Network approaches for expert decisions in sports

Human Movement Science 31 (2012) 318–333

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Human Movement Science

journal homepage: www.elsevier .com/locate/humov

Network approaches for expert decisions in sports

Andreas Glöckner a,⇑, Thomas Heinen b, Joseph G. Johnson c, Markus Raab b

a Max Planck Institute for Research on Collective Goods, Bonn, Germanyb German Sport University Cologne, Institute of Psychology, Cologne, Germanyc Miami University, Department of Psychology, Oxford, OH, USA

a r t i c l e i n f o a b s t r a c t

Article history:Available online 27 July 2011

Keywords:Parallel constraint satisfactionAccumulator modelHandballDecision makingExpertise

0167-9457/$ - see front matter � 2011 Elsevier B.Vdoi:10.1016/j.humov.2010.11.002

⇑ Corresponding author. Address: Max Planck InsBonn, Germany. Tel.: +49 (0)2 28 9 14 16 857; fax

E-mail address: [email protected] (A. Glöc

This paper focuses on a model comparison to explain choices basedon gaze behavior via simulation procedures. We tested two classesof models, a parallel constraint satisfaction (PCS) artificial neuronalnetwork model and an accumulator model in a handball decision-making task from a lab experiment. Both models predict action inan option-generation task in which options can be chosen from theperspective of a playmaker in handball (i.e., passing to anotherplayer or shooting at the goal). Model simulations are based on adataset of generated options together with gaze behavior measure-ments from 74 expert handball players for 22 pieces of video foot-age. We implemented both classes of models as deterministic vs.probabilistic models including and excluding fitted parameters.Results indicated that both classes of models can fit and predictparticipants’ initially generated options based on gaze behaviordata, and that overall, the classes of models performed aboutequally well. Early fixations were thereby particularly predictivefor choices. We conclude that the analyses of complex environ-ments via network approaches can be successfully applied to thefield of experts’ decision making in sports and provide perspectivesfor further theoretical developments.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

At first glance, making decisions in sport seems to be a complex task, because they are generallymade under high pressure, limited time, and restricted resources. For the experienced athlete, often

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A. Glöckner et al. / Human Movement Science 31 (2012) 318–333 319

the best and the most quickly recognized option in a given situation comes to mind and subsequentbehavior is aligned with this option (Brisson, 2003). Such apparently intuitive decisions by experts,based on current information intake, are not outside the realm of mathematical modeling and canbe formalized by different models of varying complexity (Johnson, 2006; see also Glöckner &Witteman, 2010). The goal of the present research is to compare two classes of models that allowpredicting action from attention in complex sport decisions such as the playmakers’ choice to passto different team members or to shoot at the goal.

Models of different complexity exist to explain how attention is associated with mental processesand behavior in a wide range of domains, including everyday activities (Rayner, 1998), consumerchoice (Reisen, Hoffrage, & Mast, 2008), and sports (Memmert & Perl, 2009; Williams, Janelle, &Davids, 2004). These models, applied to many fields, differ in their purposes, such as explaining theprocesses associated with reading, the influence of choice strategies on the saccade sequence, the sim-ulation of creative behavior in ambiguous situations, or explaining whether experts and novices usedifferent gaze strategies anticipating a tennis return. Here we will follow suit with the assumptionthat looking, in terms of directing gaze and shifting attention (cf., Vickers, 2007), towards informationrich areas drives and reflects preferences and importance (Shimojo, Simion, Shimojo, & Scheier, 2003)and thus we will use gaze behavior to predict choices in complex decisions in sports.

More specifically, we will apply two popular general classes of models in this field called ParallelConstraint Satisfaction (PCS) and accumulator models. Both classes can be expressed as networks(Busemeyer & Johnson, 2004) and have been used previously in many domains, and especially insports (Johnson, 2006; Raab, 2001). Both types of models are prepared to predict action, based onattention, often measured in terms of eye-tracking sequences of fixations and saccades, using key-board responses (Thomas & Lleras, 2007).

The first model we used was a specific implementation of a PCS model for decision making (Betsch& Glöckner, 2010; Glöckner & Betsch, 2008a). PCS accounts use a spreading activation mechanism tomodel the construction of coherent interpretations based on the overall constellation of availableinformation (Gestalten; see Read, Vanman, & Miller, 1997). These mechanisms rely on parallel process-ing, are extremely powerful, operate to a large degree without conscious awareness and have beensuggested as core processes underlying intuition (Betsch & Glöckner, 2010; Glöckner & Betsch,2008a). Naturally, PCS models can be used to model highly complex processes of perception(McClelland & Rumelhart, 1981), information chunking in social perception (Read & Miller, 1998),and coherence construction and interpretation (Thagard, 1989). However, PCS networks have alsobeen successfully applied to decision tasks similar to the ones considered in the current work, suchas probabilistic inferences (Glöckner, Betsch, & Schindler, 2010; Glöckner & Bröder, in press; Glöckner& Hochman, in press; Hochman, Ayal, & Glöckner, 2010; see also Glöckner & Betsch, 2008b; Glöckner,2009), legal intuition in complex tasks (e.g., Glöckner & Ebert, in press; see also Holyoak & Simon,1999; Simon, 2004), preferential decisions (Simon, Krawczyk, & Holyoak, 2004), selection of plans(Thagard & Millgram, 1995), and risky choices (Glöckner & Herbold, 2011; see also Glöckner & Betsch,2008c). Most directly, Raab (2001) has applied such a coherence-constructing network model to ballallocation decisions in basketball. In the current work, we use the amount of gaze-time directed to cer-tain options as indicator for the information speaking for selecting this option and use this informationas a cue in the previously developed PCS model for decision making (Glöckner & Betsch, 2008a).

