Small Feedback-based Decisions and Their Limited Correspondence to Description-based Decisions

The present paper explores situations in which the information available to decision makers is limited to feedback concerning the outcomes of their previous decisions. The results reveal that experience in these situations can lead to deviations from maximization in the opposite direction of the deviations observed when the decisions are made based on a description of the choice problem. Experience was found to lead to a reversed common ratio/certainty effect, more risk seeking in the gain than in the loss domain, and to an underweighting of small probabilities. Only one of the examined properties of description-based decisions, loss aversion, seems to emerge robustly in these ‘feedback-based’ decisions. These results are summarized with a simple model that illustrates that all the unique properties of feedback-based decisions can be a product of a tendency to rely on recent outcomes. Copyright # 2003 John Wiley & Sons, Ltd. key words Probability learning; feedback-based decisions; reinforcement learning; prospect theory; Allais paradox Many common activities involve ‘small’ decision problems. Driving, for example, requires repeated selection among routes, speeds, and various other options. Although little time and effort is typically invested in these and similar small decisions, they can be consequential. The estimated cost of traffic accidents in the USA is more than 100 billion dollars a year (see e.g. Blincoe, 1994), and many of the accidents are at least partially products of ex-post unwise decisions. The current paper focuses on an important subset of the small decision problems exemplified above that can be referred to as ‘small feedback-based’ decisions. These problems are defined by three main properties. First, they are repeated; decision makers face the same problem many times in similar situations. Second, Copyright # 2003 John Wiley & Sons, Ltd. * Correspondence to: Ido Erev, Faculty of Industrial Engineering and Management, Technion, Haifa, Israel. E-mail: [email protected] Contract/grant sponsors: USA–Israel Binational Science Foundation; Technion VPR; Max Wertheimer Minerva Center. The 1994 estimate is 150.5 billion. This estimate represents the present value of lifetime economic costs for 40,676 fatalities, 5.2 million non-fatal injuries and 27 million damaged vehicles. each single choice is not very important; the alternatives tend to have similar expected values that may be quite small. Finally, the decision makers do not have ‘objective’ prior information concerning the payoff distributions. In selecting among the possible options, they have to rely on the immediate feedback obtained in similar situations in the past. Notice that small feedback-based decision problems are quite distinct from the problems studied in mainstream decision research. Mainstream research tends to focus on ‘big’ decisions that are made based on a careful evaluation of the possible payoff distributions. These decisions are naturally studied in a ‘descriptionbased’ paradigm. In a typical study (e.g. the studies used by Kahneman & Tversky, 1979, to validate prospect theory) the participants are presented with a complete description of a non-trivial choice problem and are asked to make one selection among the possible options. One reasonable approach to modeling feedback-based decisions assumes that decision makers use the obtained feedback to discover the underlying choice problem (to estimate the payoff distributions of the different alternatives). Under this assumption, the large structural differences between feedback-based and description-based decision problems may not lead to large behavioral differences. In both cases the decision makers are expected to respond to the underlying payoff distributions. This interesting assumption was supported in research conducted by Benartzi and Thaler (1995). They show that prospect theory, the popular and elegant summary of the main results obtained in studies of description-based decisions, can account for reallife feedback-based investment decisions. In particular, it explains the suggestion that most people are under-invested in the stock market relative to the market’s past performance. According to this account, loss aversion, one of the main concepts in prospect theory, leads investors to prefer safe prospects that eliminate losses even at a very high cost (in terms of expected return). This explanation was later supported by the findings of a laboratory experiment (Thaler et al., 1997; and see Gneezy & Potters, 1997) that shows the pattern predicted by prospect theory emerging in 200 repetitions of a controlled investment task. However, there are also good reasons to predict important differences between feedback-based and description-based decisions. The most obvious reason involves the assertion that immediate feedback is likely to lead to adaptive learning that moves behavior toward expected value maximization. This assertion is consistent with certain interpretations of the law of effect (Thorndike, 1898) and with the view that ‘practice