Acta Psychologica 152 (2014) 166–176

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Attention in risky choice☆ Eduard Brandstätter a,⁎, Christof Körner b a b

Department of Social and Economic Psychology, Johannes Kepler University Linz, Linz, Austria Department of Psychology, University of Graz, Graz, Austria

a r t i c l e

i n f o

Article history: Received 4 April 2014 Received in revised form 23 August 2014 Accepted 26 August 2014 Available online 16 September 2014 PsycINFO classification: 2340 2346 Keywords: Risky choice Attention Decision making Eye tracking Process tracing Verbal protocols

a b s t r a c t Previous research on the processes involved in risky decisions has rarely linked process data to choice directly. We used a simple measure based on the relative amount of attentional deployment to different components (gains/losses and their probabilities) of a risky gamble during the choice process, and we related this measure to the actual choice. In an experiment we recorded the decisions, decision times, and eye movements of 80 participants who made decisions on 11 choice problems. We used the number of eye fixations and fixation transitions to trace the deployment of attention during the choice process and obtained the following main results. First, different components of a gamble attracted different amounts of attention depending on participants' actual choice. This was reflected in both the number of fixations and the fixation transitions. Second, the last-fixated gamble but not the last-fixated reason predicted participants' choices. Third, a comparison of data obtained with eye tracking and data obtained with verbal protocols from a previous study showed a large degree of convergence regarding the process of risky choice. Together these findings tend to support dimensional decision strategies such as the priority heuristic. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Importance attracts attention. Be it a shattering event like the 2008 economic crisis, a famous movie star, or the smartphone you intend to buy—people pay more attention to important events, persons, or goods than to unimportant ones. When buying a car, it is the favored Volvo one plans to purchase rather than the Dacia one expects to decline that captivates the mind during the buying process. Intuitions such as these receive empirical support from sophisticated laboratory experiments showing that people allocate more attention to the alternative they will later choose than to the one they will decline (Glaholt & Reingold, 2009; Krajbich, Armel, & Rangel, 2010; Shimojo, Simion, Shimojo, & Scheier, 2003; Stewart, Hermens, & Matthews, 2013). This is not surprising, because the favored alternative is usually more important than the non-favored one. The underlying assumption here is that important aspects attract more attention and thus determine the choice. But what if the chain “importance–attention–choice” does not hold? This would be disturbing, since one would have to seriously question attention as a valid measure of importance. Suppose a participant must choose between an expensive, high-quality smartphone with a high-

☆ We are grateful to Michael Achorner and Ralph Pail for collecting the data and to Markus Kemmelmeier for helpful comments. ⁎ Corresponding author at: Department of Social and Economic Psychology, Johannes Kepler University Linz, Altenbergerstr. 69, 4040 Linz, Austria. E-mail address: [email protected] (E. Brandstätter).

http://dx.doi.org/10.1016/j.actpsy.2014.08.008 0001-6918/© 2014 Elsevier B.V. All rights reserved.

resolution display and an inexpensive, low-quality phone with a lowresolution display. Further suppose that process-tracing measures, such as those used in Mouselab or eye-tracking studies, reveal that resolution attracts most of the participant's attention during the choice process, which suggests that the choice will be the high-resolution phone. Contrary to this prediction, the participant chooses the phone with the lower resolution. A finding such as this would call into question the validity of process-tracing data. Linking process-tracing data to choice, we assert, is essential to taking that data seriously as a valid measure of the process. Apart from studies finding that people allocate more attention to the alternative they will choose than to the one they will decline, research on the attention–choice link is scarce. And when such studies have been done, the evidence is, at best, equivocal. In their 1993 landmark book The Adaptive Decision Maker, Payne, Bettman, and Johnson investigated accuracy and effort of various choice strategies in different environments. These researchers predicted and confirmed that the superior performance of a particular strategy (e.g., a lexicographic rule) in a particular condition (e.g., time pressure) triggered processes (e.g., search by attribute rather than alternative) that were compatible with that strategy.1 Within this elegant paradigm, the authors found different processes under different conditions—but the essential link 1 A lexicographic choice strategy searches by attributes and selects the alternative that performs best on the most important attribute (all other attributes are ignored). If two or more alternatives are equal on the most important attribute, it selects the alternative that performs best on the second most important attribute, and so on.

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to choices was missing (see also Arieli, Ben-Ami, & Rubinstein, 2011; Ayal & Hochman, 2009; Brandstätter, Gigerenzer, & Hertwig, 2006, 2008a,b; Hilbig, 2008; Johnson, Schulte-Mecklenbeck, & Willemsen, 2008; Katsikopoulos & Gigerenzer, 2008; Pachur, Hertwig, Gigerenzer, & Brandstätter, 2013; Su et al., 2013).2 This is not to say that choices were always neglected—in fact, in some of these studies researchers extensively tested the predictive accuracy of various decision strategies (e.g., Brandstätter et al., 2006; Su et al., 2013). The essential point is that models of choice can be tested on two different levels: the level of outcome and the level of process—or on both. Predicting choices with measures of the process – and measures of attention in particular – has rarely been done. When measures of attention were used to predict risky choices, results were often disappointing: Koop and Johnson (2013) investigated choices between simple gambles and summarized that “eye-tracking data demonstrate that the majority of acquisitions on each trial were of task-critical information” (p. 174). This means that participants looked more often at the gambles than they looked at the screen beside the gambles. No relation between choice and eye-tracking data was found (see their Fig. 11). Stewart et al. (2013) concluded from their eye-tracking data that people “look a little more at larger attributes and choose the gamble they look at more” (p. 26)—which is the wellknown finding that the chosen alternative attracts more attention. Orquin and Mueller-Loose (2013) reported that “attempts to classify heuristics based on attention are largely unsuccessful” (p. 202). Pachur et al. (2013) found that acquisition frequencies were inconsistent with the two models they tested and further concluded that “acquisition frequencies are not predictive of people's choices” (p. 13). The astonishing observation was that people paid attention to specific pieces of information but used other information for choosing. In particular, researchers found that maximum gains (and their probabilities) attracted more attention than minimum gains (and their probabilities; Pachur et al., 2013; Su et al., 2013). However, when predicting choices, a lexicographic strategy that prioritized minimum gains (and their probabilities) performed much better than a strategy that prioritized maximum gains (and their probabilities). Findings like this seriously call into question the appropriateness of attention as a valid measure of importance. Taken together, research on risky choice seems to suggest that measures of attention may at best show the familiar finding that the chosen alternative attracts more attention than the non-chosen one. This is surprising, since choice models such as the priority heuristic (Brandstätter et al., 2006) and models relying on weighting and summing of information such as expected utility theory (von Neumann & Morgenstern, 1947), subjective expected utility theory (Savage, 1954), prospect theory (Kahneman & Tversky, 1979; Tversky & Kahneman, 1992), disappointment theory (Bell, 1985; Brandstätter, Kühberger, & Schneider, 2002; Loomes & Sugden, 1986), the transfer of attention exchange model (Birnbaum & Chavez, 1997), and decision affect theory (Mellers, 2000) directly lend themselves to testable predictions. The latter models are rooted in Bernoulli's expected utility framework. Interpreted as process theories, they predict that people value payoffs with a utility function, multiply the utilities by decision weights, sum the products, and finally select the gamble with the higher sum of weighted utilities. These theories assume examination within gambles and predict that all pieces of information will receive the same amount of attention. The priority heuristic represents an instance of a different class of models—those requiring examination between gambles (Brandstätter et al., 2006). To illustrate the heuristic, consider a choice between two simple gambles of the type “a probability p of winning amount x; a probability (1 − p) of winning amount y”. A choice between two such 2 To avoid misunderstanding: Payne, Bettman, and Johnson (1993) used a measure called “GAIN” that captures the accuracy of a particular strategy in relation to the weighted additive difference rule and to random choice (p. 128 & p. 158). However, they did not investigate if different process measures could predict people's choices. The latter is the focus of the present article.

