ScholarWorks @ Georgia State University ScholarWorks @ Georgia State University Paradoxes and Mechanisms for Choice under Risk Paradoxes and Mechanisms for Choice under Risk

: Experiments on choice under risk typically involve multiple decisions by individual subjects. The choice of mechanism for selecting decision(s) for payoff is an essential design feature that is often driven by appeal to the isolation hypothesis or the independence axiom. We report two experiments with 710 subjects. Experiment 1 provides the first simple test of the isolation hypothesis. Experiment 2 is a crossed design with six payoff mechanisms and five lottery pairs that can elicit four paradoxes for the independence axiom and dual independence axiom. The crossed design discriminates between: (a) behavioral deviations from postulated properties of payoff mechanisms; and (b) behavioral deviations from theoretical implications of alternative decision theories. Experiment 2 provides tests of the isolation hypothesis and four paradoxes. It also provides data for tests for portfolio effect, wealth effect, reduction, adding up, and cross-task contamination. Data from Experiment 2 suggest that a new mechanism introduced herein may be less biased than random selection of one decision for payoff.


INTRODUCTION
Most experiments on individual choice under risk involve multiple decisions by individual subjects and random selection of one choice for payoff. Such random selection is intended to avoid possible: (a) wealth effects from paying all choices sequentially during the experiment; and (b) portfolio effects from paying all choices at the end of the experiment. Random selection, however, also has a serious disadvantage first discussed by Holt (1986): In case that the reduction of compound lotteries axiom holds, random selection provides incentives for truthfully reporting preferences only if the independence axiom of expected utility theory is satisfied. This means that random selection is theoretically appropriate for testing expected utility theory (EU) but not for testing numerous prominent alternatives to EU such as rank-dependent utility theory (Quiggin, 1981(Quiggin, , 1982Luce 1991Luce , 2000Luce and Fishburn 1991), cumulative prospect theory (Starmer and Sugden, 1989;Tversky and Kahneman, 1992;Wakker and Tversky, 1993), or utility theories with the betweenness property (Chew, 1983;Gul, 1991). These theories -which all satisfy the reduction of compound lotteries axiomcan only be tested in modified variants by additionally assuming some kind of narrow bracketing similar to the isolation hypothesis in original prospect theory (Kahneman and Tversky, 1979). 2 In case of violations of theoretical predictions under random selection, however, one cannot discriminate between inconsistencies with (non-EU) decision theories and inconsistencies with the isolation hypothesis.
Isolation is an extreme hypothesis that supposes that a subject views each decision task independently of all other tasks in an experiment. Our Experiment 1 provides what we believe to be the first simple test of the isolation hypothesis. Experiment 1 data imply rejection of the isolation hypothesis. A "simple test" is distinguished from a joint test of a compound hypothesis consisting of the isolation hypothesis and one or more hypotheses from decision theory. Our Experiment 2 provides data that, like previous literature, support joint tests of compound hypotheses that include isolation. Data from Experiment 2 imply rejection of joint hypotheses that include isolation.
Unless isolation holds as a behavioral phenomenon, there is no generally-applicable payoff mechanism for use in conducting multi-decision experiments. In that case questions about the behavioral properties of alternative mechanisms become central to critical evaluation of experimental methods for studying alternative theories of decision under risk.
We experiment with the properties of six mechanisms; some of them are commonly used, some rarely used, and two of which are new. The six mechanisms are: (i) "pay one decision randomly" at the end of the experiment; (ii) "pay all decisions sequentially" during the experiment; (iii) "pay all decisions independently" at the end of the experiment; (iv) "pay all decisions correlated" at the end of the experiment; (v) "pay 1/n (of the amount) of all decisions correlated" at the end of the experiment, where n is the number of tasks; and (vi) the "one task" design in which each subject makes only one decision (and is paid for the outcome). Mechanisms (iv) and (v) are the new ones. As we show below, they are incentive compatible if preferences satisfy the dual independence axiom (Yaari, 1987).
Use of six different payoff mechanisms with the same five lottery pairs yields insight into relative effectiveness of alternative mechanisms in solving the problems that are inherent in experiments involving multiple decisions under risk. We use data from Experiment 2 to provide an empirical assessment of experimental methods including tests for the isolation effect, reduction, portfolio effect, wealth effect, and cross-task contamination.
While isolation is one extreme hypothesis, reduction of compound lotteries is an opposing extreme hypothesis that supposes that a subject views a whole experiment as one compound lottery. For Experiment 2, the reduction hypothesis implies that the proportion of risky choices in two of our lottery pairs should be the same. This implication of reduction is rejected by the data.
Another extreme hypothesis in opposition to isolation is the hypothesis that a subject would consider the whole experiment as one lottery and construct a portfolio using the choice pairs. The data are consistent with existence of a portfolio effect. Yet another extreme hypothesis in opposition to isolation is adding-up, that subjects consider all choice tasks of the experiment simultaneously and add up outcomes for each state of the world. The data are consistent with adding-up. Wealth changes during an experiment if earlier choices are paid before subsequent choices are made. This is exactly what happens with the pay all sequentially mechanism. This opens the possibility that changes in wealth can affect choices with this mechanism. The data provide support for a wealth effect.
Our data reveal that a portfolio effect and a wealth effect can occur with mechanisms that pay all decisions. Random selection of one decision for payoff was introduced to eliminate biases in risk preference elicitation experiments with multiple decisions by individual subjects. But our data reveal that random selection of one decision for payoff is not unbiased; instead, data from the treatment that uses this mechanism reveal two types of cross-task contamination. The design of Experiment 2 "crosses" the six preference elicitation mechanisms with five pairs of lotteries that can elicit four paradoxes: the common ratio effect; the common consequence effect; the dual common ratio effect; and the dual common consequence effect. The common ratio effect and common consequence effect have been researched with many experiments testing expected utility theory. They provide direct tests of the independence axiom of that theory. In contrast, there are few if any experiments with the dual common ratio effect and dual common consequence effect that provide direct tests of the dual independence axiom. We report experiment treatments on all four of these paradoxes.
Most previous experiments with the common ratio effect (CRE) and common consequence effect (CCE) have been conducted either with hypothetical payoffs or with multiple-choice experiments in which one choice was randomly selected for money payoff.
Random selection of one out of many choices for payoff raises issues of weak incentives (Harrison, 1994) and possible cross-task contamination (section 6 below). Only one previous experiment avoided both weak incentives and all possible types of cross-task contamination by having each subject make only one decision that paid non-trivial money payoff (Cubitt, Starmer, and Sugden, 1998b). In their "classic test" (p. 1376), the Cubitt, et al. data do not show a CRE. We ask whether the finding that CRE is insignificant is robust to our experimental design and procedures.
The crossed design of our Experiment 2 provides insight into ways in which elicitations of risk preferences can be contaminated by payoff mechanisms. This approach allows us to (partly) disentangle (a) behavioral deviations from postulated properties of elicitation mechanisms from (b) behavioral deviations from theoretical implications of alternative decision theories.
The paper is organized as follows. In the next section we review some of the related literature on payoff mechanisms and experiments on paradoxes. Section 3 reports Experiment 1, the simple test of the isolation hypothesis. In section 4 we present six payoff mechanisms and relate their properties to alternative decision theories. Section 5 explains the design of Experiment 2. Section 6 summarizes hypotheses we test. The results from Experiment 2 are reported in section 7. Section 8 summarizes implications for experimental methods, paradoxes of choice under risk, and subjects' revealed risk preferences.

