THE ROLE OF SALIENCE AND ATTENTION IN CHOICE UNDER RISK:

AN EXPERIMENTAL INVESTIGATION

MILICA MORMANN AND CARY FRYDMAN*

May 2016

We conduct two experiments that test a recently proposed theory of context-dependent choice under risk called salience theory. The theory predicts that a decision maker’s attention is drawn to precisely defined salient payoffs, and these payoffs are overweighted in the choice process. In our first experiment, subjects make a series of highly incentivized choices between a risky lottery and a certain option while their eye-movements are recorded. In the second experiment, subjects choose between a risky lottery and a certain option, but we exogenously manipulate the visual salience of the risky lottery. We find three main results. First, the data on lottery choices is broadly consistent with salience theory as risk-taking increases when the upside of the risky lottery becomes more salient. Second, using eye-tracking data, we find that subjects do not pay more attention to the state that is theoretically salient, which is inconsistent with the theory; instead, subjects pay more attention to the risky lottery compared to the certain option when the risky lottery’s upside is salient. Third, risk taking is reduced when a lottery’s downside becomes visually salient. More generally, our results provide evidence that attention is an important and causal factor in determining choice under risk.

JEL Codes: C91, D81, D87

*Authors contributed equally to this work. Mormann is at the Department of Finance, University of Miami School of Business, [email protected]; Frydman is at the Department of Finance and Business Economics, USC Marshall School of Business, [email protected]. We are thankful for comments and suggestions from Kenneth Ahern, Nicholas Barberis, Pedro Bordalo, Ben Bushong, Tom Chang, Nicola Gennaioli, Alison Harris, Lawrence Jin, Chad Kendall, Ian Krajbich, Yaron Levi, Andrei Shleifer, Neil Thakral and from seminar participants at Harvard University, Maastricht University, University of Miami, University of Southern California, Society for Neuroeconomics Conference, and Southern California Finance Conference. We thank Mike Lupp and Brad Swain for research assistance.

1 I. INTRODUCTION

A key tenet of expected utility theory is that preferences are stable over time and are independent of the context in which choices are presented. These preferences guarantee that choices are consistent across environments and are not affected by context-specific features such as visual cues or the choice set itself. However, starting with Allais (1953), a large body of evidence documents robust violations of context-independent choice1. For example, when confronted with a choice between two lotteries with similar expected values, experimental subjects often exhibit a preference reversal: they choose the less risky lottery but are willing to pay more for the riskier lottery (Lichtenstein and Slovic 1971). The systematic nature of these context-independent choice violations suggests that preferences are malleable and influenced by the environment.

Theorists have thus begun to formalize a new class of models where the choice set can distort the relative weights that a decision-maker attaches to attributes of an alternative. Bordalo,

Gennaioli and Shleifer (2012) (henceforth BGS) propose the first economic model in this spirit, salience theory, to explain context-dependent risk preferences. BGS assume that attention is directed towards salient lottery payoffs, and the decision maker then overweights these payoffs when computing the lottery’s utility. Bordalo, Gennaioli and Shleifer (2013a, 2013b, 2015) use this same core intuition – that attribute weighting is a function of the choice set – to explain context-dependencies in consumer choice, asset pricing, and industrial organization. Other recent models share this intuition but assume that the range of utilities across an attribute distort the associated attribute weight (Koszegi and Szeidl 2013; Cunningham 2013; Bushong, Rabin, and

Schwartzstein 2016)2.

While the salience model of BGS can deliver many classic violations of context- independent choice, its core assumptions represent a departure from the standard economic model

1 See Camerer (1995) for a review. 2 Earlier work by Loomes and Sugden (1982) is also driven by attribute-based comparisons, although the interpretation in that work is based on anticipatory regret and rejoicing. 2 and remain untested. In this paper, we conduct two experiments that allow us to perform novel tests of the key psychological mechanism that drives decision-making under risk according to salience theory. In the first experiment, subjects make highly incentivized decisions between a risky lottery and a certain option while we collect eye-tracking data. This combination of data on lottery choices and eye-tracking enables us to test both the model’s assumption about the psychological mechanism and the model’s predictions about risk taking.

To understand how the eye-tracking data is used to test the model’s assumption, we briefly describe the intuition of the BGS model here. A lottery payoff is defined as salient if “it is very different in percentage terms from the payoffs of the other available lotteries (in the same state of the world)” (pg. 1244). A decision-maker’s (DM) attention is then drawn to the salient payoff, and the DM subsequently overweights the state in which the salient payoff is delivered.

When a lottery’s upside is salient, the DM overweights the lottery’s upside and exhibits risk- seeking behavior; when its downside is salient, the DM exhibits risk-averse behavior. The key restriction of the BGS model is that it precisely defines which state of the world is salient as a function of the choice set. Therefore, we are able to test (i) whether a subject’s attention is drawn to the state that is predicted to be salient, and (ii) whether risk taking is greater when the lottery’s upside is salient.

Overall, the choice data reveal that risk preferences are systematically unstable across choice sets, and this instability is largely consistent with salience theory. We find that the basic prediction of the BGS model is upheld: risk taking is greater when the lottery’s upside is salient compared to when the downside is salient. We then compare salience theory with the leading behavioral model of choice under risk, cumulative prospect theory (CPT) (Tversky and

Kahneman 1992). After structurally estimating each model, we find that both can fit the data reasonably well, with salience theory providing a slightly better fit than CPT. Our structural estimation reveals that much of the variation in risk taking is explained by the context-dependent

3 weighting function of salience theory, while CPT relies heavily on a large degree of concavity in the value function.

At the eye-tracking level, the results are less supportive of the precise psychological mechanism that is assumed to generate context-dependent risk preferences in salience theory. The model’s key assumption implies that subjects should pay more attention to the state that is theoretically salient. We do not find support for this prediction at the eye-tracking level.

However, because of the richness of the eye-tracking data, we conduct additional exploratory analyses to better understand what factors do govern the attention allocation process. We find that when the gain state is theoretically salient, subjects allocate more attention overall to the risky lottery compared to the certain option. Moreover, the amount of attention that subjects pay to the risky lottery relative to the certain option is a strong predictor of the ultimate choice. This result is consistent with a recent model from the neuroeconomics literature that predicts that the computation of decision values is modulated by attention (Krajbich, Armel, and Rangel 2010).

In our second experiment, we examine another dimension of salience and its impact on risky choice. The salience function in BGS that maps payoffs into a salience measure is motivated by the literature on visual perception3. In the same manner that BGS propose that extreme economic payoffs are salient, the literature on visual perception identifies those attributes of visual stimuli that are salient and attract a DM’s attention (Itti and Koch 2001; Wolfe and

Horowitz 2004; Knudsen 2007). For example, the color, brightness, or shape of an object can make it “pop-out” against the environment. To make the distinction between these two sources of salience clear, we refer to the salience driven by economic payoffs in BGS as “economic salience” and the salience driven by simple visual features of objects in the environment as

“visual salience.” A lottery payoff that is extreme compared to the magnitude of other payoffs is economically salient; however, this same payoff may not be visually salient if it is displayed in

3 Woodford (2012) also argues for motivating the assumptions of behavioral economic models from the literature on visual perception.

4 faint gray font on a dark gray background compared to another payoff displayed in bold black font on the same gray background.

While the motivation for economic salience in BGS is firmly rooted in the well-known properties of visual salience, BGS do not explicitly incorporate visual salience into their model.

However, recent research in neuroeconomics shows that visual salience interacts with economic value in riskless choice (Towal, Mormann, and Koch 2013). We therefore hypothesize that incorporating visual salience into any model of risky choice can provide extra explanatory power.

As a first step in formally testing this hypothesis, we generalize the BGS model to explicitly incorporate visual salience, which we call the full salience model.

In this second experiment, we exogenously manipulate the visual salience of lotteries to test the full salience model. In the control condition, as in Experiment 1, subjects make decisions between a risky lottery and a certain option, where all information is displayed in the same font.

In the “visually salient” condition, we increase the visual salience of the state of the world that is economically salient by displaying it in bold font. We then test whether risk appetites are systematically different in the visually salient condition compared to the control condition, in the precise manner predicted by our full salience model.

The results from our second experiment show that exogenously manipulating the visual salience of lottery payoffs causes a shift in risk taking. Specifically, making a lottery’s downside payoff visually salient can reduce risk-taking by nearly 50% for some lotteries. We then provide a structural estimation of the full salience model, and show that it significantly outperforms the restricted model that does not incorporate visual salience. Our results suggest that visual salience and economic salience interact to systematically generate context-dependent risk preferences.

These results are also valuable for understanding the direction of causality between attention and preferences. Because we exogenously manipulate visual salience and find that it systematically changes risk taking, we demonstrate that attention causally affects preferences.

5 Our paper contributes to a large literature in mathematical psychology, and more recently in behavioral economics, on modeling the relationship between attention and economic choice.

Busemeyer and Townsend (1993) propose a model called decision field theory, where random fluctuations in attention bias the decision weight attached to an attribute. Diederich (1997) extends this model to provide more structure on transitions of attention between attributes, while

Krajbich et al. (2010) propose the attentional drift diffusion model which makes predictions about the joint distribution of eye-movements, choice outcomes, and reaction times. One common feature of these cognitive models of decision-making is that while attention is assumed to influence economic choice, the process that governs attention is taken to be exogenous.

Fortunately, several recent models from the behavioral economics literature have begun to endogenize the attention process as a function of the economic environment. The BGS model that we test here is the first to provide this endogenous ex-post attention allocation structure4.

