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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).

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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).

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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

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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), experience-weighted 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 Ai

j(0), with initial weight N(0), and are updated each period after receiving choice feedback, according to (1) Ai

j(t+1)= ϕN(t)A ij(t)/N(t+1) + [δ+(1-δ)I(si

j,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 Pi

j(t+1)=exp(λA ij(t))/(∑k

exp(λA ik(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(t-1),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(t-1), 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(si

j,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

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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.

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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(s-

i(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

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[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 Nash-type 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

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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

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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 equal-sized 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).

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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.

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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.

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Figure 1. An example of the experimental task display.

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0.405

0.454 0.460

0.4000.429 0.425

0.4740.499 0.505

0.420 0.4250.444

0.2

0.3

0.4

0.5

0.6

Re EWA stEWA C1 C2 Nash

Geo

met

ric M

ean

Like

lihoo

d eye-tracked all

Figure 2. Geometric mean predicted probabilities of actual choices for various learning models. Models fit on trials 1-7; results shown are out-of-sample cross-validation predictions. Standard errors computed across subjects.

Figure 3. Mean Linear Measure Scores for Different Learning Models

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