MECHANISMS AND EXPERIMENTS ON PARADOXES
The pay one decision randomly (POR) mechanism previously has been given several names including random lottery incentive mechanism. POR is a standard procedure in much literature involving multiple decisions under risk. It gets its theoretical justification from the independence axiom of expected utility theory and is therefore theoretically justified for experiments on the common ratio effect and common consequence effect.
The literature contains a large number of papers that report experiments on theories, such as cumulative prospect theory, that do not assume the independence axiom. Many of these experiments involve multiple decisions and use POR for determining salient payoffs to the subjects. Although there is no theoretical justification for use of this payoff mechanism in such experiments, many authors invoke the isolation hypothesis to justify the procedure.
Many prominent experimental studies relied upon POR (e.g. Smith, 1976;Grether and Plott, 1979;Reilly, 1982;Hey and Orme, 1994;Harrison, List, and Towe, 2007). In some studies (e.g. Camerer, 1989) subjects were allowed to change their choices after the question played out for real was determined by POR. Although this design may be fruitful in some cases, it can be used only once because initial choices become cheap talk if subjects know that there is a chance to change choices afterwards. Moreover, it should be noted that in the case of violation of independence people will possibly not change their choice even though they would have revealed a preference for the other alternative in a single-choice decision task not embedded in the POR mechanism. More precisely, only "naïve" decision makers would change their choices whereas "resolute" ones would not do so (see Machina, 1989).
In view of the above mentioned problems and our negative empirical results on isolation (reported in sections 3 and 7), it is unclear whether POR is the best of the available payment mechanisms for experiments with decisions under risk. We discuss five alternatives to POR and experiment with the properties of all six mechanisms. First we consider two new mechanisms that have not previously been discussed in the literature. In one of these mechanisms -referred to as the pay all correlated (PAC) mechanism -at the end of the experiment one state of nature is randomly drawn and then all (comonotonic) lotteries chosen in the experiment are paid out for this state of nature. We show that this mechanism is incentive compatible if the utility functional representing risk preferences is linear in payoffs, as in the dual theory of expected utility (Yaari, 1987) or linear cumulative prospect theory (Schmidt and Zank, 2009). The same is true for the PAC/n mechanism where the payoff of PAC is divided by the number of tasks (so that subjects face the same expected value of payoff incentives as in POR).
It turns out that simple tests of hypotheses that use within-subjects data can only be conducted for utility theories with functionals that are linear in probabilities (expected utility theory with POR) or linear in outcomes (dual theory or linear cumulative prospect theory with PAC and PAC/n). If utility is nonlinear in both probabilities and outcomes (e.g. rankdependent utility theory and cumulative prospect theory), there exists no known incentive compatible mechanism for multiple-decision experiments and, hence, joint hypotheses including isolation are what are actually tested. It is, however, an empirical question which of the mechanisms, POR, PAC, or PAC/n, will minimize distortions coming from cross-task contamination when isolation fails.
There is quite some evidence that people are risk averse even for small stakes (Holt andLaury 2002, 2005;Harrison et al. 2005). A central question for decision theory is whether such risk aversion can be modeled with nonlinear utility or nonlinear probability weighting or both. Some experimental studies have reported observations of nonlinear probability weighting and some have also reported data consistent with linear utility for small stakes (Lopes, 1995;Fox, Rogers, and Tversky, 1996;Kilka and Weber, 2001). Recent econometric analysis of risky decision data from experiments using POR yields a reported conclusion that the data are characterized by nonlinearity in both payoffs and probabilities (Harrison and Rutström, 2008). But such empirical analysis seems dissonant because: if (a) risk preferences are actually characterized by nonlinearity in both payoffs and probabilities then (b) POR is not theoretically incentive compatible for use in experiments to generate data for use in the econometric analysis. Validity of the seemingly-dissonant data analysis depends on POR being behaviorally unbiased even though it is not theoretically incentive compatible.
PAC is also not theoretically incentive compatible for experiments on theories with utility that is nonlinear in both payoffs and probabilities. But there is no theoretical basis for thinking that POR is a better mechanism than PAC for use in experiments on such theories.
Our Experiment 2 produces data that can be used to compare the empirical performance of POR and PAC. Since the expected payoff of subjects is larger in PAC than in POR we also experiment with PAC/n: here subjects get the payoff of PAC divided by the number of tasks, which makes the expected payoff equal to that of POR.
Many experiments with market institutions and game theory use the pay all sequentially (PAS) mechanism in which subjects accrue changes in their earnings from the experiment after each decision is made and before undertaking the subsequent decision tasks.
The main problem with PAS is the possibility of wealth effects, i.e. earnings on previous rounds may influence behavior in later rounds. However, studies using PAS in experiments with decisions under risk that tested for wealth effects reported they were insignificant (Cox and Epstein, 1989;Cox and Grether, 1996;Laury, 2006).
An obvious alternative to PAS is to pay subjects independently for each task at the end of the experiment, which we will call the pay all independently (PAI) mechanism. PAI has recently been used in experiments with role reversal in trust games that involve risks from defection (Burks, Carpenter, and Verhoogen 2003;Chaudhuri and Gangadharan, 2007). An obvious disadvantage of PAI is the possibility of portfolio effects: playing out several lotteries independently and paying subjects for all of them reduces the risk involved in the single lotteries and, thus, should induce less risk averse behavior. Laury (2006) did not find a significant difference between PAI and POR for the same payoff scale.
An alternative protocol that eliminates all potential distortions of risk preferences by the payoff mechanism is to ask each subject to make only one decision. Use of this one task (OT) "mechanism" requires that hypothesis tests be conducted with a between-subjects approach. Data from Experiment 2 reported below allow us to compare the implications of between-subjects tests of hypotheses using data elicited with OT with implications of both between-subjects and within-subjects tests of hypotheses using data elicited with POR, PAC, PAC/n, PAI, and PAS.
There is previous literature that addresses some of our questions. Cubitt, Starmer, and Sugden (1998b) investigate whether OT yields different risk preferences than POR in a treatment with pre-commitment. This joint test for isolation implies rejection at 10% but not at 5% significance. Cubitt, et al. (1998b) also report a test for ("classic") CRE using OT data that is insignificant; that is, CRE is not observed. Three other studies compare an impure form of OT to POR. Starmer and Sugden (1991), Beattie and Loomes (1997), andSugden (1998a, 2001) integrated OT into a series of hypothetical choices, i.e. there were first hypothetical choice questions and at the end of the experiment one question appeared which was played out for real. This procedure could be behaviorally unbiased; however if cross-task contamination effects exist (see section 6 below) then hypothetical questions may also contaminate the responses to the real question. Along with many treatments with hypothetical payoffs, Conlisk (1989) also reported a pilot experiment with hypothetical practice rounds followed by one choice task with real payoffs. He reported Allais paradox behavior for the hypothetical treatments but that, with the real payoff OT choices, "Allais behavior disappeared" (Conlisk, 1989, p. 401).
Other studies focus on whether experiments with POR generate data that are significantly different from data generated from PAS or PAI. Laury (2006) reports a payoff scale effect but for small payoffs no significant differences between POR data and PAI data. Lee (2008) looks at PAS versus POR and concludes that POR does better than PAS in controlling for wealth effects. Isolation under PAS implies myopic behavior, which has previously been reported in experimental studies (Thaler et al., 1997;Gneezy and Potters, 1997;Gneezy, et al., 2003;Haigh and List, 2005;Langer and Weber 2005;Bellemare et al., 2005;Fellner and Sutter, 2008 3 For studies analyzing the POR or analogous mechanisms in other fields than risky decision making see Bolle (1990), Sefton (1992)