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gambles contains four reasons for choosing: the maximum gain, the minimum gain, and their respective probabilities; because probabilities are complementary, three reasons remain: the minimum gain, the probability of the minimum gain, and the maximum gain. For choices between gambles having two non-negative outcomes (all outcomes are zero or positive), the heuristic consists of the following steps: Priority rule: Go through reasons in the order of minimum gain, probability of minimum gain, maximum gain. Stopping rule: Stop examination if the minimum gains differ by 1/10 (or more) of the maximum gain; otherwise, stop examination if probabilities differ by 1/10 (or more) of the probability scale. Decision rule: Choose the gamble with the more attractive gain (probability). We refer to the one-tenth of the maximum gain as the aspiration level for gains, and to .1 as that for probabilities. Note, the aspiration level for gains is not fixed but changes with the maximum gain of the problem. For probabilities, which are bound between 0 and 1, the aspiration level of .1 is fixed. This is a simple hypothesis and empirical evidence suggests that people typically do not make more finegrained differences (Albers, 2001). The term “attractive” refers to the gamble with the higher (minimum or maximum) gain and to the lower probability of the minimum gain. For gambles involving losses, the term “gain” is replaced by “loss.” The priority heuristic (a) makes predictions whether gamble A or B will be chosen, (b) assumes examination between gambles, and (c) predicts that different pieces of information will receive different amounts of attention. To demonstrate the different process predictions for both classes of models consider the problem taken from Kahneman and Tversky (1979) between the safe gamble S ($2400 with p = .34) and the risky gamble R ($2500 with p = .33). This problem was devised to support prospect theory, not the priority heuristic. The priority heuristic predicts that people start by comparing the minimum gains. Since they are equal, the heuristic predicts that they attend to the next reason, which is the probabilities of the minimum gains (p = .67 and p = .66; or their logical complements of .33 and .34). Because the difference of .01 falls short of the aspiration level of .1, people are predicted to turn to the maximum gains of 2400 and 2500. The higher maximum gain thus decides choice, and the prediction is that people will select R, which is the majority choice. The prediction therefore is that maximum gains are relatively more important than probabilities for participants who choose R compared to those who choose S. Consequently, maximum gains are expected to attract relatively more attention than probabilities for risk seekers compared to risk avoiders. However, whenever there is a majority (choosing R) there is a minority (choosing S). Participants choosing the minority choice S either have a different order of reasons (i.e., probability before maximum gain) or use different aspiration levels. In both cases probabilities are predicted to be relatively more important than maximum gains for participants who choose S compared to those who choose R. This is because within a lexicographic strategy no other reason than probability would favor S. Consequently, probabilities are expected to attract relatively more attention than maximum gains for risk avoiders compared to risk seekers. Models that rely on weighting and summing assume that there is no relation between attention and choice. That is, regardless of whether participants choose gamble S or R, these models predict no differences in attentional allocation between risk avoiders and risk seekers for each of the four reasons. This does not mean that all four reasons get the same amount of attention; in fact, the two minimum outcomes of 0 might get less attention than the two non-zero, maximum outcomes, but the essential point is that attention and choice are unrelated. If differences are found across choices, thus, it is quite likely that participants used a dimensional rather than a weighting and summing strategy.

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In this article, we set out to investigate the relation between attentional allocation and choice. Specifically, we will test the following hypotheses for simple binary choices between S (€x, p) and R (€y, q) with p N q and x b y :

of them, we expected to find a meaningful, if often overlooked, relation between process and choice.

H1. The gamble that attracts the most attention will determine the choice.

2.1. Participants

H2. Verbal protocols will reflect the absolute amount of attention given to a reason: The decisive reason mentioned in the verbal protocols will receive most of the attention. H3. Verbal protocols reflect the relative amount of attention given to a reason: A reason mentioned in the verbal protocols that favors a specific choice will receive more attention from participants who make this choice than from those who do not. H4. If people make choices by weighting and summing, attention will be equally distributed across risky and safe choices. H5. The gamble that is fixated last will be chosen. H6. The reason that is fixated last will determine the choice. All the hypotheses relate process to choice. H1 and H5 were aimed at checking the validity of the current data by replicating known findings. H2, H3, and H6 were intended to test the priority heuristic and H4 the weighting and summing models; H3 differs from H2 in that it does not predict that the reason mentioned in the verbal protocols always gets the most attention. Instead, H3 predicts that the reason mentioned in the verbal protocols will get more attention for those participants who choose the gamble that is supported by that reason compared to the those participants who choose the gamble supported by the other reason. To test H2 and H3, we compared choices and measures of attention with previously published findings from verbal protocols (Brandstätter & Gußmack, 2013). In the experiment reported below, we tried to overcome some of the methodological shortcomings of earlier studies that might have been responsible for the inconclusive results on process and choice mentioned earlier. In particular, we tried to accomplish the following: (a) Use of a neutral task format: All pieces of information have equal distances both between and within gambles. This is crucial and has often been violated (e.g., Fiedler & Glöckner, 2012; Glöckner & Betsch, 2008; Glöckner & Herbold, 2011; Koop & Johnson, 2013; but see Stewart et al., 2013; Su et al., 2013). Attention is controlled to a large degree by visual properties of the stimulus, such as salience of or distance between pieces of information. Therefore, task format can strongly influence the number and sequence of fixations and stimulus construction must take such properties of the oculomotor system into account; (b) counterbalancing of stimulus layout: Gambles are presented vertically and horizontally to control for attentional effects such as reading direction; (c) minimizing the effect of short-term memory: Visual attention and short-term memory are closely linked. Some studies presented only four pieces of information that could be held in short-term memory (Koop & Johnson, 2013; Stewart et al., 2013). In this case there is no need for refixations and one should not expect that participants pay different amounts of attention to the different pieces of information, since each piece could be remembered. To overcome this problem, we used eight pieces of information; (d) concentration on two opposing reasons: In all problems used in our experiment the minimum gains (losses) were zero, leaving two opposing reasons—each favoring a different gamble (see also Koop & Johnson, 2013; Stewart et al., 2013). This allowed for an unambiguous test of the process; (e) comparison of three validity measures: We related eye-tracking data to choices and verbal protocols and thereby tested the validities of these measures against each other. While different studies employed one or the other of these requirements, we are not aware of any study that used all of them. By using all