Kozsegi and Szeidl (2013) provide a model where agents focus on attributes that are most different across alternatives, Schwartzstein (2014) studies an agent whose selective attention leads to biased belief updating and Gabaix (2014) builds a general and tractable theory of an agent who chooses attention weights to build a “sparse” model of the world. While not motivated directly by the psychology of attention, Cunningham (2013) and Bushong, Rabin, and

Schwartzstein (2016) provide related formal models where the choice set endogenously influences decision weights. Our experimental data is useful in guiding the development of this theoretical literature as we provide tests of both the attention allocation process (through the eye- tracking data) and of the effect of attention on choice under risk (through the data on lottery choices).

4 There is also a large literature in economics on models of rational inattention (Sims 2003). However, these models are fundamentally different in that they assume agents rationally (ex-ante) allocate their scarce attentional resources. We expand on this point later in the paper when interpreting the results from our second experiment as evidence of ex-post attention allocation. 6 At a methodological level, our paper responds to a recent call by Caplin (2016, pg. 5) that

“advances in measurement are absolutely required for the next stage of attentional research.” He argues that in light of the growing number of both rational and behavioral economic models of inattention, collecting non-standard choice process data is essential for disciplining this new class of theories. Eye-tracking is one example of this type of non-standard choice process data, and while it has been used for decades to study cognition in the psychology literature (Kahneman and

Beatty 1966; Just and Carpenter 1976; Hayhoe and Ballard 2005; Gloeckner and Herbold 2011), only recently has it been used to investigate choice processes in economics (Wang, Spezio,

Camerer 2010; Reutskaja, Nagel, Camerer, and Rangel 2011; Arieli, Ben-Ami and Rubinstein

2011; Hu, Kayaba, Shum 2013; Polonio, Guida, and Coricelli 2015) 5. Our paper builds on this work by providing an example of how eye-tracking data can be used as a measure of attention to directly test the assumption of a behavioral economic theory.

II. EXPERIMENT 1: EYE-TRACKING EXPERIMENT

In our first experiment, we incentivize subjects to make a series of choices under risk while we record their eye-movements with an eye-tracker. We first describe our experimental design, then develop the theoretical predictions under the BGS model, and finally present results.

II.A. Experimental Design

Each subject with normal or corrected-to-normal vision enters the lab one at a time, is seated in front of a computer with eye-tracking equipment, and is given instructions about the experimental protocol, which are provided in the appendix (17 participants, 1 omitted due to poor calibration, 10 female, mean age = 24.73 years, SD = 5.51). Each subject then completes several practice trials to familiarize herself with the task. Next, a nine-point eye-tracking calibration is

5 Other types of choice-process data include mousetracking (Gabaix, Laibson, Moloche, and Weinberg 2006; Brocas, Caririllo, Wang, and Camerer 2014) and provisional choice data (Caplin, Dean and Martin 2011).

7 performed immediately before the main task begins in order to ensure a precise recording of the eye-tracking data. During the main task, each subject makes a series of thirty-five choices between a risky lottery and a certain option while her eye-movements are recorded.

For each of the thirty-five trials, the risky lottery � = (�, �; �, 1 − �) is a mean- preserving spread of the certain option, � = �, 1 where � > � > � ≥ 0 and

� ∈ 0,1 . There are thus two states of the world, � ∈ {����, ����}. We define Pr � = ���� =

� because with probability p, the risky lottery delivers a gain, g, relative to the certain option, c.

The parameter values for all thirty-five choice sets are given in Table I.

A screenshot from one of the trials is shown in Figure I. The two alternatives are always displayed one above the other, with the certain option randomly placed on top in some trials and on the bottom in others. The lotteries are named using random letters (e.g., D vs. A) to avoid any ordering effects (e.g., using 1 vs. 2, or A vs. B). Furthermore, within the risky lottery, the display location of the gain state is randomized across trials so that it is displayed on the left in some trials and on the right in others. This randomization is important because it controls for any left- right or top-bottom bias in eye-movements that subjects may have if they are accustomed to reading in this direction.

Subjects are instructed to decide which of the two alternatives they prefer on each trial and to indicate their choice by looking at the preferred alternative and pressing the space bar.

Between trials, a fixation cross is placed on the screen between the two lotteries and subjects are asked to look at the cross. This ensures that the initial eye fixation does not fall on either of the two lotteries when the trial begins, which could potentially bias the initial fixation data.

During the entire experiment, the eye-tracker (Tobii X2-60, Tobii Technology, Sweden) records subjects’ viewing patterns as they examine the lotteries and make their choices. Specifically, we collect the location, duration, and pupil diameter of each eye-fixation sixty times per second.

Figure I shows sample eye-fixations for one subject on a typical trial.

8 Upon completion of the thirty-five choices, the experimenter selects one trial using a random-number generator. If the subject chooses the certain option on the selected trial, she is paid the stated amount of the certain option. If instead she chooses the risky lottery, the subject rolls a 10-sided die twice to determine the stochastic outcome of the risky option6. We use a 1:1 exchange rate between experimental dollars and USD, and subjects are instructed that they can earn as much as $2,040 (the largest upside of one of the lotteries) or as little as $0 (the lowest downside on some lotteries), and earnings are paid in cash at the end of the experiment. The potential large upside of earnings is used to provide an incentive compatible method for eliciting preferences over positively skewed lotteries. In addition to their earnings, subjects are paid a $5 participation fee, which generates an average total pay of $5 + $36.88 = $41.88 (SD = $16.82).

II.B. Theoretical Predictions for Behavior

Under salience theory, risk preferences will be systematically context-dependent. In other words, a subject will exhibit risk-seeking behavior on some trials and risk-averse behavior on other trials. We use the BGS model to generate sharp predictions about the specific trials in which subjects will exhibit risk averse compared to risk seeking behavior. We review this model now.

When confronted with a choice between two lotteries, a subject who is prone to choosing according to salience theory will direct her attention towards the most salient state of the world.

The key behavioral implication of this attention allocation is that the subject overweights those states of the world that are most salient. Because there are only two states of the world in our experiment, this means that one state will be overweighted relative to the objective state probability, and one state will be underweighted.

6 Specifically, rolling the 10-sided die twice generates two integers, � ∈ [0, 9] and � ∈ [0, 9]. We use these two numbers to construct a variable � = 10� + � , which takes on integer values in the range [0, 99]. If � ≥ 100(1 − �), (where p is the probability of the gain state), then the subject receives the payoff from the gain state; otherwise, she receives the payoff from the loss state.

9 To determine which state is theoretically salient, we rely on the definition provided by

BGS. In particular, BGS define a salience function to be a continuous and bounded function that maps payoffs into salience measures and that satisfies two properties: ordering and diminishing sensitivity. Under ordering, the salience of a state is higher when payoff levels within the state are further from the reference level; under diminishing sensitivity, salience decreases as payoff levels

7 ! ! rise . In what follows, we specialize our analysis to a specific salience function, � �! , �! =

! ! !! !!! ! ! ! ! , where �! denotes the risky lottery payoff in state � ∈ ����, ���� and �! denotes the |!! ! |!! certain option payoff in state � ∈ ����, ���� . While the specific salience function we use here places tighter restrictions on the choice data than are implied by the general properties of ordering and diminishing sensitivity, it provides useful structure in formulating our predictions.

Given the salience function, we say that the gain state is more salient than the loss state if

! ! ! ! � �!"#$, �!"#$ > � �!"##, �!"## . Salience distorts the objective probability of each state, where we define the distorted probability of the gain state by �∗. As in the previous section, the objective probability of the gain state is given by �. The subject distorts the odds ratio of the gain state to the loss state as follows:

! ! !∗ ! ! !!"##, !!"## ! = × (1) !!!∗ ! !! , !! !!! ! !"#$ !"#$

Rearranging terms allows us to explicitly solve for the distorted probability �∗:

! ! ! ! ! ! , ! !! ! , ! ! ! !"## !"## !"#$ !"#$ × ∗ !!! � = ! ! ! ! (2) ! ! , ! !! ! , ! ! !!( ! !"## !"## !"#$ !"#$ × ) !!!

7 ! ! ! ! Formally, suppose there are two states s and s’. Ordering implies that if [�! , �! ] is a subset of [�!! , �!!] ! ! then state s’ is more salient than state s. Diminishing sensitivity implies that if �! > 0 and �! > 0, then for ! ! any � > 0, state s becomes less salient if � is added to each payoff �! and �! . 10 In this expression, the parameter � ∈ 0,1 captures the degree to which salience distorts objective probabilities, where this distortion is a smooth increasing function of the difference in salience across states. When � = 1, there is no probability distortion and �∗ = �. When � < 1 the distorted probability of the gain state, �∗, is larger than the objective probability of the gain state,

! ! ! ! p, if and only if � �!"#$, �!"#$ > � �!"##, �!"## . In other words, when � < 1, the subject overweights the gain state whenever it is more salient than the loss state.

Given this distortion algorithm, we can now assess how salience impacts decision- making under risk. We assume that the utility of a payoff is given by the value function8,

� � = �!. Subjects compute the value of each alternative under the distorted probabilities, and choose the risky lottery if its value, � � exceeds that of the certain option, � � . In particular, a subject will choose the risky lottery if and only if:

� � − � � > 0

⇒ (�∗ × �! + 1 − �∗ × �!) − �! > 0. (3)

We refer to the left-hand side of this inequality as the decision value of the risky lottery, which represents the relative value of choosing the risky lottery compared to the certain option. Finally, we assume that the value of each alternative in the choice set is subject to an independent and additive random shock, which leads the subject to exhibit stochastic choice (McFadden 2001)9.

This leads to our first prediction regarding the distribution of choices as a function of the � generated decision values.