Experimental Design
In this experiment 284 subjects were randomly assigned to five groups, referred to below as Groups 1, 2.1, 2.2, 3.1, and 3.2. Payoffs in this experiment were in Euros. The choice options received by all groups are reported in Table 1. No group of subjects was shown Table 1; instead, they were given only their own decision tasks as follows. Group 1 subjects had a one-stage decision task in which they were asked to choose either Option A or Option B. Group 2.1 subjects had a two-stage decision task in which they were asked to choose between Options C and D in the first stage and between Options A and B in the second stage.
Group 3.1 subjects were asked to choose between Options E and F in the first stage and between Options A and B in the second stage. Groups 2.2 and 3.2 differed from Groups 2.1 and 3.1, respectively, only by the order in which the choices were presented, i.e. the choice between Options A and B was presented in the first stage for Groups 2.2 and 3.2.  Huber et al., 1982;Simonson and Tversky, 1992;Hsee and Lecrerc, 1998;Bhargava et al., 2000), this could increase attractiveness of Option A and decrease attractiveness of Option B, leading to a higher proportion of A choices compared to Group 1. 4 Experiment 1 was run at the University of Kiel. In each group subjects received one page of paper which contained instructions and all tasks; details are shown in Appendix 2.
The way in which lotteries were presented to the subjects is illustrated in Figure 1, using one of the lottery pairs as an example.

Results
The results of Experiment 1 are presented in  Conlisk, 1989). As expected, Option A turns out to be more attractive (Option B less attractive) in Group 2 than in Group 1, leading to a significant violation of isolation and, therefore, to a failure of the POR mechanism to elicit true preferences.
There are order effects in Groups 2.1 and 2.2 as well as Groups 3.1 and 3.2 which are all in the expected direction. However, only one of these effects (the difference between choice of F in Groups 3.1 and 3.2) is significant at the 5% level. The relatively small order effects can be explained by the fact that all alternatives were presented prior to elicitation of any responses from the subjects.
We conclude from this experiment that isolation can be significantly violated. This questions the suitability of using POR for testing theories that do not include the independence axiom. Hence it may be fruitful to consider alternative mechanisms and explore their properties.

THEORETICAL PROPERTIES OF MECHANISMS
For convenience, we restrict attention to experiments involving binary choices between lotteries. Lotteries will often be represented by (X 1 , p 1 ; …; X m , p m ), indicating that outcome X s is obtained with probability p s , for s = 1,2, …, m. Outcome X s can be a monetary amount or a lottery. Consider experiments that include n questions in which the subject has to choose between options A i and B i , for i = 1,…,n. The choice of the subject in question i will be denoted by C i .

The Pay One Randomly (POR) Mechanism
Here each question usually has a 1/n chance of being played out for real. Suppose a subject has made all her choices apart from question i. Then her choice between A i and B i determines whether she will receive (1/n)A i + (1-1/n)C or (1/n)B i + (1-1/n)C, where C = (C 1 , 1/(n-1); …; C i-1 , 1/(n-1); C i+1 , 1/(n-1); …; C n , 1/(n-1)) is the lottery for which the subject receives all her previous choices with equal probability 1/(n-1). Consequently, a subject whose preferences satisfy the independence axiom has an incentive to reveal her preferences truthfully because under that axiom A i  B i if and only if (1/n)A i + (1-1/n)C  (1/n)B i + (1- The above result, discussed by Holt (1986), shows that POR is not appropriate for testing alternatives to expected utility theory (which do not include the independence axiom) if subjects consider all choice problems simultaneously and employ the reduction of compound lotteries axiom. A simple example -referred to as Example 1 in the subsequent analysis -illustrates this for rank dependent utility theory. Consider a rank dependent expected utility maximizer with utility function u(x) = x and f(p) = p 0.9 choosing between Option A ($30 for sure) and Option B (a coin-flip between $100 and $0); she would choose Option A. Now suppose that the choice task would be offered twice. Under POR and the reduction of compound lotteries axiom, Option A would be chosen in one task and Option B would be chosen in the other task because the resulting lottery ($100, 0.25; $30, 0.5; $0, 0.25) has a higher utility than $30 for sure. It has been argued that it is quite unlikely that subjects behave according to this reduction hypothesis as it requires too much mental effort. The opposite extreme is the isolation hypothesis: here, subjects evaluate each choice problem independently of the other choice problems in the experiment. Under this isolation hypothesis, POR is incentive compatible also for preferences violating the independence axiom. In between these two extremes is the hypothesis of cross-task contamination where responses to one choice problem may be influenced by the other choice problems in an experiment but not to such an extent as the reduction hypothesis would imply. A psychological foundation for such contamination effects is range-frequency theory (Parducci, 1965). As in the case of reduction, POR is not incentive compatible for non-expected utility preferences if contamination effects exist. In the following we will discuss four incentive mechanisms where, unlike POR, all chosen options are played out for real.