makes perfect’. Since studies of description-based decisions reveal robust deviations from maximization, the ‘adaptive learning’ assertion implies higher maximization rates in feedback-based decisions. The hypothesis of higher maximization rates in feedback-based decisions can also be supported based on previous studies of decisions among repeated play of gambles (see Lopes, 1981; Keren & Wagenaar, 1987; Keren, 1991; Bockenholt & Wedell, 1990). In a typical study, decision makers were asked to select which of two gambles they prefer to play 10 times. Comparison of the results with a control condition in which the selected gamble was played only once showed that the expectation of playing the gambles repeatedly increased the tendency to select the gamble with the highest expected value. Although feedback-based decisions differ from these ‘repeated-play’ conditions in many respects, they involve repeated play. Thus, a similar trend can be predicted. A very different type of difference is predicted under a variant of the assertion, made above, that decision makers simply use feedback to estimate the payoff distributions. If the samples used to derive the estimates are relatively small, the estimates will not be accurate. Inaccurate estimates that can lead to large differences between feedbackand description-based decisions are particularly likely when rare events are important and the payoff distributions have large variances. The availability of immediate and unbiased feedback distinguishes the problems studied here from the problems used by Herrnstein and his colleagues (see e.g. Herrnstein et al., 1993) to study melioration. Although real-life investment decisions may not be a good example of small decisions, Thaler et al.’s experiment did study small decisions. The maximal expected payoff per trial was ‘only a few cents’ (Thaler et al., 1997). Tversky and Bar Hillel (1983) note that this effect can be a result of the change in the implied payoff distributions. 216 Journal of Behavioral Decision Making Copyright # 2003 John Wiley & Sons, Ltd. Journal of Behavioral Decision Making, 16: 215–233 (2003) A fourth reason to predict unique properties of feedback-based decisions comes from previous studies of decision making in signal detection tasks (see review in Erev, 1998). These studies show regularities that seem very different from the regularities observed in description-based decisions. For example, Barkan et al. (1998) documented robust underweighting of rare events (that tend to be overweighted in description-based studies). Since signal detection tasks are similar to feedback-based tasks (with the exception of the availability of trial specific information), it is natural to assume that this pattern will be observed in feedback-based decisions. Finally, the study of choice behavior in repeated games reveals high sensitivity to recent feedback (see e.g. Erev & Roth, 1998; Cheung & Friedman, 1999; Camerer & Ho, 1999; Selten & Buchta, 1998; Daniel et al., 1998; Cooper et al., 1997; Chen & Tang, 1999). This observation is inconsistent with the assumption that the feedback is effectively used to estimate the relevant distributions. The main goal of the current research is to improve our understanding of the differences between one-shot description-based and small feedback-based decisions. To achieve this goal we chose to focus on four of the best-known properties of choice behavior in one-shot description-based tasks and examined if similar properties emerge when decision makers have to base their choices on the obtained feedback. In an attempt to study simple small feedback-based decisions, the current study examines binary choice tasks where the only relevant information available is the obtained payoff from previous decisions. In each trial of the current studies (with the exception of Experiment 5) the decision makers were asked to choose between the two keys of ‘a computerized money machine’ (see the Appendix). Each selection yielded a payoff in points—a draw from the selected key’s payoff distribution. This payoff and the accumulated earnings were presented on the screen and constituted the only information available to the decision makers. The decision makers were told that their goal was to maximize their number of points and that these points would determine their final payoff. They neither received prior information concerning the payoff distributions, nor did they know that the payoffs were random draws from static payoff distributions. This paper is organized as follows. Experiment 1 presents a replication of Thaler et al. (1997). It shows that the loss aversion tendency they documented can be observed in the minimalist paradigm used here. Experiments 2, 3, and 4 show that, unlike the loss aversion phenomenon, other properties of one-shot description-based decisions do not necessarily emerge when decision makers rely on feedback. Interestingly, the exact opposite tendencies were observed. Experiment 2 reveals that experience with feedback-based decisions can make an outcome provided with certainty less attractive. This trend is a mirror image of the common ratio/certainty