8 This assumes that subjects narrowly frame payoffs, instead of computing utility by first merging the payoff with their total wealth level. We assume that subjects use a reference point of zero when evaluating payoffs. 9 The theoretical foundation of stochastic choice is an active area of research. Besides the random utility model interpretations, other theories of stochastic choice include deliberate randomization (Fudenberg, Iijima, and Strzalecki (2015); Agranov and Ortoleva (2015)) and bounded rationality, which can be derived from neurobiological constraints on the choice process (Fehr and Rangel (2011); Webb 2015)).

11 PREDICTION 1: If subjects make decisions according to salience theory, then the probability of choosing the risky lottery should be increasing in the decision value of the risky lottery.

II.C. Theoretical Predictions for Eye-Tracking Data

In the previous section, we described how salience is computed and how it impacts a subject’s decision calculus. Salience theory, when taken literally as a description of the decision- making process, also makes predictions about psychological mechanism that generates risk- taking. Because attention is the key psychological variable in salience theory, we collect data on eye-movements, which are known to be a good measure of overt attention (Kustov and Robinson

1996). Moreover, eye-movement data has been used in recent experimental work as a measure of attention in order to test a model of the influence of attention on economic choice. This attentional drift diffusion model (aDDM) makes predictions about the joint distribution between eye-movements, reaction times, and choice outcomes (Krajbich et al. 2010). We briefly review this model now, as it will guide some of our eye-tracking predictions.

A key feature of the aDDM is that controlling for the subjective preference of the alternatives in a choice set, attention to an alternative (as measured through eye-fixations on the alternative) biases economic choice. The model predicts that, all else equal, the longer a subject looks at alternative � ∈ {�, �}, the higher is the probability that � is ultimately chosen from the choice set {�, �}. This effect has been documented using eye-tracking data in experiments on consumer choice (Krajbich et al. 2010, Krajbich and Rangel 2011, Krajbich at el. 2012).

Moreover, there is evidence that the link from attention to choice is causal, as exogenous manipulation of attention has been shown to systematically bias economic choice (Armel,

Beaumel, and Rangel 2008, Hare, Malmaud, and Rangel 2011; Milosavljevic et al. 2012, and

Frydman and Rangel 2014).

Given the growing empirical evidence on the link between attention and economic choice, it becomes critical to understand what drives the attention allocation process. This is

12 because any observables that govern the attention process should therefore also have a systematic impact on economic choice. The aDDM takes the attention process as exogenous, in the sense that the amount of time a subject fixates on an alternative is not correlated with the preference for that item10. BGS provides a theory of how attention is endogenously allocated in risky choice settings, and thus when combined with the aDDM, we can generate predictions from the BGS model about the eye-tracking data.

Before proceeding with the predictions, it is important to emphasize two limitations in applying the aDDM in our experiment. First, we do not know whether the aDDM extends to multi-attribute choice, in which attributes of an alternative that receive excess attention will receive excess weight in the decision process.11 Second, the aDDM has thus far only been applied to deterministic choice settings, and it is unclear whether the same attentional mechanisms are deployed during choice under risk. In particular, the aDDM does not provide guidance on how payoffs and probabilities are integrated, though recent theoretical work has begun to model this process (Johnson and Busemeyer 2016). With these two limitations in mind, we now turn to generating the eye-tracking predictions.

In order to generate predictions that are testable using the eye-tracking data, we need to construct an empirical measure of the amount of attention that a subject pays to each state.

Because the certain option delivers the same payoff in both states, we cannot identify which state a subject is attending to when he looks at the certain option. In contrast, when the subjects looks at the risky lottery, we can identify which state he is attending to since each payoff is state-

10 Towal et al. (2013) provide a model that endogenizes the attention process for consumer choice, by assuming that attention is driven by both the preference for an alternative and its “visual salience.” However, it is unclear how this model would extend to risky choice settings, or more generally, to settings where alternatives have multiple attributes. 11 A recent paper by Fisher (2016) provides evidence that the aDDM does extend to multi-attribute choice settings, although his setting does not examine choice under risk. Krajbich, Hare, Bartling and Fehr (2015) use aDDM parameters estimated from a food choice task to predict response times and choices in a multi- attribute social decision-making setting, although they are not able to test predictions about attention since they do not collect eye-tracking data in the social decision-making task. Hare, Malmaud, and Rangel (2011) provide evidence that exogenous cues to attributes of food items can influence attribute weights and bias consumer choice. 13 dependent. Therefore, we partition the experimental display screen into six separate areas of interest (AOIs) as shown in Figure II. We construct an AOI that contains each of the three payoffs, and each of their associated probabilities.

!"#$$!!"##$ We define ������������$ = , which provides a measure of the amount of !"#$$!!"##$ attention a subject pays to the gain state compared to the loss state12. Under the BGS theory,

������������$ should therefore be larger as the gain state becomes more salient and attracts more attention. This leads to our first eye-tracking prediction:

PREDICTION 2A: Under the BGS theory, as the gain state becomes more salient, subjects will focus more attention on the gain dollar amount of the risky lottery compared to the loss dollar

! ! ! ! amount of the risky lottery. Therefore, � �!"#$, �!"#$ − � �!"##, �!"## should positively correlate, across trials, with ������������$.

Importantly, if this attention allocation has a systematic impact on choice, then the relative amount of time a subject focuses on the gain state compared to the loss state should correlate with the choice. This prediction is motivated by the aDDM, and is independent of the endogenous attention allocation process that is predicted by BGS. In other words, even if attention is exogenously allocated across the gain state and loss state, these random fluctuations in attention should still bias economic choice. This leads to our second eye-tracking prediction:

PREDICTION 2B: If state-specific attention biases risky choice, then as ������������$ gets larger, the probability of choosing the risky lottery will increase.

12 An implicit assumption of this interpretation is that when the subject looks at the Certain$ AOI, the proportion of time he attends to the gain state relative to the loss state is also given by ������������$. We expand on this assumption in the discussion section.

14 II.D. Results from the Choice Data

II.D.i Basic Choice Results and Structural Estimation of Salience Theory

Overall, subjects choose the risky lottery on 41.2% of trials. Table II.A presents the average risk taking results disaggregated by trial, where each cell corresponds to a different trial13. Each column characterizes the probability of the gain state and each row characterizes the expected value of the risky lottery (and, by design, of the certain option). The shaded cells correspond to those trials in which the gain state is salient. Under salience theory, when the gain state is salient, a subject will distort the probability of the gain state upward, which should induce risk-seeking behavior.

We find that subjects are more willing to take risk when the gain state is salient (48.0%) compared to when the loss state is salient (35.1%). To formally test whether this difference is significant we estimate a linear probability model that regresses the probability of choosing the risky lottery on a dummy variable that equals one if the gain state is salient, and we cluster standard errors by subject. Column (1) of Table A.1 shows that the coefficient on the dummy is significantly positive (p=0.048), and columns (2) and (3) show that the result is robust to including subject fixed effects, and using a logistic specification. Column (4) shows that the result remains significant when estimating a mixed effects logistic regression, which is useful in accounting for heterogeneity in average risk taking across subjects.

One other basic pattern that is evident from Table II.A is that for each of the seven probabilities of the gain state (each column of the table), subjects exhibit nearly a 100% increase in risk taking as the expected value increases from $20 to $60. Salience theory can deliver this prediction because the probability weighting function is context-dependent. In other words, for two trials that have the same objective probability of the gain state, the distorted probabilities can still differ if the expected values of the lottery differ.

13 Due to a software error, we failed to record data on two trials for eight of our sixteen subjects, and thus these trials are excluded from the subsequent analyses in this experiment.

15 Figure III shows this context-dependent property of the weighting function for the five different levels of expected values in our design, generated with the parameter value � = 0.2114.

The dashed black line denotes the 45-degree line, which crosses each weighting curve at a different point, indicating that the shift from overweighting to underweighting objective probabilities is a function of average payoff levels. It is important to emphasize that in Figure III, c is not a free parameter that generates different shapes of the weighting function; on the contrary, c is endogenous and is fully pinned down by the choice set on each trial in our experiment. For example, on trials in which the expected value is $20, the downside of the lottery is always $0

(because losses are fixed at $20 by design), which means that the loss state is maximally salient and probabilities are therefore always underweighted. In contrast, when the expected value of the lottery is $60, objective probabilities are overweighted in the range (0, 0.4). In order to investigate how these context-dependent weighting curves generate variation in the model- predicted choice probabilities of choosing the risky lottery, we need to compute the decision value for each trial.

Because the DV of each trial is a function of the parameters (�, �), we estimate (�, �) using maximum likelihood estimation. In our estimation, we assume that all subjects have the same preference parameters15, �, � . We also assume that choices are stochastic, where each alternative in the choice set is subject to an independent and additive random shock. We assume the shocks are extreme value type I distributed, which implies that the probability that subject s

! accepts the risky lottery on trial t is, �(� | �, �, �) = . We define � = 1 if subject ! !!!!!×!"! !,! !,! s chooses the risky lottery on trial t, and �!,! = 0 if subject s chooses the certain option on trial t.

The parameter � controls the degree of stochasticity of choice (the scale of the logistic

14 The same comparative statics hold for all values of � ∈ 0,1 ; we choose this value because it is the best fitting parameter for our data as we describe below. 15 The assumption of identical preferences across subjects is unrealistic, but we pool trials across subjects given the limited number of trials per subject. One advantage of this aggregation, however, is that we estimate the preference of the “representative agent.”

