2.2.1 Experimental Tasks

Each subject attempted three different tasks a number of times, where the visual arrangement of the elements on the screen was simple and identical for all tasks. This was to ensure that any between-task variation in eye-activity could not be attributed to differences in visual appearance, and to eliminate any potential pupil foreshortening error (Reference Hayes and PetrovHayes and Petrov, 2016). In particular, as illustrated in Figure 1, subjects were shown two two-digit integers, displayed in two of four possible square fields. The fields in which the numbers were shown, as well as the numbers themselves, changed randomly in each round. This made it impossible for subjects to memorize the exact visual structure of the tasks, and thus inducing them to constantly explore the screen (which provided us with the required eye-data). Throughout the whole study, divided into Stage One and Stage Two, subjects were asked to select one of the two numbers, based on varying rules and criteria.

Figure 1: The visual arrangement of elements on the screen, the same across all tasks (only elements in black were visible to subjects).

Stage One: playing against a computer player.

In the first of two stages of the study, we presented every subject with 40 rounds of play against two types of a computer player (20 rounds for each type, in random order, where subjects were always told before the trial which type of computer player they would face).

Unpredictable computer player. Subjects were told that the computer would randomly select one of the numbers with equal probability. They were told that choosing the same number as the computer would earn two points. — every point scored in the experiment was converted to a monetary payoff at a known fixed rate — but choosing a different number would earn zero points. A subtle exception to this rule occurred if a subject was assigned to play the matching-pennies game at Stage Two of the study, and rewarded for choosing a different number than the other human player (see below). In that case, the subject was also told to select a different number than the computer player at Stage One.

Predictable computer player. Subjects were told that the computer would pick the number closer to X, where X was an integer larger than one of the two numbers subject to choice but smaller than the other (and closer to one of the two numbers). For example, given a choice between 12 or 31 and X=14, the computer would select 12. The value of X varied between rounds and was disclosed just before displaying the choice screen presented in Figure 1. Subjects were told they would receive one point if they select the same number as the computer, and zero points otherwise. Once again, the exception was that subjects assigned to play the matching-pennies game at Stage Two of the study, and rewarded for choosing a different number than the other human player, were also told to select a different number than the computer player at Stage One.

Stage Two: playing against another human player.

Upon completion of Stage One, each subject was then asked to repeatedly play one of two strategic games with the subject she was paired with for a total of 20 rounds. In each round, both subjects would now see the same pair of two distinct two-digit numbers, placed in the same two slots/fields for both subjects (i.e., they would both face exactly the same choice, of which they were made aware).

The numbers and their positions would continue to randomly change between rounds, while ensuring that no number that has already appeared in Stage One appears during Stage Two, and that no number appears more than once. The two alternative strategic games that we considered were as follows:

Coordination Game. The subject was told that she and the counterpart would each receive two points if they independently select the same number. Otherwise, they would each receive zero points.

Matching Pennies Game. One of the subjects was told that she would receive two points if she independently selects the same number as the counterpart (and zero points otherwise). The subject was also told that the counterpart has an opposite objective and would receive two points if she independently selects a different number than the rival player (and zero points otherwise). In accordance with this, the other of the two subjects was told the exact opposite.

Our choice of games was dictated partly by the fact that they were both very simple to present visually, facilitating our eye-tracking analysis. In addition, the matching pennies game is a classic example of a competitive constant-sum game in the sense of (Reference SchellingSchelling, 1960), one in which it is essential to conceal one’s actions from the counterpart. In contrast, in the coordination game it is essential to be as predictable as possible in order to coordinate on the mutually-beneficial outcome. Note that we require each pair of subjects to play only one of the two games, and do not alternate the roles in matching-pennies, so as to make it easier to learn the game, the strategic properties of which remain unchanged between rounds ((Reference Grimm and MengelGrimm and Mengel, 2012) showed that taking turns to play strategically distinct games impedes convergence to the equilibrium over time).

The other feature of the Stage Two strategic task that we manipulate is the availability of information about the counterparts’ previous choices. Specifically, we consider two possibilities (experimental factor levels), with every pair of subjects playing according to one of the two throughout the entire Stage Two:

No Feedback. Every subject is told that she would only learn the counterpart’s choices (and her own and the other player’s payoffs) upon completion of the study.

Full Feedback. Each subject is told that, in every trial of the task, after she and the other player make their choices, they will each learn which number was selected by the counterpart, as well as the resulting payoffs that each of them receives (this information was displayed after the conclusion of each trial but before starting the next one and displaying the corresponding choice screen).

2.2.2 Experimental Conditions

There were two between-group factors affecting Stage Two of the experiment: the type of strategic game and the presence of feedback, giving rise to a total of four experimental conditions: Coordination × No Feedback, Coordination × Full Feedback, Matching-Pennies × No Feedback and Matching-Pennies × Full Feedback. We assign twelve pairs of subjects to each of the four conditions, testing a total of 96 subjects of which 4 were later removed from data analysis due to poor calibration.

