Preference Reversals: The Impact of Truth-Revealing Incentives * July 2005

Transcription

1 Preference Reversals: The Impact of Truth-Revealing Incentives * by Joyce E. Berg, ** John W. Dickhaut *** and Thomas A. Rietz **** July 2005 * We thank Colin Camerer, Jim Cox, Robyn Dawes, Philip Dybvig, Ido Erev, Mike Ferguson, William Goldstein, Glenn Harrison, Charles Holt, Joel Horowitz, Jack Hughes, Forrest Nelson, Charles Plott, Paul Schoemaker, Amos Tversky, Nathaniel Wilcox, and workshop participants at the Economic Science Association, University of Chicago, University of Iowa and Cornell University for thought provoking comments and conversation in the development of this paper. ** Department of Accounting, Henry B. Tippie College of Business, University of Iowa, Iowa City, Iowa *** Department of Accounting, Carlson School of Management, University of Minnesota, th Avenue South, Minneapolis, MN **** Department of Finance, Henry B. Tippie College of Business, University of Iowa, Iowa City, Iowa 52242

2 Preference Reversals: The Impact of Truth-Revealing Incentives Abstract We re-examine preference reversal data from previously published studies to (1) identify the effects of different incentive treatments and (2) determine the models that best explain data patterns across incentive treatments. Contrary to the folk-wisdom that incentives do not affect behavior in preference reversal experiments, we find that different incentive treatments do have a clear impact on the pattern of data observed in these experiments. While truth revealing incentives do not eliminate reversals, responses become consistent with stable preferences reported with error. We demonstrate this in both a model-free context and by using maximum likelihood estimation to test specific behavioral models of choice. Without incentives, models based on taskdependent evaluations of gambles are necessary to explain the data. However, under truthrevealing incentives, models of stable preference expressed with error explain behavior as well as any model possibly could. Thus, our results suggest that incentives change both the overall response pattern and the underlying decision processes of subjects.

3 Preference Reversals: The Impact of Truth-Revealing Incentives I. Introduction An extensive literature documents the robustness of preference reversals, the inconsistency in stated preference when two different elicitation methods are used. According to this literature, monetary incentives have little effect on overall reversal rates. But, reversal rates do not capture all aspects of behavior in preference reversal experiments. In this paper, we reexamine studies in which researchers elicited preferences over monetary gambles (bets) using direct comparison (choices) and pricing. We document a previously undetected incentive effect. Though rates of reversal are stable across incentive environments, the pattern of responses changes dramatically when truth-revealing incentives are used. Without incentives, subjects reveal preferences that seem affected by whether they are revealed through prices placed on bets or direct comparisons. With clear, truth-revealing incentives, responses become consistent with subjects having systematic preferences between the bets, but revealing these preferences with random errors in the two tasks. We show that this change is consistent with a fundamental change in the model underlying behavior. There is a current debate on the effects of incentives in experiments (e.g., Camerer and Hogarth, 1999; Gneezy and Rustichini, 2000; and Kahneman, 2003). Preference reversal and incentives research naturally overlap. The preference reversal literature was one of the first lines of research to suggest that incentives made no difference (e.g., Lichtenstein and Slovic, 1971 and 1973; and Grether and Plott, 1979). Recently, interest in the combination of the two areas resurfaced (e.g., Camerer and Hogarth, 1999; Selten, Sadrieh and Abbink, 1999; and Berg, Dickhaut and Rietz, 2003). Apart from Berg, Dickhaut and Rietz (2003) researchers generally conclude that incentives make little difference in standard preference reversal experiments. Why? Incentives seldom make - 1 -

4 reversal rates fall much. The apparent conclusion is that incentives do not matter. Here, we show that this conclusion does not follow. We document clear effects of incentives by studying prior preference reversal literature. We show that experiments with incentives lead to behavior decidedly more consistent with an assumption that subjects have systematic preferences across the two bets, but make random errors in revealing them. We arrive at our conclusion using two different analyses of how incentives alter the entire response pattern, not just the overall reversal rate, in preference reversal data. First, we simply present the observable changes in the aggregate patterns of behavior across experiments with no incentives, indeterminate incentives (those that do not necessarily lead expected utility maximizing subjects to reveal the truth) and truth-revealing incentives (those that induce expected utility maximizing subjects to reveal the truth). We show that these differences are significant and argue that they are consistent with truth-revealing incentives leading to more stable preferences across bets (i.e., preferences that appear unaffected by the elicitation task). Then, we develop formal models of behavior and test them explicitly. The formal models and tests validate the initial aggregate analysis. Taken together, our results are striking. Without incentives, preferences revealed through choice tasks indicate approximate indifference across the bets in a pair. However, subjects price bets as if they strongly prefer the more risky bet. Further, differences in conditional reversal rates (reversal rates conditional on which bet is chosen and reversal rates conditional on which bet is priced higher) cannot be reconciled with stable preferences, random errors in revelation, and error corrections. Consequently, in experiments without incentives, the response patterns can only be explained with theories that assume that evaluations are affected by the task. Models that assume stable preferences revealed with random errors (e.g., expected utility with error) cannot fit the data. Choice patterns under truth-revealing incentives differ dramatically. Subjects consistently prefer the high variance bets in both the choice and pricing tasks. And, the conditional reversal rates are consistent with subjects preferring high variance bets on average, revealing these - 2 -