The Pay All Independently (PAI) Mechanism
In the PAI mechanism, at the end of the experiment all tasks are played out independently which seems to be the most obvious alternative to POR. However, PAI has a serious problem, well known as portfolio effect in the finance literature: the risk of a mixture of two independent random variables is less than the risk of each variable in isolation.

The Pay All Sequentially (PAS) Mechanism
The PAS mechanism could be superior to PAI if portfolio diversification effects significantly distort single choice preference revelation. However, it is easy to see that PAS is not incentive compatible in the case of expected utility of terminal wealth. If we would play the lotteries of Example 1 under PAS two times, the optimal strategy for the given utility Option B is chosen. Since the latter alternative has the higher utility, PAS would not elicit true preferences. However, since each choice is paid out directly, assuming isolation seems to be particularly intuitive in the case of PAS. Isolation under PAS would imply myopic behavior, as for instance discussed in the literature on myopic loss aversion (Benartzi and Thaler, 1995).

The Pay All Correlated (PAC) and PAC/n Mechanisms
If the independence axiom is violated, the mechanisms discussed above incentivize revelation of true preferences only under additional assumptions such as narrow bracketing. In contrast, the preference revelation properties of PAC and PAC/n depend on the dual independence axiom. With these mechanisms, preferences are revealed truthfully if dual independence is satisfied, otherwise additional assumptions are required.
For the PAC and PAC/n mechanism, states of the world need to be defined (e.g. tickets numbered from 1 to 100) and all lotteries need to be arranged in the same order such that they are comonotonic. More formally, there are m states indexed by s = 1, 2, …, m and lotteries are identified by A i = (a i1 , p 1; …; a im , p m ) and is the outcome of lottery A i (B i ) in state s and p s is probability of that state. We arrange lotteries such that a is ≥ a is+1 and b is ≥ b is+1 for all s = 1, …, m-1 and all i = 1, …, n. At the end of the experiment one state is randomly drawn and the outcomes of all choices in this state are paid out under PAC. Suppose as above that a subject made all choices apart from choice i.
Then her choice between A i and B i will determine whether she will receive either A i * = (a i1 +  j≠i c j1 , p 1 ; …; a im +  j≠i c jm , p m ) or B i * = (b i1 +  j≠i c j1 , p 1 ; …; b im +  j≠i c jm , p m ) as reward before the state of nature is determined. This shows that PAC is incentive compatible under Yaari´s (1987)  When we wish to compare PAC with POR we have to keep in mind that the expected total payoff from the experiment is n times higher under PAC. This may have significant effects on behavior. In particular one can expect lower error rates under PAC as wrong decisions are more costly (see Laury and Holt, 2008). Therefore, we also include PAC/n in our experimental study where the payoff of PAC is divided by the number of tasks. PAC/n has the same theoretical properties as PAC and is incentive compatible under the dual theory and linear cumulative prospect theory.

The One Task (OT) Mechanism
So far we can conclude that payment mechanisms for binary choice are incentive compatible only under narrow bracketing or if utility is linear in probabilities or in outcomes. This is not true for the OT mechanism. In this mechanism each subject has to respond to only one choice problem which is played out for real. Besides being rather costly, this mechanism has one obvious disadvantage in the context of decision making under risk: since a test of independence conditions requires at least two binary choice questions, OT allows only for between-subjects tests of these conditions. OT is nevertheless very interesting because it is the only mechanism that is always (i.e. for all possible preferences) incentive compatible. Given this fact it is somewhat surprising that OT in its pure form has been used to test the independence axiom in only one study (Cubitt, Starmer, and Sugden, 1998b).

Lottery Pairs
Experiment 2 includes the five lottery pairs that are presented in Table 4. Payoffs in Experiment 2 are in dollars. Each lottery pair consists of a relatively safe and a relatively risky lottery. There is a bingo cage with twenty numbered balls out of which one ball is drawn in the presence of the subjects to determine the payoff of a lottery. The five lottery pairs contain a common ratio effect (CRE, Pairs 2 and 3), a common consequence effect (CCE, Pairs 3 and 4), a dual common ratio effect (DCRE, Pairs 1 and 3), a dual common consequence effect (DCCE, Pairs 2 and 5), a dominated pair (Pair 1), and a dominating pair (Pair 5).
A CRE consists of two lottery pairs where the lotteries in the second pair (Pair 3 in our design) are constructed from the lotteries in the first pair (Pair 2 here) by multiplying all probabilities by a common factor (1/4 in our study) and assigning the remaining probability to a common outcome (in our study $0). It is easy to see that according to EU either the safe lottery should be chosen in both pairs or the risky lottery should be chosen in both pairs.
Many empirical studies report evidence that subjects tend to violate the independence axiom of EU in CREs by choosing the safe lottery in the high EV pair (Pair 2 in our study) but the risky lottery in the low EV pair. A prominent explanation of this effect is the fanning out hypothesis (Machina, 1982) which states that the degree of risk aversion is increasing with the attractiveness of lotteries (measured in terms of stochastic dominance). A CCE consists of two lottery pairs. Here, the lotteries in the second pair (Pair 4 in our design) are constructed from the lotteries in the first pair (Pair 3 here) by shifting probability mass (75% in our study) from one common outcome ($0 in our study) to a different common outcome ($12 in our study). EU again implies that the subjects will either choose the safe lottery in both pairs or the risky lottery in both pairs. While the empirical evidence against EU reported in studies relying on CCEs is as extensive as for CREs the direction of violations is less clear cut. Some studies involving a sure outcome mostly reported violations in the direction of fanning out which means in our example choosing the risky lottery in Pair 3 and the safe one in Pair 4. There exist, however, also several studies observing the majority of violations in the opposite direction (termed fanning in): see Conlisk (1989), Prelec (1990), Camerer (1992), Wu and Gonzalez (1998), Birnbaum (2004), Birnbaum and Schmidt (2010), As far as we know, however, the dual paradoxes have never been investigated in a systematic empirical test with a theoretically correct incentive mechanism.