The second model class we used was the accumulator models (see Ratcliff & Smith, 2004, for anoverview and comparison). These models have also been very successful in accounting for robusttrends across a variety of domains, including early applications to basic processes such as perceptualdiscrimination (Laming, 1968; Link & Heath, 1975) to more recent work on higher-order cognitive pro-cesses such as preferential choice (see Busemeyer & Johnson, 2004, for a survey). These models havealso been cast in neural network architectures (Busemeyer, Jessup, Johnson, & Townsend, 2006; Roe,Busemeyer, & Townsend, 2001) and have been validated in neuroscientific studies (see Gold &Shadlen, 2000; and Ratcliff, Cherian, & Segraves, 2003 for two among many examples). Generally,accumulator models suggest that one or more ‘‘counters’’ keep track of the relative preference for eachoption in a set of options, where the position of the counter(s) is updated by momentary fluctuationsin attended information. Our specific model in this class is based on the simplified assumption thatgaze to a specific region of interest provides evidence in favor of options in the corresponding region.

320 A. Glöckner et al. / Human Movement Science 31 (2012) 318–333

We state that, as gaze shifts over time, evidence accumulates for options in the corresponding region—i.e., gaze in a particular region increments the preference ‘‘counter’’ for the options in that region.More specifically, our model implies a direct and positive relationship between gaze duration and evi-dence for a specific option.

Thus, our conceptual approach suggests that, when people are faced with an ill-defined decisiontask, their attention (in terms of directing gaze to a specific region), during information assessment,provides a strong indication of their corresponding behavior, which can be formalized with differentmodels of varying complexity. There is concluding evidence for the existence of such a direct and po-sitive relationship in team handball. Raab and Johnson (2007), for instance, examined expertise-baseddifferences in visual search and option-generation strategies. Their results show interactions betweengaze behavior and option generation strategies, as well as interactions between the two factors andexpertise of handball players. The authors conclude that option generation strategies are strongly af-fected by gaze strategies or information search in a complex situation. Further evidence comes fromcurrent studies in cognitive psychology (Glaholt & Reingold, 2009), risky choices (Glöckner & Herbold,2011), and other decisions (Armel, Beaumel, & Rangel, 2008; Innocenti, Rufa, & Semmoloni, 2010;Shimojo et al., 2003), demonstrating a clear gaze bias effect toward preferred options.

In this paper, we applied both general classes of PCS and accumulator models to a playmaker deci-sion task. For the PCS model, we used variants of a general decision model for probabilistic inferences(Glöckner & Betsch, 2008a) for predicting choices in sports. For the accumulator model, we relied on apreviously established simple linear accumulator model for sports decisions (Johnson & Raab, 2003;Raab & Johnson, 2007). With both general classes of models, we simulated data from a previous studyon passing decisions by handball players. Because previous applications of PCS often use a determin-istic outcome (e.g., Glöckner & Betsch, 2008; Glöckner & Bröder, in press) and accumulator models of-ten use a probabilistic outcome (e.g., Johnson, 2006), we will model probabilistic and a deterministicversions of both model classes in order to allow direct comparisons.

We predicted that both classes of models can model participants’ choices based on sequences offixations. As this was the first attempt to compare models of these two classes in complex environ-ments in sports, we were open to the question which model would fit and predict best. One rationalewas also to test different implementations of these models with and without free parameters.

The implementation of accumulator and PCS models used in this study implies in the first instancethat fixations have the same meaning over the whole inspection period. Because it might be criticizedthat this assumption does not hold particularly in complex playmaker decisions, we additionally ap-plied both classes of models to analyze the development of fixations over time in order to capture thedynamics of gaze behavior across time to predict participant’s choices.

2. Method

2.1. Description of the dataset

We used gaze behavior and option-generation data of 74 participants (handball players) who wererecruited from the state training center and clubs in north Germany and were successful in theirrespective divisions. They formed a sub-sample (wave 1 and wave 2) from the study of Raab andJohnson (2007). Participants provided informed consent before participating in the study, whichwas carried out according to the ethical guidelines and with the approval of the University ofFlensburg.

The participants’ task was to list possible options in realistic game situations in handball (see Raab& Johnson, 2007, for methodological details and descriptive results). On each of the 11 trials at two testintervals, a video clip was frozen after approximately 10 s of play development. At that moment theball was in the possession of one of the players (the playmaker). The frozen frame was held for 6 sduring which participants were asked to generate all plausible courses of action that the playmakershould consider. The response data of participants was reduced to a discrete value and was codedby spatial location, such that each response could be classified as an action directed to the left third,middle third, or right third of the playing field (again, see Raab & Johnson, 2007, for rationale).


During the decision-task, we collected eye-tracking data by means of a video-based head-mountedinfrared eye tracker from BioMed Jena (2003, version 2.0). These gaze behavior data were classifiedinto three regions of interest. To do this, the eye tracker was calibrated to a large video screen for eachtrial of the decision-making video test, providing a realistic visual situation from the perspective of thebackcourt facing the goal, near the position of the playmaker. The position of the eyeball was trackedin real time (25 Hz) and the gaze direction was calculated with regard to the video screen. The eye-tracking data were reduced to a sequence of fixation regions (left third, middle third, and right third)and their respective durations.

Basically, the data consists of information on eye-fixation durations towards the left, the middle, orthe right side of the playing field and information on participants’ initial passing option to the left, themiddle, or the right side of the field, which should be predicted by our models. In the following, forsimplicity, we refer to this initially generated passing option also as the playmakers’ choice.