16 distribution), and ��! �, � denotes the decision value of the risky lottery on trial t. We estimate

(�, �, �) by pooling all trials and subjects and maximizing the following log likelihood function:

�� �, �, � �) = ! ! �!,! log �(�! | �, �, �) + (1 − �!,!) log 1 − �(�! | �, �, �) (4)

We find that the log likelihood is maximized at � = 0.21, � = 0.62, and � = 0.35. Table

II.B provides the decision values for each trial, conditional on these estimated parameters16.

When fixing the probability of the gain state, we find the decision value of the risky lottery does increase with the expected value of the risky lottery. This is driven by two factors. First, the concavity of the value function will generate an increase in risk taking as average payoff levels rise. Second, there is probability weighting effect that is independent of the value function.

Because diminishing sensitivity of salience gets weaker at higher average payoff levels, the loss state becomes less salient at higher average payoff levels, which leads to a larger value of the distorted probability �∗ (Figure III). In other words, salience generates variation in �∗ even for a fixed level of �. This implies that, even with linear utility, the context-dependent property of the salience function is sufficient to generate the positive relationship between average payoff levels and risk taking.

In order to compute the model implied probability of choosing the risky lottery on each trial, the decision value is passed through a stochastic choice function, which is a logistic function parameterized by �. For illustration purposes, in Figure IV we plot the empirical probability of choosing the risky lottery as a function of the model implied probability. These results are consistent with Prediction 1, as the probability of choosing the risky lottery is increasing in the

16 One potential concern is that a majority of the estimated decision values in Table II.B. are negative, even for those choice sets where the gain state is salient. However, our stochastic choice function imposes the restriction that, when the decision value equals zero, subjects choose the risky lottery exactly 50% of the time. Thus, because we rarely observe subjects choosing the risky lottery more than 50% of the time, it is to be expected that a majority of the estimated decision values are negative.

17 decision value, and more generally, the propensity to take risk is significantly higher when the gain state is salient compared to when the loss state is salient.

II.D.ii An Alternative Behavioral Theory: Cumulative Prospect Theory

While our results in the previous section are consistent with the BGS model, they may also be explained by other behavioral theories of decision-making under risk. In this subsection, we examine whether cumulative prospect theory (CPT) (Tversky and Kahneman 1992) can explain the observed patterns of risk taking. To do so, we repeat the same logic as in the previous section, and compute the decision value for each trial under CPT. We use the value function given by:

(� − �)! � ≥ � � � = (5) −�(−(� − �))! � < �

The parameter � describes the curvature of the value function. The parameter � governs the degree of loss aversion, which controls the extent to which subjects are more sensitive to losses than gains. In this specification, we assume that the reference point, �, is given by $0, which is the same assumption we made for salience theory in the previous section. Because all payoffs in our experiment are positive, the $0 reference point implies that all payoffs are perceived as gains, and thus we cannot estimate the loss aversion parameter in this specification. Finally, CPT posits that subjects use decision weights instead of objective probabilities. Following Tversky and

Kahneman (1992), the mapping from the objective cumulative probability distribution to these decision weights is captured by the probability weighting function17:

!! � � = ! (6) (!!!(!!!)!)!

17 For simplicity we assume the weighting function is identical for gains and losses.

18 Therefore, under CPT with a $0 reference point, the decision value of the risky lottery is given by:

!"# ��! = � � − � �

= � � × � � + 1 − � � × � � − � �

= � � × � ! + 1 − � � × � ! − �!

(7)

As in the previous section, we assume subjects exhibit stochastic choice, where the probability

! that subject s accepts the risky lottery on trial t is, �(� | �, �, �) = . To estimate the ! !!!!!×!"! !,! parameters, (�, �, �), we pool all trials and subjects and maximize the log likelihood function:

�� �, �, � �) = ! ! �!,! log �(�! | �, �, �) + (1 − �!,!) log 1 − �(�! | �, �, �) (8)

We find that the best fitting parameter values are � = 0.33, � = 0.18 , � = 0.55, which characterizes a value function with a very high degree of curvature. Table II.C provides the estimated decision values, and Figure V displays the relationship between the empirical choice probabilities and the CPT model implied probabilities. Upon visual comparison of Figure IV and

Figure V, we see that both salience and CPT with a zero reference point do a reasonably good job explaining variation in risk taking across trials. To formally compare the models, we compute the

AIC for both: 711.34 for salience and 712.44 for CPT, which implies that the models provide similar fits, with the salience model being slightly more likely to fit the data.

It is worth emphasizing, however, that in order for CPT to provide this fit and explain the increase in risk taking as payoff levels rise, the value function must contain a large amount of concavity (� = 0.33). In contrast, salience theory can provide this result with a more moderate

19 level of value function concavity (� = 0.62) because the distorted probability �∗ is allowed to vary with average payoff levels (Figure III).

One feature that may seem unusual about the previous CPT specification is that we do not estimate the loss aversion parameter, �. Recall that this is because all payoffs in our experimental design are positive, and thus when the reference point is zero, all payoffs are perceived as gains. An alternative specification that sets the reference point equal to the certain option will generate some outcomes that are perceived as gains and others that are perceived as losses, which allows us to estimate �. The decision value under this CPT specification, where

� = �, is defined as:

!"# ��! = � � − � �

= � � × � � + 1 − � � × � � − � �

! = � � × � − � ! + 1 − � � × −� − � − � (9)

Estimating this model using MLE we find that the best fitting parameters are � = 0.56, � =

0.37 , � = 1.55, and � = 0.05; however, the AIC is 723.6, indicating that it provides a much worse fit to the data than the CPT specification with a zero reference point. To see why this CPT specification fits so poorly, notice that for any � > 0, the trial characterized by the choice set

�, � = �, �; �, 1 − � , (�, 1) , will have the same decision value as the trial characterized by

�!, �! = � + �, �; � + �, 1 − � , (� + �, 1) . In other words, adding a positive constant to every payoff will leave the decision value and choice probability unchanged because both the outcome and reference point are increased by the same amount. Thus, in our design, this CPT specification generates only seven distinct decision values (one for each level of �) and it therefore cannot explain the systematic increase in risk taking as average payoff levels rise.

20 In sum, we find that the BGS theory can explain variation in risk-taking as subjects are significantly more likely to take risk when the gain state is salient. CPT with a zero reference point can also provide a comparable fit to the data, but the value function must be sufficiently concave in order to generate the positive relationship between risk-taking and average payoff levels. Finally, CPT where the reference point is set to the certain option cannot explain the data.

In the next section, we examine the psychological mechanism that generates the choice data.

II.E. Eye-Tracking Results

We now turn to the eye-tracking data, which can provide insight on the underlying psychological mechanism that generates the choice data we analyzed in the previous section. We begin by presenting an overview of the average total time that a subject looks at each of the six

AOIs defined in Figure II18. In addition to these six AOIs, we also compute average total fixation durations for the “salient” state and “non-salient” state, which are defined by the BGS salience function. Upon visual inspection of Figure VI, the data indicate that subjects look at dollar amounts more than probabilities, and they spend only a very small amount of time, approximately

200 milliseconds, looking at the probability associated with the certain option. We use a subset of this data to begin constructing tests of Predictions 2A and 2B.

Prediction 2A is concerned with testing whether subjects allocate their attention to those states that are predicted to be salient. In particular, we check whether, as predicted by BGS, subjects spend more time attending to the state that is salient compared to the state that is not

!"#$$!!"##$ salient. To test this prediction we construct the variable ������������$ = , where !"#$$!!"##$

Gain$ and Loss$ denote the total amount of time a subject looks at the payoff that is delivered by the risky lottery in the gain and loss state, respectively. Under the BGS model, the theoretical

18 We first pre-process the eye-tracking data using the same method as in Krabjich et al. (2010). For all measurements following the first fixation and preceding the last fixation, we define a non-ROI fixation as a time point in which the eye-tracker does not record a fixation on any of the six AOIs (e.g., because of an eye-blink or because the subject looks outside one of the AOIs). If a non-ROI fixation is recorded between fixations on the same ROI, we change the non-ROI fixation to that ROI.

21 analogue of this variable is the difference between the salience of the gain state and that of the

! ! ! ! loss state: � �!"#$, �!"#$ − � �!"##, �!"## . We run a linear regression of ������������$ on

! ! ! ! [ �!"#$, �!"#$ − � �!"##, �!"## ] but find that there is no significant relationship between these two variables (p=0.36). Table A.2 provides the regression results and also shows there is no significant effect if we include subject fixed effects or estimate a mixed effects linear model.

However, it is important to note that an implicit assumption of this test is that when the subject looks at the Certain$ AOI, the proportion of time she attends to the gain state relative to the loss state is given by ������������$. Under this assumption, our data reject prediction 2A, and in the discussion section we speculate why we do not find this result.

Even though we find that subjects do not attend to the gain state longer when it is more salient, it is still possible that the longer subjects attend to the gain state compared to the loss state, the more likely they are to choose the risky lottery. This prediction is motivated by the aDDM framework, which takes the attention process as exogenous, but still predicts that attention biases economic choice. Testing whether state-based attention influences risky choice can therefore still provide evidence on the decision-making process of choice under risk. We therefore proceed with testing Prediction 2B, which says that if state-based attention influences risky choice, then ������������$ should positively correlate with the probability of choosing the risky lottery. Using a linear probability model, we regress the probability of choosing the risky lottery on ������������$ but find that there is no significant relationship between the two variables (p=0.33, Table A.3). Table A.3 provides the regression results, and also shows that there is no significant effect if we use a logistic or mixed effects logistic specification.

