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The time horizon of risky choice: Effects of delayed outcomes
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Jonathan J Rolison*1, Vittorio Girotto2, Paolo Legrenzi3 and Yaniv Hanoch4
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Queen’s University Belfast, UK
University IUAV of Venice, Italy
University Ca’ Foscari of Venice, Italy
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University of Plymouth, UK
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*Send correspondence to
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Jonathan Rolison
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David Keir Building
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School of Psychology
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Queen’s University Belfast, UK
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Belfast, BT7 1NN
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Northern Ireland, UK
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E-mail: j.rolison@qub.ac.uk
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KEY WORDS:
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Risky choice, prospect theory, cumulative prospect theory, time discounting, time delay,
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intertemporal choice, expected utility theory, preferential choice, time horizon, standard
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economic theory, watchful waiting
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ABSTRACT
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Studies of risky choice centre on the decision maker’s evaluations of choice options at the
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neglect of their time horizon. We demonstrate the importance of time horizon for risky choice
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behavior by manipulating the delay of choice outcomes. Two hundred fifteen participants
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made hypothetical choices between a sure amount and a lottery, offering either gain or loss
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prospects. The sure amount, lottery outcome, both, or neither was delayed by one year to
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investigate effects of time horizon on risky choices. Delaying a lottery outcome with respect
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to a sure outcome reduced risk taking for gains and increased risk taking for losses by
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reducing participants’ sensitivity to values and probabilities of outcomes, whereas delaying a
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sure outcome with respect to a lottery outcome had opposite effects on choices by instead
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increasing sensitivity to values and probabilities. Our findings demonstrate the importance of
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time horizon for choices under uncertainty, and highlight a need for further research to
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investigate effects of time horizon on risky choice behavior.
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Imagine finding that your life savings invested in stocks have slumped in value.
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Your financial advisor informs you of the odds that you might recover your losses
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in one year from now if you hold onto your investments, but warns that you could
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stand to lose an even larger amount. Your alternative is to sell your stocks
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immediately, realising a loss to your savings (Problem 1).
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There are two key aspects of Problem 1 relevant to the study of risky choice. First, the choice
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options differ in their potential outcomes. Holding onto the stocks avoids a sure loss but
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exposes the risk of losing an even larger amount. Second, the options differ in their time
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horizon. Selling the stocks means realising an immediate loss to your savings, whereas
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holding onto the stocks defers a potential loss to some point in the future.
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Studies of risky choice centre on the decision maker’s evaluations of choice options
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at the neglect of their time horizon. The standard economic view proposes that the decision
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maker, a rational agent, endorses the choice option that yields the maximum expected utility
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(von Neumann & Morgenstern, 1947; Savage, 1954). A number of findings inconsistent with
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standard economic theory have led to alternative accounts of risky choice. Among the most
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influential is cumulative prospect theory (CPT; Kahneman & Tversky, 1979; Tversky &
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Kahneman, 1992), which proposes that prospects of gains and losses are evaluated separately
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and in comparison with a reference point, such as current wealth. According to CPT, the
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magnitude and probabilities of outcomes undergo transformations when the decision maker
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evaluates choice options, where evaluations depend on subjective perceptions of the values
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and probabilities of outcomes rather than the objective amounts.
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A key advance of CPT is the coupling of decision weights for probabilities with
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diminishing sensitivity to gains and losses, described by a value function. The decision
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weights cause small probabilities to be overweighted and medium to large probabilities to be
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underweighted. The value function, which is concave for gains and convex for losses, is
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steeper for losses, displaying loss aversion. As a result, in situations similar to Problem 1,
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people show a tendency to gamble on a riskier option (e.g. holding onto the depreciated
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stocks) to avoid a sure loss, even when the safer option has a higher expected value
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(Kahneman & Tversky, 1979).
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There is good reason to believe, however, that time horizon is also involved in risky
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choice behavior. Time horizon is commonplace in our everyday risky choices. Problem 1
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illustrates how choice options can differ in their time horizon, such that a delayed outcome
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might by evaluated differently in comparison to one that yields an immediate result. In the
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same way that a savings account grows at a fixed rate of interest, standard economic theory
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assumes that a delayed reward increases in value by fixed proportions as it approaches in
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time (Fishburn & Rubinstein, 1982). Psychological research, however, has found the
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subjective value of a delayed reward to increase at proportionally larger amounts as it
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approaches in time (Green & Myerson, 2004; Kirby & Marakovic, 1996), and as a result,
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preference for a larger delayed reward can switch to an immediate smaller reward as the
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delay of the larger reward reduces.