effect captured by prospect theory. Experiment 3 shows that feedback-based decisions reflect a stronger risk aversion tendency in the loss domain than in the gain domain. This pattern is a reversal of the pattern observed in description-based studies. Experiment 4 demonstrates that when decision makers rely on feedback, they tend to underweight rare outcomes. This tendency is the opposite of the overweighting of rare outcomes observed in description-based studies. The relationship of these results to the decision-making literature is clarified in Experiment 5 and with a simple model. Experiment 5 suggests that the effect of feedback is quite distinct from the effect of the expectation of playing the selected gamble repeatedly. The model demonstrates that the unique properties of feedback-based decisions can be a product of a tendency to rely on recent outcomes. EXPERIMENT 1: LOSS AVERSION (A REPLICATION OF THALER ET AL., 1997) A comparison of two of the conditions examined by Thaler et al. (1997) provides evidence for loss aversion in a feedback-based decision task. In Condition 1 (‘Monthly’ in their scenario) subjects repeatedly divided a The main cost of this minimal information setting involves the difficulty in deriving ‘rational behavior’ (any behavior can be rational under certain prior beliefs). Since the current research is not designed to ask if people are ‘rational’, we are not concerned by this cost (see Budescu et al., 1997, for a discussion of the value of ‘rationality free’ decision research). G. Barron and I. Erev Feedback-based and Description-based Decisions 217 Copyright # 2003 John Wiley & Sons, Ltd. Journal of Behavioral Decision Making, 16: 215–233 (2003) portfolio between a risky and a safe investment. The risky investment (‘stocks’) produced high-yield returns with large variability (a normal payoff distribution with a mean of 1 and standard deviation of 3.54). The safe investment (‘bonds’) produced low-yield returns but insured small variability and positive outcomes (a normal distribution with a mean of 0.25 and standard deviation of 0.177 that was truncated at 0). Subjects were not informed as to the actual payoff matrix and were told only that they were dividing their portfolio between two unnamed funds. In Condition 2 (‘Inflated monthly’) payoffs from both investments were increased by a constant so that all payoffs were positive. The results reveal that in Condition 1 subjects consistently preferred the low-risk (and low expected value) bonds. Interestingly, 200 trials with immediate feedback did not lead subjects to prefer the stocks and to maximize expected return. Over the 200 trials, the proportion of expected-value maximizing choices was 0.42. When returns were all in the gain domain (Condition 2), the maximization rate increased to 0.56. Thaler et al. note that these results are consistent with the prediction of a natural generalization of prospect theory to repeated decision tasks. To derive the implications of these results for small feedback-based decisions it is important to distinguish between two possible interpretations. Under the first, the observed loss aversion pattern is facilitated by the investment scenario and is robust to the availability of feedback. Under a second interpretation, loss aversion emerges from feedback. Experiment 1 was designed to evaluate the second interpretation. It is a simplified replication of the conditions described above, with the addition of a condition in which the variance of both distributions was small. Subjects partook in a repeated binary task where they chose between two buttons without receiving prior information as to the payoff distribution of either button. This minimalist design was implemented to evaluate the pure effect of feedback. Design Three payoff conditions were compared. The basic task in all three conditions was a binary choice between two approximately normal payoff distributions referred to as L and H. This basic task was performed 200 times (with immediate feedback) in each of the conditions. The conditions differed with respect to the two distributions. Following Thaler et al., in the baseline Condition (Problem 1), distribution L was centered at 25, had a standard deviation of 17.7, and was truncated at 0 to prevent negative returns. The implied mean was 25.63. Distribution H had a mean of 100 points with a standard deviation of 354. In the second condition (Problem 2), all payoffs were transformed to the gain domain by adding 1200 points to the means of both distributions while the standard deviations remained unchanged. The third condition (Problem 3) was identical to the second except that the standard deviation of both distributions equaled 17.7. At the conclusion of the experiment, points were converted to monetary payoffs according to the exchange rate: 100 points1⁄4 0.05 Shekels (about 1.25 US cents). Subjects Thirty-six Technion students served as paid participants in the experiment. Most of the participants in this and the other experiments described in this paper were secondand third-year industrial engineering and economics majors who had taken at least one probability and economics course. They were randomly assigned to one of the three experimental conditions. In addition to the performance contingent payoff, described above, subjects received 5 Shekels for showing up. Final payoffs ranged between 6.25 and 15 Shekels ($2–$4). The draws were rounded to the nearest integer. The other payoff distributions were not truncated. In Problems 2 and 3 subjects were only paid for points above 240,000 (1200 points 200 trials). 