2.2. Model Descriptions

In the PCS as well as in the accumulator models, we made the simplifying assumption that fixationto a specific area of the visual field indicated evidence for the options in the corresponding portion ofthe field. Specifically, the total duration of fixation to any region was used as an indicator of thestrength or amount of evidence for the respective part of the field. We simulated eight variants ofmodels in total, four versions of accumulator models and four versions of PCS models. Half of themwere deterministic model implementations (i.e., the model predicts one specific choice); the other halfwere probabilistic model implementations (i.e., the models predict the likelihood for choosing thefavored option). For half of the simulations, we do not fit any parameter; for the other half we do(although to a different degree; see below). So the total modeling design is a 2 (PCS vs. accumulatormodels) � 2 (probabilistic vs. deterministic) � 2 (parameter fitting yes vs. no) (Table 1).

2.2.1. Parallel constraint satisfaction (PCS) modelThe PCS network model for decision making by Glöckner and Betsch (2008a) was adapted to take

into account fixations as cues (Fig. 1). The model consists of a layer of option (i.e., pass options) andcue nodes (i.e., cues for passing to the respective side). Connections between option and cue nodesrepresent cue values, that is, the total amount of information in support of selecting the respective op-tion. Connections between the general weighting node and the cue nodes represent the cue weighting,that is, the weighting or importance of cues.

Although this network structure is the core of the PCS model by Glöckner and Betsch (2008a), theoriginal model additionally describes the interaction between automatic-intuitive spreading activa-tion mechanisms and deliberate processes. The latter are assumed to be activated if the network doesnot find a sufficiently coherent interpretation and can, for instance, be used for double-checking. Forpragmatic reasons, in this work we consider the network part of the model only.

PCS models are simulated by setting up the constraints and links among units, and then allowingthe model to converge to a stable pattern of activation after some number of iterations. Activation ofoption nodes represents their evaluation; that is, activation of the node for Option A is interpreted assupport for the hypothesis that ‘‘passing to Option A is good.’’ The activation of cue nodes representssubjective trust in the cue. In other words, higher activation corresponds to more faith or ‘‘weight’’given to the corresponding cue, which is referred to as cue weighting. To start with, activation spreadsfrom the general weighting node (i.e., the activation/driver node) in the network to the cue nodes and

Table 1Overview of Simulated Models.

Without parameter fitting With parameter fitting

Deterministic Probabilistic Deterministic Probabilistic

PCS Basic PCSdet Basic PCSprob Fitted PCSdet Fitted PCSprob

Accumulator Summation Modeldet Summation Modelprob Individual Modeldet Individual Modelprob

Pass right ao1

Pass middle ao2

Pass left ao3

Cues right ac1

Cues middleac2

Cues left ac3

general weight

wv1 wv2wv3

wc1-o1wc3-o3wc2-o2

wo1-o2 wo2-o3

wo1-o3

cue values

cue weights

Fig. 1. Parallel Constraint Satisfaction model for passing decisions.


then onto the option nodes. All links are bidirectional, such that increased activation in a node pro-duces an increase of activation for any connected node with a positive link, and a decrease in activa-tion for those connected nodes with negative links. Interconnections between options are allinhibitory, suggesting a competitive structure where only one option can be selected.

Connections between cues ck ðk 2 f1;2;3gÞ and the general weighting node (i.e., wvk) are referred toas cue weights, which represent a priori judgments about the predictive power or trustworthiness ofcues from the different regions. We set these to a constant in one version of the model and fit it inanother. Connections between option nodes ol ðl 2 f1;2;3gÞ and cue nodes ck represent the cue value,that is, the amount of evidence in region k in favor of selecting option l. We calculated the respectivelink weights wck-ol between cues and options (i.e., cue values) based on aggregated fixation durationsto the respective part of the field by dividing the absolute aggregated fixations time (in s) by 10. Due tothe fact that the overall fixation time was about 6 s, this scales cue weights down to a conventionallyused range (i.e., they add up to .6).

Once the general weighting or ‘‘driver’’ node is activated, activation spreads through the network initerative steps (details below). This continues until a stable level of activation (consistency maximiza-tion) is reached which is operationalized by a threshold for changes in activation. Once this state isachieved, the option with the highest activation is chosen. Activation in the network is spread usingthe standard iterative activation function proposed by McClelland and Rumelhart (1981; see also Read& Miller, 1998):

aiðt þ 1Þ ¼ aiðtÞð1� decayÞ þif inputi < 0 inputiðaiðtÞ � floorÞif inputi P 0 inputiðceiling � aiðtÞÞ

�ð1Þ

with

inputiðtÞ ¼X

j¼1!n

wijajðtÞ ð2Þ

The parameter ai(t) represents the activation of the node i at iteration t. The parameters floor andceiling stand for the minimum and maximum possible activation (in our model, set to a constant valueof -1 and + 1). Inputi(t) is the amount activation node i receives at iteration t, which is computed bysumming up all products of activations and connection weights wij for node i. Decay is a constantdecay parameter. According to Eq. (1), the activation of all nodes is repeatedly updated in each


time-step t + 1, on the basis of the incoming activation from other nodes (i.e., input). This incomingactivation is weighted by the difference in activation of the respective node from the maximum(i.e., ceiling) or the minimum (i.e., floor) activation level. Additionally, a proportional decay of previousactivation in each iteration step is assumed. In simple networks as the ones considered in this paper,the activation of nodes approaches stable levels most often after less than 500 iterations. Activationchanges become very small and stability is considered to be reached after activation changes over sev-eral iterations are below a certain threshold. Specifically, it is determined whether changes in theoverall consistency (Energy) in the network are below a certain threshold over several iterations. En-ergy is calculated by:

1 In cone con

2 Notthe othvalid ob

EnergyðtÞ ¼ �X

i

Xj

wijaiaj ð3Þ

The iterative algorithm minimizes energy and maximizes consistency under parallel considerationof all constraints.1 From a more abstract perspective, this can be understood as the best explanation (orinterpretation) given the overall constellation of hypotheses and evidence.

