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\title{A novel approach to studying strategic decisions with eye-tracking and machine learning}

\author{Michal Krol\thanks{Department of Economics, The University of Manchester, Oxford Rd., Manchester M13 9PL, UK.  Email:
    michal.krol@manchester.ac.uk.}
 \and Magdalena Krol\thanks{University of Social Sciences and Humanities in Wroclaw, Aleksandra Ostrowskiego 30b, 53-238 Wrocław, Poland. Email: mkrol1@swps.edu.pl 
}
}

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\newcommand{\jhead}{Judgment and Decision Making, Vol.~12, No.~6}
\newcommand{\jdate}{November 2017}
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  \jdate \hfill Strategic decisions, eye tracking, and machine learning \qquad}
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\href{\jref}{\jhead}, \jdate, pp.\ 596-609
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\begin{onecolabstract}
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We propose a novel method of using eye-tracking to study strategic decisions. The conventional approach is to hypothesize what eye-patterns should be observed if a given model of decision-making was accurate, and then proceed to verify if this occurs. When such hypothesis specification is difficult a priori, we propose instead to expose subjects to a variant of the original strategic task that should induce processing it in a way consistent with the postulated model. It is then possible to use machine learning pattern recognition techniques to check if the associated eye-patterns are similar to those recorded during the original task. We illustrate the method using simple examples of 2x2 matching-pennies and coordination games with or without feedback about the counterparts' past moves. We discuss the strengths and limitations of the method in this context. 

\smallskip
\noindent
Keywords: task recognition, eye-tracking, strategic games, machine learning
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{\renewcommand{\thefootnote}{}
\footnotetext{ % note blank lines above and below acknowledgment

The authors would like to thank two anonymous reviewers and the editor, Prof.\ Andreas Gl\"ockner for their insightful and helpful comments.

This work was supported by the Polish Ministry of Science and Higher Education under grant BST/WROC/2016/A/6.

Copyright: \copyright\ 2017.
The authors license this article under the terms of the
\href{http://creativecommons.org/licenses/by/3.0/}{Creative Commons
  Attribution 3.0 License.}
}}

\saythanks

\section{Introduction}

Recently, there has been growing interest in using the eye-tracking method,
already well established in psychology, to study economic decision-making. The
purpose is to gain insights about behavior that would be unavailable based on
observed choice data alone. At the same time, in contrast with more expensive
and complex procedures, like fMRI, eye-tracking makes it possible to test
subjects under exactly the same conditions as those that they encounter in
standard behavioral experiments.

To date, the prevalent approach has been to formulate models of the decision
process, together with the corresponding characteristic gaze patterns that
should occur if each of the alternative models was accurate. One can then
select the best model based on: a) the fit between the postulated and observed
gaze patterns; and b) the fit between the choices predicted by the model for
the observed gaze patterns and the choices that actually occur. Such a
potential use of process tracing techniques in studying decisions has long
been acknowledged \cite{Schulte-Mecklenbeck2011}.

For instance, \citeA{Krajbich2010} and \citeA{Krajbich2011} evaluate a number of
alternative drift-diffusion models in the context of choosing between two or
between three food items, and report, among other things, that: a) as
predicted by the theory, eye-fixations immediately prior to choice are
relatively short; and b) some of the models, but not others, correctly predict
that food items that were looked at more often are also more likely to be
selected. In a similar context (but for a larger number of alternative choice
items), \citeA{Reutskaja2011} compare an optimal costless search model,
a satisficing search model and a hybrid search model, while \citeA{Arieli2011}
consider choosing between lotteries. Relatedly, \citeA{Rubaltelli2012} report
that eye-movement patterns differ across two variants of the same task,
depending on whether subjects evaluate gambles on an attractiveness scale or
via specifying the monetary amount one would pay to partake in or avoid the
gamble. This suggests that the impact of decision consequences (even
hypothetical ones) on the underlying decision process could be reflected in
the accompanying eye-movements --- an idea that will be important for this
study. Interestingly, \citeA{Glockner2011} report that eye-movement patterns
recorded while choosing between gambles ``conflict with the idea that necessary
calculations are deliberately conducted by computing weighted sums'', while
\citeA{Venkatraman2014} use eye-data to argue that people readily switch
between different decision strategies.\footnote{Recent evidence also suggests
that findings from eye-tracking studies on valuation of risky prospects extend
to choices between ``ordinary'' consumer goods, see, e.g., \citeA{Ashby2015}.}