5 preferences with random errors and error corrections. These overall patterns are consistent with models in which subjects have stable preferences across bets, but reveal them with noise. We show that in typical experiments with truth-revealing incentives, a model based on stable preferences revealed with error not only fits the data statistically, but fits the data as well as any model possibly could. Overall, the underlying model driving choice appears to change with the kind of incentives used in the experiment. In the next section, we discuss preference reversal experiments and the nature of the data we analyze. Then, we present the differences in the patterns of aggregate data. In particular, we show the change in patterns is consistent with incentives shifting subjects toward more stable preferences. In Section IV, we develop formal models and tests. In Section V, we provide our estimation and test results. In Section VI, we conclude. II. Preference Reversal Experiments and Data A. General Description of Preference Reversal Experiments So that our analysis isolates the impact of incentives, we consider only preference reversal experiments that are similar to the original work of Lichtenstein and Slovic (1971) and not experiments that introduce significant changes in the tasks performed by subjects. A typical Lichtenstein and Slovic type experiment consists of two tasks: comparing and pricing bets presented in monetary or abstract (e.g., points as the experimental currency) terms. Subjects evaluate pairs of bets with approximately equal expected value, but unequal variance. One bet (the P-bet") has a high probability of winning a small amount of money. The other bet (the $bet") has a low probability of winning a large amount of money. For example, Grether and Plott (1979) contains the following typical preference reversal bet pair: a bet with a 32/36 chance of winning $4.00 but a 4/36 chance of losing $0.50 (the P-bet), and a bet with a 4/36 chance of winning $40.00 but a 32/36 chance of losing $1.00 (the $-bet). While the bets have similar expected values ($3.50 versus $3.56), the P-bet has significantly lower variance (2.00 versus - 3 -

6 166.02). Approximately equal expected values and a large difference in variance characterize all of the bet pairs in the experiments we study. 1 Each subject participates in two tasks designed to elicit preferences for the paired bets: a paired choice task and a pricing task. In the paired choice task, the subject is presented with the two bets simultaneously and asked which one is preferred. In the pricing task, the subject is shown each bet independently and asked to state a minimum selling price for the bet. The preference ordering implied by the choice task is compared to the ordering implied by the pricing task to determine whether the subject s orderings are consistent. A preference reversal occurs when the choice and pricing tasks imply different orderings for the two bets. 2 The data from preference reversal experiments are typically analyzed at the aggregate level rather than at the individual subject level and are presented as frequencies in four response cells (when indifference is not allowed as a response) or nine cells (when indifference is allowed). We use the same unit of analysis here. Figure 1 shows a typical data set (from Grether and Plott, 1979, experiment 1b). We focus our analysis on Cells a through d. 3 Cells a and d represent stable preferences: the bets within a pair are assigned the same ordering by both the choice and pricing decisions. Cells b and c represent reversals: the less risky bet chosen but priced lower (Cell b) or the more risky bet chosen but priced lower (Cell c). 4 Using the variables a through d to represent the frequencies of observations in Cells a through d, the reversal rate is (b+c)/(a+b+c+d). 1 We note that because of the way we screen experiments included in this paper, the differences in bets across data sets is small and, hence, the results we present cannot be attributed to differences in bets across experiments. 2 More details of the experiments analyzed here are found in the appendix. 3 We do this for both data limitation and theoretical reasons. These are discussed later in this paper. 4 Cell b reversals, where the P-bet is chosen and the $-bet priced higher, are often called "predicted" reversals. Cell c reversals, where the $-bet is chosen while the P-bet is priced higher are often called unpredicted reversals. See Lichtenstein and Slovic (1971) for further elaboration on this point

7 Figure 1: Typical pattern of Preference Reversal Responses (from Grether and Plott, 1979, Experiment 1b, 276 observations) P-bet priced higher P-bet chosen Cell a % $-bet chosen Cell c % Indifference in choice task Cell I % $-bet priced higher Cell b % Cell d % Cell I % P- and $-bets priced the same Cell I % Cell I % Cell I % B. Classification by Incentives Our study re-evaluates the effects of incentives in preference reversal experiments. To select the data to be included in our analysis, we applied the following criteria. First, the data must be published and the published paper must present the cell frequencies a through d in Figure 1 (which we need to estimate and differentiate the underlying models of behavior that we will discuss later). 5,6 Second, the experiments must involve either hypothetical or real monetary rewards and use individual choice tasks with bets viewed simultaneously and individual selling price elicitation tasks with bets viewed individually. We exclude studies where the experimental design dramatically changed the tasks. This allows us to focus cleanly on incentive effects. 7 Even with these restrictions, our data set contains a large variety of incentive designs conducted by experimenters with a variety of backgrounds and interests. An overview of the experimental 5 We do not include working papers for two reasons: (1) we have no way of knowing that we have selected all available working papers, and (2) the working papers have not satisfied the rigor of peer review. 6 Since most preference reversal papers report only aggregate data in the form of overall response frequencies, our analysis is on the consistency of the aggregate response frequencies with various models of behavior. In related work, Berg, Dickhaut and Rietz (2003) show that, under induced risk-preference incentives, analysis of individual behavior directly is also consistent with noisy maximization. 7 Examples include: (1) context rich, multidimensional environments (e.g. Tversky, Sattath and Slovic, 1988), (2) changes in value elicitation response modes (e.g., Cox and Epstein, 1989, Bostic Herrnstein and Luce, 1990, Casey, 1991, and Bohm, 1994, Cox and Grether, 1996), (3) arbitraging irrational choices (e.g., Berg, Dickhaut and O Brien, 1985, arbitrage treatments), (4) process tracing, in which subjects cannot see all potential payoffs to gambles simultaneously, and may never actually see them all sequentially (e.g., Schkade and Johnson, 1989) and (5) risk preference induction (e.g., Selten, Sadrieh and Abbink, 1999, induced risk preference experiments, and Berg, Dickhaut and Rietz, 2003), among others. Such factors are important and deserve study, but confound the focus of our study: pure incentive effects