PCS models for the playmaker decision can be constructed on different levels of abstraction. Similarto PCS models proposed for social perception (Read & Miller, 1998), an overall PCS model with multi-ple layers of nodes with increasingly complex meanings would be possible. For the sake of simplicityand considering the limited amount of data, we used the PCS model introduced in Fig. 1 only.

Although the PCS model is rather simple, it basically allows for identifying individual parametersconcerning the (a priori) weighting of cue evidence wvk. However, in the data set the number of dif-ferent scenarios is at a maximum of 22,2 and therefore we have restricted data for reliable individualdata fitting (cf. Geman, Bienenstock, & Doursat, 1992). Multiple observations per scenario might reducethe danger of over-fitting. We therefore decided to fit PCS parameters over the total data set of individualobservations (i.e., including data from all subjects) instead of fitting parameters individually for each per-son. This, of course, makes the model hard to compare to the individual accumulator model which usesindividual parameter fitting and is introduced below.

We simulated a basic PCS that used all equal cue weights and parameters from previous simulations(Glöckner et al., 2010). No parameter was fitted for this model; we hence applied this model withoutrelying on any free parameters. We compared this basic PCS model with a fitted PCS in which we fixedthe weight of cue 1 (wv1) and estimated weights for cues 2 and 3 by determining optimal weights forwv2 and wv3 over all participants. The parameters are summarized in Table 2. In contrast to previoussimulations, we used a lower stability criterion for pragmatic reasons to find quicker convergence inparameter optimization simulations.

In the basic PCS simulation, we used an equal weighting standard parameter for all weights:wv1 = wv2 = wv3 = .10. For each choice of each participant, we used the aggregated fixation times toeach region to predict the playmakers’ choice. For the deterministic implementations basic PCSdet

and fitted PCSdet, the option that received the highest activation in the network was used as determin-istic choice prediction of PCS. To evaluate the model, the proportion of predictions in line with play-makers’ choices pcorr was determined for each participant. For probabilistic implementations basicPCSprob and fitted PCSprob, we calculated a choice probability for the favored option based on an expo-nential choice rule that transforms the activation of option nodes into a choice probability (see Appen-dix). For these models, pcorr was determined for each participant as the average predicted probabilityof the chosen option.

In the fitted PCS model simulation, we set wv1 = 0.5 (i.e., the middle of the scale) and used a com-plete-search algorithm to find optimal wv2 and wv3 weights in the parameter space [0.1, 0.2, 0.3 . . .

0.9]. In detail, for each set of parameters, we calculated the proportion of correct choice predictionsfor the training set (even trials) and identified the parameter set that maximized it. A best set was

omplex networks, local instead of global minima of energy might be reached by the algorithm. In simple networks as thesidered here, both usually fall together.e that some missing values were also encountered if passing decisions could not be classified as clearly belonging to one orer category. We a priori excluded two participants from the initial total of 76 participants for which the overall number ofservations was low (subject # 6, 56).

Table 2Model Parameters for PCS Simulations.

Value/Function Comment

Decay .05 Decay parameter for node activation; influences theoverall activation level of the nodes, the higher the valuethe lower the final activation level.

wo1-o2 -.20 Inhibitory connection between options; influences thesize of information distortions (coherence shifts); thestronger the inhibitory connection the stronger thecoherence shifts.

wo1-o3

wo2-o3

wc1-o1 w = (fix_time/10) CUE VALUES: Connection between cues and optionsrepresenting the strength of the evidence.wc2-o2

wc3-o3

wv1 constant or fitted CUE WEIGHTS: Links between general weighting nodeand cues representing a priori cue weighting.wv2

wv3

ceiling/floor 1/-1 Upper and lower limit for cue activations.Stability criterion 10 trials energy change < 10�4 Stopping criterion for the iterative updating algorithm.


identified: wv1 = 0.5, wv2 = 0.8 and wv3 = 0.4, and used to predict choices in the test trials (i.e., odd tri-als). The set of parameters was determined for the probabilistic version fitted PCSdet and the sameparameters were used for fitted PCSprob.

The resulting PCS model and parameters can be interpreted as follows. The aggregated fixationduration, measured by eye-tracking, is used to determine the cue values in the model; this is thestrength of the evidence for the option in the given region (see Table 2). For instance, if half of thefixation time was in the left region, and the remaining half was divided evenly among the middleand right region, then we assumed that there was twice as much evidence for the left options ascompared to the options in the middle or right regions, and the latter evidence would be equal(i.e., wo1-c1/2 = wo2-c2 = wo3-c3). The final (fit) values of the cue weights suggest that evidence in themiddle region was, on average, weighted most (wv2 = 0.8), followed by the left (wv1 = 0.5), and thenby the right region (wv3 = 0.4). Stated differently, there was a bias towards the center in that even ifthere was the same sum of fixation times to all three regions, the center region was more likely tobe chosen (see also the baseline model below).