2. Method

Eighty participants (39 females) volunteered to participate in the eye-tracking experiment. All of them had at least university entrance qualification and most of them were students of the University of Graz, Austria. Their average age was 28.5 years (SD = 8.7; range 18 to 62 years); five participants were age 50 or over. With respect to the dependent measures, we did not find any major deviations from the rest of the sample for this group. All participants had normal or corrected-to-normal vision and gave written informed consent. 2.2. Stimuli and design We used a subset of 11 problems selected from the classic decision problems used by Kahneman and Tversky (1979). Specifically, we selected all two-outcome, one-stage problems, that is, their Problems 2, 3, 3′, 4, 4′, 7, 7′, 8, 8′, 14, and 14′. These problems consist of two outcomes per gamble and their respective probabilities, and they produce such prominent violations of expected utility theory as the common ratio effect, the possibility effect, and the reflection effect. These violations led to the development of the non-linear probability weighting and the S-shaped value function for money. For a common presentation format, outcomes with a value of zero were explicitly denoted as “0” and a probability of one as “1.0”. The gambles were presented in white on a black background. We presented the problems in two different layouts to counterbalance layout effects. In the horizontal layout, the reasons (outcomes and probabilities) to choose a gamble were displayed in a row with one gamble above the other. In the vertical layout, the reasons to choose a gamble were displayed in a column with one gamble next to the other. The designations Alternative A and Alternative B labeled the rows (or columns, respectively). Outcomes were presented to the left of (or above) their respective probabilities (see Fig. 1). Half of the participants worked on the problems in horizontal layout and the other half in vertical layout. Special care was taken to ensure that (a) participants fixated the values (i.e., outcomes and probabilities) properly and (b) fixations could later be assigned to values. This was accomplished as follows. The gambles' values were presented in Arial font at a height of 0.24° visual angle. Outcomes and probabilities were presented with the same number of digits by using dummy digits; if, for example, a gamble contained the 4-digit outcome “5000” then a 1-digit outcome of “5” would be presented as “xxx5.” Thus, an outcome with fewer digits could not be identified in peripheral vision by the length of the respective string. Each value was surrounded by a quadratic frame whose edge was 1.67° long and 0.45° thick to provide a clear saccade target. This also minimized peripheral identifiability of adjacent values (e.g., Bouma, 1970). The (horizontal/vertical) center-to-center distance between any two values was 3.0°. These measures, together with corresponding pilot experimentation (see Körner & Gilchrist, 2007), ensured that values had to be fixated properly and that during fixation of a value peripheral identification of an adjacent value was impossible. This is important as it is known that information can be (pre-) processed peripherally, that is, without proper fixation (e.g., Rayner, 1998). 2.3. Procedure At the beginning of each trial, a fixation disc was presented at the center of the display; this also served as a drift correction for the eye tracker. When fixation on that disc was registered, the experimenter triggered the presentation of a decision problem. Participants could respond by pressing one of two buttons. They were instructed to press

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Fig. 1. Examples of the presentation of a decision problem. The left panel shows Problem 3 in horizontal layout and the right panel Problem 3′ in vertical layout. (See Table 1 for a description of the problems.) The stimuli are shown in inverted color and enlarged for readability. Dummy digits (“xxx”) were used to prevent peripheral processing (see the Stimuli and design section for details).

the left button to indicate a decision for the top (left) gamble or to press the right button to indicate a decision for the bottom (right) gamble. There was no time limit, and the screen was cleared and the trial ended once the participant pressed a button. We recorded participants' choices, manual response (decision) times, and eye movements, measured from the onset of a display with a decision problem until the button press. Participants were seated in a dimly lit, acoustically shielded booth in front of the monitor with a viewing distance of 63 cm. Before the start of the experiment, participants practiced on two sample problems under supervision of the experimenter. The 11 problems were randomly interspersed among a series of decision problems that also contained gambles with two or three outcomes. This series was organized in up to three blocks of maximally 48 problems. The eye tracker was calibrated before each block. A block lasted maximally 16 min, on average, and participants paused several minutes between blocks.

standard deviations from the mean. The 863 choices were aggregated across the 11 problems and the two choice outcomes (i.e., choice of gamble R or S).3 Mean decision time was 9526 ms (SD = 877). Decision times for gains, M = 9416 ms (SD = 878), and losses, M = 9658 ms (SD = 903), were similar, F (1, 20) = 0.40, p = 0.53, η2 = .02—as were the decision times for majority choices, M = 9395 ms (SD = 846), and minority choices, M = 9656 ms (SD = 927), with F(1, 20) = 0.48, p = .50, η2 = .02 [using a one-way analysis of variance (ANOVA)]. A participant's choice was denoted a “majority choice” if the participant selected the same gamble as the majority in Kahneman and Tversky (1979). The priority heuristic made 10 correct predictions out of 11. Consistent with Brandstätter and Gußmack (2013), we could not replicate the majority choice R in Problem 14, which is the choice between R (5000; .001) and S (5, 1.00). As shown in the second column of Table 1, most participants (43) chose the sure gain S. Selecting low-probability gains, thus, seems less robust than the other choice phenomena.

2.4. Apparatus 3.2. Fixation frequencies We recorded two-dimensional eye movements using an EyeLink II eye tracker (SR Research, Canada). We recorded with a sampling rate of 500 Hz and analyzed the data from the eye that produced the better spatial resolution, which was typically better than 0.30°. Displays were presented on a 21-inch monitor running at a refresh rate of 75 Hz with a resolution of 1152 × 864 pixels. To minimize head movements, a chin rest was used. Thresholds for saccade detection were set at a velocity of 35°/s and an acceleration of 9500°/s2. The eye tracker was calibrated using a 9-point calibration procedure. A drift correction (operated by the experimenter) was performed before each trial. A custom-made two-button response box was placed in front of the participant. Participants operated this box with their thumbs. Stimulus presentation was controlled with custom software written in C++.

3.1. Decision times and choices

To identify the value fixated, we calculated for each fixation the Euclidean distance between that fixation and each value in the display of a trial. The fixation was then assigned to the value with the smallest distance. This is an established method in eye-movement research (see, e.g., Zelinsky, 1996). Given the careful construction of our displays (see the Stimuli and design section), this method has the advantage that all fixation data can be used and each fixation can be assigned uniquely to an attribute value. Consecutive fixations of the same value were collapsed into a single fixation. Fixation frequencies are a direct measure of the extent of attention paid to the different pieces of information. Following previous research, we assumed that attention (as measured by the number of fixations) reflects importance: The more important an attribute was, the more attention we expected. Participants fixated on average 20.73 pieces of information (SD = 1.68) per problem. A problem contained eight pieces of information;

Overall, 17 of the 880 choices (1.93%) were excluded from the analyses (leaving 863 choices) because the decision time was longer than 3

3 This led to a data matrix consisting of 22 rows, in which each problem, separated by choice (i.e., risky or safe gamble chosen), constitutes the unit of analysis.

3. Results

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Table 1 Verbal protocols and fixation frequencies. Problem

Gambles (n)

2

Max

pmax

Min

pmin

Congruence?