8 designs is contained in our Appendix; design details are in the original papers. Table 1 shows the data used in this analysis. It includes author and year, experiment number, data arranged according to the four cells we use in our analysis, reversal rate and incentives classification (described next). To begin our analysis, we first classify each experiment according to the form of incentives used: no monetary incentives, indeterminate incentives, and truth-revealing incentives. Details supporting the classification of individual experiments are contained in the Appendix. "No-incentives experiments" use hypothetical bets. We analyze four such experiments, all of which use flat participation fees, but have no performance-based rewards in their experimental design. Because no differential reward is given for responding truthfully, any response is optimal for an expected utility maximizing subject who cares only about the monetary payoffs for the experiment. Included in this category are Lichtenstein and Slovic (1971) experiments 1 and 2, Goldstein and Einhorn (1987) experiment 2a, and Grether and Plott (1979) experiment 1a. We label the data sets L&S1, L&S2, G&E2a, and G&P1a, respectively. 8 "Indeterminate-incentives experiments" use bets that have real monetary payoffs, but the designs do not strictly induce truthful revelation for utility maximizing subjects. We include these experiments to determine whether monetary incentives alone affect behavior in preference reversal experiments or whether truth-revealing incentives are necessary. We classify and analyze them separately to provide the cleanest test of incentive effects. These experiments include Lichtenstein and Slovic (1971) experiment 3, two Lichtenstein and Slovic (1973) experiments conducted in Los Vegas, and four experiments from Pommerehne, Schneider and Zweifel (1982). We label the data sets L&S3, L&SLV+ (this session used positive expected value bets), L&SLV- (this session used negative expected value bets) and PSZ1.1, PSZ1.2, PSZ2.1, and PSZ2.2, respectively. 8 There are other preference reversal experiments that do not use monetary incentives, for instance, Schkade and Johnson (1989). We do not include that experiment here because their design does not guarantee that subjects see all parameters of the bets in the choice or pricing task. However, we note that including this data set would strengthen the results we report

9 Incentives Category None Indeterminate Truth- Revealing Table 1: Datasets Analyzed Data Data set Short Reversal number Paper and Experiment Name N * a b c d Rate 1 Lichtenstein and Slovic (1971), Exp. 1 L&S % 2 Lichtenstein and Slovic (1971), Exp. 2 L&S % 3 Goldstein and Einhorn (1987) Exp. 2a G&E2a % 4 Grether and Plott (1979) Exp. 1a G&P1a % 5 Lichtenstein and Slovic (1971), Exp. 3 L&S % 6 Lichtenstein and Slovic (1973) Positive EV L&SLV % 7 Lichtenstein and Slovic (1973) Negative EV L&SLV % 8 Pommerehne, Schneider and Zweifel (1982) Grp. 1 Run 1 PSZ % 9 Pommerehne, Schneider and Zweifel (1982) Grp. 1 Run 2 PSZ % 10 Pommerehne, Schneider and Zweifel (1982) Grp 2 Run 1 PSZ % 11 Pommerehne, Schneider and Zweifel (1982) Grp 2 Run 2 PSZ % 12 Grether and Plott (1979) Exp. 1b G&P1b % 13 Grether and Plott (1979) Exp. 2b (Selling Prices) G&P2SP % 14 Grether and Plott (1979) Exp. 2b ($ Equivalents) G&P2DE % 15 Reilly (1982) Stg.1, Grp. 1 R % 16 Reilly (1982) Stg.1, Grp. 2 R % 17 Berg, Dickhaut, and O'Brien (1985), Exp. 1, Ses. 1 BDO % 18 Berg, Dickhaut, and O'Brien (1985), Exp. 1, Ses. 2 BDO % 19 Berg, Dickhaut, and O'Brien (1985), Exp. 2, Ses. 1 BDO % 20 Berg, Dickhaut, and O'Brien (1985), Exp. 2, Ses. 2 BDO % 21 Chu and Chu (1990) Reilly Replication C&C % 22 Selten, Sadrieh and Abbink (1999) Money Payments w/o Summary Stats. SSA % 23 Selten, Sadrieh and Abbink (1999) Money Payments w/ Summary Stats. SSA % * N is the total number of choices in cells a, b, c and d

10 "Truth-revealing-incentives experiments" incorporate unambiguous incentives for truthfully revealing preferences. These experiments all use a paired choice task and a pricing procedure that should elicit truthful revelation of prices (generally the Becker, DeGroot, Marschak, 1964, pricing procedure). We analyze twelve experiments in this category: three from Grether and Plott (1979), two from Reilly (1982), four from Berg, Dickhaut, and O'Brien (1985), one from Chu and Chu (1990) and two from Selten, Sadrieh and Abbink (1999). These are denoted G&P1b, G&P2SP, G&P2DE, R1.1, R1.2, BDO1.1, BDO1.2, BDO2.1, BDO2.2, C&C, SSA1 and SSA2, respectively. III. Model Free Effects of Incentives in Preference Reversal Data Patterns Table 2 presents summary statistics for each of the datasets analyzed in this paper. In the preferences columns, we show the percentage of time the P-bet is revealed as preferred in each task and the difference in these two percentages. The next column repeats the overall reversal rate (from Table 1) for convenience. The last six columns present conditional reversal rates and the differences in these rates. We discuss each item in detail in the following subsections