2.2.2. Accumulator modelsIn order to evaluate the quantitative fit of the model, we compared the PCS models to two

accumulator models in deterministic and probabilistic implementations, respectively. These modelsused the same data (eye-tracking fixations) to predict the same outcome (choices), but in a differ-ent framework. The PCS models used fixation duration in the aggregate to determine the cuevalue—how much evidence speaks for passing to region k? The PCS models also fit the weightingof each cue as free parameters (or assume that weights are all equal). Weights indicate howstrongly one should consider cue information (fixations) contained in region k in the decision. An-other interpretation, formalized below, is to use the fixation duration to ‘‘hard-code’’ the relativeattention to each region instead, under the assumption that visual attention to a region producesconsideration of, or ‘‘weight to’’, the evidence in that region. Hence, the proportion of weight to aregion is simply proportional to the fixation time spent in the region. Then, in the accumulatormodel, the link between attention and evidence was assumed to be holistic, which means that,given that attention is in region k, attention is complete to every single option in that region.For instance, if gaze shifts to the right for 800 ms and then to the middle for 1600 ms, this modelwould expect more evidence for options in the middle. Note that the same result is reached in PCSby determining cue values from gaze durations (see Table 2).

We used an additive model, implying a direct and positive relationship between gaze duration andevidence for a specific option. Thus, spreading activation in an accumulator network and spreading


activation in a PCS network are conceptually different, such that in the accumulator model, gazebehavior is accumulated dynamically, whereas in the PCS network it is aggregated a priori to serveas a static cue for inference. From this point of view, the accumulator model suggests evidence Efor a specific region r after a specific stream of n gaze fixations can be formalized as follows:

3 Ek(n

ErðnÞ ¼ b � Erðn� 1Þ þ trðnÞ ð4Þ

As can be seen from Eq. (4), Er(n) is equal to a weighted sum of the previous evidence Er(n-1) andthe current fixation duration to the corresponding region tr(n). The parameter b can be used to incor-porate growth or decay of evidence for a specific region. If b departs from zero, Er(n) reflects either pri-macy (b > 1) or recency (b < 1) effects. If the parameter b is equal to 1, then evidence is equal to thetotal fixation duration to the region. b = 0 suggests extreme decay such that evidence is based onlyon the most recent fixation. We used an individual’s stream of gaze fixations and durations on a giventrial to compute evidence for all three regions (left, middle, right), reflecting the potential options inthese areas. In the next step, we used a ratio choice rule to compute the probability that the choice(i.e., the initially generated option for the trial) would be generated within each region:3

Pr½Initial option in region k� ¼ Ekðn�ÞPErðn�Þ

ð5Þ

In the first step, we incorporated no primacy or recency effects in our model (b = 1 for each indi-vidual). We refer to this model as a summation model. In the second step, we adjusted the parameterb to maximize the average probability of a correct prediction across trials for each participant. We re-fer to this model as the individual model (b adjusted).

For the probabilistic implementations summation modelprob and the individual modelprob, we calcu-lated pcorr as the probability (Eq. (5)) that the model correctly predicted the region on each trial andthen averaged across trials, for each individual. For the deterministic implementations summationmodeldet and the individual modeldet, the prediction was that the option with the highest probabilityis chosen. We calculated pcorr as the proportion of correct predictions per person.

Baseline model. To evaluate further the suggested models, we created two simple baseline mod-els as a benchmark. The first model was a deterministic baseline model. Most choices were for themiddle region; hence, it had the highest base-rate probability. So the best prediction without datawould be to predict choices for the middle region. For each person we calculated the proportion ofcorrect predictions with this simple rule as baseline. The second baseline model had a probabilisticstructure. It referred solely to participants’ global proportion of choices across trials and for differ-ent regions. For example, if 70% of a participant’s choices across trials were for the left regioncompared to other regions, then the baseline model would predict choices for the left region of.70 for every trial.

2.3. Model comparison procedure

In order to correct for overfitting, we used a cross-validation procedure. Models with free param-eters were fitted to half of the data according to an even–odds method and the resulting parameterswere used to predict the other half of the data. Hence, the comparison of pcorr between models in thecross-predicted test trials is the most informative one. The overall fit is, in contrast, only partiallyinformative and the comparison between fitting and test trials provides information concerning theprevalence of overfitting. Note that for all models (except for baseline) predictions are made on anindividual level, that is, predictions are derived for each choice situation and each participant sepa-rately taking into account persons’ individual fixation data. Note also that the fitted PCS models douse two free parameters to fit the total sample, whereas the fitted individual model uses one param-eter per person (which means 74 parameters for the total dataset). The fitted models are thereforeonly partially comparable.

⁄) refers to evidence for (fixations to) the chosen option in region k; Er(n⁄) refers to evidence for all options.


3. Results

In order to compare the models, we calculated the overall prediction performance of choice fromgaze behavior for each of the models in both the deterministic and the probabilistic version. In a sec-ond step, we performed cross-validation tests on all models, incorporating only the even trials(n = 485; i.e., participants X trials) for model fitting, and predicting the odd trials from the correspond-ing model equations (n = 529). In a third step, we conducted an additional analysis on the develop-ment of fixations over time in order to capture the dynamics of gaze behavior across time.

3.1. Performance in prediction

The performance of the two PCS models and the two accumulation models are shown along withthe performance baseline in Table 3. The baseline model achieved a mean accuracy across participantsof .436, which is better than chance performance of one-third (t-test against l = .33: t(73) = 7.13,p < .001). Accuracy in the even trials was higher than in the odd trials, indicating that there were fewerchoices for the middle region in the odd trials.

Overall, all considered models performed better than chance and better than the baseline in theirpredictions (test trials). The deterministic implementations of the models predicted better than theirprobabilistic counterparts: by about 4 to 5%. The performance differences in predictions between thedifferent types of models (PCS vs. accumulator models) and fitted vs. unfitted implementation weresmall. The performance of the deterministic models was essentially equal with differences of less thanhalf a percent. The individual modeldet was the best model. The almost equal performance is therebydue to the extremely high overlap of predictions instead of aggregation. Basic PCSdet and summationmodeldet, for example, make the same predictions for all but three cases in which the PCS model didnot converge.