218 Journal of Behavioral Decision Making Copyright # 2003 John Wiley & Sons, Ltd. Journal of Behavioral Decision Making, 16: 215–233 (2003) Apparatus and procedure Subjects were informed that they were playing on a ‘computerized money machine’ (see a translation of the instructions in the Appendix) but received no prior information as to the game’s payoff structure. Their task was to select one of the ‘machines’’ two unmarked buttons (see the figure in the Appendix) in each of the 200 trials. Participants were aware of the expected length of the experiment (30 minutes to an hour), so they knew that it included many rounds. To avoid an ‘end of task’ effect (e.g. a change in risk attitude), they were not informed that the experiment included exactly 200 trials. Payoffs were contingent upon the button chosen; they were drawn from the distribution associated with the selected button, described above. Two types of feedback immediately followed each choice: (1) the payoff for the choice, that appeared on the selected key for the duration of 1 second, and (2) an update of an accumulating payoff counter, which was constantly displayed. Results The graph in the left-hand cell of Figure 1 presents the proportion of maximization (H) choices in 20 blocks of 10 trials over the 12 participants in each condition (the right-hand cell presents the predictions of the model discussed below). The learning curves show non-monotonic trends in Problems 1 and 2. The proportion of H choices decreased initially, and then increased. We return to this pattern below. The aggregated results in Problems 1 and 2 are practically identical to those of the corresponding conditions in Thaler et al. In line with the counter-intuitive prediction made by a generalization of prospect theory, the average subject in Problem 1 did not learn to prefer the alternative that maximizes expected value. Over the 200 trials, subjects’ aggregate proportion of H choices was 0.30 (STD1⁄4 0.22) in Problem 1, 0.51 (STD1⁄4 0.21) in Problem 2, and 0.85 (STD1⁄4 0.08) in Problem 3. The differences are significant (t[22]1⁄4 2.43, p< 0.05 for Problems 1 and 2, and t[22]1⁄4 5.28, p< 0.001, for Problems 2 and 3). Notice that the difference between Problems 2 and 3 is an example of the payoff variability effect (see Myers et al., 1965; Busemeyer & Townsend, 1993): an increase in payoff variability impairs maximization. Not knowing the length of the experiment also prevents subjects from using probability-based reasoning (the focus on the likelihood of achieving a particular aspiration level) (Lopes, 1996). This type of reasoning bases choice on the probability of coming out ahead, which is a function of the number of choices to be made. A second reason for not telling subjects the game’s length is that this better approximates the real-world small decisions that interest us. In such situations, the number of future choices to be made is often unknown. Figure 1. Replication of Thaler et al. (Experiment 1). Each curve in the left-hand column shows the observed proportion of maximization (H) choices in 20 blocks of 10 trials in one of the three problems. The problems are presented on the left: Alternative N(x, y) yields a draw from a normal distribution with mean of y and standard deviation x. In TN(x, y) the payoff is truncated at 0 (0 is the worst possible payoff ). The right-hand column presents the predictions of the VA model G. Barron and I. Erev Feedback-based and Description-based Decisions 219 Copyright # 2003 John Wiley & Sons, Ltd. Journal of Behavioral Decision Making, 16: 215–233 (2003) This difference suggests that there are two reasons for the low maximization rate in Problem 1: loss aversion and high payoff variability. The observation that an increase in payoff variability impairs maximization is, of course, not surprising; even optimal procedures designed to find the maximum are less accurate when the variability increases. Nevertheless, the behavioral payoff variability effect has many interesting implications. One set of implications is concerned with the effect’s magnitude and its robustness to experience. As Figure 1 shows, the large effect does not seem to decrease with time. Experiment 2 evaluates another nontrivial implication. EXPERIMENT 2: THE REVERSED COMMON RATIO/CERTAINTY EFFECT Allais (1953) has noticed that the more risky of two prospects becomes relatively more attractive when the probability of winning in both prospects is multiplied by a common ratio. This ‘common ratio’ effect is important because it represent a robust violation of expected utility theory’s substitution principle. Kahneman and Tversky (1979) demonstrate this effect with an experimental