11 Incentives Category Data Set None Indeterminate Truth Revealing Average Preferences For the P Bet According to Choices (a+b) Table 2: Preferences, Reversal Rates and Conditional Reversal Rate Asymmetries Average Preferences For the P Bet According to Prices (a+c) Absolute Difference Between P Bet Preference Measures Conditional (on Choice) Reversal Rates Conditional (on Pricing) Reversal Rates Reversal Rate P-Bet $-Bet P-Bet $-Bet (b+c)/(a+b+c+d) (b/(a+b)) (c/(c+d)) Difference (b/(b+d)) (c/(a+c)) Difference L&S % L&S % G&E2a % G&P1a % L&S % L&SLV % L&SLV % PSZ % PSZ % PSZ % PSZ % G&P1b % G&P2SP % G&P2DE % R % R % BDO % BDO % BDO % BDO % C&C % SSA % SSA %

12 A. Effects on Reversal Rates Table 2 shows why prior researchers concluded that incentives have little effect on reversal rates. Reversal rates are remarkably robust across experiments. They ranged from 35% to 46% (average 40%) in experiments without incentives, 27% to 43% (average 34%) in experiments with indeterminate incentives, and 22% to 41% (average 33%) in experiments with truth-revealing incentives. While some truth-revealing incentives experiments exhibit lower reversal rates and the lowest on average, many reversal rates are higher than those observed with no incentives or indeterminate incentives. The difference across treatments is only marginally significant. A Wilcoxon rank-sum statistic comparing the rates under no-incentives to rates under truth-revealing incentives using each experiment as a data point is (p-value=0.0523). Such data leads some researchers (e.g., Camerer and Hogarth, 1999, Table 1) to conclude that incentives do not affect behavior (or may even make behavior less rational) in preference reversal experiments. However, the overall reversal rate is not the only metric for measuring effects of incentives. B. Effects on Preference over Bets Induced value theory (Smith, 1976) implies that preferences are induced and revealed through appropriate incentive structures. In experiments without incentives, monetary payments, if any, do not depend on a subject s choices or prices. As a result, a subject s choices and prices may not reveal distinct preferences in the experimental tasks. This could arise either because subjects do not have preferences across the bets (as would be the case for a subject with preferences over monetary rewards alone) or because subjects do not have an incentive to reveal their preferences truthfully. In experiments with indeterminate incentives, the way monetary payments are connected to choices sometimes gives incentives for subjects to reveal their preferences truthfully. But, in other cases, the payment mechanisms actually give subjects incentives to misrepresent their preferences. In experiments with truth-revealing incentives, risk neutral subjects remain indifferent across bets with the same expected value. However, risk

13 averse subjects will prefer the P-bet and risk seeking subjects will prefer the $-bet. As a result, incentives may lead to clearer preferences across bets as well as providing a reward for truthful revelation of these preferences. Table 2 and Figure 2 (horizontal axis) both show the percentage of P-bet choices in the experiments. Here, an incentives effect becomes apparent. P-bet choices ranged from 49% to 53% and averaged 51% in experiments without incentives. Subjects appear to have little, if any, preference for one bet-type over another on average. P-bet choices ranged from 38% to 68% and averaged 52% in experiments with indeterminate incentives. Across the varying incentive mechanisms in the indeterminate incentive group, preferences seem to change, but not in a systematic way. However, when truth-revealing incentives are introduced, P-bet choices ranged from 32% to 50%, with an average of 41%. Subjects appear decidedly more risk seeking on average under truth-revealing incentives. The difference across treatments is significant. A Wilcoxon rank-sum statistic comparing the rates in no-incentives experiments to rates in truthrevealing incentives experiments using each experiment as a data point is (p-value=0.0053). Thus, under truth-revealing incentives, clear, systematically risk-seeking preferences are revealed

14 Figure 2: Overall Reversal Rates and Percentages of P-Bet Choices 50% 45% 40% Overall Reversal Rate 35% 30% 25% 20% 15% No Incentives Indeterminate Clear Incentives 10% 5% 0% 0% 10% 20% 30% 40% 50% 60% 70% 80% Percentage of P-Bet Choices Another effect of truth-revealing incentives on revealed preferences over bets is a remarkably higher consistency in preferences according to the two tasks. Table 2 shows the absolute difference between the percentage of times subjects prefer the P-bet according to the choice task and the percentage of times subjects prefer the P-bet according to the pricing task. These statistics represent the degree of consistency (stability) of preferences across tasks at an aggregate level. Notice the dramatic increase in consistency under truth-revealing incentives. The average absolute difference without incentives is 25%. Under indeterminate incentives it is 21%. Under truth-revealing incentives it drops to 12%. Again, the difference across treatments is significant. A Wilcoxon rank-sum statistic comparing the rates in no-incentives experiments to rates in truth-revealing incentives experiments using each experiment as a data point is (p

15 value=0.0153). Thus, incentives seem to create both clear and consistent preferences across the two tasks. C. Effects on Conditional Reversal Rates We examine whether subjects choices and prices are more likely to reveal stable preferences with error when they operate under truth-revealing incentives (in Berg, Dickhaut and Rietz, 2003, we call this noisy maximization). In such a case, we should see a systematic pattern in conditional reversal rates. In particular, reversals that represent errors should be uncommon while those that represent error corrections should be common. Here, we show how conditional reversal rates are affected by preferences. In the next section, we show how this is tied to noisy maximization as a behavioral model of choice. Conditional reversal rates and their signed differences are presented in Table 2. These differences are also shown in Figure 3. Clearly, truth-revealing incentives have an impact, primarily on the asymmetry in reversal rates conditional on the pricing task ranking. While the asymmetry in reversals conditional on choice differs little across treatments, incentives change the sign of the asymmetry conditional on the price-ranking. A Wilcoxon rank-sum statistic comparing the latter asymmetry under no-incentives to the asymmetry under truth-revealing incentives using each experiment as a data point is (p-value=0.0076). Thus, incentives have a significant effect on conditional reversal rates. We next show that this effect is consistent with a noisy maximization explanation of behavior