These results are important, given our focus on a particular type of
decisions, namely those between different strategies, each of which
will yield payoffs contingent on the unknown choice of another
party. Because of this strategic uncertainty, such decisions may also
be thought of as gambles. In this context, eye-tracking becomes
technically more challenging to conduct on more than one computer
terminal at the same time. Hence, many studies use a related but less
complex mouse-tracking procedure, which entails hiding parts of the
decision screen containing various pieces of information behind boxes
that reveal their contents only as long as the subject hovers the
mouse cursor over them. This has some limitations, such as the fact
that subjects might be reluctant to explore the screen to collect data
when this requires more effort, and need not look at the data under
the cursor at all. An example of this approach is the work of
\citeA{Johnson2002}, who reject backward induction reasoning in a
sequential bargaining game based on the fact that subjects do not look
at the information sufficiently far ahead. A number of studies focus
on subject heterogeneity. For example, \citeA{Costa-Gomes2001} and
\citeA{Costa-Gomes2006} group subjects into a priori-defined types,
depending on their choices and the accompanying mouse lookup pattern,
and find that deviations of observed choices from equilibrium can be
largely explained by subjects belonging to types based on various
degrees of k-level thinking (see also \citeNP{Stewart2016}). More
recently, \citeA{Brocas2014} and \citeA{Polonio2015} use cluster
analysis to group subjects into types without having to define them
beforehand (thereby requiring less model structure to be imposed a
priori). Note that the latter study is based on eye-tracking (rather
than mouse-tracking), and reports that the subjects' types can remain
stable over time and across different strategic tasks. Our aim is
similar, in that we want the analysis to be as model-free as possible,
but with the focus on heterogeneity of cognitive processing across
different tasks, rather than between subjects.

More specifically, in many situations it may be difficult to specify a priori
what gaze patterns should accompany a given decision rule. For instance,
suppose we have subjects playing a coordination game in which they each
independently choose one of two numbers shown on-screen, and get rewarded only
if they both choose the same number. Suppose we want to ascertain how
confident they are of the counterpart's expected choice, and consequently of
their own decision. As shown by \citeA{Rutstrom2009616}, asking subjects to
state their beliefs can interfere with game play, while inferring the beliefs
from counterparts' past observed actions requires data from several decision
trials and feedback from previous trials being available to subjects. In
contrast, eye-tracking is completely noninvasive (does not interfere with the
choice problem), and makes it possible to analyze single decision trials in
isolation, or scenarios in which feedback on counterparts' past actions is unavailable.

At the same time, however, it is difficult to specify a priori what
eye-movement patterns would be a marker of making confident
predictions of others' choices, as opposed to being unsure about
them. In the former case, we might expect subjects to look carefully
at both options in turn in order to compare them and form a
well-grounded expectation. However, switching one's gaze between the
options could also be a sign uncertainty and indecision
\cite{Patalano2009}. Similarly, focusing on one of the options could
mean that one is sure the counterpart would choose it, or that, on the
contrary, one thinks that to predict the choice of the other player is
impossible and hence chooses the first option she happens to look at.

\renewcommand{\baselinestretch}{1.02} \large\normalsize

Instead of making assumptions about how players' decision strategies
dictate the distribution of eye fixations between options, it would
therefore be prudent to establish this relationship empirically. For
example, we might ask subjects to play two additional variants of the
original task against a computer player: (1) where they are told
that the computer will ``flip a coin'' to select one of the numbers with
equal probability; and (2) where the computer will select the number
according to a pre-specified rule. Thus, in the latter case subjects
should be considerably more certain of the counterpart's choice than
in the former case. We can then check which of the two variants in
question is more similar to the original strategic task (playing
against a human player) in terms of the accompanying eye-tracking
data. This should allow us to gain insight into decision-making
strategies involved in the original task.

Of course, as a prerequisite for this analysis, we need to have a
method of measuring similarity that would allow us to distinguish
between the two variants of the task with good accuracy. In other
words, given eye-data of a subject playing a single trial of the game
against the computer, we should be able to predict whether the trial
was an instance of variant (1) or (2) of the original task.

This alone is a considerable challenge, much more so than simply
documenting the statistical differences between the two variants of
the task. An early encouraging result is the work of \citeA{Day2010},
who uses the Needleman-Wunsch algorithm to predict which of a number
of decision-strategies a given sequence of eye-movements corresponds
to, with accuracy significantly above chance. However, this is done by
comparing the sequence to a hypothetical model sequence, specified a
priori for each strategy by the researcher. In this sense, the study
resembles the bulk of existing literature, as described above, in
theorizing in advance as to what gaze patterns we should find in order
for a given hypothesis to be confirmed.  Furthermore, the decision
strategies themselves are imposed on the subjects, who are first
extensively trained and then explicitly instructed to use them.  This
puts into question the ability of the method to identify the decision
strategy when people can choose it freely, and when eye-movements
actually accompanying each strategy are different from the
hypothetical ones specified by the researcher (as was in fact observed
by the author).

Indeed, \citeA{Greene2012} reported a negative result, suggesting that it is
impossible to identify the observer's task by comparing the accompanying eye
movements with those of other subjects pursuing each different type of task.
However, \citeA{Borji2014} showed that applying non-parametric machine learning
classification techniques can change this state of affairs. In particular,
subjects' gaze patterns were recorded while performing one of a number of
image-viewing tasks (like estimating the ages of people shown in the picture
or their material circumstances). In the end, the classification algorithm was
able to guess which task was performed based on eye-data with an accuracy
significantly above chance.