Direction of examination

Reason for choice

Attention to Max, M (SD)

Attention to pmax, M (SD)

Attention to Min, M (SD)

Attention to pmin, M (SD)

Max pmax Max, pmax Certainty Max pmax Max pmax Max pmax Max Certainty

7.52 (4.12) 6.29 (2.79) 5.83 (5.38) 5.92 (3.76) 6.70 (4.47) 7.17 (3.52) 5.88 (3.21) 5.67 (3.40) 6.29 (3.70) 6.43 (5.66) 6.12 (3.31) 7.33 (3.70) 6.43 (0.62)

7.37 (3.60) 7.64 (2.78) 5.56 (3.40) 6.18 (3.31) 6.84 (4.03) 7.78 (3.25) 5.82 (3.17) 6.17 (2.88) 5.65 (2.64) 6.27 (3.56) 5.91 (2.78) 6.98 (3.30) 6.51 (0.78)

3.54 (1.79) 3.86 (1.78) 4.17 (3.71) 3.39 (1.79) 3.77 (2.42) 4.69 (2.45) 3.53 (2.32) 3.84 (2.24) 3.33 (1.48) 4.03 (2.67) 4.56 (2.26) 3.81 (2.52) 3.88 (0.43)

4.04 (2.60) 4.12 (2.31) 3.94 (3.02) 2.93 (2.05) 3.56 (1.93) 4.67 (2.63) 3.18 (2.19) 3.75 (2.30) 3.31 (1.54) 3.67 (1.86) 3.29 (1.55) 3.93 (3.33) 3.70 (0.49)

Yes Yes Yes Yes No Yes Yes Yes Yes No Yes No

pmax pmax Max pmax Max pmin, pmax Max pmax Max

6.45 (2.93) 6.64 (4.18) 5.42 (3.34) 7.22 (4.13) 7.00 (3.53) 6.33 (3.72) 6.44 (3.53) 6.81 (3.37) 6.88 (4.00) 6.76 (3.95) 6.56 (0.45)

5.49 (2.76) 5.16 (3.75) 5.18 (2.83) 6.26 (3.88) 6.12 (2.80) 5.07 (3.36) 6.00 (3.92) 6.11 (3.86) 6.03 (3.82) 5.09 (3.19) 5.65 (0.49)

3.94 (1.88) 4.28 (2.75) 2.97 (1.38) 4.54 (2.22) 4.00 (2.24) 4.44 (2.46) 3.56 (2.04) 4.42 (2.14) 4.82 (3.06) 3.80 (1.82) 4.07 (0.54)

4.91 (2.89) 5.04 (3.77) 4.21 (2.30) 5.85 (3.80) 4.98 (3.02) 4.11 (2.97) 4.60 (2.63) 4.32 (2.23) 4.67 (2.26) 4.64 (2.63) 4.71 (0.45)

No No Yes No Yes No Yes No Yes

R (52) S (28) 3 R (18) S (61) 4 R (43) S (36) 7 R (17) S (63) 8 R (49) S (30) 14 R (34) S (43) Means for gains

2500 2400 4000 3000 4000 3000 6000 3000 6000 3000 5000 5

.33 .34 .80 1.00 .20 .25 .45 .90 .001 .002 .001 1.00

0 0 0 0 0 0 0 0 0 0 0 0

.67 .66 .20 .00 .80 .75 .55 .45 .999 .998 .999 .00

Within reasons

3′

−4000 −3000 −4000 −3000 −6000 −3000 −6000 −3000 −5000 −5

.80 1.00 .20 .25 .45 .90 .001 .002 .001 1.00

0 0 0 0 0 0 0 0 0 0

.20 .00 .80 .75 .55 .45 .999 .998 .999 .00

Within reasons

R (53) S (25) 4′ R (33) S (45) 7′ R (50) S (27) 8′ R (25) S (53) 14′ R (33) S (45) Means for losses

Eye-tracking: fixations

Verbal protocols

Within reasons Within reasons Within reasons Within reasons Within reasons

Within reasons Within reasons Within reasons Within reasons

Note. First column refers to problem number as reported in Kahneman and Tversky (1979). Max = maximum gains or losses; pmax = probabilities of the maximum gains (losses); Min = minimum gains (losses); pmin = probabilities of the minimum gains (losses). Results from verbal protocols were taken from Brandstätter and Gußmack (2013). Hyphen (-) indicates that no data are available. Bold numbers indicate the highest value within each row. Congruence = yes, if bold number indicates the same reason that was mentioned in the verbal protocols.

thus, each piece was fixated 2.59 times on average. H1 states that the gamble that attracts the most attention will determine the choice. In support of H1, the chosen gamble received 2.99 more fixations (SD = 1.27) than the non-chosen one, F(1, 20) = 30.42, p = .001, η2 = .60 (using a one-way ANOVA with choice as independent variable and the difference between fixations on the chosen and non-chosen gamble as dependent variable), supporting previous findings (e.g., Shimojo et al., 2003). Table 1 further shows that maximum gains and their probabilities received more attention than minimum gains and their probabilities, F(3, 9) = 130.19, p = .001, η2 = .98 (obtained with a repeated-measures analysis using fixations on maximum gains, minimum gains, and their probabilities as the four levels of the within-subject factor); this is not surprising, given that the two minimum gains were zero. The same pattern holds for the loss domain, F(3, 7) = 177.56, p = .001, η2 = .99. Between gains and losses, Table 1 further reveals two significant differences concerning probabilities, F(4, 17) = 19.67, p = .001, η2 = .99 [obtained with a multivariate analysis of variance (MANOVA) using maximum gains, minimum gains, and their probabilities as dependent variables and domain as betweensubjects factor]. The probabilities of the maximum gains, M = 6.51 (SD = 0.78), received more attention than the probabilities of the maximum losses, M = 5.65 (SD = 0.49), F(1, 20) = 9.30, p = .006, η2 = .32, whereas the probabilities of the minimum losses, M = 4.71 (SD = 0.45), received more attention than the probabilities of the minimum gains, M = 3.70 (SD = 0.49), F(1, 20) = 25.13, p = .001, η2 = .56. For gains this suggests that the probabilities of maximum gains were relatively more important than the probabilities of the minimum gains; for losses, the probabilities of minimum losses seem to have been relatively more important than those of maximum losses. Both patterns suggest that in each domain the probabilities of the better outcome received relatively more attention than those of the worse outcome; we return to this finding below. Do the findings obtained from fixation frequencies converge with those obtained from verbal protocols? To test H2, we looked at the findings obtained from verbal protocols by Brandstätter and Gußmack

(2013) for the 11 problems from Kahneman and Tversky (1979; see Table 1). These protocols reveal that participants examined information across gambles and within reasons, and they show the reason that stopped examination and determined the choice. For each problem, the reason mentioned in the verbal protocols corresponded to the reason that determines the majority choice. For Problem 2, for example, verbal protocols demonstrated that participants who selected gamble R referred to its higher maximum gain (e.g., “2500 is higher than 2400; I take R”), while the minority who selected S referred to its higher probability of winning (e.g., “.34 is more than .33, so I take S”). Note that both statements imply comparisons across gambles and within reasons. If participants referred to the higher maximum gain in the verbal protocols, H2 predicts that (a) maximum gains would receive more attention than any other of the three remaining reasons, and (b) participants would choose the gamble with the higher maximum gain. If probability of the maximum gain was mentioned in the verbal protocols, probability should attract more attention and determine the choice. To test these predictions, we counted the number of problems for which this was true (the reason that received the most fixations is marked in bold in Table 1). The last column in Table 1 (“Congruence?”) shows that verbal protocols and fixation frequencies converge in 13 of 21 choices. Theoretically, any of the four reasons could get the most fixations, but Table 1 clearly shows that only the maximum gains (losses) and their probabilities are fixated most often. Assuming, thus, a stricter probability of p = .5 that either the maximum gain (loss) or its probability gets the most fixations (rather than the more lenient probability of p = .25 for any of the four reasons), the probability of finding 13 or more of 21 congruencies is .21 (derived from the binomial distribution). The troubling observation about Table 1, however, is that incongruencies are systematic, since maximum losses always received the most fixations, even when verbal protocols showed that participants chose gamble S because of its more attractive probability. Together these findings do not support H2. To test H3 and H4, which address the relative distribution of attention depending on the chosen gamble, we calculated the difference of