Eye-tracking Setup.

The experiments were computerized — our stimulus presentation software was programmed in C# using Microsoft Visual Studio Express 2015. Each of the paired subjects was seated at a separate computer terminal and both were asked not to communicate with each other. One of the computer terminals had attached underneath the screen a RED250mobile eye-tracking device, the other a RED-m device (both manufactured by SensoMotoric Instruments and set to 60Hz frequency). Immediately prior to the commencement of the study we conducted a standard five-point semi-automatic calibration and validation procedure. The average deviation was below 0.5^∘ for 92 subjects, the other 4 subjects’ data was removed from the analysis.

Choice protocol.

We considered it essential to avoid asking subjects to use keyboard to make their choices. This would result in taking eyes off the screen, potentially even for the entire duration of the task (when playing an unpredictable computer player subjects could simply press one of the two choice keys at random without looking at the screen). Using the computer mouse instead would similarly result in a distortion of the recorded eye-data, since some of it would be an artefact of subjects seeking out and then following the cursor with their gaze.

In order to address these issues, we ask subjects to first decide which number they wish to select, then press any keyboard key (which they can do without taking eyes off the screen), and finally to look at their number of choice. The eye-tracking device would detect this instantly, but to prevent choices being made accidentally, the subject had to hold her gaze on the chosen number for a further two seconds before the choice was finalized. This kind of eye-control is widely used in “augmentative and alternative communication” (to enable people with disabilities to navigate their computer with their eyes), and is adapted here for our research purposes. Subjects found the protocol straightforward and intuitive, and had no difficulty using it.

2.4.1 Pre-processing the eye-tracking data

For every trial/choice made by every subject, a total of 92×( 40+20) trials, we gather the eye-tracking data corresponding to the time period starting with showing the choice screen (Fig. 1), and ending with the subject finalizing her choice. For each trial, we then calculate and record the values of the following three variables.

Reaction time.

This we calculate as the difference (in milliseconds) between the time of the subject pressing a key to signal the readiness to submit her choice (by looking at the chosen number), and the time of the choice screen being displayed.

Gaze dispersion.

This we define and compute as the proportion of the total time of the trial which the subject spent looking at the square field (as in Fig. 1) containing the number she did NOT subsequently choose. Note that in this case we include the time after the key has been pressed, because it takes a further two seconds since pressing it to finalize the choice (recall Section 2.3). The decision on whether to use data until pressing the key or until finalizing the choice is not important for the measurement of reaction times, because the latter option simply adds a constant to each value. However, considering the total duration of the trial for the measurement of the other two variables slightly increases the accuracy of distinguishing between the Stage One tasks, while having no effect on the sign and significance of the differences between the four experimental conditions.

Pupil deviation.

Pupil dilation can serve as a measure of both cognitive workload and emotional response (see (Reference WangWang, 2011) for a review of the related literature). However, pupillary responses have so far rarely been used to study economic decisions, although there are a few notable exceptions (Reference Wang, Spezio and CamererWang, Spezio and Camerer, 2010; Glöckner et al., 2012; Reference Fiedler, Glöckner and NicklischFiedler, Glöckner and Nicklisch, 2012; Reference Franco-Watkins and JohnsonFranco-Watkins and Johnson, 2011). A significant difficulty associated with this kind of analysis is that pupil dilation varies periodically in response to changes in the intensity of light that falls on the retinal ganglion cells of the eye, creating undesirable noise in the signal. We solve this issue using the procedure introduced in (Reference MarshallMarshall, 2007). Specifically, we conduct a discrete wavelet transform of the pupil dilation (measured in mm) time series using a Daubechies wavelet. The coefficient outputs of the wavelet decomposition are subsequently submitted to a unit threshold to remove noise. The remaining high frequency details are then extracted from the convolution and inverse wavelet transform used to construct a residual time series. This gives the sudden unexpected changes (deviations) in pupil dilation that cannot be explained by adjustments to changes in lumination, and are therefore indicative of cognitive or emotional activity. We record the maximum value of the residual during each trial, again including the time after pressing the key.

2.4.2 Classifying individual Stage One trials

We investigate if it is possible to predict, based on the values of reaction time, gaze dispersion and pupil deviation corresponding to an individual trial, whether that single trial is of “predictable” or “unpredictable” type.

To this end, we collect each subject’s data corresponding to the 40 instances of Stage One trials. Of these, we drop the (chronologically) first 5 instances of both the predictable and unpredictable type, because the subjects need to learn the general structure of the problem at hand before the data becomes representative of the underlying cognitive processes.

We then conduct within-subject standardization of the remaining data (using the mean and standard deviation of each subject’s remaining 30 trials). This removes between-subject differences not related to the specific task types. As a result, it becomes feasible to train and validate a classification model based on data from all 92 subjects, instead of having 92 individual models. We find the former preferable, as it will allow us to extract the overall features of the entire tested population, as well as providing the classification model with more training and validation data.

Training the neural-network classifier.