Protocol
Experiment 2 was run in the laboratory of the Experimental Economics Center at Georgia State University. Subjects in groups OT i , i = 1, 2, …, 5, just had to perform one binary choice between the lotteries of Pair i which was played out for real. Subjects in other (payoff-mechanism) treatments saw all five lottery pairs in advance on five separate, randomly-ordered (small) sheets of paper. Each subject could display his or her five sheets of paper in any way desired on his or her private decision table. Subjects entered their decisions in computers. In all treatments, including OT, the top or bottom positioning of the two lotteries in any pair was randomized by the decision software. In all treatments other than OT, the five lottery pairs were presented to individual subjects by the decision software in randomly-drawn orders.
Subjects in group POR had to make choices for all five lottery pairs and at the end one pair was randomly selected (by drawing a ball from a bingo cage) and the chosen lottery in that pair was played out for real (by drawing a ball from another bingo cage). Also in groups PAI, PAC, PAC/5, and PAS subjects had to make choices for all five pairs but here the choice from each pair was played out for real by drawing ball(s) from a bingo cage. In group PAI the five choices were played out independently at the end of the experiment whereas in groups PAC and PAC/n the five choices were played out correlated at the end of the experiment (i.e. one ball was drawn from the bingo cage which determined the payoff of all five choices). In group PAS the chosen lotteries were played immediately after each choice was made (by drawing a ball from a bingo cage after each decision).
Before any ball was drawn from a bingo cage, subjects were permitted to inspect the cage and the balls. Each ball drawn from a bingo cage was done in the presence of the subjects. Lotteries were presented in a format illustrated by the example in Figure 2 which shows one of the two ways in which the lotteries of Pair 4 were presented to subjects. As

Risk Preferences Hypotheses
We presuppose that the OT groups elicit true preferences whereas choices in the other groups could be biased by the incentive mechanism. First we can use the responses of the OT groups to analyze behavior with respect to CRE and CCE. The null hypothesis that follows from the independence axiom is that the proportion of choices of the risky option in Pair 3 should be the same as the proportions of choices of the risky options in Pairs 2 and 4. An alternative hypothesis that follows from fanning out (Machina, 1982) is: Hypothesis 1 (fanning out): The proportion of choices of the risky option is higher in group OT 3 than in OT 2 (CRE) and also higher in group OT 3 than in OT 4 (CCE).
All payoff amounts in Pair 3 are two times corresponding payoff amounts in Pair 1.
All payoff amounts in Pair 5 are $12 higher than corresponding payoffs in Pair 2. Responses of the OT groups can be used to analyze behavior with respect to DCRE and DCCE. The null hypothesis that follows from the dual independence axiom (which implies linearity in payoffs) is that the proportion of choices of the risky option should be: (a) the same in Pairs 1 and 3; and (b) the same in Pairs 2 and 5. The null hypothesis of choices in Pairs 1 and 3 coming from the same distribution also follows from a power function for payoffs, with or without linearity in probabilities. On the other hand, the null hypothesis of choices in Pairs 2 and 5 revealing the same distribution is consistent with an exponential function for payoffs.
Alternative hypotheses are that choices correspond to DRRA or DARA.

Hypothesis 2 (DRRA): The proportion of choices of the risky option is higher in group
OT 3 than in OT 1 .
Hypothesis 3 (DARA): The proportion of choices of the risky option is higher in group OT 5 than in OT 2 .

Elicitation Mechanism Hypotheses
We now compare data from the OT treatment to data from five multi-decision treatments. The first hypothesis in this context is isolation. If isolation holds, subjects tackle each choice task independently of the other choice tasks. In that case, data from all treatments should reveal the same distribution and thus conform to the following null hypothesis. Type 1 cross-task contamination by POR is narrowly defined and specific to lottery Pairs 2 and 5. In contrast, type 2 cross-task contamination reflects the straightforward view that if POR is unbiased then it should elicit the same preferences for safe vs. risky lotteries as does OT. Hypothesis 8 addresses type 2 contamination.

Hypothesis 8 (type 2 cross-task contamination): In the POR treatment the proportions of choices of the risky options are the same as in the OT treatment.
The new mechanism, PAC is incentive compatible for utility that is linear in outcomes, i.e. DEU, the model of Yaari (1987), and since we only have gains as outcomes, also for linear cumulative prospect theory (Schmidt and Zank, 2009). Suppose now that utility is not linear as we can infer from responses to DCRE and DCCE in the OT groups (see Hypotheses 2 and 3). In the extreme case, subjects could again consider all choice tasks of the experiment simultaneously and add up outcomes for each state of the world. In this case choosing the safe option in Pair 2 and the risky option in Pair 5 leads to the same statedependent sums of payoffs as choosing the risky option in Pair 2 and the safe option in Pair 5.
Similarly to the above (in case of the reduction and portfolio hypotheses), this implies similar proportions of patterns (R 2 ,S 5 ) and (S 2 ,R 5 ) even if the proportions of safe option choices in Pairs 2 and 5 differ according to true preferences in the OT groups (i.e., DCCE is violated). .

Hypothesis 9 (adding-up): In the PAC and PAC/n treatments the proportions of patterns (R 2 ,S 5 ) and (S 2 ,R 5 ) are the same.
Finally, recall that in PAS subjects accrue earnings between decisions during the experiment and that different subjects face different randomly-generated orders of lottery pairs across rounds. If there is a wealth effect on risk preferences, subjects' payoffs between rounds are expected to induce less risk averse behavior (given DARA).

Hypothesis 10 (wealth effect): The proportion of choices of safe options is lower in the
PAS treatment than in the OT treatment.

RESULTS
A first overview of our results can be taken from the left panel in Table 5 which presents for each choice problem i (=1,…,5) and each elicitation mechanism the percentage of subjects who chose the less risky (or "safe") lottery denoted by S i . There are some striking differences across mechanisms. For example, only 15.5% of subjects chose S 2 with OT whereas 50%, 52.6%, and 52.6% of choices were S 2 with POR, PAC, and PAI respectively.
With OT, 28.95% and 38.5% of choices were S 4 and S 5 whereas the S 4 and S 5 choice figures were 10.26% and 17.95% with PAS. Figures for the root mean-squared deviations between figures in rows 2-6 and row 1 (the OT row) in Table 5 are: 18.6, 10.2, 17.2, 14.5, and 18.4 for POR, PAC/5, PAC, PAS, and PAI respectively. PAC/5 has the smallest root mean-squared error whereas POR has the largest one (which is 1.8 times the figure for PAC/5). 6 By this measure, PAC/5 introduces less bias in risk preference elicitation than POR (or any of the other mechanisms). Statistical tests of differences among risk preferences elicited by the six mechanisms are presented below in tests of the isolation hypothesis. We next report results concerning the paradoxes for expected utility theory (CRE and CCE) and dual theory of expected utility (DCRE and DCCE). After that we report tests of hypotheses on risk preferences and elicitation mechanisms.