In this paper, we extend this approach, for the first time, to economic and
particularly strategic decisions. Furthermore, we show that it can be used to
analyze tasks that are visually similar, but where the underlying decision
strategy is unknown, in order to gain meaningful insights into the impact of
factors such as the game's payoff structure or the availability of feedback on
the players' decision process.

More specifically, our demonstration of the proposed method using simple 2x2
games proceeds as follows. Depending on the experimental condition, we ask
each subject to play either a coordination game or a matching pennies game a
number of times. Initially, the subject plays against a computer player, who
behaves according to variant (1) above in some trials, and otherwise according
to variant (2), where the subject is always told which scenario occurs in a
given trial. Next, the subject plays the same game against another human
player, where (depending on the second experimental condition) feedback about
the counterpart's past decisions is either available or not.

To analyze the process tracing data, we use a neural-network classification
algorithm with reaction time, pupil dilation and gaze dispersion as inputs. In
other words, we refrain from using more complex identifiers, such as the
various similarity or distance measures between scanpath sequences
\cite{Cristino2010,Dewhurst2012}, or scanpath representation
matrices \cite{Hayes2011}. This is because, with only two numbers required
to convey all information about the game (and hence only two areas of
interest), we considered the complete scanpath sequence information
superfluous. At the same time, we conjectured that the main problem would be
that different people might use different viewing strategies to approach the
same variant of the task. Such a tendency would result in considerable noise
and possibly non-linear relationships between the variables. This motivated
our use of the machine-learning neural network technique.

We find that we are able to predict, with cross-validated accuracy
significantly above chance, whether eye-data from a given trial played against
the computer corresponds to the latter being known to behave according to
variant (1) or (2) of the task. In addition, the accuracy rate is slightly,
but significantly higher than when using the canonical logistic
regression model instead of a neural network. This demonstrates that
eye-tracking, combined with machine-learning, can be used to identify the
information that players have about the counterparts' play in strategic games.

We therefore proceed to use the trained neural network algorithm to
investigate \textit{unknown} decision-processes involved in playing
against a human player, and investigate how these are influenced by
the experimental conditions. As might be expected, we find that the
availability of feedback results in more trials being classified as
instances of variant (2) of play against the computer (where the
behavior of the latter is fully predictable).  More surprisingly, the
effect of a more competitive payoff structure is
similar. Specifically, when playing the matching-pennies game,
compared with the coordination game, subjects' eye-movements are more
similar to those observed when playing a predictable computer
player. This holds despite subjects being incentivized to hide their
mode of play from the rival in the matching-pennies game, and to make
it transparent and predictable in the coordination game.

To further illustrate the strengths and limitations of the proposed
method, we conclude by discussing the implications of the results of
our demonstration for existing theories, particularly the
interpretation of the mixed-strategy Nash Equilibrium concept, and
existing research on mixed-strategy play by ``amateur'' subjects in
the lab vs.\ ``professional'' subjects in the field
\cite{Wooders2010,STEVEND.LEVITTJOHNA.LIST2010}.

\section{Method} % example of a heading
\subsection{Subjects}

The experiment was conducted at the University of Social Sciences and
Humanities in Wroclaw (SWPS), Poland. 96 subjects were
recruited from the local population of undergraduate and postgraduate students
(mean age $=23.66$, SD $=6.7$, $51$ females). Subjects did not previously take
part in any Economics or related courses, nor in any experiments on strategic
interactions. The data of four subjects who were not able to successfully
undergo the eye-tracking calibration procedure (see Section \ref{setup}) was
removed from the analysis, leaving $92$ subjects.

Every point scored in the study was converted to 0.50 PLN (local currency,
equivalent to 13 US cents). The experiment took around 30 minutes and the
average payoff was 29.67 PLN (7.55 USD). This was in line with the rewards
typically offered to subjects in similar studies in Poland.

\renewcommand{\baselinestretch}{1.00} \large\normalsize

\subsection{Stimuli and Design}

The experiment adapted the fixed-pair protocol (subjects randomly
matched into pairs and interacting only with the other subject from
the same pair).  This was first used in the classic study by
\cite{neillo1987}, and subsequently by most related experimental
studies mentioned in the previous section. The tasks were presented in
visual form with instructions communicated in words (both verbally
and on-screen), rather than as a matrix. In addition, the procedure
was adapted to enable the use of eye-tracking.

\subsubsection{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 \cite{Hayes2016}. In particular, as illustrated in Figure
\ref{fig1}, 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.\bigskip

\begin{figure}[t!]
\includegraphics[width=\columnwidth,clip=true]{fig1.eps}
\caption{The visual arrangement of elements on the screen, the same across all
tasks (only elements in black were visible to subjects).}
\label{fig1}
\end{figure}

\paragraph{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).

\textbf{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.