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Preference reversal for delayed rewards has its parallels in risky choice behavior
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(Green, Myerson, & Ostaszewski, 1999; Prelec & Loewenstein, 1991; Rachlin, Raineri, &
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Cross, 1991). As the probability of a gamble increases, initial preference for a larger reward
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at a lower probability (e.g. a lottery) switches to preference for a smaller sure amount when
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its riskier alternative increases in probability. This tendency occurs because decision weights
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applied to probabilistic outcomes cause small probabilities to be overweighted, and medium
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to large probabilities to be underweighted. Loewenstein and Prelec (1992) identified a
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number of commonalities between risky and intertemporal choice, and the striking
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similarity between discounting of probabilistic and temporal rewards has led some theorists
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to propose that a single underlying psychological mechanism is involved (Rachlin et al.,
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1991; Rachlin, Logue, Gibbon, & Frankel, 1986). Keren and Roelofsma (1995) found that
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preference reversal was reduced by introducing uncertainty to choice options. When
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rewards were made probabilistic participants preferred a larger delayed reward even
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when a smaller reward was immediate. The authors concluded that delay implies
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uncertainty that a reward may never be received, and as such, that delay and
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uncertainty are inter-changeable1. The gain/loss asymmetry in risky choice, in which
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probabilistic gains are discounted more steeply than probabilistic losses, also has its
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counterpart in preferential choice. While individuals would prefer a larger delayed payment
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than make an immediate smaller payment, and would prefer a smaller immediate reward than
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a larger delayed reward, losses are discounted less steeply than gains as a function of delay
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(Estle, Green, Myerson, & Holt, 2006; Mitchell & Wilson, 2010).
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Similarities between risky choice and time discounting suggest that the time horizon
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of uncertain outcomes may influence choice behavior, even though studies of risky and
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preferential choice have typically studied time delay and uncertainties in isolation (e.g.
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Green & Myerson, 2004; Green et al. 1999; Tversky & Kahneman, 1992). In the present
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study, we present participants choices between risky options whilst manipulating the time
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horizon of outcomes. In some sets, participants choose between a sure gain (or loss) and a
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lottery gain (or loss) in a similar form to Problem 1, with either the sure amount, lottery
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outcome, both, or neither delayed. If individuals discount the value of delayed outcomes, then
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we should expect that time horizon will (a) exacerbate risk taking behavior when a delay is
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applied to the prospect of a loss, and (b) heighten risk aversion when the prospect of a gain is
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delayed.
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METHOD
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Participants
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Two hundred fifteen US participants (110 males, 105 females; mean age= 37.06; SD=13.32)
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recruited via Amazon’s Mechanical Turk completed one of four versions of an online survey
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designed to measure their choices under risk. The reliability of the participant pool provided
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by Amazon’s Mechanical Turk has been established by comparisons with other methods of
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human data collection (Paolacci, Chandler, & Ipeirotis, 2010). The majority of participants
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indicated either high school (N=62 of 215; 28.8%) or college (N=113 of 215; 52.6%) as their
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highest educational attainment, and just under half (N=102 of 215; 47.4%) indicated an
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annual household income above 40,000 US dollars.
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Design and Materials
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Participants were presented a choice between a sure amount and a lottery for 115 gambles
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divided into five gambling sets. Two of the gambling sets presented a choice between a sure
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gain and a lottery gain of $100 with a low probability (p=.10) for one set and a high
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probability (p=.90) for the other set. The sure amount increased in increments of $5 for each
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of 21 gambles from $0 to $100. Two further sets were identical with the exception that the
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sure amount and lottery referred instead to losses rather than to gains. In a fifth set gain and
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loss prospects were mixed. Across 31 gambles participants were offered the option of a
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lottery with a 50% chance of losing $50 and a 50% of gaining an amount, x, that incremented
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on each gamble in steps of $20 from $0 to $600, and the option of receiving nothing. The
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five gambling sets were presented in a random order for each participant.