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two sums – the sum of fixations on the two maximum gains (losses) minus the sum of fixations on their corresponding probabilities – and called this measure Dfmax. For the choice between S (€x; p) and R (€y; q) Dfmax is: f

Dmax ¼ ½ f ðxÞ þ f ðyÞ−½ f ðpÞ þ f ðqÞ;

ð1Þ

where f denotes the number of fixations on the respective piece of information. Dfmax is positive if the two maximum gains (losses) got more attention than their probabilities; it is negative if the two probabilities got more attention than their maximum gains (losses). The measure Dfmin was calculated in the same way for the two minimum gains (losses) and their probabilities. We then performed a 2 (domain: gains, losses) × 2 (choice: risky gamble or safe gamble chosen) MANOVA with Dfmax and Dfmin as dependent variables and found a significant main effect of domain, F(2, 17) = 22.75, p = .001, η2 = .73, and a significant interaction effect between domain and choice, F(2, 17) = 3,97, p = .039, η2 = .32. We first turn to losses. Fig. 2 shows that Dfmax was positive for losses, M = 0.91 (SD = 0.45), and Dfmin was negative, M = − 0.64 (SD = 0.59). That is, maximum losses received more attention than their probabilities, and probabilities of minimum losses received more attention than minimum losses. This suggests that participants focused on the worst outcomes (maximum losses) while simultaneously being concerned with the probabilities of obtaining the best outcomes (minimum losses). Pairwise comparison tests revealed a quantitatively different but qualitatively analogous pattern for gains (all ps b .003). As shown in Fig. 2, Dfmax was slightly negative for gains, M = − 0.08 (SD = 0.54), and Dfmin was positive, M = 0.17 (SD = 0.45). That is, probabilities of maximum gains received slightly more attention than the maximum gains, and minimum gains received more attention than their

f

D min

f

D max

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probabilities. This suggests that participants tended to focus on the probabilities of the best outcomes (maximum gains) while simultaneously being concerned with worst outcomes (minimum gains). We conclude from this pattern that participants were “cautiously optimistic”: In both domains they were concerned with the worst outcome while concurrently focusing on the probability of obtaining the best one; this effect was stronger for the loss than for the gain domain. A simple effects analysis of the interaction showed significant differences for the two Dfmax, but the two Dfmin did not reach significance (both Fs b 0.7, ns). In particular, for the gain domain, Fig. 2 shows that participants who chose the safe gamble (e.g., 2400; .34 rather than 2500; .33) paid more attention to the two probabilities of the maximum gains than to the maximum gains, while participants who chose the risky gamble (e.g., 2500; .33) paid more attention to the two maximum gains than to their probabilities, F(1, 10) = 4.41, p = .031, η2 = .31. Taking the gamble with the higher of the two maximum gains is consistent with risk seeking, while taking the gamble with the higher probability of winning is consistent with risk aversion. In the loss domain the opposite pattern emerged: Participants who chose the safe gamble (e.g., − 2400; .34 rather than − 2500; .33) paid relatively more attention to the maximum losses than participants who chose the risky gamble (e.g., −2500; .33), F(1, 8) = 3.63, p = .04, η2 = .31. Avoiding the larger of the two maximum losses is consistent with risk aversion. The latter finding highlights an apparent paradox in risky choice: Participants focused on maximum losses (see also Pachur et al., 2013), which should have led to a choice of the safe gamble with the lower maximum loss. The majority, however, chose the risky gamble with the larger maximum loss. How can this paradox be resolved? Our findings suggest that maximum losses attracted most of the attention, but more so for risk avoiders than for risk seekers. We have to think in relative rather than absolute terms, which may explain the paradox. Although the difference between the minimum gains (losses) and their respective probabilities reached statistical significance in neither the gain nor the loss domain, their trends support the same conclusion. If we start from the reasonable assumption that comparisons between the two minimal gains (losses) of 0 and 0 are uninformative, Fig. 2 shows that the lines for Dfmin have the same trends as those for Dfmax. For gains the slight upward trend of Dfmin suggests that the probabilities of the minimum gains (i.e., p = .66 and p = .67 for the example above) are slightly more important for risk avoiders than for risk seekers. The better of these two probabilities (p = .66) implies risk aversion. For losses, Fig. 2 shows that the probabilities of the minimum losses (i.e., p = .66 and p = .67) are more important for risk seekers than risk avoiders. The better of these two probabilities (p = .67) implies risk seeking. Overall, we found consistent patterns of attention that are meaningfully related to people's choices and that support H3 while refuting H4. Attention, thus, varies as a function of choice.

f

D max

3.3. Transitions

f

D min

Fig. 2. Dfmax represents the difference of two sums: the sum of fixations on the two maximum gains (losses) minus the sum of fixations on their corresponding probabilities. If Dfmax is positive, the two maximum gains (losses) got more attention than their probabilities; if Dfmax is negative, the two probabilities got more attention than their maximum gains (losses). Dfmin was built in the same way for the two minimum gains (losses) and their probabilities.

Similar to fixation frequencies, consecutive fixations of two values within a reason (transition) also capture the degree of attention given to a reason. Table 2 shows that most transitions occurred between the maximum gains and between their probabilities, F(3, 9) = 33.55, p = .001, η2 = .92, and this also holds for transitions between the maximum losses and between their probabilities, F(3, 7) = 50.49, p = .001, η2 = .96 (obtained with a repeated-measures analysis using transitions between the two maximum gains, between the two minimum gains, and between each of their probabilities as the four levels of the within-subject factor). Do verbal protocols and eye tracking yield similar findings? We expected similar results to those for the fixation frequencies; that is, we expected most of the transitions to occur within the reason that was mentioned in the verbal protocols. The last column in Table 2 (“Congruence?”) reveals that verbal protocols and transition frequencies again converged in 13 of 21 choices (p = .21). However,

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Table 2 Verbal protocols and transitions. Problem

2 3 4 7 8 14 Means for gains 3′ 4′ 7′ 8′ 14′

Gambles (n)

Max

pmax

Min

pmin

Verbal protocols

Eye-tracking: within-reason transitions

Congruence?