So far in this section we found that the salience function in BGS cannot explain variation in attention as measured through our eye-tracking data. In order to explore what factors do govern attention allocation in our experiment, we present additional post-hoc analyses of the eye-tracking data. As discussed above, a key prediction from the aDDM of consumer choice is that when a

22 subject looks more at an appetitive alternative (an alternative for which he has a positive willingness to pay) then she is more likely to ultimately choose this alternative from the choice set (Krajbich et al. 2010). Because all payoffs in our experiment are positive, and because we assume a zero reference point, every payoff is perceived by the subject to be a gain. Thus, a plausible extension of the aDDM applied to risky choice would predict that the longer a subject looks at any payoff in our experiment, the higher is the probability she chooses the alternative associated with that payoff.

For example, in our setting, the longer a subject looks at the payoff from the risky lottery delivered in the gain state, the higher the probability she chooses the risky lottery. Similarly, because the payoff delivered by the risky lottery in the loss state is also perceived to be a gain

(because it is positive), this predicts that the longer a subject attends to the loss payoff, the higher is the probability she chooses the risky lottery. Finally, the longer the subject looks at the payoff of the certain option, the higher is the probability she chooses the certain option. These basic predictions describe the hypothesized relationship between the attention process and risky choice.

We test these predictions by constructing a variable that represents the total amount of time subjects spend fixating on the risky lottery payoffs relative to the certain option payoff. We

!"#$$!!"##$!!"#$%&'$ define this variable as �������������$ = , where Gain$ and Loss$ denote !"#$$!!"##$!!"#$%&'$ the total amount of time a subject looks at the payoff that is delivered by the risky lottery in the gain and loss state, respectively. Certain$ denotes the total amount of time a subject looks at the certain option payoff. �������������$ has only a small correlation of 0.06 with ������������$, suggesting that the relative amount of time a subject spends fixating on the gain state relative to the loss state is independent of the relative time she spends looking at the risky lottery compared to the certain option.

We cannot directly test whether this variable correlates with risky choice because subjects are instructed to fixate on their chosen alternative when they are ready to execute a

23 decision, which would introduce a bias in the test. Therefore, we construct a modified version of this variable, which measures the relative amount of time that a subjects looks at the risky option compared to the certain option in the early stages of the decision process, a period in which recent work has shown that subjects selectively encode attributes of stimuli (Glaholt and Reingold

2009a, Glaholt and Reingold 2009b, Schotter et al. 2010, Glaholt and Reingold 2012).

Specifically, we replace ����$, ����$, �������$ with ���������$, ���������$, ������������$, which denote the duration of the first fixation on the gain payoff, the loss payoff and the certain payoff, respectively. We then construct

!"#$%#&'($!!"##!"#$%$!!"#$%&'(&#)$$ ������������������$ = , and use this as our measure of the !"#$%#&'($!!"##$%&#'$!!"#$%&'(&#)$$ amount of time a subject spends looking at the risky option compared to the certain option. As expected, this variable is positively correlated with �������������$ (r=0.42), since any time advantage towards looking at the risky option in the early stage will impact the overall time advantage of looking at the risky option.

To test whether relative attention to the risky lottery correlates with choice, we estimate a linear probability model that regresses the probability of choosing the risky lottery on

������������������$ and we cluster standard errors by subject. Column (1) of Table A.4 shows that the correlation between relative attention to the risky lottery and actual choices is positive and strongly significant (p=0.002), and columns (2) and (3) show that the result is robust to including subject fixed effects, and using a logistic specification. It is also worth noting that the mean value of ���������$ + ���������$ + ������������$ across subjects and trials is 1307 ms

(SD = 840 ms), which indicates that eye-movements from the first two seconds of the trial can successfully predict the upcoming choice.

We can also explore if there are any factors from the choice set that explain variation in attention to the risky lottery relative to the certain option. This is analogous to the test of

Prediction 2A, which was concerned with the endogenous allocation of attention at the attribute

24 level. One plausible hypothesis is that when the gain state is salient, this biases the subject towards choosing the risky lottery, but it does so by biasing attention towards all payoffs of the risky lottery. Similarly, when the loss state is salient, this biases the subject towards choosing the certain option, but it does so by biasing attention towards the payoff of the certain option. One interpretation of this hypothesis is that it describes a fundamentally different psychological mechanism compared to the one assumed by BGS, but at the same time, the inputs to the model

(the salience measures of the gain and loss states) still govern the endogenous attention process.

We can test this prediction by running a linear regression of ������������������$ on

! ! ! ! ! ! � �!"#$, �!"#$ − � �!"##, �!"## . Table A.5 shows that the coefficient on � �!"#$, �!"#$ −

! ! � �!"##, �!"## is 0.126, which is highly significant (p=0.002), and columns (2) and (3) demonstrate the result is robust to different model specifications. Figure VII provides a scatterplot of the relationship between ������������������$ and the degree to which the gain state is more salient than the loss state.

In sum, we find that the eye-tracking data does not support the specific endogenous attention allocation that is predicted in the BGS theory. However, in post-hoc analyses, we find evidence of a different psychological mechanism: relative attention to the risky lottery payoffs compared to the certain option payoff strongly predicts risky choice. Importantly, the salience function of BGS explains the attention allocation between the risky lottery and the certain option.

III. EXPERIMENT 2: VISUAL SALIENCE AND ITS IMPACT ON RISKY CHOICE

While the salience function, � ∙ , is motivated by the literature on visual perception (Itti and Koch 2001), visual salience is not explicitly modeled in the BGS theory. To illustrate the distinction between economic salience (modeled in BGS) and visual salience (not explicitly modeled in BGS), recall that a lottery payoff that is extreme compared to the magnitude of other payoffs is economically salient; however, this same lottery payoff may not be visually salient if it

25 is displayed in faint gray font on a dark gray background compared to another payoff displayed in bold black font on the same gray background. In this second experiment, we extend the BGS model and test whether visual salience is also an important driver of risky choice. We describe the theory, experimental design, and results in the next three subsections.

III.A. Theoretical Background and Predictions

A key assumption from the BGS model is that the objective probability distribution

(����, �; ����, 1 − �) is distorted to ����, �∗; ����, 1 − �∗ , given by:

! ! !∗ ! ! !!"##, !!"## ! = × (10) !!!∗ ! !! , !! !!! ! !"#$ !"#$

Because BGS only model economic salience in their theory, we refer to the first term on the right- hand side as the “economic salience distortion factor.”

However, it is well known that two types of factors attract attention: top-down and bottom-up factors. Top-down factors are cognitive and are related to goal-directed behavior

(Connor, Egeth, Yantis 2004; Pieters and Wedel 2004; Knudsen 2007; Milosavljevic and Cerf

2008), such as obtaining monetary payoffs. Therefore, the economic salience that BGS model belongs to this first set of factors. On the other hand, bottom-up or “stimulus-based” factors, such as accessibility or visual salience of information, can involuntarily attract attention. These bottom-up factors are also important for economic choice because it has been shown that they can override one’s voluntary goals and intentions (Milosavljevic, Navalpakkam, Koch and Rangel

2012) and influence economic choice (Towal, Mormann and Koch 2013). We make the distinction between “top-down” economic salience (which is affected exclusively by monetary outcomes), and “bottom up” visual salience, and propose that both types of salience influence information processing and risky choice.

26 To formalize this argument, we provide an extension of the BGS model by adding a

“visual salience distortion factor” to the distortion equation in (10). In particular, if visual salience of the gain state is increased, e.g., by making the gain payoff bold, we expect this to distort the objective probability, p, upwards. Conversely, if the visual salience of the loss state is increased, we expect this to distort � downwards. The following equation generalizes the distortion algorithm to include both economic salience and visual salience, where �∗∗ now reflects the distorted probability of the gain state taking into account both sources of salience:

! !! , !! !∗∗ !!×(�!"##) ! !"## !"## ! = × × (11) !!!∗∗ !×(�!"#$) ! !! , !! !!! ! ! !"#$ !"#$

We refer to this model that takes into account both sources of salience as the full salience model. Here, �!"#$ is an indicator function taking on the value 1 when the gain state is more visually salient than the loss state. In contrast, �!"## is an indicator function taking on the value 1 when the loss state is more visually salient than the gain state. � ∈ [−1, 1] is a parameter that controls the degree to which visual salience distorts objective probabilities19. When � = 0, the first term on the right side vanishes and the distortion is exclusively a function of economic salience. Similarly, when neither state is more visually salient, both indicator functions take on the value 0, and the first term on the right side vanishes. Note that the full distortion equation (11) implies that visual salience and economic salience interact with each other. Both sources of salience, � (visual) and � (economic) are mediated through the parameter �.

The decision value of the risky lottery under the full salience model is computed just as before, except we now use the fully distorted probability distribution (����, �∗∗; ����, 1 − �∗∗).

We again assume a value function given by, � � = �!, which implies that the fully distorted

19 We constrain −1 ≤ � ≤ 1 because the salience function, �, is constrained to [0, 1], and thus the range of ! ! ! ! [� �!"##, �!"## − � �!"#$, �!"#$ ] is [-1, 1]. 27 decision value of the risky lottery, when both visual and economic salience are taken into account, is given by:

∗∗ ∗∗ ��!"## = � × �(�) + (1 − � ) × �(�) − �(�)

= �∗∗ × �! + (1 − �∗∗) × �! − �!

(12)

If visual salience operates through our hypothesized full salience model, and if the gain

∗∗ ∗ state is made visually salient, then � > � . This will increase ��!"## and lead to an increase in risk taking. Conversely, when the loss state is made visually salient, �∗∗ < �∗, which will decrease ��!"## and lead to a decrease in risk taking. This leads to our third and final prediction:

PREDICTION 3: If visual salience does not impact risk taking, then � = 0. If increasing the visual salience of the gain (loss) state leads to increased (decreased) risk taking, then � > 0.