study of the following choice problems (the outcomes represented hypothetical payoffs in thousand Israeli Lira): Problem 4. Choose between: L: 3 with certainty H: 4 with probability 0.8; 0 otherwise Problem 5: Choose between: L: 3 with probability 0.25; 0 otherwise H: 4 with probability 0.2; 0 otherwise Notice that Problem 5 was created by dividing the probability of winning in Problem 4 by four. Kahneman and Tversky’s results revealed that while 80% of the subjects preferred L in Problem 4, only 35% preferred L in Problem 5. In prospect theory the common ratio effect is an example of the certainty effect, and it is captured by the assumption of a nonlinear probability weighting function that overweighs outcomes that occur with small probabilities (and underweighs outcomes that occur with high probabilities). Specifically, Alternative H is unattractive in Problem 4 because the ‘0’ outcome ( p1⁄4 0.2) is overweighted, but it is attractive in Problem 5 because the ‘4’ outcome ( p1⁄4 0.2) is overweighted. Interestingly, MacDonald et al. (1991) found a similar pattern in feedback-based decisions by rats. The current experimental paradigm appears to fall between the paradigms studied by Kahneman and Tversky and Macdonald et al. We focus on human subjects (like Kahneman & Tversky) and on feedback-based decisions (like MacDonald et al.). Since Kahneman and Tversky and MacDonald et al. found the same pattern, interpolation predicts that a similar pattern will be found here. However, a different prediction can be derived from the payoff variability effect. Payoff variability (relative to expected payoff difference) is larger in Problem 4 than in Problem 5. Thus, the robust payoff variability effect found in Experiment 1 predicts a higher maximization rate in Problem 4 than in Problem 5. The current study compares these competing predictions. Design Problems 4 and 5 were compared. The translation from points to monetary payoffs was according to the exchange rate: 1 point1⁄4 0.01 Shekel (0.25 US cents). Subjects Forty-eight Technion students served as paid participants in the experiment. They were randomly assigned to one of the two experimental conditions. Subjects received an initial (showing-up) fee and additional payoff 220 Journal of Behavioral Decision Making Copyright # 2003 John Wiley & Sons, Ltd. Journal of Behavioral Decision Making, 16: 215–233 (2003) contingent upon performance. The total payoff ranged between 20 and 30 Shekels ($5–7). The showing-up fee was 12.60 Shekels in Problem 4 and 21.90 Shekels in Problem 5; the difference was implemented to equate the expected earnings in both conditions assuming random choice. Apparatus and procedure The procedure was identical to Experiment 1 with the exception that the experiment was run for 400 trials. Results The experimental data are presented in Figure 2. The mean proportion of H choices (high EV but more risky alternative), over subjects and blocks, was 0.63 (STD1⁄4 0.24) when L provided ‘3 with certainty’, it decreased significantly to 0.51 (STD1⁄4 0.10) when payoff probabilities were divided by four (t[46]1⁄4 2.35, p< 0.05). Thus, as predicted by the payoff variability hypothesis, the aggregated choice proportion deviated from the predictions of identical choice proportion in the two problems in the opposite direction of the deviation observed in description-based decisions. We refer to this deviation as the ‘reversed common ratio/certainty’ effect. Notice that a comparison of the current results to the results reported by MacDonald et al. appear to suggest that there are differences between humans and rats. However, the difference in subject population is only one of many differences between the two experiments. For example, MacDonald et al. used a within-subject design and allowed the decision makers to immediately consume their rewards. EXPERIMENTS 3a AND 3b: THE REVERSED PAYOFF DOMAIN EFFECT Experimental studies of decision making in one-shot description-based tasks reveal a ‘payoff domain effect’: Decision makers tend to be more risk averse in the gain domain than in the loss domain. A particularly strong example, also known as the ‘reflection effect’, was documented by Kahneman and Tversky (1979) in a comparison of Problem 4 (see above), with: Figure 2. The reversed certainty effect (Experiment 2). Each curve in the left-hand column shows the observed proportion of maximization (H) choices in 40 blocks of 10 trials in one of the two problems. The problems are described on the left: Alternative (V, p) yields payoff of V with probability p, and zero otherwise. Alternative (V) yields payoff of V with certainty. The right-hand column presents the predictions of the VA model G. Barron and I. Erev Feedback-based and Description-based Decisions 221 Copyright # 2003 John Wiley & Sons, Ltd. Journal of Behavioral Decision Making, 16: 215–233 (2003) Problem 6. Choose between: L: 4 with probability 0.8; 0 otherwise H: 3 with certainty Kahneman and Tversky’s results revealed deviation from expected value maximization (preference of L) in both problems: 80% of the subjects preferred L in Problem 4, and 92% preferred L in Problem 