Concerning probabilistic implementations of the models, the basic PCS was the best model in pre-diction, which was about 2% better than the worst model, the individual model. The individual modelseemed to suffer from overfitting, in that there was a particularly large difference between fitting andprediction performance. The low number of observations that were fitted with one parameter seemedto have induced the problem. Therefore, the data do not speak against the argument that individual

Table 3Simulation Results.

Proportion Correct (pcorr)

Baselinemodel

Basic PCS (equalweighting)

Fitted PCS(fitted weighting)

Summationmodel (b = 1)

Individual model(b adjusted)

Probabilistic ModelsFitting/Training (even trials) .457 (.013) .523 (.017) .514 (.016) .512 (.016) .605 (.017)Prediction/Test (odd trials) .320 (.013) .509 (.014) .498 (.013) .497 (.012) .483 (.022)Overall .388 (.010) .516 (.013) .506 (.011) .505 (.011) .570 (.013)

Deterministic ModelsFitting/Training (even trials) .475 (.021) .596 (.026) .597 (.024) .597 (.026) .691 (.020)Prediction/Test (odd trials) .404 (.021) .545 (.022) .548 (.023) .549 (.022) .549 (.027)Overall .436 (.015) .570 (.018) .572 (.019) .572 (.018) .666 (.015)

Notes: The baseline (deterministic) model predicts choices based on base-rates only. Because most choices were for the middleregion, it predicts the middle region for all choices. The basic PCS model predicts choices from cue values that are estimatedfrom fixation durations to the respective areas using equal weights for all fixations. The fitted PCS model uses optimizedweighting of cue values. The data is best explained by a middle-region choice bias in that choices for the middle region are morelikely for the same fixation times to this region (cue weights: wv1/right = 0.5, wv2/middle = 0.8 and wv3/left = 0.4; weights were fittedfor fitted PCSdet and the same weights were used for the fitted PCSprob). The summation model uses Eq. (4) with b = 1, and theindividual model uses Eq. (4) with most accurate b value for each individual. Fitting/training refers to the even trials that wereused for data fitting. Prediction/Test refers to the models prediction performance in the odd trials which are most informativefor model comparisons (bold). Overall values refer to all trials. Standard errors are provided in parentheses. We used data fromN = 74 experts including n = 485 even trials and n = 529 odd trials.


differences in weighting play a role as suggested by the individual model. However, more observationsseem to be necessary to fit these parameters.

3.2. Development of fixations over time

The implementation of accumulation and PCS models used in this study implies that fixations havethe same meaning over the 6 s inspection period. PCS models aggregates over all fixations a priori andthe accumulation models add them up to overall preference values after weighting them. It might becriticized that this simplifying assumption does not hold particularly in complex playmaker decisions.According to an initial-scanning hypothesis, one could assume that in order to maximize informationinput initial fixations are used for scanning all options. Hence, initial fixations might be unrelated topreferences for specific option. Initial scanning would lead to a flat distribution of fixations over alloptions in the first seconds. After a preliminarily favored option has been identified, fixations mightshow a J-shaped distribution in that people strongly focus on the favored option, which then mightrepresent preferences.

Alternatively, it might be possible that cues in the scene direct fixations automatically and instantlyto the favored option without a necessity for scanning all options. One might therefore also predict theexact opposite, namely a relatively early focusing on the favored options, which has also been found inrecent studies (Innocenti et al., 2010). According to the PCS model by Glöckner and Betsch (2008a) andother default-interventionist models for dual processing (see Evans, 2008, for an overview), it mightbe argued, furthermore, that these quick automatic processes are followed by deliberate checking pro-cesses which include checking and comparing with non-favored options. Hence, the second predictionfrom this perspective would be that there are more fixations to the non-preferred options in the laterpart as compared to the earlier part of the inspection period.

To investigate this issue, we divided fixations into the categories ‘‘early’’, ‘‘middle’’ and ‘‘late’’,according to whether fixations started in the first, middle or last two seconds of the inspection period.We find that the proportion of fixation time towards the favored option decreased over time (Fig. 2,left). A regression of the proportion-of-fixation-time-index on a linear time-period variable (with val-ues 1, 2, 3 for ‘‘early’’, ‘‘middle’’ and ‘‘late’’ fixations) correcting for clusters in observations and forfixed effects of trial and training session due to repeated measurement (Rogers, 1993) turned out sig-nificant, b = -.045, t = 7.30, p < .0001.

Furthermore, if we aggregate fixation times separately for the three time periods, the predictiveperformance of, for example, the basic PCSdet model is highest when using data from the first 2 sand it decreases over time (Fig. 2, right). As mentioned above, the other considered models makeessentially the same predictions and therefore show the same development. Note that consideringthe fixations from the first 2 s only, the predictions of the model would be almost 10% better thanwhen using fixations from the last two seconds and even 2% better than when considering allfixations.

Both findings support the prediction of PCS and other default-interventionist models that there is aquick intuitive reaction that might later on be checked by further inspections of other areas. The dataspeak against the initial-scanning hypothesis and later fixation to the favored region. The resultsmight also indicate that playmakers are able to identify the favored option very quickly and thatchoices might not be due to a summation of preference values over the whole inspection time.

4. Discussion

The goal of the present research was to compare two classes of models with varying complexityand formalization to predict individuals’ courses of actions from gaze behavior data, representingplayers’ visual attention. We were especially interested in the extent to which models with varyingcomplexity and in different implementations can predict specific choices. We approached this goalby using eye-tracking and option-generation data from expert handball players and modeled choices(in terms of courses of action) from gaze behavior (in terms of fixation sequences) in a realisticdecision-task.