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Participants were randomly assigned to receive one of four versions of the survey
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that asked for hypothetical choices between a sure amount (S) and a lottery (L), of which the
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outcomes were either immediate (I) or delayed (D) by one year. For one group of participants
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the five gambling sets referred to a sure amount and a lottery outcome that were both
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available immediately (IS:IL; N=49). For a second group, both the sure amount and lottery
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outcome were delayed by one year (DS:DL; N=55), and for a third and fourth group either
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the sure amount (DS:IL; N=55) or lottery outcome (IS:DL; N=56) was delayed by one year.
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Since the survey was administered online, we included an instructional manipulation
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check developed by Oppenheimer and colleagues (2009) as a final item of the questionnaire.
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This item identifies unmotivated participants who failed to pay close attention to instructions.
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The item, which followed the demographic items, explained the importance of carefully
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reading the study instructions for the experimental manipulations to be effective (see
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Oppenheimer et al., 2009). The final section of the item requested that participants ignore the
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first three choice options provided below the statement (which included “never”, “once per
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week”, “more than once per week”), and instead write “yes” in the text box that followed the
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choice options, alongside the label “other”. Data from the five participants (of 220
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participants recruited) who failed to enter “yes” in the text box were not included in the
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present study, confirming that participants generally were motivated to provide diligent
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choices on the survey.
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Procedure
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We used cumulative prospect theory (CPT) to model participants’ risky choices
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simultaneously across all five sets of gambles. Details of the CPT model are provided in the
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Appendix. For each choice made by participants we calculated the probability, L, that they
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would choose the lottery over the sure amount, S, (or the option of receiving nothing in the
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mixed prospects set), where
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and V equals the subjective expected value of each choice option (Luce, 1959). In Equation 1,
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 is a consistency parameter that estimates choice consistency, and could range in value from
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0 to 10 (Pachur, Hanoch, & Gummerum, 2010 ; Rieskamp, 2008). A high value for the 
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parameter would indicate that participants were switching back and forth inconsistently
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between choice options within gambling sets. We applied the maximum likelihood
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approach for our model fitting and used the G2 measure to assess the goodness of fit of
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the CPT model (Rieskamp, 2008), where
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In Equation 2, fiP(L,S) = P(L,S) if the lottery was chosen on choice trial i and 1 - P(L,S)
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if the sure option was chosen, based on the CPT model parameters. The best fitting CPT
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parameter values were those that minimized Equation 2, where smaller values of G2
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indicate better fit. We fitted the CPT model to the data separately for each of the four groups
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of participants.
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RESULTS
For the gain and loss sets we calculated the amount of the sure option that was equivalent to
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participants’ preference for the lottery. These certainty equivalents (CEs) measure risk taking
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attitudes and are determined by the mid-point between gambles where a participant switches
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from preferring the sure amount to the lottery for loss sets, or vice versa for gain sets2. The
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mean group CEs are provided in Figure 1. The expected value of the lottery for each set
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is indicated by the dotted horizontal line. Observing Figure 1, when the expected value
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of the lottery was a $10 gain participants switched from the lottery to the sure amount
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at $21.94, and when the expected value of the lottery was a $90 loss participants
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switched from the sure amount to the lottery at $70.66, exhibiting risk seeking behavior
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for low-probability gains and high-probability losses (Figure 1). In contrast, when the
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expected value of the lottery a $90 gain participants switched from the lottery to the
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sure amount at $65.25, and when the expected value of the lottery was a $10 loss
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participants switched from the sure amount to the lottery at $19.30, exhibiting risk
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averse behavior for high-probability gains and low-probability losses (Figure 1). Thus,
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participants displayed the four-fold pattern predicted by prospect theory (Kahneman,
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2011; Tversky & Kahneman, 1992), and this was the case for all four groups (Figure 1).
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We conducted a four way mixed analysis of variance (ANOVA) on participants’
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CEs, including the time horizon of the sure outcome (immediate, delayed) and lottery
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outcome (immediate, delayed) as between subjects factors, and the domain (gain, loss)
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and probability (low probability, high probability) of outcomes as repeated measures.