III.B. Design

To test Prediction 3, we collect data from 202 subjects through an online data panel

(Amazon Mechanical Turk; Buhrmester, Kwang, and Gosling 2011). The subjects make decisions over the same set of thirty-five lotteries from Experiment 1 and are paid $1 for completing the task. We randomly assign subjects to one of two conditions. The control condition (N = 103) is identical to the design used in Experiment 1, where all text is displayed in regular font. In the

“visually salient” condition (N = 99), shown in Figure VIII, subjects complete the same task, but we increase the visual salience of one state of the world by displaying it in bold font. For each trial t=1,…,35 we increase the visual salience of the state that is most economically salient, according to the � salience function.

! ! ! ! For example, if � �!"#$, �!"#$ > � �!"##, �!"## , then we highlight the lottery payoff and probability in the gain state in bold. In terms of equation (11), our design implies that

28 ! ! ! ! whenever � �!"#$, �!"#$ > � �!"##, �!"## , then �!"#$ takes on the value 1. Conversely,

! ! ! ! whenever � �!"#$, �!"#$ < � �!"##, �!"## , then �!"## takes on the value 1. In both conditions, the lotteries are displayed next to each other, and as in Experiment 1, we randomize the location of each alternative to avoid order effects.

III.C. Results

Before turning to a formal test of Prediction 3, we first present the average levels of risk- taking across both experimental conditions. Table III.A provides the proportion of subjects who take risk on each of the thirty-five trials in the control condition. The overall pattern of risk-taking is very similar to the results we obtain in the lab during Experiment 1. First, and crucially, in the control condition, the probability of choosing the lottery increases from 30.4% when the loss state is salient to 40.2% when the gain state is salient (p < 0.001). Moreover, risk-taking dramatically increases as the expected value of the lottery increases, which is consistent with diminishing sensitivity of salience getting weaker at higher payoff levels20. In the visually salient condition, the probability of choosing the risky lottery increases from 28.3% when the loss state is salient to

40.6% when the gain state is salient (p < 0.001).

Table III.B provides the key changes in risk taking when moving from the control condition to the visually salient condition. The shaded cells denote trials where the gain state is exogenously made visually salient, and non-shaded cells denote trials where the loss state is exogenously made visually salient. Because increasing the visual salience of the loss (gain) state should reduce (increase) risk taking, we expect to see negative numbers in non-shaded cells and positive numbers in shaded cells. The table clearly shows that there is a large amount of heterogeneity in the size of the treatment effect across trials. For example, in the bottom row of the table where the risky lottery delivers a payoff of $0 in the loss state, we find that exogenously

20 This result can also be driven by diminishing sensitivity of the value function.

29 increasing the visual salience of the loss state leads to a reduction of nearly 50% in risk taking for each level of probability. In other trials, the magnitude of the treatment effect is much smaller or has a sign that is opposite to what is expected.

To investigate this more systematically, we run logistic regressions pooling the data from both conditions. The dependent variable equals one if the subject chooses the risky lottery. We

! ! ! ! include the variable EconSal defined by � �!"#$, �!"#$ − � �!"##, �!"## , which measures the degree to which the gain state is economically salient (EconSal is increasing in the economic salience of the gain state). We also include the dummy VisSal which equals 1 if the subject is in the visually salient condition, and finally we include the key interaction EconSal*VisSal.

Column (1) of Table IV shows that EconSal is a strong predictor of risky choice, which means that as the gain state becomes more salient, subjects are more likely to take risk. However, neither the coefficient on VisSal nor EconSal*VisSal is significantly different from zero, indicating that across the full sample, visual salience does not have a systematic impact on choice.

To test whether the impact of visual salience on the gain state is different from its impact on the loss state, we partition the sample into trials where the gain state is visually salient and trials where the loss state is visually salient. Column (3) shows that in trials where the loss state is visually (and economically) salient, both the coefficients on EconSal and EconSal *VisSal are significantly positive (p < .01). This indicates that variation in economic salience predicts risky choice, and that this effect is amplified when the loss state is made visually salient. Column (5) shows that these effects are not present in trials in which the gain state is visually salient. Taken together, these reduced form results suggest that visual salience has an asymmetric impact on risk taking.

To formally test Prediction 3, we structurally estimate the full salience model by maximum likelihood. We again assume that subjects exhibit stochastic choice where the

30 probability of choosing the risky lottery is a logistic function of ��!"##. Specifically, we assume that the probability that subject s accepts the risky lottery on trial t is, �(�! | �, �, �, �) =

! , where � controls the degree of stochasticity of choice and �� �, �, � is the !!!!!×!"! !,!,! ! decision value of the risky lottery under the full salience model on trial t. We define �!,! = 1 if subject s chooses the risky lottery on trial t, and �!,! = 0 if subject s chooses the certain option on trial t. We estimate (�, �, �, �) by pooling all trials and subjects across both the control condition and the visually salient condition, and maximize the following log likelihood function:

�� �, �, �, � �) = ! ! �!,! log �(�! | �, �, �, �) + (1 − �!,!) log 1 − �(�! | �, �, �, �) (13)

We find that the log likelihood function is maximized at (�, �, �, �) = (0.3, 0.56, 0.09, 0.6).

To test whether � > 0, note that the BGS model is a special case of our full salience model. In particular, when � = 0, the two models are identical, and thus the BGS model is a nested model of the full salience model. We therefore run a likelihood ratio test by first estimating the restricted model (� = 0) on the pooled data. We find that

(�, �, �) = (0.27, 0.56, 0.6) and Figure IX.A plots the empirical choice probability of each trial against the model implied choice probability conditional on the estimated parameters of the restricted model. Figure IX.B provides the analogous plot for the full salience model. The maximum of the log likelihood function of the restricted model is -4085.54, and the maximum of the log likelihood of the unrestricted model is -4083.56. This yields a likelihood ratio test statistic of ��� = 2× 1.987 , which has a chi-square distribution with one degree of freedom under the null hypothesis that � = 0. We can reject this null hypothesis at the 5% significance level, which consistent with Prediction 3, suggests that including visual salience in the model provides a significantly better fit to the risky choice data.

31 IV. CONCLUSION

In this paper, we use a combination of data on lottery choices and eye-tracking to test the

BGS model. We then generalize the BGS model to account for perceptual processes that are in play during risky choice, with a specific focus on the role of visual salience. In our first experiment we find that risk taking is broadly consistent with salience theory: subjects exhibit unstable risk preferences and are significantly more likely to take risk when the gain state is salient compared to when the loss state is salient. Much of this variation in risk taking is captured by the context-dependent weighting function of salience theory, which endogenously increases the decision weight attached to the gain state as average payoff levels rise (Figure III). We also find that CPT can provide a good fit to the data as long as there is a large amount of concavity in the value function.

While the data on risk taking is broadly consistent with salience theory, the eye-tracking data does not support the precise psychological mechanism that is assumed in BGS. We find that subjects do not pay more attention to the salient state compared to the non-salient state. In an effort to understand what does drive the attention allocation process in our experiment, we conduct post-hoc analyses of the eye-tracking data and find two pieces of evidence in favor of a different psychological mechanism. First, when the gain state is salient, subjects attend to the risky lottery payoffs more than they attend to the certain option payoff. Second, the amount of attention subjects pay to the risky lottery compared to the certain option is a strong predictor of risky choice. Taken together with the choice data, these results suggest that the BGS salience function can explain variation in choice, and it can also explain some variation in attention across alternatives.

As noted in the previous paragraph, we do not find evidence that the BGS salience function can explain variation in attention across attributes (states). One potential reason why we do not find evidence for this is driven by the experimental display of the lotteries. Recall that the certain option delivers the same payoff in both states and so we cannot identify which state a

32 subject attends to when she is looking at the certain option. In our analyses, we assume that when a subject attends to the certain option payoff, she is attending to the salient and non-salient state in the same proportion that she does when looking at the risky lottery. If this assumption is not valid, it will decrease the precision of our state-based attention estimates and make it harder to detect differences in attention across states.

The experimental display is therefore a critical variable because it may influence not only the researcher’s ability to accurately measure state-specific attention, but it may also influence the subject’s perception of the state space. More broadly, the “consideration set” of alternatives that is allowed to influence decision weights in our experiment is purposefully unambiguous: it contains only the risky lottery and the certain option. Yet in more applied settings, the decision- maker’s perception of the consideration set and state space is likely to be much more complex.

While this can make the theory harder to apply in some instances, it can also be seen as a feature of the model as different consideration sets will systematically lead to different decision weights and choices. For example, BGS show how different perceptions of the state space can modulate a decision-maker’s propensity to exhibit the Allais paradox. The perception of the consideration set is therefore an important topic for future research, as it is crucial for all theories that endogenize decision weights through consideration sets.

In our second experiment, we explore a different dimension of salience. We demonstrate that manipulating the visual salience of specific lottery attributes has a causal effect on risky choice. We exogenously increase the visual salience of a lottery’s downside payoff and observe a reduction in risk-taking; however, we do not observe a similar boost in risk taking when increasing the visual salience of a lottery’s upside. One potential explanation for this asymmetric treatment effect is that the visual salience distortion factor (see equation (11)) is itself asymmetric. In particular, if attention is differentially sensitive to increasing visual salience of losses compared to gains, then this may generate the observed effect. However, this explanation

33 is clearly post-hoc, and further research is needed on the properties of the visual salience distortion factor.