\textbf{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 \ref{fig1}. 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.

\paragraph{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:

\textbf{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.

\textbf{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 \cite{shelling}, 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 (\cite{Grimm2012} 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:

\textbf{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.

\textbf{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).

\subsubsection{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 $\times$ No Feedback,
Coordination $\times$ Full Feedback, Matching-Pennies $\times$ No Feedback and
Matching-Pennies $\times$ 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.

\subsection{Procedure}

\paragraph{\label{setup}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$^\circ$ for
92 subjects, the other 4 subjects' data was removed from the analysis.

\paragraph{\label{choiceProt}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.

\subsection{Data Analysis}

\subsubsection{\label{encodeData}Pre-processing the eye-tracking data}

For every trial/choice made by every subject, a total of
$92\times\left( 40+20\right)$ trials, we gather the eye-tracking data
corresponding to the time period starting with showing the choice
screen (Fig.\ \ref{fig1}), and ending with the subject finalizing her
choice. For each trial, we then calculate and record the values of the
following three variables.

\paragraph{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.

\paragraph{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.\ \ref{fig1})
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
\ref{choiceProt}). 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.

\paragraph{Pupil deviation.}
Pupil dilation can serve as a measure of both cognitive workload and emotional
response (see \cite{Wang2011} 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 \cite{Wang2010,glockner2012processing,fiedler20122,franco2011applying}. 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
\cite{Marshall2007}. 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.

\subsubsection{\label{compareAlgs}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.

\paragraph{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
\ref{figNet}. 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
\citeNP{stathakis2009many} 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 \cite{nair2010rectified}.

\begin{figure}[t!]\centering
\includegraphics[width=3in,clip=true]{fig2.eps}
\caption{The structure of the neural network used in the study.}
\label{figNet}
\end{figure}

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 ($30\times92$ 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.

\paragraph{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 \cite{Dietterich1998} to assess the statistical
significance of the latter comparison.

\subsubsection{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 $92\times2$ 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.

\subsubsection{\label{analyzeStrat}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
\ref{compareAlgs}.

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{We repeat the training and prediction procedure $100$
  times and consider the average prediction results across these $100$
  repetitions. This is to minimize the effect of random factors on the
  training process (the neural network is randomly initialized on each
  occasion).} 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 $\times$ Full Feedback and Matching-Pennies $\times$ 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).

\section{Results}

\subsection{Stage One}

\subsubsection{Comparing the overall characteristics of Stage One tasks}

To indicate how our three variables may be used to classify trials, we
compared the average value of each variable for every subject, separately for
the 20 Stage One trials of each type (thus obtaining $3\times92$ matched pairs
of values), and the resulting median value of each set of $92$ averages. To
this end, we used non-parametric matched-pair Wilcoxon tests.
\begin{itemize}
\item The reaction time for ``predictable rival'' trials (Mdn $=2.86$s) was
significantly longer than for unpredictable trials (Mdn $=1.72$s),
$W=302,p<.001$.

\item The gaze dispersion for predictable trials (Mdn $=12.5\%$) was
significantly larger than for unpredictable trials (Mdn $=8.9\%$),
$W=451,p<0.001$.

\item The pupil deviation for predictable trials (Mdn $=0.176$mm) was
significantly larger than for unpredictable trials (Mdn $=0.105$mm),
$W=156,p<0.001$.
\end{itemize}

Thus, compared with unpredictable rival trials, during predictable
trials subjects' reaction times are longer, their gaze is more evenly
distributed across the two options, and they exhibit greater
unexpected increases in pupil dilation.

\subsubsection{Evaluating the classification accuracy}

\paragraph{Single-trial model.}
\label{res1model}Based on reaction time, gaze dispersion and pupil
deviation, the neural network model can predict whether a single Stage
One trial is played against a predictable or unpredictable computer
player with an accuracy of $0.67$. This value is significantly above
chance level (binomial test $p$-value $<0.001$), and is also
significantly higher than the $0.65$ accuracy achieved by the logistic
regression model (5-by-2 paired $t(5)=3.80,p=0.012$).

\paragraph{Multi-trial model.}
\label{res2model}Based on the average reaction time, gaze dispersion
and pupil deviation from all Stage One trials of the same type of task
by the same subject, the neural network model can predict whether
those trials are played against a predictable or unpredictable
computer player with an accuracy of $90\%$. This value is
significantly above chance level (binomial test $p$-value $<0.001$),
and is significantly higher than the $0.88$ accuracy achieved by the
logistic regression model (5-by-2 paired $t(5)=3.94,p=0.011$).