16 Figure 3: Asymmetry in conditional reversal rates when the rates are conditioned on the type of bet chosen versus the type of bet priced higher 40% Asymmetry in Reversal Rate Conditional on Choices in Pricing Tasks (b/(b+d)-c/(a+c)) 30% 20% 10% 0% 0% 10% 20% 30% 40% 50% 60% -10% -20% -30% Asymmetry in Reversal Rate Conditional on Choices in Choice Tasks (b/(a+b)-c/(c+d)) No Incentives Indeterminate Clear Incentives D. Noisy Maximization Explanation The combination of revealed preferences and conditional reversal rates in the truthrevealing incentives experiments can be explained by noisy maximization: preference for one bettype over another, revealed with random errors. Here, we argue the point in a model-free manner. In the next section, we model noisy maximization formally and test it as an explanation of behavior against other behavioral models in the literature. To see how noisy maximization works, suppose a subject prefers the $-bet. This is largely consistent with average revealed preferences under truth-revealing incentives (a+b 50% in all cases and a+c 50% in all but one case). If the subject chooses the $-bet in the choice task and makes relatively few errors in the pricing task, then we should see a low reversal rate conditional

17 on choosing the $-bet (c/(c+d) should be low). On the other hand, if the subject chooses the P-bet, this is an error. Again, relatively few mistakes in the pricing task implies that prices will reflect true preferences more often than not in the pricing task and we should actually see a high reversal rate conditional on (mistakenly) choosing the P-bet (b/(a+b) should be high). This high conditional reversal rate is consistent with error correction. Similarly, if the subject prices the $-bet higher (the subject s true preference) the reversal rate conditional on pricing should be low (b/(b+d) should be low). If the subject prices the P-bet higher (in error), the reversal rate conditional on this price ordering should be high (c/(a+c) should be high), representing error correction. 9 Table 2 shows that with no incentives subjects are largely indifferent between bet-types according to the choice task, but tend to prefer the $-bet in the pricing task. The reversal rate conditional on choosing the P-bet is always higher than the rate conditional on choosing the $-bet (b/(a+b) is high while c/(c+d) is low). But, the reversal rate conditional on pricing the P-bet higher is always lower than the rate conditional on pricing the $-bet higher (c/(a+c) is low while b/(b+d) is high). This pattern is inconsistent with noisy maximization where subjects are assumed to have stable preferences across bets regardless of the task. Instead, it is consistent with the compatibility hypothesis underlying expression theory (Goldstein and Einhorn, 1987) or task dependent preferences (Tversky, Slovic and Kahneman, 1990). Both models imply that apparent preferences will vary with the task. We call such models task dependent evaluation models. The pattern of preferences and conditional reversal rates changes dramatically under truthrevealing incentives. On average, subjects prefer the $-bet according to both tasks. Consistent with noisy maximization, the reversal rate conditional on choosing the P-bet is higher than the rate conditional on choosing the $-bet. In all but two cases, the reversal rate conditional on pricing the 9 Similarly, if a subject actually prefers the P-bet, the opposite relationships on conditional reversal rates should hold. The rates conditional on choosing the P-Bet and conditional on pricing the P-Bet higher should be low (b/(a+b) and c/(a+c) should be low). The rates conditional on choosing the $-Bet and conditional on pricing the $-Bet higher should be high (c/(c+d) and b/(b+d) should be high), each representing error corrections

18 P-bet higher is higher than the rate conditional on pricing the $-bet higher. This is also largely consistent with noisy maximization. In summary, without incentives, the aggregate data can only be explained by task dependent evaluation. With truth-revealing incentives, the data is largely consistent with noisy maximization. In the next section, we develop formal models of these ideas and test them. IV. Modeling the Effects of Incentives in Preference Reversal Experiments In this section, we analyze behavior in preference reversal experiments in three steps. First, we show how cell frequencies and maximum likelihood estimation can be used to estimate parameters for underlying models of behavior in each data set. Then, for any given data set, we define a best-fit benchmark model that gives the maximum likelihood attainable by any conceivable underlying model of behavior. No model can ever fit the data better than this benchmark. 10 This benchmark model is a statistical representation of fit, not a specific behavioral model. However, in context, restrictions on this model yield models that correspond to common behavioral models including strict expected utility theory, noisy maximization, expression theory and an extreme form of task dependent preferences. Finally, we test these actual behavioral models against the benchmark to determine whether each of these behavioral models attains the best possible fit (and, therefore, we can never find a model that explains the data better) or whether each model fits significantly worse (and, therefore, there may be models that can explain the data better). The patterns discussed in the last section suggest that, without incentives, we should see data consistent with expression theory, but not with noisy maximization. With truth-revealing incentives, the data should be consistent with noisy maximization. 10 Again, we are fitting a model to the aggregate-level data, not each individual subject s responses