6. The deviation from maximization in the gain domain (Problem 4) can be described as risk aversion, and the deviation in the loss domain (Problem 6) can be described as risk seeking. Prospect theory captures both deviations with the assumption of an S-shaped value function. The present experiments examine if a similar pattern emerges in feedback-based decisions. Experiment 3a Design Two choice problems were compared: Problem 4 from Experiment 2, and Problem 6 (note that new subjects partook only in Problem 6). As in Experiment 2, the exchange rate was 1 point1⁄4 0.01 Shekel (0.25 US cents). To insure similar objective incentive structures in the two conditions, the initial showing-up fee in Problem 6 was 37.4 Shekels. Subjects, apparatus, and procedure As in Experiment 2. Results The experimental results for the two conditions, presented in the top panel of Figure 3, show maximization rates of more than 0.5. Thus, feedback did not lead to more risky choices in the loss than the gain domain. In fact, the opposite pattern was observed. The proportion of risky choices was 0.63 (STD1⁄4 0.24) when prospects were positive, and it decreased significantly to 0.40 (STD1⁄4 0.19) when prospects were negative (L in Problem 4) (t[46]1⁄4 3.64, p< 0.05). The difference between the maximization rates is insignificant (t[46]1⁄4 0.58, NS). Experiment 3b Although the results of Experiment 3a are a mirror image of the famous demonstration of the payoff domain effect, they do not prove that feedback can reverse this effect. Recall that the low rate of H choices in the description-based decisions in Problems 4 and 6 implies a deviation from maximization in the direction of the payoff domain effect. The higher rate of H choices observed in feedback-based decisions can, therefore, be interpreted in two ways: a stronger tendency to maximize earning, and/or a reversed payoff domain effect. To evaluate if a clear reversed payoff domain effect can emerge, we chose to study choice among gambles with equal expected values. Design Two binary choice problems were compared (see lower panel in Figure 3). Problem 7 involves a choice between a safe prospect (S) that promises ‘9 with certainty’ and a risky prospect (R) that yields ‘10 with p1⁄4 0.9; 0 otherwise’. In Problem 8 the safer prospect (S) yields ‘ 9 with certainty’ and the risky prospect (R) yields ‘ 10 with p1⁄4 0.9; 0 otherwise’. As in Experiment 2, the exchange rate was 1 point1⁄4 0.01 Shekel. To insure similar objective incentive structures in the two conditions, the initial showing-up fee in was set at 7 Shekels in Problem 7 and at 43 Shekels in Problem 8. Subjects, apparatus, and procedure As in Experiment 2 with the exception that 30 subjects participated in each condition and the experiment lasted only 200 trials. 222 Journal of Behavioral Decision Making Copyright # 2003 John Wiley & Sons, Ltd. Journal of Behavioral Decision Making, 16: 215–233 (2003) Results The results are presented in the lower panel of Figure 3. Over trials and subjects the proportion of risky choice (preferring the gamble over the sure payoff ) was 0.56 (STD1⁄4 0.30) in the gain domain and 0.37 (STD1⁄4 0.27) in the loss domain. The difference between the two proportions is significant (t[58]1⁄4 2.65, p< 0.01). Thus, in the current setting, decision makers appear to take more risk in the gain than in the loss domain. We refer to this pattern as the ‘reversed payoff domain effect’. EXPERIMENT 4: UNDERWEIGHTING OF SMALL PROBABILITIES In one-shot description-based tasks decision makers behave ‘as if’ they overweight small probabilities. For example, Kahneman and Tversky (1979) found that most people prefer the gamble (5000, with p1⁄4 0.001; 0 otherwise) over a sure payoff with the same expected value. These findings can be captured by with the assumption of a nonlinear weighting function that implies the overweighting of small probabilities Figure 3. The reversed payoff domain effect (Experiments 3a and 3b). (a) Observed (left) and predicted (right) proportion of maximization (H) choices in 40 blocks of 10 trials. (b) Observed (left) and predicted (right) proportion of risky (R) choices in 20 blocks of 10 trials G. Barron and I. Erev Feedback-based and Description-based Decisions 223 Copyright # 2003 John Wiley & Sons, Ltd. Journal of Behavioral Decision Making, 16: 215–233 (2003) (in prospect theory), and with the idea that the rare events ‘tax’ the more likely events (in the TAX model, see Birnbaum & Navarrete, 1998). Careful quantification of this effect shows that the average decision maker behaves as if she overweighs probabilities below 0.25 (see a recent quantification of this effect in Gonzalez & Wu, 1999). To examine whether feedback-based decisions lead to a similar pattern, we examined 400 trials of play in three problems containing small probabilities. Design The three choice problems presented in Figure 4 were examined. As in Experiments 2 and 3, the exchange rate was 1 point1⁄4 0.01 Shekel. The initial showing-up fee was 12.60 Shekels in Problem 9, 21.90 Shekels in Problem 10, and 37.40 Shekels in Problem 11. Subjects, apparatus, and procedure As in Experiment 2.