Direction of examination

Reason for choice

Transitions Max, M (SD)

Transitions pmax, M (SD)

Transitions Min, M (SD)

Transitions pmin, M (SD)

Max pmax Max, pmax Certainty Max pmax Max pmax Max pmax Max Certainty

3.04 (2.29) 2.14 (1.67) 1.44 (1.89) 1.57 (1.77) 2.09 (2.23) 2.06 (1.92) 1.35 (1.00) 1.49 (1.62) 1.96 (2.57) 2.23 (3.21) 1.85 (1.74) 1.91 (1.61) 1.93 (0.46) 1.57 (1.45) 2.28 (2.00) 1.48 (1.56) 2.04 (1.67) 1.96 (1.80) 2.11 (1.91) 1.72 (1.34) 2.26 (1.60) 1.24 (1.25) 2.00 (2.00) 1.87 (0.35)

1.98 (2.13) 2.25 (1.86) 0.78 (1.06) 1.30 (1.84) 1.56 (1.91) 2.08 (2.21) 0.71 (1.05) 1.27 (1.47) 1.10 (1.45) 1.40 (1.79) 0.88 (1.70) 1.12 (1.34) 1.37 (0.51) 1.41 (1.17) 1.56 (1.33) 1.72 (1.31) 2.09 (2.02) 1.60 (1.26) 1.59 (1.50) 2.16 (2.34) 1.89 (1.87) 0.94 (1.32) 1.27 (1.18) 1.62 (0.37)

0.96 (0.99) 0.82 (0.77) 0.50 (0.79) 0.74 (0.85) 0.79 (0.86) 0.92 (0.77) 0.71 (0.99) 0.78 (0.92) 0.73 (0.70) 1.00 (1.80) 0.91 (0.90) 0.72 (1.26) 0.80 (0.14) 0.66 (0.76) 1.16 (1.03) 0.55 (0.56) 1.16 (0.98) 0.80 (0.86) 1.30 (1.35) 0.84 (0.98) 0.98 (0.97) 0.82 (1.07) 0.62 (0.71) 0.89 (0.25)

1.50 (1.43) 1.39 (1.32) 0.89 (1.08) 0.66 (0.96) 1.19 (1.35) 1.39 (1.38) 0.76 (1.09) 0.98 (1.18) 0.94 (0.92) 1.10 (1.27) 0.59 (0.70) 0.98 (1.79) 1.03 (0.29) 0.55 (0.95) 0.68 (1.38) 0.64 (1.03) 1.33 (1.87) 1.02 (1.45) 0.93 (1.30) 0.68 (0.95) 0.62 (0.88) 0.36 (0.63) 0.53 (0.90) 0.73 (0.28)

R (52) S (28) R (18) S (61) R (43) S (36) R (17) S (63) R (49) S (30) R (34) S (43)

2500 2400 4000 3000 4000 3000 6000 3000 6000 3000 5000 5

.33 .34 .80 1.00 .20 .25 .45 .90 .001 .002 .001 1.00

0 0 0 0 0 0 0 0 0 0 0 0

.67 .66 .20 .00 .80 .75 .55 .45 .999 .998 .999 .00

Within reasons

R (53) S (25) R (33) S (45) R (50) S (27) R (25) S (53) R (33) S (45)

−4000 −3000 −4000 −3000 −6000 −3000 −6000 −3000 −5000 −5

.80 1.00 .20 .25 .45 .90 .001 .002 .001 1.00

0 0 0 0 0 0 0 0 0 0

.20 .00 .80 .75 .55 .45 .999 .998 .999 .00

Within reasons

Within reasons Within reasons Within reasons Within reasons Within reasons

Within reasons Within reasons Within reasons Within reasons

pmax – pmax Max pmax Max pmin, pmax Max pmax Max

Means for losses

Yes Yes Yes No Yes Yes Yes No Yes No Yes No No Yes No No Yes Yes Yes No Yes

Note. First column refers to problem number as reported in Kahneman and Tversky (1979). Max = maximum gains or losses; pmax = probabilities of the maximum gains (losses); Min = minimum gains (losses); pmin = probabilities of the minimum gains (losses). Results from verbal protocols were taken from Brandstätter and Gußmack (2013). Hyphen (-) indicates that no data are available. Bold numbers indicate the highest value within each row. Congruence = yes, if bold number indicates the same reason that was mentioned in the verbal protocols.

participants' tendency to produce transitions between maximum gains (losses) more often than within any other reason yielded systematic incongruencies when probability was reported as the decisive reason for choice. To test H3 and H4 regarding the relative distribution of attention depending on the chosen gamble, we performed a 2 (domain: gains, losses) × 2 (choice: risky gamble or safe gamble chosen) MANOVA with Dtmax and Dtmin as dependent variables. As it did for fixation frequency, Dtmax denotes the difference of the number of transitions between the two maximum gains (losses) minus the number of transitions between the probabilities of the maximum gains (losses). For the choice between S (€x; p) and R (€y; q) Dtmax is t

Dmax ¼ ½ f ðx and yÞ−½ f ðp and qÞ;

t

D max

t

D min

ð2Þ t

where f denotes the number of transitions between two pieces of information (between outcomes or between probabilities in either direction). If Dtmax is positive, there are more transitions between the two maximum gains (losses) than between their probabilities; if Dtmax is negative, there are more transitions between the probabilities of the maximum gains (losses) than between the maximum gains (losses). Dtmin was calculated in the same way for the two minimum gains (losses) and their probabilities. The results again support H3 while refuting H4. In particular, we found a significant main effect of domain, F(2, 17) = 11.78, p = .001, η2 = .58, and a significant interaction effect between domain and choice, F(2, 17) = 4,90, p = .02, η2 = .37. Fig. 3 shows that Dtmax was larger for gains than for losses, while Dtmin was smaller for gains than for losses, which resulted in a larger difference between Dtmax and Dtmin in the gain compared to the loss domain. The main effect (i.e., the larger difference between Dtmax and Dtmin for gains than for losses) together with the findings shown in Fig. 2 basically reveals that Dtmin always tends to be closer to zero than Dtmax. That is, attention tends to be more equally distributed between the minimum

D max

t

D min

Fig. 3. Dtmax denotes the difference of the number of transitions between the two maximum gains (losses) minus the number of transitions between the probabilities of the maximum gains (losses): If Dtmax is positive, there were more transitions between the two maximum gains (losses) than between their probabilities; if Dtmax is negative, there were more transitions between the probabilities of the maximum gains (losses) than between the maximum gains (losses). Dtmin was built in the same way for the two minimum gains (losses) and their probabilities.

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gains (losses) and their probabilities than between the maximum gains (losses) and their probabilities. While Figs. 2 and 3 may look superficially different, they share the same underlying pattern, rising trends in the gain domain and falling trends in the loss domain, and these trends are always stronger for Dmax than for Dmin. As before, simple effects analysis of the interaction showed significant differences for the two Dtmax, while the two Dtmin did not reach significance (both Fs b 0.6, ns). In particular, in the gain domain Fig. 3 shows that all participants made more transitions between the maximum gains than between their probabilities (Dtmax N 0)—and switching between maximum gains was stronger for participants who chose the risky gamble than for those who chose the safe gamble, F(1, 10) = 6.22, p = .02, η2 = .38. In the loss domain participants also made more transitions between the maximum losses than between their probabilities (Dtmax N 0), but now switching between the maximum losses was more frequent for participants who chose the safe gamble than for those who chose the risky gamble, F(1, 8) = 4.20, p = .04, η2 = .34. Taking the gamble with the lower maximum loss implies risk aversion. The trends of both Dtmin support the conclusions drawn from Dtmax, but neither of them reached statistical significance (ps N .2). The present analysis highlights again that absolute levels yield little information and may often produce contradictory results. Take the gain domain in Fig. 3. The maximum gain attracts most of the transitions— irrespective of the choice. The important finding is that this observation is stronger for risk seekers than risk avoiders, and the reverse holds for the loss domain. 3.4. Last fixations H5 and H6 concern participants' last fixation. We surmise that participants' last fixation would predict which gamble they would choose. Particularly, H5 predicts that the last fixation rests more often on the chosen than on the non-chosen gamble; H6 tests the stopping rule of the priority heuristic and states that the last fixation rests on a reason that favors the chosen gamble. We found strong support for H5 and a slight tendency for H6. In the gain domain, the chosen gamble and the last-fixated gamble coincided in 73.20% of the choices, χ2 (1, N = 474) = 102.62, p = .001; in the loss domain the respective number was 77.89%, χ2 (1, N = 389) = 121.20, p = .001. To test if the lastfixated reason predicts the chosen gamble, we condensed gamble's four reasons into two: (a) the maximum gain predicts the risky gamble,