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We reversed the negative values for the CEs in the loss sets in our ANOVA in order to
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make group comparisons between the gain and loss sets. The certainty equivalent of the
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lottery was significantly higher in participants’ choices when the lottery outcome was
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likely (M=67.96) than when it was unlikely (M=20.62), F(1,155)=748.82, p<.001, but did
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not differ between gain and loss sets (gains, M=43.60; losses, M=44.98), F(1,155)=0.73,
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p=.394. However, the domain and probability of outcomes interacted, F(1,155)=10.99,
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p=.001. Follow-up ANOVA’s, conducted separately for low and high probability sets,
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revealed that the certainty equivalent of the lottery was higher for potential losses
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(M=70.66) than for potential gains (M=65.25) when the lottery outcome was likely,
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F(1,174)=7.20, p=.008, but not when the lottery outcome was unlikely (gains, M=21.94;
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losses, M=19.30), F(1,179)=0.88, p=.348. This indicates that participants were more risk
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seeking for likely losses than they were risk averse for likely gains.
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Time horizon affected participants’ risky choices. The certainty equivalent of
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the lottery was reduced by delaying its outcome (immediate, M=49.94; delayed,
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M=39.06), F(1,155)=31.67, p<.001, and was increased by delaying the sure amount
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(immediate, M=37.91; delayed, M=51.09), F(1,155)=41.37, p<.001. The time horizon of
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the lottery outcome interacted with choice domain, F(1,155)=7.91, p=.006. Follow-up
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ANOVAs, conducted separately for the gain and loss sets, confirmed that the certainty
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equivalent of the lottery was reduced by delaying its outcome for both gain sets,
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F(1,185)=29.99, p<.001, and loss sets, F(1,168)=8.39, p=.004. However, follow-up
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ANOVAs, conducted separately for immediate and delayed lottery outcomes, revealed a
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significant difference between gain and loss sets for the delayed lottery outcome,
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F(1,86)=9.66, p=.003, but not immediate lottery outcome, F(1,69)=1.36, p=.248,
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indicating that delaying the lottery outcome had a greater effect on gains (immediate,
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M=51.23; delayed, M=36.39) than on losses (immediate, M=48.65; delayed, M=41.74).
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In the mixed prospects set participants were offered the option of a lottery with
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a 50% chance of losing $50 and a 50% of gaining an amount, x, that incremented on
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each gamble, and the option of receiving nothing. All four groups indicated an amount
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that was significantly greater than $50 (IS:IL, M=154.13, t(45)=4.55, p<.001; DS:DL,
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M=125.10, t(50)=3.83, p<.001; IS:DL, M=146.40, t(49)=3.85, p<.001; DS:IL, M=139.80,
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t(49)=4.81, p<.001), displaying loss aversion for mixed prospects. A two-way
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independent ANOVA conducted on participants’ CEs for the mixed prospects lottery,
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including the time horizon of the sure outcome (immediate, delayed)3 and lottery
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outcome (immediate, delayed) as between subjects factors, revealed that the amount of x
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at which point participants found the lottery equally attractive to the option of receiving
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nothing was unaffected by delaying the sure outcome, F(1,193)=0.68, p=.412, or lottery
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outcome, F(1,193)=0.27, p=.606, and there were no significant interactions.
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Provided in Table 1 are the best fitting cumulative prospect theory (CPT) parameter
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values for each group. In order to make comparisons between groups, we held one of the
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parameters constant for each between-group comparison in a second round of model fitting.
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The reduction in model fit that result from constraining a parameter indicates differences
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between the two groups on the respective parameter, and can be tested for significance using
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the chi-square statistic (Table 1). This analysis revealed that time horizon affected
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participants’ risk taking behavior by increasing their sensitivity to changes in the payoffs (α
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and β parameters) and changes in the probabilities (γ and δ parameters) of gains and losses
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when the sure outcome was delayed, and by decreasing their sensitivity to changes in payoffs
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and probabilities of gains and losses when the lottery was delayed (Table 1). This pattern is
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illustrated in Figure 2, which plots the probability that the lottery is chosen over the sure
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amount based on the best fitting CPT parameter values for each group. Delaying the sure
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amount made the lottery a more attractive option when the prospect was a potential gain, but
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reduced its appeal when it presented a potential loss (Figure 2).