.46

.48

.5.5

2.5

4.5

6p

early middle latestarting time of fixation

Proportion of Fixation Time to the Favored Option

.46

.48

.5.5

2.5

4.5

6p(

corr)

early middle latestarting time of fixation

Proportion correct for PCS

Fig. 2. Development of the proportion of fixations to the favored option over time (left) and proportion of correct predictions ofthe basic PCSdet model over time. Each category on the x-axis represents fixations from consecutive 2 s time-frames.


4.1. Performance in prediction

We found that the models we calculated were better than chance level and a baseline model in pre-dicting option-generation from eye gaze. For deterministic implementations of PCS and accumulationmodels, we found essentially equal performance in prediction between all models with a slight advan-tage for the individual modeldet. Interestingly, this similarity in performance was not due to aggrega-tion, but caused by the fact that the models essentially make the same predictions for almost alldecision tasks for which they can be calculated. The predictive accuracy for probabilistic modelswas in general 4 to 5% lower than for deterministic models and showed some differences betweenmodels. Considering probabilistic models only, the basic PCSprob model performed best in predictionand was 2% better than the worst-performing individual modelprob. We thereby took the problem ofoverfitting seriously and followed the cross-validation procedure (cf. Mosier, 1951; see Browne,2000, for discussion), instead of taking into account the overall accuracy. Performance of modelswas evaluated on the basis of performance in the predicted (test) trials only.

In our small sample of observations, fitting additional model parameters did not increase predictiveperformance of the models substantially. The fitted PCS did not outperform the basic PCS, nor did theindividual accumulation model outperform the summation model in predictions. Hence, the highercomplexity did not lead to higher performance. This finding, should, however, only be interpreted withcaution. It might be partially due to the fact that the fitting sample was too small.

Taking a broader perspective, the reasonable fit of PCS models as well as accumulation models pro-vide further evidence supporting the notion that the attention given to certain cues that are measuredby perceptual information acquisition can predict subsequent choice behavior. Theoretically, it isinteresting to compare the nature of the explanations for the different models. The PCS models incor-porated visual attention (fixation duration) to represent cue values, that is, how much evidence sup-ports the option in that region, and fits the weight of each cue as free parameters (or sets it to aconstant). In contrast, the evidence accumulation models used visual attention to determine howmuch relative weight was given to each region, using the data rather than fitting this construct as freeparameters (or assuming a uniform constant). The only free parameter in the evidence accumulation


models was used to measure the dynamic influence of the fixation stream. This also highlights a keydistinction between the current simplified implementation of PCS and evidence accumulation ap-proaches. Passage of time in the PCS models occurs only in terms of activation updating iterations,and the dynamic nature of the input data is lost (fixation durations were entered in the model afterbeing aggregated over time per choice). However, in the evidence accumulation models, the dynamicinput is central to the function of the model, and the idiosyncratic use of this information over time isembodied in the single free parameter b. This allows the accumulation models to make specificpredictions about the dynamics of processing that the PCS models in the simplified implementationused in this paper do not (but see below). In contrast, the fitted parameter in the PCS model mightincorporate something like an overall choice bias. Higher values for certain areas mean that choicesare more likely for this area given the same amount of fixations. Note that the fitted parameter washighest for the middle region, for which also the majority of choices was observed.

4.2. Development of fixations over time

Taking a closer look at the development of fixations over time indicates that expert playmakersshow an early focus of attention towards the chosen option within the first 2 s. Afterwards, however,they shift attention more strongly towards the non-favored options. One explanation for this finding isthat expert playmakers have learned to react intuitively to certain cues that shift attention to thefavored option. Later on, more deliberate information search processes might be activated todouble-check this automatic orientation reaction, which include comparisons with other options. Thisexplanation would be in line with default-interventionist dual-process models (see Evans, 2008, for anoverview). It would also accord with the full dual processing PCS model by Glöckner and Betsch(2008a) from which in this paper only the (first layer) network part was modeled. The findings are alsoin line with results demonstrating quick fixations to the preferred option (Innocenti et al., 2010), anover-time increase in between gamble comparisons in risky choices (Glöckner & Herbold, 2010), aswell as the finding that people are able to generate complex intuitive responses within lessthan 2 s (Glöckner & Betsch, 2008b; Glöckner & Hodges, in press).

4.3. Relation to the quiet-eye concept

Our results extend empirical evidence regarding the well-elaborated quiet eye concept (Vickers,2007). Quiet eye is a stabilization of the retinal picture during which the gaze is directed at a specificposition in the visuomotor workspace for a minimum of 100 ms. It occurs prior to the final movementof a task, and its onset and offset depend on specific movements in the task. In other words, it is seenas the last piece of visual information before performing the final critical movement in a specific skillsuch as putting or shooting a basket. This piece of visual information is thought to be the most impor-tant information an athlete needs in order to perform the skill successfully (quiet eye duration). In linewith this point of view, most often the quiet eye duration is analyzed and shifts in attention over timeprior to this final fixation are neglected. However, instead of analyzing only the last piece of visualinformation prior to the final critical movement in a specific skill, our research clearly demonstrates,that—at least in our experimental task—shifts in attention over time need to be acknowledged in orderto predict playmakers’ choices and their success. The systematic shifts in attention over time indicatethat the six-second period in our experimental task should not be interpreted as an unweighted accu-mulation process, which leads to a choice at the end. Choice preferences seem to be reached much ear-lier and—according to the above-suggested interpretation—early vs. late fixations might haveconceptually very different meanings for handball players. Perceptual expertise in handball seemsto depend on the recognition of information about a specific pattern of play in a given situation,and on the exploration of possible choices and their consequences (Gobet, 1997).