19 A. Maximum Likelihood Estimation of Behavioral Models of Preference Reversal We use maximum likelihood estimation to find parameters that maximize the likelihood of the observed data in Cells a, b, c, and d in Figure 1 subject to restrictions imposed by the model under investigation. 11 In fitting models, note that the cell frequencies in Figure 1 represent a multinomial distribution. The joint log likelihood function based on predicted cell frequencies is: L = ln(n!) ln(a!) ln(b!) ln(c!) ln(d!) + Aln[m a (θ)] + Bln[m b (θ)]+ Cln[m c (θ)]+ Dln[m d (θ)] where n is the number of observations in the dataset; A, B, C and D are the total numbers of observations in Cells a, b, c and d; m a (θ), m b (θ), m c (θ) and m d (θ) are the model s predictions of frequencies for cells a, b, c and d based on the model s underlying parameters, θ. Estimates and their variances are found using standard maximum likelihood techniques (see Judge, Hill, Griffiths, Lutkepohl and Lee, 1982). The value of the likelihood function given the estimated parameters indicates the model s ability to explain the data. We use likelihood ratio tests to distinguish between the behavioral models that we study. B. The Best-Fit Benchmark Model The multinomial nature of reported preference reversal data allows us to define a best-fit benchmark. If the predicted cell frequencies could be set freely, then the (global) maximum likelihood is attained by matching the predicted cell frequencies to the observed cell frequencies. That is, a = m a (θˆ ), b = m b (θˆ ), c = m c (θˆ ), and d = (θˆ ). Because this solution results in the global maximum of the likelihood function, it provides the best possible fit that any model can possibly attain in explaining these cell frequencies. m d 11 Allowing for errors and applying maximum likelihood to test between competing models of behavior is similar to Harless and Camerer (1994) and Hey and Orme (1994). We focus on only these four cells for several reasons. First, not all experiments permitted indifference as a response. Second, indifference responses do not have unambiguous implications for preference orderings. For example, a subject may rationally state the same price for two gambles while choosing one gamble over the other if the difference in preference is not sufficiently high to be observed in the price grid used in the experiment. Third, the models generally predict continuous preferences over the gambles, so the chances of being truly indifferent are vanishingly small according to theory

20 This benchmark is useful for two reasons. First, any model that can always be parameterized to match observed frequencies will always attain the global maximum likelihood and, as a result, can never be rejected by aggregate cell frequency data. That is, such a model has no testable restrictions in the context of this data: any two such models are observationally equivalent given the structure of the data. We will show that, a priori, one model proposed in the literature will always match cell frequencies. As a result, while this model always attains the best-fit benchmark likelihood, it can never be rejected nor distinguished from other models that also always attain the best-fit benchmark. Second, using a best-fit benchmark allows us to test whether models that introduce meaningful restrictions on predicted frequencies are rejected by the data. If a restricted model does attain the global maximum likelihood of the best-fit benchmark for a particular data set (i.e., the restrictions are not binding for that data set), then no other model can do any better in explaining that data than the restricted model. If a restricted model does not achieve the benchmark, but a likelihood ratio test does not reject the restrictions, then no other model can do significantly better in explaining that data than the restricted model. If the likelihood ratio test does reject the restrictions, then there may be models that explain the data significantly better than the restricted model being tested. 12 This is similar in spirit to developing "fully saturated" models for testing against in the sense that we find and test against models that can always be parameterized to predict exactly the observed cell frequencies. 13 We define the Best Fit Benchmark Model as a simple extension of either Goldstein and Einhorn s (1987) expression theory or Lichtenstein and Slovic s (1971) expected-utility-based twoerror-rate model. We do not argue that it corresponds to actual behavior. It is developed purely as a means for nesting the behavioral models we study and to provide a benchmark likelihood against 12 In fact, since we know that one model always attains the best-fit benchmark for aggregate preference reversal data, there is always a model that fits significantly better in this case. 13 However, the intuition from the typical log linear case does not always apply here because of the nonlinearities in the models. For example, the two-error-rate model developed below might appear fully saturated because it has three parameters and must fit three free cell frequencies. In fact it cannot always fit all three observed cell frequencies because of its quadratic form

21 which we test other models. 14 The way in which these models nest is shown in Figure 4 and described in the rest of this section. Figure 4: Relationships between Models Studied* Task Dependent Expression Models 1. Benchmark Model ** parameters q, r, s P, s $ r=0 (II) s P = s $ :=s Noisy Maximization Models 2. Expression Theory ** (I) 1-s P =s $ :=s (III) q=0 & re-label 4. Expected Utility with Two Error Rates (IV) r=s 5. Expected Utility with One Error Rate 3. Task Dependent Preferences r=s=0 6. Strict Expected Utility Theory *Numbered models correspond to model numbers in the text. Numbered restrictions correspond to likelihood ratio tests in Table 5. **The Benchmark Model and Expression Theory are observationally equivalent models that always attain the best-fit benchmark The parameters of the benchmark model are q, r, s P and s $. The parameter q corresponds to the percentage of subjects who have an underlying preference for the P-bet. In the choice task, subjects make errors and, at the rate r, report the wrong preference ordering. 15 Reported prices also sometimes change the apparent ordering of the bets. Denote by s P the rate at which 14 We note that purely random behavior or mixed strategies in selecting the two gambles under the two tasks can always be parameterized to achieve the best fit benchmark and, as a result, could be used as best fit benchmark models. However, we choose to parameterize the best fit benchmark model in the way we do so that it nests other behavioral models developed in the literature. As a result, we can test the ability of these other models to explain the data with simple likelihood ratio statistics. 15 This is the extension of Goldstein and Einhorn s (1987) expression theory, which assumes subjects do not make errors in reporting their preferences in the choice task