The design of our second experiment is also useful for investigating the direction of causality between attention and preferences. The BGS theory assumes that salient payoffs attract the decision-maker’s attention, and this attention allocation affects preferences through the distorted decision weights. An alternative hypothesis is that preferences have a causal effect on attention (Sims 2003). Under this alternative hypothesis, attention is allocated to stimuli for which the decision-maker has an “ex-ante” preference. In our second experiment, we exogenously manipulate the visual salience of lottery payoffs, without changing the magnitude of payoffs. This manipulation therefore shifts attention while holding constant the economic content of the choice set. Because we observe that the visual salience manipulation systematically shifts choice, the data suggest that attention causally affects preferences. This interpretation is consistent with BGS, and is also supported by previous experimental work that demonstrates attention causally affects choice21. It is important to note, however, that we do not provide any data that explicitly rejects the hypothesis that preferences causally affect attention. Therefore, it is certainly possible that in our experiments, the causality between attention and preferences can run in both directions.

Finally, our motivation for incorporating visual salience into the BGS model is driven by the fact that, at a very basic level, there is a similar psychological mechanism that characterizes visual salience and economic salience. A salient attribute – whether it is visually salient or economically salient – will attract the decision-maker’s attention, and will receive disproportionate weight in the decision-making process. As a first pass at incorporating visual salience into a model of risky choice, it therefore seems natural to incorporate it into the BGS model. However, we emphasize that one could just as easily incorporate visual salience into an

21 See, for example, Shimojo et al. (2003), Armel, Beaumel, and Rangel (2008), Hare, Malmaud, and Rangel (2011), Milosavljevic et al. (2012), and Frydman and Rangel (2014).

34 alternative model of risky choice, such as prospect theory. In the prospect theory case, it is entirely plausible that visual salience could interact with the value function or the probability weighting function to generate systematic shifts in risk-taking. Fortunately, a similar methodology to the one used in this paper can be employed to test for such an interaction between visual salience and alternative models of risky choice.

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39 Table I. Each row represents a single trial in the experiment. In each trial a subject is given the choice between a risky lottery (x, p; y, (1-p)) or a certain option with payoff c, where x > c > y. At the end of the experiment, subjects are paid according to their decision and the outcome of one randomly chosen trial. All payoffs are in US dollars with a 1:1 exchange rate.

Trial Number x y c p 1 2000 0 20 0.01 2 400 0 20 0.05 3 100 0 20 0.2 4 61 0 20 0.33 5 50 0 20 0.4 6 40 0 20 0.5 7 30 0 20 0.67 8 2010 10 30 0.01 9 410 10 30 0.05 10 110 10 30 0.2 11 71 10 30 0.33 12 60 10 30 0.4 13 50 10 30 0.5 14 40 10 30 0.67 15 2020 20 40 0.01 16 420 20 40 0.05 17 120 20 40 0.2 18 81 20 40 0.33 19 70 20 40 0.4 20 60 20 40 0.5 21 50 20 40 0.67 22 2030 30 50 0.01 23 430 30 50 0.05 24 130 30 50 0.2 25 91 30 50 0.33 26 80 30 50 0.4 27 70 30 50 0.5 28 60 30 50 0.67 29 2040 40 60 0.01 30 440 40 60 0.05 31 140 40 60 0.2 32 101 40 60 0.33 33 90 40 60 0.4 34 80 40 60 0.5 35 70 40 60 0.67

40 Table II. Table II.A provides the proportion of subjects who choose the risky lottery over the certain option in Experiment 1. Each row of the table corresponds to the expected value of the lottery (and of the certain option). Each column corresponds to the probability of the gain state. Table II.B provides the decision value for each of the thirty-five risky lotteries under the BGS theory, conditional on the estimated parameters. Table II.C provides the decision value for each of the thirty-five risky lotteries under CPT where the reference point is equal to zero, conditional on the estimated parameters.

Table II.A.

Proportion(of(Risk(Seeking(Subjects

$60 0.56 0.88 0.56 0.56 0.69 0.56 0.44 $50 0.50 0.50 0.25 0.44 0.56 0.25 0.56 $40 0.38 0.31 0.56 0.31 0.56 0.31 0.50 $30 0.38 0.25 0.25 0.31 0.25 0.31 0.31 $20 0.31 0.25 0.25 0.13 0.19 0.38 0.13 0.01 0.05 0.20 0.33 0.40 0.50 0.67

666666666666Expected6value6c Probability6of6gain6p

Table II.B.

Salience(Decision(Values(of(Risky(Lottery

$60 0.39 0.96 0.13 (0.15 (0.24 (0.29 (0.26 $50 (0.02 0.72 0.02 (0.26 (0.36 (0.41 (0.36 $40 (0.67 0.22 (0.24 (0.50 (0.60 (0.64 (0.55 $30 (1.85 (0.88 (0.95 (1.12 (1.19 (1.19 (1.02 $20 (5.33 (4.63 (4.16 (4.03 (3.98 (3.84 (3.37 0.01 0.05 0.20 0.33 0.40 0.50 0.67

777777777777Expected7value7c Probability7of7gain7p

Table II.C.

CPT$Decision$Values$of$Risky$Lottery

$60 0.04 &0.29 &0.42 &0.44 &0.45 &0.45 &0.46 $50 &0.02 &0.36 &0.49 &0.51 &0.52 &0.53 &0.53 $40 &0.12 &0.47 &0.61 &0.63 &0.64 &0.65 &0.65 $30 &0.34 &0.69 &0.84 &0.86 &0.87 &0.88 &0.88 $20 &1.97 &2.35 &2.51 &2.55 &2.55 &2.56 &2.56 0.01 0.05 0.20 0.33 0.40 0.50 0.67

777777777777Expected7value7c Probability7of7gain7p 41 Table III. Table III.A provides the proportion of subjects who choose the risky lottery over the certain option in the control condition in Experiment 2. Table III.B provides the change in proportion of risk seeking subjects in the “visually salient” treatment compared to the control condition in Experiment 2. Shaded cells denote those trials in which the gain state is made visually salient, non-shaded cells denote trials in which the loss state is made visually salient.

Table III.A.

Proportion(of(Risk(Seeking(Subjects

$60 0.46 0.47 0.44 0.44 0.53 0.51 0.33 $50 0.38 0.42 0.41 0.38 0.40 0.39 0.38 $40 0.39 0.39 0.42 0.39 0.36 0.35 0.36 $30 0.31 0.41 0.33 0.27 0.30 0.32 0.32 $20 0.20 0.22 0.20 0.14 0.17 0.16 0.16 0.01 0.05 0.20 0.33 0.40 0.50 0.67

666666666666Expected6value6x Probability6of6gain6p

Table III.B.

Change'in'Proportion'of'Risk'Seeking'Subjects

$60 $16% 10% 13% 34% 10% 15% 20% $50 $6% $2% 1% 18% 13% 22% 4% $40 $13% $3% $10% 0% 20% 21% 4% $30 $4% $20% 10% $44% 4% 7% $16% $20 $67% $72% $68% $78% $69% $42% $60% 0.01 0.05 0.20 0.33 0.40 0.50 0.67

888888888888Expected8value8c Probability8of8gain8p

42 Table IV. Logistic regressions of risky choice on both sources of salience from Experiment 2. EconSal denotes the difference in economic salience between the gain state and loss state. VisSal is a dummy that takes on the value 1 if the state that is economically salient is also made visually salient. For the logit specifications, standard errors are clustered at the subject level. For the mixed effects logit specifications, a random intercept is included as well as a random slope on the EconSal coefficient. ***, **, * denote statistical significance at the 1%, 5%, and 10% levels, respectively.

Full Sample Loss State Salient Gain State Salient (1) (2) (3) (4) (5) (6)

Mixed Effects Mixed Effects Mixed Effects Logit Logit Logit Logit Logit Logit dependent variable: risky choice EconSal 0.791 1.48 1.60 2.99 -0.05 -0.467 (5.16)*** (5.15)*** (5.14)*** (6.40)*** (-0.24) (-0.80) VisSal -0.06 0.012 0.169 0.275 0.206 0.462 (-0.35) (0.04) (0.92) (0.98) (0.95) (1.07) VisSal*EconSal 0.199 0.106 1.58 1.725 -0.546 -1.11 (0.91) (0.26) (3.36)*** (2.75)** (-1.65) (-1.37) Constant -0.66 -1.207 -0.514 -0.779 -0.347 -0.737 (-5.08)*** (-5.40)*** (-3.81)*** (-3.88)*** (-2.16)*** (-2.39)*

Observations 6679 6679 3624 3624 3055 3055 Pseudo R^2 0.019 0.039 0.002

43 FIGURE I: Sample Choice Trial With Sample Recorded Eye-Fixations. The location of each gray circle indicates the subject’s eye-fixation location. The size of each circle indicates the duration of each fixation with larger circles signifying longer looking times. The circles are for analysis purposes only; subjects do not see these circles when they are making choices.

44 FIGURE II: Experimental Display is Partitioned into Six Areas of Interest. Each gray box represents a different area of interest. “Certain” indicates the certain option, “Gain” indicates the gain state and “Loss” indicates the loss state. The “$” and “%” labels identify the payoff and probability, respectively. These boxes are for analysis purposes only; subjects do not see them when making their decisions.

Certain($( Certain(%( Lo#ery''S':'''$40''for''sure'

Loss($( Loss(%( Gain($( Gain(%( Lo#ery''G':'''$20''with''80%''chance,'''$120''with''20%''chance'

45 FIGURE III: Context-dependent probability weighting function. Black dashed line is the 45- degree line. Each different colored line is the probability weighting function conditional on the average payoff level in our experiment. The probability weights are generated using the salience ! ! ! ! !! !!! function, � �! , �! = ! ! , and the two properties that characterize choice sets in our |!! ! |!! experiment: 1) the choice set contains a risky lottery with a two-outcome support and a mean- preserving certain option and 2) the downside of the risky lottery is fixed at (c-20), where c denotes the payoff of the certain option. The probability weighting function is generated using � = 0.21.