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The effects of delaying a sure outcome on participants’ choices did not mirror
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effects of delaying its riskier lottery alternative. Figure 2 suggests that delaying the prospect
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of a sure gain (or loss) had a greater effect on participants’ preferences for the lottery than did
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delaying the prospect of a lottery gain (or loss). To test this, we compared the ∆G2 values for
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each parameter—the reduction in model fit between groups when a parameter is held
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constant—for when a sure outcome was delayed with the ∆G2 values when the lottery was
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delayed. These comparisons revealed that delaying the sure outcome had a greater impact on
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increasing participants’ sensitivity to changes in payoffs (gains, χ2(1)=.61.17, p<.001; losses,
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χ2(1)=.122.47, p<.001) and probabilities (gains, χ2(1)=.86.21, p<.001; losses, χ2(1)=.12.15,
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p<.001) of outcomes than delaying the lottery outcome reduced them. Waiting for a sure loss
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rather than risk losing more immediately on a lottery was a more attractive option to
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participants than paying the sure amount now rather than wait for the outcome of the lottery
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(Figure 2).
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DISCUSSION
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Despite similarities between risky choice and time discounting, studies of choice behavior
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have typically studied the uncertainty and time horizon of choice options in isolation
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(e.g. Green & Myerson, 2004; Green et al. 1999; Tversky & Kahneman, 1992). We found
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presently that the time horizon of prospects influenced participants’ evaluations of choice
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options. Delaying a lottery outcome with respect to a sure outcome reduced risk taking for
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gains and increased risk taking for losses by reducing participants’ sensitivity to values and
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probabilities of prospects, whereas delaying a sure outcome with respect to a lottery outcome
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had opposite effects on risky choice by increasing sensitivity to the values and probabilities
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of outcomes. This innocuous sounding result has important consequences for theoretical
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accounts of risky choice.
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Problem 1 presented a choice between selling depreciated stocks at an immediate
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loss, and gambling on the prospect that the loss might be recovered, if the stocks are invested
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for a longer term. Our results suggest that delaying the outcome of a riskier option (holding
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onto depreciated stocks) could elicit extreme risk taking behavior. We found that participants
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were more willing to bet on a gamble to avoid a sure loss when the outcome of the gamble
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was deferred to the future. Thus, studies of risky choice using standard economic theory (von
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Neumann & Morgenstern, 1947) and CPT (Tversky & Kahneman, 1992) may underestimate
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risk seeking behavior for losses and underestimate risk aversion for gains by neglecting the
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time horizon of prospects.
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Participants displayed the four-fold pattern predicted by CPT, exhibiting risk
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seeking behavior for low-probability gains and high-probability losses, and risk averse
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behavior for low-probability losses and high-probability gains (Figure 1). Effects of time
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horizon on risky choice behavior varied as a function of both the domain and
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probabilities of outcomes (Figure 2). Participants showed a preference for a lottery win
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over a smaller sure amount when the lottery outcome was unlikely, and this was
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especially the case when the sure outcome was delayed (Figures 1 and 2). When a lottery
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win was likely participants showed a preference for a smaller sure amount, but this
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preference was almost entirely eliminated by delaying the sure amount. When the
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lottery offered an unlikely loss participants preferred to lose a smaller amount with
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certainty, which was heightened by delaying the sure loss. Instead, preference for a
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lottery loss over a certain loss when the lottery outcome was likely was almost entirely
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eliminated by delaying sure loss (Figure 1). These findings imply that effects of time
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horizon depend on both the likelihood of outcomes and whether they refer to potential
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gains or to potential losses.
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Our current results suggest that time horizon is a key to risky choice behavior by
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altering people’s evaluations of risky options. Preference reversal for delayed and
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probabilistic rewards (Prelec & Loewenstein, 1991), and the gain/loss asymmetry observed
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for preferential and risky choice (Mitchell & Wilson, 2010), has led some theorists to propose
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that a single underlying mechanism is involved in discounting of probabilistic and temporal
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rewards (Rachlin et al., 1991). Our results provide a demonstration that time horizon should
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be integrated into models of risky choice behavior.