22 orderings are reversed for subjects who actually prefer the P-bet. Denote by s $ the rate at which orderings are reversed for subjects who actually prefer the $-bet. 16 Under the benchmark model, data will conform to the pattern shown in Figure 5. This model can always be parameterized to attain the best-fit benchmark likelihood. This is true because for any observed a, b, c and d, one can find valid parameters for the model such that a = m a (θˆ), b = m b (θˆ ), c = m c (θˆ ) and d = m d (θˆ ). To see this, set r=0 and maximize the (global) likelihood function by matching observed frequencies to predictions. Setting the model predictions equal to the observed frequencies shows that the estimated parameters must solve a = qˆ (1 sˆ ), b = qs ˆ ˆ P, c = ( 1 qˆ) sˆ $ and d = ( 1 qˆ)(1 sˆ $ ). Simple algebra shows that q ˆ = a + b, sˆ P = b /( a + b) and s ˆ$ = c /( c + d ) solve the system and represent valid fractions and probabilities. Hence, the benchmark model always achieves the (global) best-fit benchmark. By definition, no model can ever fit the data better than the benchmark model. P Figure 5: Pattern of Responses for the Benchmark Model P-bet priced higher P-bet chosen a = (q)(1-r)(1-s p ) + (1-q)(r)(s $ ) $-bet chosen c = (q)(r)(1-s p ) + (1-q)(1-r)(s $ ) $-bet priced higher b = (q)(1-r)(s p ) + (1-q)(r)(1-s $ ) d = (q)(r)(s p ) + (1-q)(1-r)(1-s $ ) where: q = percentage of subjects whose underlying preference ordering ranks the P-bet higher r = error rate in the paired-choice task s p = error rate in the pricing task when P-bet preferred s $ = error rate in the pricing task when $-bet preferred C. Task Dependent Evaluation Models A single restriction on the benchmark model gives Goldstein and Einhorn s (1987) expression theory (Model 2 in Figure 4). Expression theory argues that subjects have a preference 16 This is the extension of Lichtenstein and Slovic s (1971) two-error-rate model, which assumes s $ =s P =s

23 ordering over bets that is accurately reported in the choice task. However, in the pricing task, valuations are influenced by an anchor and adjust mechanism based on the compatibility of the units of measure used in prices and bet outcomes. Because of this, price evaluations of bets may differ from choice evaluations. Further, Goldstein and Einhorn (1987) conjecture that the rate of inconsistencies will depend on whether the subject prefers the P-bet or the $-bet in the choice task. Starting with the benchmark model let q represent the fraction of subjects who prefer the P- bet in the choice task. Because this preference is accurately reported in the choice task, set r=0. Denote the conditional reversal rates by the fractions s P for reversals conditional on the P-bet being chosen and s $ for reversals conditional on the $-bet being chosen. Given this notation, expression theory predicts the cell frequencies given in Figure 6. This pattern results from applying the restriction that r=0 to the response pattern in Figure 5. Figure 6: Pattern of Responses According to Expression Theory P-bet priced higher $-bet priced higher P-bet chosen a = (q)(1-s p ) b = (q)(s p ) $-bet chosen c = (1-q)(s $ ) d = (1-q)(1-s $ ) where: q = percentage of subjects whose underlying preference ordering ranks the P-bet higher s p = the fraction of the time that pricing biases reverse apparent preference orderings when the P-bet is actually preferred s $ = the fraction of the time that pricing biases reverse apparent preference orderings when the $-bet is actually preferred Like the benchmark model, the expression theory model can explain any pattern of aggregate preference reversal behavior. To see this, use exactly the same argument as used to show that the benchmark model is a best-fit model. Both the benchmark model and expression theory always achieve the best-fit benchmark and are observationally equivalent on aggregate preference reversal data. Further, neither model has testable implications for aggregate data

24 An alternative interpretation of parameters in this model gives task dependent preferences (Tversky and Thaler, 1990) where there is some systematic preference, but the elicitation task may change the preferences in some cases. Define q as the percentage of the time subjects prefer the P-bet when preferences are elicited using the choice task. Some percentage of them, s P, change their preference in the pricing task. A percentage of subjects who prefer the $-bet change their preference in the pricing task. Denote this by s $. This model differs from expression theory only in interpretation. It is identical mathematically and cannot be distinguished by the data. However, another restriction (Restriction I in Figure 4) leads to an extreme form of task dependent preferences where preferences are constructed purely through the elicitation task and preference orderings implied by responses are entirely task specific (Model 3 in Figure 4). In an extreme form this would imply independence of the rows and columns in Figure 1. To achieve this, simply restrict 1-s P =s $ =s. This is a two parameter model with q representing the fraction of choices for the P-bet and s representing the fraction of times the P-bet is priced higher. Testing whether task dependent preferences explains the data significantly worse than the benchmark is a simple χ 2 test of independence of the rows and columns of Figure 1. D. Noisy Maximization Models We define noisy maximization in the same manner as Berg, Dickhaut and Rietz (2003). Expected utility maximization would lead subjects to have invariant preferences over the bets. Camerer and Hogarth (1999) argue that even if this was so, the labor cost of optimization and accurate reporting may result in errors. The error rates should depend on the difficulty of optimization relative to the payoffs for optimizing. In the case of preference reversal experiments, the choice and pricing tasks may have different difficulties and different error costs, so the two tasks could have different error rates. Lichtenstein and Slovic s (1971) two-error-rate model (Model 4 in Figure 4) has exactly these properties. It results from a single restriction on the

25 benchmark model (Restriction II in Figure 4 that s P =s $ =s) and results in the response pattern given in Figure 7. Figure 7: Pattern of Responses Generated by the Two-Error-Rate Model P-bet priced higher P-bet chosen a = (q)(1-r)(1-s) + (1-q)(r)(s) $-bet chosen c = (q)(r)(1-s) + (1-q)(1-r)(s) $-bet priced higher b = (q)(1-r)(s) + (1-q)(r)(1-s) d = (q)(r)(s) +(1-q)(1-r)(1-s) where: q = percentage of subjects whose underlying preference ordering ranks the P-bet higher r = error rate in the paired-choice task s = error rate in the pricing task Unlike the restriction that yields expression theory, the restrictions in the two-error-rate model are meaningful and make the model refutable. In particular, matching the frequencies and solving for q, r and s shows that, when a solution exists, the following relationships maximize the global likelihood function: ad - bc q ˆ (1 - q)= ˆ, (1) (a+ d) - (b+ c) rˆ = (a+ b - q)/(1 ˆ - 2q) ˆ (2) and s = (a+ c - q)/(1 ˆ - 2q) ˆ. ˆ 17 (3) 17 Due to the quadratic form, there are actually two equivalent sets of parameters that satisfy these equations because q and 1-q are interchangeable. The resulting estimates of r and s are each one minus the original estimate. We do not take a stand on which set of estimates is correct because it is irrelevant to the likelihood function (both sets give the same likelihood) and, hence, to the likelihood ratio tests discussed below. We let the data choose which set we display in the tables by looking at the consistent choice cells a and d. If a>d, we display the solution with q>0.5 and if a<d, we display the other solution