1"

c=$20" 0.8" c=$30" c=$40" c=$50" c=$60" 0.6"

0.4" Decision(Weight(

0.2"

0" 0" 0.2" 0.4" 0.6" 0.8" 1" Objec0ve(Probability(

46 FIGURE IV: Relationship between salience model implied probability and empirical probability of choosing the risky lottery. Each point represents a single trial. Model implied probabilities are conditional on the estimated parameters, � = 0.21, � = 0.62, and � = 0.35.

1"

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0.4" Choosing)Risky)Lo8ery))

Empirical)Probability)of) 0.3"

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0" 0" 0.1" 0.2" 0.3" 0.4" 0.5" 0.6" 0.7" 0.8" 0.9" 1" Salience)Model)Implied)Probability) of)Choosing)Risky)Lo8ery)

47 Figure V. Relationship between CPT model implied probability and empirical probability of choosing the risky lottery. Each point represents a single trial. In this specification of CPT, the reference point is $0. Model implied probabilities are conditional on the estimated parameters, � = 0.33, � = 0.18 , � = 0.55.

1"

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Choosing)Risky)Lotery) 0.3" Empirical)Probability)of))

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48 Figure VI. Average total fixation duration for each of the areas of interest on the experimental display screen. “Gain” and “Loss” refer to the upside and downside of the risky lottery, respectively. “Salient” refers to the risky lottery payoff that is predicted to be salient by BGS, and “Non-Salient” refers to the risky lottery payoff that is predicted not to be salient by BGS. Average total fixations are computed by computing individual subject means across trials, and then taking the mean across subjects.

2500" Dollar" Probability"

2000"

1500"

1000" Average'Total'Fixa/on'Dura/on'(ms)' 500"

0" Gain" Loss" Salient" Non2Salient" Certain"

49 FIGURE VII: Correlation between salience and relative attention to the risky lottery using first fixation durations.

0.45"

0.4"

0.35"

0.3"

0.25"

0.2" r=0.46' p=0.006' Rela%veRiskyFirst$/ 0.15"

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0" )1" )0.8" )0.6" )0.4" )0.2" 0" 0.2" 0.4" 0.6" 0.8" 1" Salience(Gain/state)/6/Salience(Loss/state)/

50 FIGURE VIII: Sample Choice Trial in Visually Salient Condition. All text uses the same black font on light gray background, except the economically salient state in the risky lottery is displayed in bold to make it visually salient.

51 FIGURE IX: Relationship between observed behavior and estimated choice probabilities. The x-axis measures the model implied probability of choosing the risky lottery, the y-axis measures the empirical probability of choosing the risky lottery. Panel A: Estimated choice probabilities from the restricted salience model. Panel B: Estimated choice probabilities from the full salience model. There are twice as many distinct estimated decision values in panel B compared to panel A because the full salience model can distinguish between the same choice set across the control condition and visually salient condition.

PANEL A

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PANEL B

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52 ONLINE APPENDIX FOR “THE ROLE OF SALIENCE AND ATTENTION IN CHOICE UNDER RISK: AN EXPERIMENTAL INVESTIGATION” BY MILICA MORMANN AND CARY FRYDMAN

EXPERIMENTAL INSTRUCTIONS

Welcome to the experiment.

The experiment takes around 30 minutes to complete and pays $5 plus another amount ranging from $0 to $2,040 depending on your answers during the experiment. You can quit the experiment at any point for any reason.

Your task is to make a choice between two options. You can either select an option which gives you a certain amount of money, or you can select the other option where you can win one of two different amounts, each with a different probability. For example, you can select to receive $40 for sure, or you can select the other option where you have 80% chance of getting $20 and 20% chance of getting $120.

Take as much time as you need to make your choice. Once you have decided on which option you prefer, please look at that option on the screen and press the spacebar to indicate that this is your choice.

You will make this kind of decision 35 times and each time the amounts that you can receive will vary.

At the end of the experiment, we will select one of the 35 trials at random. We will then check what choice you made on that trial. If you selected a certain option, you will receive that amount of money in addition to your $5 payment. If you selected the other option, you will then roll a ten-sided die twice. The first number will represent tens and the second roll will represent units. This will give us a number that we will compare to the probabilities from that trial. This will determine which of two amounts you receive in addition to your $5 payment. In the example above, you would need to roll a number between 80 and 99 in order to receive the larger amount of $120. If you roll a number between 00 and 79 you will receive the smaller amount of $20.

About half way through the task, a “break” screen will appear. You can then rest your eyes for a bit if needed. To proceed please press the spacebar when ready.

While you make your choices the eye-tracker will record your eye movements. So, before we begin with the experiment we will set up the eye-tracker. It will only take about 1-2 minutes to do so. You can sit normally at the desk and use the laptop as you usually do. Please keep looking at the laptop screen during the entire experiment.

Do you have any questions?

53 SUPPLEMENTAL TABLES FOR UNIVARIATE REGRESSIONS

Table A.1. Relationship between risky choice and salience of the gain state. The dependent variable takes on the value one if the subject chooses the risky lottery. “SalientGain” is a dummy that takes on the value one if the gain state is salient, and zero otherwise. Standard errors are clustered by subject for specifications (1) – (3) and t-statistics are provided in parentheses. ***, **, and * represent statistical significance at the 1%, 5%, and 10% levels, respectively.

(1) (2) (3) (4)

Mixed Effects OLS OLS Logit Logit dependent variable: risky choice SalientGain 0.13 0.215 0.537 0.668 (2.15)** (2.21)** (2.14)** (2.21)** Constant 0.351 0.408 -0.616 -0.736 (5.43)*** (242.15)*** (-2.17)** (-2.11)**

Observations 544 544 544 544 Subject Fixed Effects X (Pseudo) R^2 0.017 0.013

54 !"#$$!!"##$ Table A.2. Dependent variable is ������������$ = , which provides a measure of the !"#$$!!"##$ attention that subjects pay to the gain state relative to the loss state. “EconSal” is defined as ! ! ! ! � �!"#$, �!"#$ − � �!"#!, �!"## , and provides the theoretical difference in salience between the gain state and loss state. Standard errors are clustered by subject for specifications (1) and (2) and t-statistics are provided in parentheses. ***, **, and * represent statistical significance at the 1%, 5%, and 10% levels, respectively.

(1) (2) (3) Mixed OLS OLS Effects dependent variable: RelativeGain$ Linear Model EconSal -0.046 -0.046 -0.046 (0.93) (-0.94) (-0.91) Constant 0.095 0.952 0.953 (4.71)*** (112.15)*** (4.62)***

Observations 544 544 544 Subject Fixed Effects X R^2 0.002

55 Table A.3. Relationship between risky choice and ������������$. The dependent variable takes on the value one if the subject chooses the risky lottery. Standard errors are clustered by subject for specifications (1) – (3) and t-statistics are provided in parentheses. ***, **, and * represent statistical significance at the 1%, 5%, and 10% levels, respectively.

(1) (2) (3) (4)

Mixed Effects OLS OLS Logit Logit dependent variable: risky choice RelativeGainFixation 0.063 0.089 0.261 0.400 (1.00) (1.58) (0.98) (1.46) Constant 0.406 0.403 -0.383 -0.458 (6.57)*** (75.65)*** (-1.48) (-1.36)

Observations 544 544 544 544 Subject Fixed Effects X (Pseudo) R^2 0.003 0.002

56 Table A.4. Relationship between risky choice and ������������������$. The dependent variable takes on the value one if the subject chooses the risky lottery. Standard errors are clustered by subject for specifications (1) – (3) and t-statistics are provided in parentheses. ***, **, and * represent statistical significance at the 1%, 5%, and 10% levels, respectively.

(1) (2) (3) (4)

Mixed Effects OLS OLS Logit Logit dependent variable: risky choice RelativeRiskyFirst$ 0.21 0.172 0.895 0.895 (3.86)*** (4.04)*** (3.72)*** (3.12)*** Constant 0.368 0.376 -0.550 -0.599 (6.40)*** (43.09)*** (-2.17)** (-1.79)*

Observations 544 544 544 544 Subject Fixed Effects X (Pseudo) R^2 0.023 0.017

57 !"#$%#&'($!!"##$%&#'$!!"#$%&'(&#)$$ Table A.5. Dependent variable is ������������������$ = , !"#$%#&'($!!"##$%&#'$!!"#$%&'(&#)$$ which provides a measure of the attention that subjects pay to the risky lottery relative to the certain option reg risk if gain==0 & replication==1 & risk!=999. “EconSal” is defined as ! ! ! ! � �!"#$, �!"#$ − � �!"##, �!"## , and provides the theoretical difference in salience between the gain state and loss state. Standard errors are clustered by subject for specifications (1) and (2) and t-statistics are provided in parentheses. ***, **, and * represent statistical significance at the 1%, 5%, and 10% levels, respectively.

(1) (2) (3) Mixed OLS OLS Effects dependent variable: RelativeRiskyFirst$ Linear Model EconSal 0.125 0.127 0.126 (3.05)*** (3.08)*** (3.03)*** Constant 0.203 0.203 0.202 (9.10)*** (284.96)*** (8.91)***

Observations 544 544 544 Subject Fixed Effects X (Pseudo) R^2 0.018

58