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Time horizon has important implications for real-world risky choice. Radical
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prostatectomy is a recommendation made by health care professionals for men diagnosed
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with early prostate cancer (Wilt et al., 2012). ‘Watchful waiting’ is an alternative to
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aggressive treatment that involves monitoring cancer growth, but implies a longer time
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horizon. Potential costs and benefits of watchful waiting over immediate treatment has
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received much recent attention, both in the medical literature (Wilt et al., 2012) and media
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reports (Parker-Pope, 2012). Time horizon is likely an important aspect of both patients’
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decisions about treatment and physicians’ recommendations. In the financial domain, time
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horizon is important for investment strategy. Individuals who have shorter investment time
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horizons (e.g. saving for a mortgage down-payment) are encouraged by financial advisors to
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invest less aggressively, and it is recommended that older adults be prudent in their
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investments, for a loss to savings in later life could take many years to recover (Bernard,
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2012). Recent studies report increased aversion to financial risk taking in older age (Rolison,
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Hanoch, & Wood, 2012), and future research may seek to understand how time discounting
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factors in financial risk taking of older adults who have shorter life time horizons than
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younger individuals.
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Time horizon is also central to policy making and may be a key to the climate
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change debate. Some policy makers advocate a “wait-and-see” strategy for dealing with
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climate change (Yohe, Andronova, & Schlesinger, 2004), and despite strong
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recommendations by the scientific community for immediate action, long term negative
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consequences of climate change are low on the list of concerns for the US general public
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(Dunlap & Saad, 2001). As illustrated in Problem 1, a wait-and-see policy defers to the
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future a potentially much larger cost that could be mitigated by immediate action. Our
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current findings indicate that a wait-and-see policy for climate change may be
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appealing because of the delayed time horizon of uncertain future costs. Immediate
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action for climate change could be made more appealing if costs of immediate action
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were re-framed in terms of long term savings.
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Effects of extending the time horizon of sure outcomes in our study did not mirror
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effects of delaying its lottery alternative, suggesting that time horizon has different effects on
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certain and uncertain outcomes. Waiting for a sure loss rather than risk losing more
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immediately on a lottery was more appealing to participants than realizing a sure loss now
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rather than wait for the outcome of a lottery (Figure 2). Our analysis suggests that this
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tendency did not result from a difficulty to interpret the time horizons of risky and riskless
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outcomes. Delaying both types of outcomes together had minimal effects on participants’
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evaluations (Figures 1 and 2). It appears that discounting rates for sure amounts are steeper
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than for risky prospects, which may reflect an adversity to uncertain events in the future with
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a greater tolerance of sure events. Managing finances to meet a fixed debit at a fixed date in
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the future is easier to accommodate than an unknown lottery outcome.
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In conclusion, time horizon of outcomes is a key to risky choice behavior. Our
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findings demonstrate the importance of time horizon for choices under uncertainty, and
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highlight a need for further research to investigate effects of time horizon on risky choices.
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Table 1
Best Fitting Cumulative Prospect Theory (CPT) parameter values for each group
CPT Parameters
Group
α
β
γ
δ
λ
IS; IL
0.684
0.719
0.575
0.636
0.889
DS; DL
0.681
0.722
0.628
0.711
0.783
IS; DL
0.586
0.637
0.485
0.611
0.675
DS; IL
1.178
1.336
2.772
0.758
0.152
φ
0.199
0.204
0.367
0.012
Mean G2
97.677
92.709
91.580
101.321
∆G2(IS; IL vs. DS; DL)
0.029
0.035
4.206
6.833
3.917
0.041
P
.864
.852
.040
.009
.048
.839
∆G2(IS; IL vs. IS; DL)
15.286
7.084
19.438
1.266
7.573
23.219
P
<.001
.008
<.001
.260
.006
<.001
∆G2(IS; IL vs. DS; IL)
76.460
129.553
105.645
13.417
67.531
140.956
P
<.001
<.001
<.001
<.001
<.001
<.001
450 Note. Α, β, γ, δ, λ, and φ, are the best fitting CPT parameters values based on participants’ choices
451 when either the lottery outcome (IS:DL), sure outcome (DS:IL), both (DS:DL), or neither (IS:IS) is
452 delayed. Α and β describe participants’ sensitivity to changes in the value of gains and losses,
453 respectively, whereas γ and δ describe participants’ sensitivity to changes in probabilities of gains
454 and losses. Λ measures loss aversion, and φ measures choice consistency. G2 is the model fit, and
455 ∆G2 is the reduction in model fit for each parameter when the parameter is held constant across
456 respective groups. P is the chi-square significance test for the ∆G2 value.