\subsection{\label{analyzeStageTwo}Stage Two}

We begin by comparing the four experimental conditions separately in terms of
each of our independent variables: reaction time, gaze dispersion and pupil
deviation. For each variable, we report all four pairwise comparisons between
the conditions and the associated Mann-Whitney test results. In each case, we
evaluate the statistical significance of the four comparisons using the
step-wise Bonferroni-Holm correction (where the successive $p-$values, in
descending order, are evaluated against respective $\alpha-$thresholds of
$0.05,0.025,0.017$ and $0.0125$).
\begin{itemize}
\item Reaction time in the Coordination $\times$ No Feedback condition
(Mdn $=1.45$) was not significantly different than in the Matching-Pennies
$\times$ No Feedback condition (Mdn $=1.80$), $U=53,p=.145$, or in the
Coordination $\times$ Full Feedback condition (Mdn $=1.47$), $U=131,p=.794.$%

\item Reaction time in the Matching-Pennies $\times$ Full Feedback
condition (Mdn $=1.86$) was not significantly different than in the
Coordination $\times$ Full Feedback condition (Mdn $=1.47$), $U=39,p=.053$, or
in the Matching-Pennies $\times$ No Feedback condition (Mdn $=1.80$),
$U=146,p=.959.$

\item Gaze dispersion in the Coordination $\times$ No Feedback condition
(Mdn $=9.0\%$) was significantly smaller than in the Matching-Pennies $\times$
No Feedback condition (Mdn $=12.0\%$), $U=153,p=.009$, but not significantly
different than in the Coordination $\times$ Full Feedback condition (Mdn
$=9.9\%$), $U=135,p=.903.$

\item Gaze dispersion in the Matching-Pennies $\times$ Full Feedback
condition (Mdn $=13.4\%$) was significantly larger than in the Coordination
$\times$ Full Feedback condition (Mdn $=9.9\%$), $U=31,p=.016$, but not
significantly different than in the Matching-Pennies $\times$ No Feedback
condition (Mdn $=12.0\%$), $U=114,p=.306.$

\item Pupil deviation in the Coordination $\times$ No Feedback condition
(Mdn $=0.04$mm) was not significantly different than in the Matching-Pennies
$\times$ No Feedback condition (Mdn $=0.06$mm), $U=64,p=.624$, but was
significantly smaller than in the Coordination $\times$ Full Feedback
condition (Mdn $=0.13$mm), $U=24,p=.005.$

\item Pupil deviation in the Matching-Pennies $\times$ Full Feedback
condition (Mdn $=0.23$mm) was not significantly different than in the
Coordination $\times$ Full Feedback condition (Mdn $=0.13$mm), $U=34,p=.026$,
but was significantly larger than in the Matching-Pennies $\times$ No Feedback
condition (Mdn $=0.06$mm), $U=14,p=.001.$
\end{itemize}

Thus, we see that pupil deviation is generally higher under full feedback than
under no feedback, while gaze dispersion is higher in the matching-pennies
game than in the coordination game. In addition, under full feedback both
reaction time and pupil deviation are almost significantly higher in matching
pennies than in the coordination game.

In other words, giving subjects access to feedback at Stage Two has a similar
effect on pupil deviation as giving them information about how the computer
player would make its choice at Stage One. It may be that sudden increases in
pupil dilation can be a sign of an ``Aha! moment'', i.e., of a subject realizing
that she has arrived at the correct answer to the problem, something that is
only possible if enough information is available.

As for the gaze dispersion, in this respect the matching-pennies game is more
similar to the predictable task than the coordination game. This could reflect
the fact that in matching pennies more effort is put into comparing the two
numbers subject to choice. In contrast, in the coordination game, without
feedback, people might tend to choose the first option that catches the eye,
hoping that the counterpart will do the same. Even when feedback is given,
subjects might quickly determine the rule based on which the counterpart makes
her choice, reducing the number of eye transitions between the numbers.

More generally, it appears that different predictor variables are best at
discriminating between different experimental factors (the presence of
feedback and type of strategic game). Thus, it is interesting to see if
combining all three variables in a neural network algorithm will result in
being able to better discriminate between all four factor combinations in
terms of their similarity to the Stage One tasks. Specifically, we consider
the proportion of Stage Two trials classified as instances of playing against
a predictable opponent (``similarity to predictable task''). We report all
pairwise comparisons between the conditions and the associated Mann-Whitney
test results, separately for single- and multi-trial classification.

\subsubsection{Single-trial Classification}
\begin{itemize}
\item The similarity to predictable task in the Coordination $\times$ No
Feedback condition (Mdn $=0.09$) was significantly smaller than in the
Matching-Pennies $\times$ No Feedback condition (Mdn $=0.32$),
$U=150.5,p=.008$.

\item The similarity to predictable task in the Coordination $\times$ Full
Feedback condition (Mdn $=0.35$) was significantly smaller than in the
Matching-Pennies $\times$ Full Feedback condition (Mdn $=0.63$),
$U=20,p=.002$.

\item The similarity to predictable task in the Coordination $\times$ No
Feedback condition (Mdn $=0.09$) was significantly smaller than in the
Coordination $\times$ Full Feedback condition (Mdn $=0.35$), $U=48,p=.002$.