Tests for Paradoxes
The right panel of Table 5 presents tests for the four paradoxes on a between-subjects basis. The reported tests use the statistics of Conlisk (1989) unless otherwise indicated. 7 These test results are discussed along with across-subjects test results in the following subsections.
6 Let s t (i) be the percentage of the safer choices in treatment t and task i; then the root mean squared error is

7.1.a Tests for a Common Ratio Effect (CRE)
The first test of the independence axiom of EU involves two pairs of binary lottery choices that preserve all payoffs and the ratio of the probabilities of the outcomes. In our experiment design, Pairs 2 and 3 have these properties. Using notation S and R for the less risky and the riskier lotteries within a pair, EU predicts only patterns SS and RR for choice in Pairs 2 and 3.
The CRE is a paradox for EU theory, so we focus on data from the OT (one task) and POR (many-tasks, pay one randomly at the end) treatments, which are the theoreticallycorrect elicitation mechanisms for EU. Results from between-subjects tests are reported in  right-most column, first POR row). We conclude that the common ratio implication of EU theory is not rejected by the OT treatment data nor by the POR treatment data.

7.1.b Tests for a Common Consequence Effect (CCE)
The second test of the independence axiom of EU involves two pairs of binary lottery choices that differ only with respect to a common consequence. Lotteries in Pair 3 differ from lotteries in Pair 4 with respect to a common consequence: a payoff of $12 with likelihood 3/4 replaces $0 in both lotteries.
The CCE is a paradox for EU theory, so here we focus again on OT and POR treatments. Results from between-subjects tests are reported in  Table A1, second POR row, the right-most column). We conclude that the common consequence implication of EU theory is rejected by the POR treatment data but not by the OT treatment data.

7.1.c Tests for a Dual Common Ratio Effect (DCRE)
This test of the dual independence axiom of DEU involves two pairs of binary lottery choices that preserve ratios of payoffs. Note that Pairs 1 and 3 provide an example of the dual common ratio problem since the ratio of (nonzero) payoffs in lotteries is the same for Pair 1 (3/5) and Pair 3 (6/10). DEU predicts that the safer option is chosen in Pair 1 if and only if the safer option is chosen in Pair 3. 8 The DCRE is constructed to test DEU theory, so we focus on OT, PAC, and PAC/5 treatments. (Recall that PAC and PAC/n are incentive compatible mechanisms for DEU).
Results from between-subjects tests are reported in   Table 7, which shows the total number of subjects for each multi-choice treatment and the numbers of subjects that exhibited relevant choice patterns. The observed percentages of subjects with S 1 S 3 , R 1 R 3 , R 1 S 3 and S 1 R 3 preferences are 21.05, 60.53, 2.63, and 15.79 in the PAC treatment and 25.0, 52.5, 10.0, and 12.5 in the PAC/5 treatment. So, we observe that 81.58% and 77.50% of patterns in the PAC and PAC/5 treatments are consistent with DEU predicted patterns. 9 We conclude that the dual common ratio implication of DEU theory is not rejected by the OT treatment data nor by the PAC and PAC/5 treatment data.

7.1.d Tests for a Dual Common Consequence Effect (DCCE)
This test of the dual independence axiom of DEU involves two pairs of binary lottery choices that preserve differences of payoffs. Recall that the dual common consequence problem in our experiment design consists of Pairs 2 and 5; DEU predicts that the risky option is chosen in Pair 2 if and only if the risky option is chosen in Pair 5. 10 The DCCE is constructed to test DEU theory, so we focus on OT, PAC, and PAC/5 treatments. Results from between-subjects tests are reported in

Tests of Risk Preferences Hypotheses
Our results for the risk preferences hypotheses follow from the analysis of paradoxes in the preceding subsection. According to Hypothesis 1 (fanning out) the riskier choice should be observed more often in Pair 3 than in either Pair 2 (CRE) or in Pair 4 (CCE). On the contrary, the riskier choice is observed more often in Pair 2 than in Pair 3: 84.48% and 72.41% respectively. Percentages of the riskier option being chosen are almost the same in Pairs 4 and 3: 71% and 72.41% respectively. Looking at Table 5, Hypothesis 1 requires positive z-values in the CRE column and negative z-values in the CCE column. The sign is correct for the CCE column (as is expected from the above figures) but the one-sided p-value is too high (0.399). Thus, we conclude that the OT data are not consistent with fanning out preferences.
10 EU and RDU with an exponential function for money make the same prediction. OT data are not consistent with this specification; p-value is 0.005 (see Table 5, the right-most column, OT row).
Hypothesis 2 (DRRA) requires that the riskier option should be observed more often in Pair 3 than in Pair 1. We observe that the riskier option was indeed chosen more often in Pair 3 than in Pair 1: exact figures are 72.41% and 60.53% for Pairs 3 and 1, respectively.
However, the difference is not statistically significant (one-sided p-value is 0.111); see Table   5, OT row and DCRE column). We conclude that the OT data are not characterized by the DRRA property. Finally, OT data are not consistent with Hypothesis 3 (DARA). Contrary to the hypothesis, the proportion of risky choices is not higher in group OT 5 than in OT 2 : we observe the riskier option being chosen in 61.54% of the decisions in Pair 5 but 84.48% in Pair 2. Referring to Table 5, Hypothesis 3 requires a positive z-value in the DCCE column; this is not what we find. Thus, we conclude that the OT data are not characterized by the DARA property either.