457
458
459
460
19
461
462
Figure 1. Mean group Certainty Equivalents (CEs) when either the lottery outcome (IS:DL),
463
sure outcome (DS:IL), both (DS:DL), or neither (IS:IL) is delayed for low (p = .10) and high
464
(p = .90) probability gain and loss sets. Dotted lines indicate the expected value of the lottery
465
for each set. Error bars represent 1 standard error above and below the mean.
466
467
468
469
470
20
471
21
472
473
Figure 2. The predicted probability that the lottery is chosen over the sure amount based on
474
the best fitting cumulative prospect theory (CPT) parameter values for each group. The
475
observed probabilities are the proportions of participants who chose the lottery over the sure
476
amount. The probabilities are plotted separately for each gambling set and separately for each
477
group, with either the lottery outcome (IS:DL), sure outcome (DS:IL), both (DS: DL), or
478
neither (IS:IL) delayed.
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
22
497
APPENDIX
498
Within cumulative prospect theory (Tversky & Kahneman, 1992), the subjective value of an
499
outcome, x, is identified by its value function
500
(A1)
501
where the curvature of the value function is determined separately for gains and losses by the
502
α and β parameters, respectively. Α and β were treated as free parameters to be fitted to
503
participants’ data, and were constrained to values greater than zero and <10, producing a
504
concave function for gains and a convex function for losses when below 1. Loss aversion is
505
captured by the λ parameter applied to the value function for losses, and was constrained to
506
values greater than 0 and <10, where values greater than 1 indicate loss aversion.
507
508
509
510
511
The decision weights, γ and 𝛿, transform the probabilities, px, for gains and losses,
respectively, and identify the individual’s sensitivity to gains and losses, where:
;
(A2)
512
.
(A3)
The γ and 𝛿 parameters were constrained to values above 0 and <10. For decision weights
513
smaller than 1, probabilities of rare events are overestimated, and moderate to likely events
514
are underestimated. The transformed probabilities, px, are used to assign subjective
515
probabilities to each choice prospect, such that the probability of an outcome,
516
517
(A4)
;
518
(A5)
519
520
.
Probability weights for prospects refer either to gains, π+, or to losses, π-, and in Equations
521
A4 and A5 are determined by the difference between the probability of an outcome at least as
522
good or better than the outcome for gains, and the difference between the probability of an
523
outcome at least as bad or worse than the outcome for losses.
23
524
The subjective value, V(choice option), for each choice option is determined by
525
combining the subjective values and probability weights of each prospect, such that
526
527
The decision maker then chooses the option with the highest subjective expected value.
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
24
549
FOOTNOTES
550
1Chapman
551
choices in separate blocks of trials that referred to either monetary amounts or health
552
outcomes. While participants’ choices corresponded in the monetary and health
553
contexts, preference reversal in the intertemporal choice sets did not correspond with
554
biases in the risky choice sets. The authors concluded that similar biases in risky and
555
intertemporal choice have different underlying psychological mechanisms.
and Weber (2006) asked participants to make uncertain and intertemporal
556
557
2Fifteen
558
sure amount and the lottery for at least one of the five gambling sets, compared with 15
559
of 55 (27%) participants in the DS:DL group, 14 of 56 (25%) participants in the IS:DL
560
group, and 21 of 55 (38%) participants in the DS:IL group. Across all four groups, 13 of
561
the total 215 (6%) participants switched inconsistently in the low probability gain set,
562
compared with 15 (7%) in the high probability gain set, 25 (12%) in the low probability
563
loss set, 26 (12%) in the high probability loss set, and 18 (8%) in the mixed prospects
564
set. Gambling sets that contained inconsistent switching were excluded from our
565
ANOVA. All data were included in our model analysis.
of 49 (31%) participants in the IS:IL group switched inconsistently between the
566
567
3In
568
immediate and delayed outcomes are equivalent for the sure option.
the mixed prospects set, the sure option is to receive nothing, and thus the
569
570
571
ACKNOWLEDGEMENT
The authors would like to thank Swiss&Global-Ca’ Foscari Foundation for its support.
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