\item The similarity to predictable task in the Matching-Pennies $\times$
No Feedback condition (Mdn $=0.35$) was significantly smaller than in the
Matching-Pennies $\times$ Full Feedback condition (Mdn $=0.63$),
$U=20,p=.002$.
\end{itemize}

The four differences are jointly significant based on the step-wise
Bonferroni-Holm correction. In addition, as seen in Figure \ref{figTemp}, the
above relationships between conditions are stable over time.

\begin{figure}[t!]
\centering\includegraphics[width=\columnwidth,clip=true]{fig3.eps}\caption{Changes in average
similarity to predictable task for each of the four experimental conditions
throughout trials $46-60$ of stage two of the study, split into $5$ time bins
of $3$ trials each (recall the first $5$ trials of stage two are dropped).}%
\label{figTemp}%
\end{figure}

\subsubsection{Multi-Trial Classification}
\begin{itemize}
\item The similarity to predictable task in the Coordination $\times$ No
Feedback condition (Mdn $=0$) was significantly smaller than in the
Matching-Pennies $\times$ No Feedback condition (Mdn $=0.51$), $U=136,p=.001$.

\item The similarity to predictable task in the Coordination $\times$ Full
Feedback condition (Mdn $=0.38$) was significantly smaller than in the
Matching-Pennies $\times$ Full Feedback condition (Mdn $=0.91$),
$U=12,p=.001$.

\item The similarity to predictable task in the Coordination $\times$ No
Feedback condition (Mdn $=0$) was significantly smaller than in the
Coordination $\times$ Full Feedback condition (Mdn $=0.38$), $U=61.5,p=.004$.

\item The similarity to predictable task in the Matching-Pennies $\times$
No Feedback condition (Mdn $=0.51$) was significantly smaller than in the
Matching-Pennies $\times$ Full Feedback condition (Mdn $=0.91$),
$U=83.5,p=.038$.
\end{itemize}

As in the case of the single-trial classifier, the four reported differences
are jointly significant based on the Bonferroni-Holm correction. Thus,
irrespective of whether we use single-trial or multi-trial classification, and
other things being equal, similarity to predictable task is significantly
larger under full feedback than under no feedback, and significantly larger in
the matching-pennies game than in the coordination game.

\subsubsection{\label{behRes}Behavioral Results}

We also report the results of two Mann-Whitney tests performed on behavioral
data, which will be useful in the interpretation of the eye-tracking findings
in the following section.
\begin{itemize}
\item The fraction of rounds of the strategic task won by the subject who
scored more points than the other in total over all rounds was not
significantly different in the Matching-Pennies $\times$ No Feedback condition
(Mdn $=0.55$) than in the Matching-Pennies $\times$ Full Feedback condition
(Mdn $=0.55$), $U=76,p=.83$.

% is "following section" correct?

\item The frequency of selecting the same number by both players in the
strategic task was significantly smaller in the Coordination $\times$ No
Feedback condition (Mdn $=0.50$) than in the Coordination $\times$ Full
Feedback condition (Mdn $=0.60$), $U=30.5,p=0.014$.
\end{itemize}

\section{Discussion}

A crucial result for the purpose of evaluating the method proposed in
this paper is that it is indeed possible to predict the Stage One task
a person attempted in a given trial based on the associated values of
the independent variables. Recall that the only difference between the
two tasks is the information about the behavior of the computer
player. Thus, eye-tracking data can potentially discriminate between
two scenarios: 1) where the human player is unsure of the
counterpart's behavior, and therefore also of her own optimal
strategy; 2) where the human player is able to work out the action
that the counterpart is likely to take, and to choose her own strategy
accordingly.

To justify the above statement, let us compare the obtained single-
and multi-trial classification accuracy rates to those reported by
related existing studies. In particular, \citeA{Marshall2007}
considered a similar problem of neural network discrimination, based
on eye-tracking data, between an arithmetic problem-solving task and a
doing-nothing task. When using data from four-second time intervals
(longer than the median duration of both of our tasks, equal to $1.72$
and $2.86$ sec. respectively), and combining data from different
subjects to build a general model (as we do here), the obtained
classification accuracy rate was $70\%$. For one-second time intervals
(slightly shorter than the duration of our tasks) the figure was
$65\%.$ When comparing this with our (similar) figure of $67\%$ in the
single-trial model, one needs to note that the current classification
problem was considerably harder, since both tasks entailed a visually
identical problem of choosing between two numbers and differed only in
terms of the information about the computer player behavior. In this
sense, the current problem bears more resemblance to the more recent
study by \citeA{Borji2014}, who distinguished between different
information extraction tasks performed while looking at the same
picture, and obtained an accuracy rate approximately $10\%$ above
chance.

Similarly, the $90\%$ accuracy of our multi-trial classifier is comparable to
the top accuracy rates reported by existing papers dealing with analogous
problems of classifying subjects' \textit{overall} performance in a study. For
example, \citeA{Richstone2010} use neural networks in a way similar to the
present study to classify surgeons into expert and non-expert cohorts based on
their eye-data gathered during simulated and live surgical environments. They
too report classification accuracy rates of around $90\%$, albeit obtained in
a small and highly unbalanced sample.