173

and (b) the sum of the fixations on the two probabilities predicts the safe gamble. For losses these predictions are reversed. Fixations on the minimum gains (losses), which are both zero, were omitted since they do not favor either of the two gambles. In the gain domain the last-fixated reason predicted the chosen gamble in 52.67% of the choices, χ2 (1, N = 408) = 1.20, p = .14; in the loss domain this number was 49.97%, χ2 (1, N = 316) = 0.18, p = .49. We conclude that it is the last-fixated gamble and not the last-fixated reason that predicts the choice. 3.5. Predicting choices The previous analyses revealed that choice and process are meaningfully related; each of these analyses considered one process measure only. This approach offers the advantage of clearly illustrating the relation between a single process measure and choice. However, this approach may come with two disadvantages: (a) we do not exactly know the contribution of each process measure when integrated into a single model, because effect sizes (i.e., η2, percentage points) were calculated separately for each process measure, and (b) data analyses were performed on an aggregate rather than individual level. To overcome both possible shortcomings, next we sequentially integrate process measures into a series of models to predict participants' choices (risky or safe gamble) using binary logistic regression implemented in a generalized linear mixed model. We then compare these models with each other. The baseline Model 1 in Table 3 contains a fixed and random intercept only (Heck, Thomas, & Tabata, 2013). In this model the variable Subject-ID served as the only (Level 2) predictor. The significant random intercept of 0.30 suggests that participants differed significantly in their propensity to choose the risky or safe gamble, which justifies using Subject-ID as a random effects predictor in further model testing. This model predicted 67.1% of 863 choices correctly. Model 2 in Table 3 additionally contains the significant process measures of the previous analyses as fixed effects (Level 1) predictors. These are the variables Dfmax, Dtmax, their respective interactions with domain (gains/losses), the last fixation on the risky gamble (yes/no) and ratio, which captures the ratio of the number of fixations on the risky compared to the safe gamble; a ratio higher than 1 means more attention to the risky gamble and a ratio lower than 1 the opposite. Table 3 shows that the prediction rate improved from 67.1 to 76.1%, while the Bayesian information criterion (BIC) deteriorated from 3694 to 3997.

Table 3 Binary logistic regression to predict choice of risky gamble. Parameter Fixed effects Intercept Dfmax Dtmax Domain = losses Dfmax ∗ domain = losses Dtmax ∗ domain = losses Last fixation = risky Ratio Random effects Subject-ID (intercept) Ratio (slope) Cov (Subject-ID/ratio) BIC Prediction rate (%)

Model 1 −0.11 (0.09)

0.30 (0.11)⁎⁎

3694.24 67.1

Model 2 −1.93 (0.31)⁎⁎ −0.03 (0.06) 0.17 (0.08)⁎ 0.33 (0.20)† 0.04 (0.11) −0.28 (0.13)⁎ 2.13 (0.21)⁎⁎ 0.39 (0.24)† 0.23 (0.12)†

3996.60 76.1

Model 3 −2.38 (0.30)⁎⁎ 0.13 (0.06)⁎ 0.30 (0.19) −0.23 (0.09)⁎ 2.08 (0.21)⁎⁎ 0.81 (0.19)⁎⁎ 1.20 (0.70)† 0.86 (0.45)† −0.95 (0.54)† 4007.31 77.6

Model 4 −1.94 (0.27)⁎⁎ 0.15 (0.05)⁎⁎ 0.32 (0.16)† −0.25 (0.08)⁎⁎ 2.11 (0.17)⁎⁎ 0.42 (0.21)⁎

864.30 75.1

Note. Standard errors are in parentheses. D f max = sum of fixations on the two maximum gains (losses) minus the sum of fixations on their corresponding probabilities; Dtmax = difference of the number of transitions between the two maximum gains (losses) minus the number of transitions between the probabilities of the maximum gains (losses); domain = gain/loss domain; last fixation = whether last fixation was on the risky or safe gamble; ratio = risky gamble/safe gamble ratio of fixations; BIC = Bayesian information criterion; prediction rate = percentage of correctly predicted choices. † p b .1. ⁎ p b .05. ⁎⁎ p b .01.

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The predictor's last fixation, Dtmax, and its interaction with domain reached significance. The interpretation for this interaction has already been given in Fig. 3. Dfmax did not turn out to be significant; this is not surprising, given that the correlation between Dfmax and Dtmax is r(863) = .68, p b .0005. Model 3 represents the best-fitting model we obtained from the following procedure: We kept all predictors with p b .1 and added each of the remaining predictors as a random effects (Level 1) predictor (slope). Each model contained only one additional random effects predictor (using both variance component and unstructured covariance matrices in model fitting). From these models only Model 3 converged at a solution, which contains ratio as a random effects variable. Prediction rate slightly improved from 76.1 to 77.6% and BIC again deteriorated to 4007. The fixed effects predictor's last fixation, ratio, and the interaction of domain with Dtmax remained significant. In Model 4 we took a radical approach and removed all random effects (i.e., the random intercept for Subject-ID and the random slope for ratio). The reason was that none of the random effects turned out to be significant (p b .05) in Models 2 and 3. BIC sharply improved from 4007 to 864, while the prediction rate deteriorated slightly from 77.6 to 75.1%. From this we infer that Model 4 best explains the data. In Model 4 the process measures Dtmax, its interaction with domain, last fixation, and ratio remained significant. The coefficient of 2.11 for last fixation shows the strongest deviation from 0 and therefore predicts choices best. That is, a last fixation on the risky rather than on the safe gamble increases the odds of choosing the risky gamble by a factor of e2.11 = 8.25. Together these findings suggest that different process measures succeed in explaining choices. 4. Discussion In this article we reviewed previous studies linking process measures and choice and found that process data (e.g., from Mouselab or eye tracking) have rarely been used to predict choices. When it has been done, results have often been somewhat disappointing — thereby calling into question if such attentional measures can reflect importance (see examples in the Introduction section). The relation between attention and risky choice is important because the question of whether making a risky choice is based on a weighting and summing process is still debated, and process data such as fixation frequencies and transitions have the potential to discriminate between opposing classes of choice models. To overcome the limitations of previous studies, we introduced a set of methodological improvements and used a simple measure based on the relative amount of attentional deployment to different components of the gambles. Thus, we established direct relations between process and choice. Our results were as follows. First, depending on participants' choices, different reasons attracted different amounts of attention, which supports H3 and dimensional strategies such as the priority heuristic. We found that maximum losses always received most of the attention (Table 1), regardless of choice and regardless of the reason mentioned in the verbal protocols (thereby refuting H2); however, participants who chose the risky gamble paid relatively less attention to maximum losses than those who chose the safe gamble. This highlights the importance of comparing process measures across choices and suggests that we have to think in relative rather than absolute terms. Second, fixations were not equally distributed across risky and safe choices, which refutes the weighting and summing models and H4. Third, a reason that favored a specific choice received more transitions from participants who made this choice than from those who did not (again contradicting H2 and H4 and supporting H3). Fourth, the last-fixated gamble (H5) and not the last-fixated reason (H6) predicted participants' choices. The latter finding contradicts the stopping rule hypothesized by the priority heuristic, which assumes that the last fixated reason determines the choice. Fifth, participants were “cautiously optimistic”: In both the gain and loss domain they