7.3.a Isolation
According (c) PAC/5 data (two-sided p-values are 0.004 and 0.005 for the proportions test and Fisher's 11 According to one-sided sample proportions tests, POR and PAI data are biased towards significantly more risk aversion compared to OT data (p-values are ½ of above). At 10% significance with one-sided tests, PAS data show less risk aversion (one-sided Fisher's exact p-value is 0.077) and both PAC and PAC/5 data show more risk aversion (one-sided p-values are 0.086 and 0.092) than OT data. test); (d) PAI data (two-sided p-values are 0.000 for both tests). Other differences between sample proportions are insignificant.
An alternative test for bias in the multi-choice mechanism data can be based on probit analysis with subject-ID clustered errors. The left hand variable is probability of choosing the risky option. Right hand variables are: a constant; differences between expected values, standard deviations and skewness of payoffs for the risky and safe options; and a dummy variable with value 1 for data from a multi-task mechanism and value zero for OT data.
Estimated coefficients for the dummy variable are negative and significantly different from 0 in the probit for POR (p-value is 0.046) and in the probit for PAI (p-value is 0.013). By this test only POR and PAI significantly bias risk preferences compared to OT.
Another comparison of data from different mechanisms can be based on the percentages reported in Table 5  hand generates data that are statistically different from other protocols for Pairs 2 and 5.
All of the above tests support a central conclusion. Data from Experiment 2 support rejection of the isolation hypothesis. As explained in section 3, data from Experiment 1 also support rejection of isolation.

7.3.b Reduction, Portfolio Effect, Wealth Effect, and Adding-Up
According to Hypothesis 5 (reduction) the safer-riskier mixed patterns for choices in Pairs 3 and 4 should be the same in the POR group. The POR data are not consistent with reduction; the p-value is 0.057 (see Table A.1, the most right column, fifth row from the bottom).
According to Hypothesis 6 (portfolio effect), the safer-riskier mixed patterns for choices in Pairs 2 and 5 should be the same in the PAI treatment. Data are consistent with this hypothesis (see Table 5; DCCE column, the bottom row). This result together with the observation that OT data reject the null hypothesis of equal choice distributions in Pairs 2 and 5 suggest that subjects' choices with PAI reveal a portfolio effect. Such an effect is consistent with EU under the PAI mechanism; indeed it is just such a possible effect that POR is supposed to eliminate. The empirical question for experimental methods is whether the POR mechanism successfully eliminates possible portfolio effects.
As predicted by Hypothesis 9 (adding-up), we do not observe significant differences in responses to Pairs 2 and 5 for PAC, and PAC/5 data (p-values are 0.291; see Table A.2, the right-most column, PAC and PAC/5 rows). We conclude that this hypothesis is not rejected.
As reported above, we observe the riskier options more often in PAS than in OT; PAS data are statistically less risk averse in Pairs 4 and 5. We conclude that the data are consistent with Hypothesis 10 on existence of a wealth effect in the PAS treatment.

7.3.c Cross-Task Contamination with POR
According to Hypothesis 7 (type 1 cross-task contamination): if the safer-riskier mixed patterns for choices in Pairs 2 and 5 are equally observed in the POR treatment but not in the OT treatment then POR exhibits type 1 cross-task contamination. Data are consistent with the hypothesis that the proportions of patterns (R 2 ,S 5 ) and (S 2 ,R 5 ) are the same in the POR treatment (see Table 5; DCCE column, the POR row). This result, combined with the observation that OT data reject the null hypothesis of equal proportions of choices of the safe option in Pairs 2 and 5, support the conclusion that subjects' choices in POR exhibit type 1 with isolation, as noted in the first row of Table 8.
Empirical failure of the isolation hypothesis is especially a problem for design of experiments to test theories such as rank dependent utility theory, cumulative prospect theory, and betweenness theories that do not include either the independence axiom or the dual independence axiom. There is no known theoretically correct payoff mechanism to use in multi-decision experiments designed to test such theories.
The finding that the data are inconsistent with the isolation hypothesis makes clear the importance of systematic study of the properties of alternative payoff mechanisms and the relationship of those properties to validity of conclusions about theory that can be drawn from data. Experiment 2 is a step in the direction of such study.  Isolation is an extreme hypothesis that supposes that a subject views each decision task independently of all other tasks in an experiment. Reduction of compound lotteries is an opposing extreme hypothesis that supposes that a subject views a whole experiment as one compound lottery. For Experiment 2, the reduction hypothesis implies that the proportion of risky choices in Pairs 3 and 4 should be the same. This implication of reduction is rejected by the data at a conventional level of 10% significance.

Isolation, Reduction, Portfolio Effect, Adding-up, Wealth Effect, and Cross-Task Contamination
Another extreme hypothesis in opposition to isolation is the hypothesis that a subject would consider the whole experiment as one lottery and construct a portfolio using the choice pairs. The data are consistent with existence of a portfolio effect. Yet another extreme hypothesis in opposition to isolation is adding-up, that subjects consider all choice tasks of the experiment simultaneously and add up outcomes for each state of the world. The data are consistent with adding-up. Wealth changes during an experiment if earlier choices are paid before subsequent choices are made, which is exactly what happens with the pay all sequentially (PAS) mechanism. This opens the possibility that changes in wealth can affect choices with PAS. The data provide some support for a wealth effect.
Our data reveal that a portfolio effect and a wealth effect can occur with mechanisms that pay all decisions. Random selection of one decision for payoff was introduced to eliminate biases in risk preference elicitation experiments involving multiple decisions by individual subjects. But our data reveal that random selection, as with the pay one randomly (POR) mechanism, is not unbiased; instead data from the POR treatment reveal two types of cross-task contamination.

Paradoxes and Mechanisms
All our test results for paradoxes are summarized in Table 9. Bold entries denote tests that use theoretically "right" mechanisms. The common ratio effect (CRE) and common show CCE (but the mechanism is not theoretically right). The PAC data are the only ones to show CRE (but the mechanism is not theoretically right).  but not in OT data although POR is a theoretically right mechanism for testing for CCE.
Perhaps the observation of CCE with POR data is a false positive coming from the cross-task contamination we observe with POR. DCCE is observed with OT data but not with PAC or PAC/n data even though PAC and PAC/n are theoretically right mechanisms for this test.
POR has been commonly used in multi-decision experiments designed to test hypotheses from non-EU theories for which there exists no known theoretically-incentivecompatible mechanism. And POR has been used to generate data used in econometric estimation of non-EU utility functionals. In both types of research, the conclusion that PORgenerated data support non-EU theories is dissonant because non-EU theories do not imply that POR is unbiased. It may be the case that biases introduced into data by POR are insignificant in some (or most) contexts but the question needs systematic exploration. One good candidate for comparison with POR is OT because the latter is incentive compatible for all theories. But OT has two problems: high cost of subject payments and limitation to between-subjects tests of hypotheses. Another good candidate for comparison with POR is the PAC/n mechanism introduced in this paper. These two mechanisms have the same expected cost to the experimenter's research budget. And the theoretical duality between the mechanisms makes them good candidates for robustness checks on conclusions from empirical applications of non-EU theories. Data from our Experiment 2 reveal much higher bias for POR than for PAC/n. It is currently unknown whether this finding is robust.  In the following you have the choice between two options. In one option (Option A) you get a certain payoff for sure, the other option (Option B) is risky. Here a coinflip will determine which of the two possible payoffs you get. All payoffs will be paid out in cash directly after the experiment.