In summary, we consider the performance of both single- and multi-trial
classification algorithms to be satisfactory.

The first part of our contribution is therefore to show that
eye-tracking data can be used to discriminate between strategic
decision tasks that differ only in terms of the information about the
counterpart's behavior that players are provided with. The second part
of the contribution is to demonstrate, through results reported in
Section \ref{analyzeStageTwo}, how the results of such analysis may in
turn be used to investigate the effect of various external factors on
players' expectations and decision-making processes when playing
against another human player. Specifically, in our example we focus on
changes in the payoff structure of the game and the availability of
feedback about the counterpart's past actions. We establish how these
factors affect the similarity of subjects' stage-two play, in terms of
the concurrent eye-tracking data, to their stage-one involvement
against predictable vs.\ unpredictable computer players.

On the one hand, the proposed method may be related to the previously
mentioned existing studies that classify subjects into groups (types)
based on their recorded gaze or lookup patterns. The types can be
either defined a priori, as in \citeA{Costa-Gomes2006}, or cluster
analysis can be performed, as in \citeA{Polonio2015}, and the obtained
clusters can then be associated with different types (e.g., the
level-2 type in the cognitive hierarchy typology of
\citeNP{Camerer01082004}) based on their characteristics.

On the other hand, the difference from these studies is that here the
types are not derived from existing models, but defined indirectly via
similarity to the visual processing of other, related tasks by many
different subjects, established by virtue of machine learning pattern
recognition analysis of their eye-data. Thus, the present approach is
more model-free in this sense, and of more practical use in some
circumstances, specifically when it is difficult to specify the
eye-movement patterns that would distinguish one mode of
decision-making from others. In the context of the present example, we
could characterize the underlying rationale as: ``We do not know what
patterns in the eye-tracking data would reflect predicting the
strategic action of the other human player. However, we do expect that
these patterns should be more similar to ones accompanying play
against a predictable computer player, than to ones accompanying play
against a computer player choosing its actions at random.''

At this point, it should be stressed that we do not see the proposed method as
a replacement for scan-path analysis, but rather as an approach that might
incorporate it (and many other techniques). For instance, suppose we used
tasks with more than two AOIs, allowing for potentially meaningful complex
scanpaths. One could then use a machine learning algorithm appropriate for
sequential scan-path data (e.g.,\ a hidden Markov model) to classify a scanpath
recorded in a given Stage Two trial as similar to playing against a
predictable or unpredictable computer player at Stage One. The reason why we
used simple tasks, and only three predictor variables, was mainly to
demonstrate the first use of the method in a relatively straightforward context.

We should also consider the potential susceptibility of the proposed
approach to p-hacking, specifically, to manipulating the set of
predictors and data analysis techniques until desired results are
obtained. In particular, a useful feature of our method is that the
accuracy of distinguishing between the different types of proxy choice
trials (at Stage One) can serve as a relatively objective standard by
which alternative ways of analyzing the data can be compared and the
optimal one selected and motivated. In our relatively simple case of
short tasks with two AOIs, the set of possible predictors is quite
narrow and the differences between Stage Two tasks are clear-cut and
intuitive (as further discussed below). Hence, we restricted to
comparing the accuracy of a neural network and a canonical logistic
regression. However, in more nuanced situations researchers can use
the proxy tasks to compare a wider range of predictors and techniques
before choosing the combination to use for analyzing the original
tasks. While the specification of the proxy tasks themselves is still
a relatively arbitrary aspect of the design, it has to be done prior
to collecting the data, thereby leaving less room for manipulation.
In contrast, the traditional approach of a priori specifying the eye
patterns that should occur if a given model was accurate leaves plenty
of space for hypothesizing after the results are known (``HARKing'').

At the same time, a caveat associated with the proposed method is that
one should be careful when interpreting the results, and not forget
that subjects' thought processes are not elicited directly. To
illustrate, in our example we found that similarity to predictable
task is higher in the matching-pennies game than in the coordination
game, even though the former is a classic example of a `competitive'
constant-sum game in the sense of \citeA{shelling}, in which it is
essential to conceal one's actions from the counterpart (consistent
with this, its only Nash Equilibrium is in mixed-strategies). In
contrast, in the coordination game it is essential to be as
predictable as possible.

The observed tendency could therefore be seen as evidence against the
classic interpretation of the mixed-strategy Nash Equilibrium (MSE)
concept due to von Neumann and Morgenstern, according to which in
competitive games with unique MSE players use randomization to prevent
rivals from predicting their actions.  Here, we see that subjects play
the matching-pennies game as if they were up against a highly
predictable computer player.

