colin Camerer (in preparation). Using
eyetracking to study learning in games. SSRN working paper.
Selten, Reinhard (1991), Properties of a measure of predictive success,
Mathematical Social Sciences, 21, 153-167.
9
Figure 1. An example of the experimental task display.
10
Geometric Mean Likelihood
0.6
eye-tracked
0.5
0.505
0.499
0.474
0.454
0.4
all
0.460
0.400
0.405
0.420
0.429 0.425
0.425
0.444
0.3
0.2
Re
EWA
stEWA
C1
C2
Nash
Figure 2. Geometric mean predicted probabilities of actual choices for
various learning models. Models fit on trials 1-7; results shown are out-ofsample cross-validation predictions. Standard errors computed across subjects.
Figure 3. Mean Linear Measure Scores for Different Learning Models
11
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