were concerned with the worst outcome while concurrently focusing on the probability of obtaining the best one; this effect was stronger for the loss than for the gain domain. Sixth, we replicated the wellknown finding that participants paid more attention to the chosen than to the non-chosen gamble (supporting H1). Seventh, verbal protocols, relative measures of attention, and choices showed a large degree of convergence regarding the process of risky choice (supporting H3 and refuting H2). All in all, findings from verbal protocols, search behavior, and choices tend to support dimensional strategies and cognitive processes similar to those postulated by the priority heuristic. The dimensional comparisons we found raise doubts about whether such prominent phenomena as the common ratio effect, the impossibility effect, and the reflection effect are best explained by (cumulative) prospect theory. Why have previous studies so often failed to establish a relation between choice and process? We surmise that a probability and its corresponding outcome represent a coherent unity, since probabilities per se are meaningless: For instance, the statement “a probability of .3” is meaningless, but “an amount of €3000” is not. Probabilities are necessarily bound to events, whereas the reverse is not true. This is why most transitions occur between outcomes and probabilities (Johnson et al., 2008; Pachur et al., 2013)—although this finding has sometimes been taken as evidence for weighting and summing models. Here we used the measures Dmin and Dmax, which operationalize differences in attention between outcomes and their corresponding probabilities. Put differently, these measures disentangle the natural unity between outcome and probability and are therefore more sensitive in capturing people's attention to different pieces of information. The two measures in combination with a number of methodological improvements (e.g., equidistance between pieces of information, no separating lines, overcoming short-term memory limit, etc.) produced more consistent and meaningful relations between attention and risky choice. Comparing process measures across choices further offers the advantage of circumventing problems such as complementarity of probabilities. In simple gambles probabilities are complementary, which might have caused participants to pay less attention to probabilities than outcomes (Table 1). Possible biases like this might be why the reason mentioned in the verbal protocols did not get the most attention. Comparing process measures across choices, however, should neutralize this possible distortion, since complementarity of probabilities holds for both the risky and the safe gamble. 4.1. Do risky gambles attract more attention? We assume that risky gambles evoke more emotions and arousal than safe ones, because a risky gamble contains both more of a good thing (e.g., high maximum gain) and more of a bad (e.g., low probability of winning). Put differently, compared to the safe gamble, the risky gamble bears a higher potential for ambivalence. Since emotional stimuli are usually more important than unemotional ones, we hypothesize that risky gambles attracted more attention than safe ones. To test this hypothesis, we used the risky gamble/safe gamble ratio of fixations. The mean ratio of 1.34 (SD = 0.27) is significantly larger than 1, t(21) = 5.82, p = .001, for the one-sample t test. This holds for participants who chose the risky gamble, M = 1.57 (SD = 0.14), t(10) = 13.36, p = .001, and for those who chose the safe one, M = 1.11 (SD = 0.14), t(10) = 2.66, p = .024. The latter is noteworthy, since H1 suggests that the chosen gamble got more attention than the non-chosen one. Problems 3, 3′, 14 and 14′ contain a safe alternative consisting of a sure gain (loss) only, which may have caused more attention to the more complex, risky gamble. To control for this possible confounding, we calculated the mean ratios for problems containing no sure gain (loss), M = 1.32 (SD = 0.27), t(13) = 4.46, p = .001, and for problems containing a sure gain (loss), M = 1.37 (SD = 0.29), t(7) = 3.62, p = .008. In each case the ratios were significantly larger than 1. Together these results once more corroborate the basic tenet

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of this paper that importance, now conceptualized as emotional ambivalence of the stimuli, attracts attention. 4.2. Prediction and explanation in model comparisons We used a generalized linear mixed model to predict choices and succeeded in 75.1% of the predictions. In a model with last fixation on a gamble as the single predictor, the prediction rate was 75.3% and the BIC sharply improved to 25.95. A finding like this could lead to the conclusion that the other predictors were unnecessary. While this is true for prediction, it is not true for explanation. When we removed last fixation on a gamble from Model 4 all other predictors remained significant; the prediction rate was 66.7% and the BIC was 940.12. That is, these other process measures, including Dtmax, predicted choices 16.7% above chance level. It is important to note that process measures, such as Dfmax and Dtmax, are theoretically related to choice models, such as the priority heuristic. We thus believe that prediction is a significant goal in model building. If it becomes the only goal, it could hide important insights that help researchers get a better understanding of the underlying process. 4.3. Limitations We obtained the present findings from choices between simple gambles of the form (€x, p; 0, 1 − p). Although this allows for an unambiguous investigation of reason and choice, these stimuli naturally limit generality. We do not know how people's processes are linked to choice for more complicated gambles. Findings from verbal protocols (Brandstätter & Gußmack, 2013) and the assumption that choices between more complicated gambles are more likely to trigger simple heuristics than the usage of complex expectation models suggest that dimensional comparisons might even be more common. Another limitation concerns counterbalancing. We counterbalanced horizontal and vertical gamble presentation but not outcomes and probabilities. This could be why maximum gains, for example, received more attention than their probabilities. If so, probability got less attention and participants might have selected the gamble with the higher maximum gain (rather than the gamble with the higher probability of winning). Lack of counterbalancing, thus, prevents all pieces of information from having the same chance of receiving equal attention, thereby diluting the hypothesized link between attention and choice. Despite this hindrance, we found significant links between attention and choice, which seems to lend credibility to our results. A third limitation is the lack of support for a lexicographic strategy. We found that the reason that determined choice received relatively more attention than the reason that did not. Similarity models would have comparable findings (Leland, 1994; Rubinstein, 1988). These models are mute, however, when two approximately equal differences exist, which is the case in Problem 7 but not in Problem 8 (Table 1). In the case of Problem 7, the priority heuristic offers the advantage that it prioritizes probabilities over maximum gains and thereby makes a clear prediction for the majority choice. 4.4. Future research We have observed a troubling trend in recent studies that used task formats that foster search within gambles (see also Brandstätter, 2011). Whereas early process tests (using Mouselab or eye tracking) usually employed neutral information display matrices in which columns represented alternatives and rows attributes (Ford, Schmitt, Schlechtman, Hults, & Doherty, 1989), recent Mouselab and eye-tracking studies have employed gambles that were visually separated: Gambles were either placed in extra boxes (e.g., Glöckner & Herbold, 2011), separated by a line (Johnson et al., 2008), or spaced widely apart (Koop & Johnson, 2013). Such formats, we think, bias attentional allocation by fostering information search within gambles and thus favor models that rely on

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weighting and summing. These studies, we think, merely support the Gestalt law of proximity, according to which participants perceive pieces of information that are close to one another as belonging together. Careful control of the display is particularly important whenever eyemovement measures are used because the oculomotor system is very sensitive to visual properties of the stimulus. Verbal protocols are less sensitive to such properties, since they operate on a higher cognitive level (Russo, 1978). Future research will profit by using neutral, counterbalanced gamble presentations. Innovative methods, such as mouse-tracing (Koop & Johnson, 2013), seem promising, although the link between model prediction and process trace needs to be better founded. We have shown that different methods can arrive at the same conclusion if attention is compared across choices in a way that considers relative instead of absolute levels. 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