26 When a solution exists, the two-error-rate model achieves the (global) best-fit benchmark. 18 Thus, it explains the data as well as the benchmark model, as well as expression theory and, in fact, as well as any model possibly could. Note however that the two-error-rate model does impose testable restrictions on the parameter estimates. In particular, this model will not achieve the best-fit benchmark when (ad-bc)/(a-b-c+d) < 0 or (ad-bc)/(a-b-c+d) > 0.25 because equation (1) cannot be solved. Further, equations (2) and (3) may not result in valid error rates in the [0,1] range even when (1) can be solved. The restrictions imposed by the two-error-rate model are stronger than one might think. Because there are three free cell frequencies to fit in Figure 1, one conjectures that any three parameter model should explain the data. However, this intuition fails for the two-error-rate model because cell frequencies are not simple linear functions of q, r, and s. Using Monte Carlo simulations, we find that out of all possible cell frequencies that can be explained by the benchmark model, only about one third can be explained as well by the two-error-rate model. In the rest of the cases, one or more restrictions keep the two-error-rate model from achieving the best-fit benchmark. 19 While simulation results are informative about the general performance of the two-error-rate model, what is more important is to consider whether the two-error-rate model is likely to fit the patterns of data typically seen in preference reversal experiments. We find that the two-error-rate model s ability to fit the data is primarily determined by the same two factors that appear to be affected by incentives: (1) whether subjects have a distinct preference for one bet-type over another, and (2) the asymmetries in conditional reversal rates. In particular, the two-error-rate model cannot fit patterns of data when subjects are approximately indifferent between the bets and there are asymmetries in the conditional reversal rates. Further, it cannot fit the data when 18 In spite of having three parameters to fit three free cell frequencies, it is not fully saturated because of its quadratic form. 19 We arrive at this number through 50,000 simulations, drawing cell frequencies randomly from the feasible set. Other simulations (details and results available upon request) show that that the two-error-rate model also frequently fails to fit the data when it is simulated from expression theory and task-dependent preferences

27 subjects have distinct preferences for one bet-type and the asymmetries in conditional reversal rates are inconsistent with those preferences. Consider the case in which subjects do not exhibit a clear preference for one bet-type over another. Empirically, this is typical of data from preference reversal experiments conducted without incentives: P-bets and $-bets are chosen with approximately equal frequency in the choice task. Figure 8 shows the two-error-rate model s ability to fit the data in such a case. We graph the asymmetry in conditional (on choices) reversal rates (a/(a+b)-c/(c+d)) and the overall reversal rate (b+c) for choices that show only a slight preference for the P-bet (a+b = , corresponding to Lichtenstein and Slovic s (1971), experiment 1). The shaded area shows combinations of reversal rates and asymmetries that can be explained by the two-error-rate model. The white areas are instances in which the two-error-rate model fails. Notice that the two-errorrate model can explain data only under quite limited conditions: (1) when there are no reversals (the bottom axis), (2) when reversals are about 50% (horizontally across the middle of the diagram, indicating true indifference between the bets) or (3) when the conditional reversal rates are roughly symmetric (vertically up the middle of the diagram, indicating that reversals for either choice are approximately the same). Simply put, when there are not strong preferences across the bet-types, asymmetric conditional reversal rates cannot be accommodated by the two-error-rate model. Instead, a model such as the anchor and adjust hypothesis of expression theory is needed

28 Figure 8: Reversal configurations that can be explained by the two-error-rate model when subjects choose the P-bet slightly more often than the $-bet (a+b= ). Now consider the case where subjects have a distinct preference for one bet-type over another. Empirically, this is typically the case in preference reversal experiments with incentives. Figure 9 shows the two-error-rate model s ability to fit the data in such a case. As in Figure 8, we graph the asymmetry in conditional reversal rates and the overall reversal rate for a particular choice pattern. In this case, we examine subjects who prefer the $-bet on average according to the choice task (a+b = , corresponding to Grether and Plott s (1979), experiment 1b). Again, the shaded area shows combinations of reversal rates and conditional asymmetries that can be explained by the two-error-rate model. Notice that the two-error-rate model can only explain data when (1) there are no reversals (the bottom axis) or (2) when the asymmetries are in

29 the direction of error correction. Patterns of reversal that are inconsistent with simple error correction (the white areas of the graph) are inconsistent with the two-error-rate model. 20 Figure 9: Reversal configurations that can be explained by the two-error-rate model when subjects strongly prefer the $-bet according to the choice task (a+b= ). We note that two simple restrictions on the two-error-rate model lead to two other maximization models of choice. First, restricting r=s (Restriction IV in Figure 4) leads to a oneerror-rate model (Model 5 in Figure 4). Though Camerer and Hogarth s (1999) capital-laborproduction theory would argue that this is unlikely, we examine the one-error-rate model in order to determine whether the less restrictive two-error-rate model proposed by Lichtenstein and Slovic (1971) is necessary to explain the data. Restricting r=s=0 leads to an expected utility model 20 Graphs based on the asymmetry in reversal rates conditional on price-ranking lead to the same conclusion: the two-error-rate model can only explain data when asymmetries are consistent with an error correction explanation