We analyze the data using a neural network classifier with one input layer of three nodes (one for each variable), one hidden layer of seven nodes and one output layer of two nodes (one for each class: predictable vs. unpredictable). The structure of the resulting “multi-layered perceptron” is illustrated in Figure 2. Using a single hidden layer is standard practice and the most common neural network architecture across various fields of research; it has been demonstrated that any continuous function can be represented by a neural network that has only one hidden layer with exactly 2n + 1 nodes, where n is the number of input nodes (see Stathakis, 2009 for more detail). We use a rectified linear hidden layer activation function instead of the logistic sigmoid function in order to improve training time and performance of the network (Reference Nair and HintonNair and Hinton, 2010).

Figure 2: The structure of the neural network used in the study.

In addition, to prevent overfitting and allow the model to generalize to the Stage Two strategic tasks in rounds 41–60, we conduct K-fold repeated cross-validation with R=5 runs and K=2 folds in the following manner.

First, we randomly split the data (30×92 trials) in half, the first of these K=2 subsamples used to train the neural network, and the second one used for testing. In addition, a fifth of the training data is set aside for validation purposes. In the process of training, each case in the training data is presented to the model, and the weights of the neural network are adjusted to fit the (known) true classes of the training cases (predictable vs. unpredictable). Once all cases have been presented, the average error between the true classes and the ones predicted by the network is calculated for the training and validation data. The entire process is repeated until the validation error starts to increase (despite continuing to decrease for the training data), indicating that overfitting begins to occur. Note that we use L2 regularization, which is a technique that reduces overfitting by adding a fraction of the sum of the squared weights to the error term.

After that, the accuracy of the trained neural network is evaluated on previously unseen testing data (the other of the two subsamples). More specifically, we compute the fraction of trials in the testing data that were correctly classified as predictable or unpredictable by the trained network.

To complete the cross-validation, we repeat the whole procedure, but this time use the first of the two subsamples for testing and the second one for training and validation. In addition, we re-run the entire cross validation R =5 times, each time randomly re-partitioning the data into the K =2 subsamples.

Evaluating model performance.

We report the overall average cross-validated accuracy of the classification algorithm. We evaluate it both against the chance level and against the accuracy achieved by the canonical logistic regression model with the same independent variables and subject to an identical cross-validation process (including an identical partitioning of the data). We use the 5-by-2 repeated cross-validation test (Reference DietterichDietterich, 1998) to assess the statistical significance of the latter comparison.

2.4.3 Classifying aggregated Stage One trials

We also wish to explore the possibility that aggregating data over time may reduce noise and improve accuracy. To this end, we proceed in the same way as described above, except that instead of treating each subject’s 30 trials, 15 for each kind of task, as 30 separate data points, we take the average values of reaction time, gaze dispersion and pupil deviation over the 15 trials of each type, turning them into a single, averaged, data point. Thus, we obtain a pair of data points for each subject: one representing the subject’s average profile during the predictable task, the other corresponding to the unpredictable task.

Using the resulting, reduced dataset of 92×2 cases, we carry out the same K-fold cross validation procedure (of the same neural network model) that was described and conducted for single-trial classification above. Once again, we report the overall average cross-validated accuracy of classification, comparing it against the chance level and against the accuracy of the logistic regression model.

2.4.4 Analyzing the Stage Two strategic tasks

For each of the 92 subjects we proceed as follows. We collect the subject’s data (reaction time, gaze dispersion and pupil deviation) corresponding to the first 40 trials of the study, drop the first 5 instances of both the predictable and unpredictable tasks, and perform within-subject standardization. In case of multi-trial classification, each subject’s data for each type of task is additionally averaged. Thus, for both single- and multi-trial classification we use the same data as in Section 2.4.2.

The difference is that we now use the entire Stage One data (rather than just one fold amounting to half of it) for the purpose of training and validation of the neural network. This is to maximize the accuracy of the trained model when dealing with the unseen Stage Two data.

As with both Stage One tasks, the first 5 trials of the Stage Two strategic task are dropped, and the rest of the data standardized using the same mean and standard deviation as the training and validation data. In case of the multi-trial classifier, the data from the remaining 15 trials of the Stage Two task is averaged to constitute a single data point for each subject. The trained neural network then predicts whether each data point in the testing data (trials 46−60) is an instance of the predictable or unpredictable class.Footnote ² Specifically, for each subject we compute the overall average proportion of Stage Two task data points classified as predictable rather than unpredictable. We will refer to this proportion as “similarity to predictable task”. Note that, in the Coordination × Full Feedback and Matching-Pennies × Full Feedback conditions, we further average the proportions for each pair of matched subjects. This is because, playing against each other, each pair of subjects then constitutes an independent observation for the purpose of statistical analysis.

We compare the values of similarity to predictable task between the conditions, using non-parametric two-tailed Mann-Whitney tests to assess significance, and report the test statistic and p-value for each comparison. We do this separately for each of the two classifiers (single- and multi-trial).