Nevertheless, to underline the point about the need for careful
interpretation, one could counterargue that what is really captured by
``similarity to predictable task'' is the overall complexity of the
task. Thus, matching-pennies resembles playing a predictable computer
player merely because both involve a higher degree of working out
the correct solution compared with the coordination game and playing
an unpredictable computer player respectively. This illustrates the
need for a careful design of the proxy (Stage One) tasks on which the
machine-learning classifier is to train itself, so that, if possible,
they do not differ in aspects other than the one subject to
investigation.

In the present case, the interpretation of the results can be
clarified by examining the impact of the other factor, namely the
availability of feedback, which we find to increase similarity to
predictable task. If the observed differences were merely due to
changes in task complexity, then they should diminish over time,
particularly when feedback is available, allowing subjects to better
learn the task. However, we see from Figure \ref{figTemp}\ that the
differences in similarity to predictable task across all four
conditions remain stable over the course of Stage Two of the
experiment.

What is more, the first of our behavioral results in Section
\ref{behRes} is robust to restricting to trials closer to the end of
the study. Thus, it appears that in the matching-pennies game subjects
persist in believing that they can use past feedback to predict the
rival's present choices. This happens even as they find themselves to
be repeatedly unsuccessful in this endeavor (the margin between the
winning and losing player is $5\%,$ the same as in the absence of
feedback). It appears that each subject thinks she can predict and
outsmart the other by using the feedback information, but ultimately
they cancel each other out and end up in the same situation as in the
absence of feedback. This is particularly interesting, given that the
two numbers subject to choice change randomly (and are re-positioned)
in each round, whilst no number appears more than once. This suggests
that people will attempt to make use of even very limited feedback
information, and persist in doing so even as they repeatedly fail to
make the right strategic choices as a result.

Similarity to predictable task also increases with feedback in the
coordination game. In contrast with matching-pennies, in this case the
apparent increase in predictability or confidence is justified by
observable outcomes of the game. Specifically, the coordination rate
is at randomness level (50\%)) in the absence of feedback, but
significantly higher (60\%) when feedback becomes available.

These findings can also add to the debate on the discrepancy between
the behavior of amateur subjects in the lab and that of professionals
in the field. It appears that the former do not exhibit the serial
independence of actions in repeated play, which is a prerequisite for
MSE \cite{rosenthal90,Mookherjee1994,Scroggin2007,Roth1998}. In
contrast, studies such as \citeA{Walker2001}, \citeA{Hsu2007},
\citeA{Chiappori2002} and \citeA{Palacios-Huerta2003} show that the
behavior of professional players in the field, such as penalty-kick
takers in soccer, can be consistent with the MSE, though the same
players no longer exhibit serial independence when tested in a lab
setting (\citeNP{Wooders2010,STEVEND.LEVITTJOHNA.LIST2010}).

Our results suggest that the specific aspect of laboratory testing
which might be causing this change is the fact that feedback about the
counterpart's past actions is readily available to both parties (in
contrast, a soccer player in the field may for instance find it hard
to recall the specific part of her penalty-kick history which could
have been observed by the rival keeper). Even though information about
past actions that we provide is limited by between-trial variation in
available choices, it still turns out to have a significant effect on
the players' processing of the task. In fact, from our comparison of
the coordination and matching-pennies games, it seems that the fact
that subjects do not share a common interest, and only a
mixed-strategy equilibrium exists, makes them not less, but more
likely to try to use any information they have to predict a rival's
likely strategy, and possibly play a best-response to it rather than
randomize. In the words of \citeA{Rubinstein1991}, ``We prefer to be
able to point to a reason for each action we take. Outside of Las
Vegas we do not spin roulettes.''

\section{Conclusion}

This paper demonstrated a new way of using eye-tracking to study strategic
decisions. The behavior observed in experiments can often be explained by
different modes of decision making. Although eye-tracking often provides an
answer to the problem, existing studies often rely on an a priori
specification of the eye-movement patterns that should accompany a given
decision-making strategy, in order to compare the observed and assumed
patterns. Such an a priori specification is usually difficult and not based on
objective criteria, while the assumed patterns do not allow for individual
differences between decision-makers or for the often difficult to anticipate
visual attention biases.

To address these problems, we propose to expose subjects to variants of the
investigated strategic task that should induce a particular manner of
processing, and then check if the original and modified tasks are similar in
terms of the accompanying process-tracing data. To measure the similarity of
tasks, we propose to analyze the eye-data using model-free machine-learning
pattern recognition techniques like neural networks. This flexible approach
can capture any unforeseen relationships in the data, while requiring less
theorizing on behalf of the researcher.

We present an example application of the method. Subjects first interact with
a predictable or unpredictable computer player, before playing a strategic
game against another human. We manipulate the features of the latter task to
see how they affect the similarity of the accompanying process-tracing data to
that accompanying play against a predictable vs.\ unpredictable computer
player. Our findings have implications for the interpretation of the
mixed-strategy Nash Equilibrium concept, and indicate that the method may be
applied to other similar problems.

\bibliographystyle{apacite}
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\end{document}