4.4. Probabilistic vs. deterministic implementations

It is interesting that the accumulator models discussed here are usually applied in a probabilisticimplementation, whereas the PCS model is often used in a deterministic one. We find, however, that


each type of model is particularly good in the implementation that is less often used. This findingmight inspire fruitful further developments of the models. The finding that deterministic models showa higher pcorr score compared to probabilistic models should not be over-interpreted because a meth-odological difference should be kept in mind: for deterministic models, pcorr was determined as theaverage proportion of correct predictions per person (i.e., correct predictions/all predictions) becausethe models’ outputs are specific choice predictions. Probabilistic models, in contrast, predict probabil-ities of choice for each option; hence, pcorr was determined as the average proportion for selecting theoption, which was actually chosen by the playmaker. Further research is needed to determine whetherthis methodological difference drives the lower performance of probabilistic models.

4.5. Possible developments and further investigations

A crucial comparison of further applications of these general types of models seems to depend onthe underlying conceptual assumptions such as whether overall gaze is used as a cue, such as in thecurrent PCS implementation, or if a more complex temporal weighting of associations between atten-tion and choice is assumed such as in the accumulator model. Interestingly, the good fit for both typesof models suggest that even conceptually different models are able to describe how gaze informschoices. Whether one model or the other may in general predict behavior in complex environmentsbetter is a matter of further empirical investigations that may use further type of models as compar-ison standards as well as demonstrated in this special issue.

In spite of its good fit, the simplifying assumption of the current implementation of PCS are cer-tainly unsatisfying from a theoretical perspective and should be refined in further versions of the mod-el. Using fixations as a proxy for cue values can only be a first step to understand experts’ (intuitive)decision making. One advantage of PCS is, however, that it can be easily extended in this direction. Ithas already been discussed elsewhere how the PCS approach to experts’ decision making could takeinto account training effects and construction of complex knowledge structures in expertise develop-ment (Herbig & Glöckner, 2009; Holyoak, 1991; Spellman, 2010). Similar to expertise development inchess, experts in handball are likely to acquire complex knowledge structures which allow chunking ofinformation, instantly recognizing increasingly complex constellations of the game and reactingappropriately. In further research, the simple cue value approximations used in the current PCS modelshould be replaced by complex information chunks that are represented in multiple layers of nodeswith increasing complexity of meaning (Holyoak, 1991). Investigations of these issues would, how-ever, necessitate considerably more in-depth analyses of mental structures, as well as much moreobservations per individual to allow a more fine-grained analysis of mental representations and fittingof PCS. In such an attempt, PCS could take into account primacy and the recency parameter as well ifthey prove to be important mediators in further work on accumulator models.

4.6. Perspective for improved practice

From a Bayesian perspective, and assuming that there are no costs for information, models shouldmaximize and use all information. However as the application to probabilistic and dynamic decisionsin sports shows, athletes may end up not using all information available (Vickers, 2007). Far from pro-viding conclusions on how to optimize decision-making in ill-defined task, we acknowledge that ex-pert decision makers are not born, but made through a combination of developmental experiences aschildren, and then through quality coaching that provides on- and off-court decision making trainingopportunities (Farrow & Raab, 2008). Coaches should be encouraged to use training programs whichimplement tasks that vary in temporal as well as spatial demands, so that the developing athlete hasthe opportunity to explore his/her individual gaze behavior and option generation strategy. Moreover,athletes could be encouraged to generate options in situation where a limited set of options are avail-able (Raab & Johnson, 2007).

We believe that our research makes an important contribution in providing further evidence for thelink between perception and cognition. In particular, it highlights the ability to predict choice behaviordirectly from perceptual inputs from different points of view and even in the absence of detailedassumptions about mental representations or transformations thereof. Nevertheless, on the basis of


our work, more complex model implementations might be developed that could take into accountmental structures as described above. An important challenge for future modeling approaches isthe observation from this and other papers (e.g., Innocenti et al., 2010) that early fixations are morepredictive for choice than later ones. Furthermore, it has to be explained how early orientation reac-tions towards the preferred option arise.

Although sports tasks provide a useful test bed for our models, it is important to generalize thefindings of the current study beyond the sports domain. We have no reason to believe that the currentresults are specific to the sports domain. Our models could be applied to many domains, situations ortasks studied within the naturalistic decision making approach (Klein, 1993) such as parking andselecting living spaces (Gettys, Pliske, Manning, & Casey, 1987) or problem-solving in chess (Gobet,1997), just to name a few. First attempts have been done to do so in risky decision (Glöckner &Herbold, 2011), but much more work in this direction is needed.

Appendix

One of the differences between PCS and the accumulation models is that the former are usuallyimplemented as deterministic models whereas the latter are implemented as probabilistic models.Hence, it might be argued that differences in predictive performance might result from this difference.Therefore we tested both models in probabilistic and deterministic implementations. For accumula-tion models the implementations are described in the main text. For PCS the probabilistic implemen-tation was done as follows:

We implemented the basic PCSprob and the fitted PCSprob as probabilistic model using an exponen-tial choice rule on the final activation of the option nodes according to

Pr½chosen k� ¼ ekaokPi¼1!3ekaoi

;

with aok indicating the network activation of the chosen option, aoi indicating the activation of allthree options (which are summed up), and k as scaling parameter that was fitted to the data (inthe interval [0, 3]). We fitted k for even trials using a complete search algorithm and used it to predictodd trials resulting in k = 2.90.

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