Appendix 1: Tests of Paradoxes
Which of the following options do you choose? In the following you have to make two choices, each consisting of two options. At the end of the experiment a coinflip will determine whether the first or the second choice problem will be paid out for real. In both choice problems there are two options. In one option (Option A in Choice Problem 1 and Option C in Choice Problem 2) you get a certain payoff for sure, the other option (Option B in Choice Problem 1 and Option D in Choice Problem 2) is risky. Here a coinflip will determine which of the two possible payoffs you get. All payoffs will be paid out in cash directly after the experiment. For all other groups instructions were the same as those for Group 2.1 except the option was changed.

Appendix 3: Subject Instructions for Experiment 2 Subject Instructions (OT)
In this experiment, you are asked to choose between two options. The example below shows two options that are similar to ones on the decision page.
In Option A you receive either $3 or $10. Your payoff is determined by drawing one ball from a bingo cage containing 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 4 is drawn then you receive $3. If a ball with number 5 to 20 is drawn then you receive $10.
In Option B you receive either $5 or $7 or $8. Your payoff is determined by drawing one ball from a bingo cage that contains 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 3 is drawn then you receive $5. If a ball with number 4 to 7 is drawn then you receive $7. If a ball with numbers 8 to 20 is drawn then you receive $8. Payoffs After you make a decision, your chosen option will be played. Your payoff in the option you selected will be determined by drawing a ball from a bingo cage that contains balls numbered 1,2,3,…,20.

Subject Instructions (PAC)
In this experiment, you are asked to choose between two options on each of five decision pages. On each decision page you will choose between a different pair of options. The example below shows two options that are similar to ones on decision pages.
In Option A you receive either $3 or $10. Your payoff is determined by drawing one ball from a bingo cage containing 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 4 is drawn then you receive $3. If a ball with number 5 to 20 is drawn then you receive $10.
In Option B you receive either $5 or $7 or $8. Your payoff is determined by drawing one ball from a bingo cage that contains 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 3 is drawn then you receive $5. If a ball with number 4 to 7 is drawn then you receive $7. If a ball with numbers 8 to 20 is drawn then you receive $8. Payoffs After you make a decision on each of the five decision pages, all your chosen options will be played as follows. One numbered ball will be drawn from a bingo cage that contains balls numbered 1,2,3,…,20. The ball drawn determines your payoff from the option you chose on all five decision pages.
Your total payoff is the sum of your payoffs from all five decision pages; all payoffs are determined by the one ball drawn.

Subject Instructions (POR)
In this experiment, you are asked to choose between two options on each of five decision pages. On each decision page you will choose between a different pair of options. The example below shows two options that are similar to ones on decision pages.
In Option A you receive either $3 or $10. Your payoff is determined by drawing one ball from a bingo cage containing 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 4 is drawn then you receive $3. If a ball with number 5 to 20 is drawn then you receive $10.
In Option B you receive either $5 or $7 or $8. Your payoff is determined by drawing one ball from a bingo cage that contains 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 3 is drawn then you receive $5. If a ball with number 4 to 7 is drawn then you receive $7. If a ball with numbers 8 to 20 is drawn then you receive $8.
Ball nr 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Option A $3 $10 Option B $5 $7 $8 Making Choices Please make your choice on each of the five decision pages by clicking on Option A or Option B Payoffs After you make a decision on each of the five decision pages, one of the pages will be randomly selected and your chosen option on that page will be played. The selection of the page is carried out by drawing a ball from a bingo cage that contains five balls numbered 1,2,3,4,5. The number on the drawn ball determines the decision page that is selected.
After the one page is randomly selected, your money payoff will be determined by playing the lottery in the option you selected on that page. Your payoff in the option you selected will be determined by drawing a ball from a bingo cage that contains balls numbered 1,2,3,…,20.

Subject Instructions (PAS)
In this experiment, you are asked to choose between two options on each of five decision pages. On each decision page you will choose between a different pair of options. The example below shows two options that are similar to ones on decision pages.
In Option A you receive either $3 or $10. Your payoff is determined by drawing one ball from a bingo cage containing 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 4 is drawn then you receive $3. If a ball with number 5 to 20 is drawn then you receive $10.
In Option B you receive either $5 or $7 or $8. Your payoff is determined by drawing one ball from a bingo cage that contains 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 3 is drawn then you receive $5. If a ball with number 4 to 7 is drawn then you receive $7. If a ball with numbers 8 to 20 is drawn then you receive $8.

Making your Choices and Determining Payoffs on Subsequent Pages
Make your choice on page 2. Then a ball is drawn to determine your payoff. The ball is returned to the bingo cage. Next, you make your choice on page 3. Another ball is drawn and then returned to the bingo cage. This process continues until your choices and payoffs have been determined for all five decision pages. Your total payoff is the sum of your payoffs from all five decision pages that are determined by the sequence of choices and independently drawn balls.

Subject Instructions (PAI)
In this experiment, you are asked to choose between two options on each of five decision pages. On each decision page you will choose between a different pair of options. The example below shows two options that are similar to ones on decision pages.
In Option A you receive either $3 or $10. Your payoff is determined by drawing one ball from a bingo cage containing 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 4 is drawn then you receive $3. If a ball with number 5 to 20 is drawn then you receive $10.
In Option B you receive either $5 or $7 or $8. Your payoff is determined by drawing one ball from a bingo cage that contains 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 3 is drawn then you receive $5. If a ball with number 4 to 7 is drawn then you receive $7. If a ball with numbers 8 to 20 is drawn then you receive $8. Payoffs After you make decisions on all five decision pages, all of your chosen options will be played as follows.
A numbered ball will be drawn from a bingo cage that contains balls numbered 1,2,3,…,20.
The ball drawn determines your payoff from the option you chose on the first page. The drawn ball is returned to the bingo cage. Next, a second ball is drawn, which determines your payoff from the option you chose on the second page. That ball is returned to the bingo cage.
This sequential procedure is continued until your payoffs are determined for all five decision pages.
Your total payoff is the sum of your payoffs from all five decision pages, each of which is determined by an independently drawn ball.

Subject Instructions (PAC/5)
In this experiment, you are asked to choose between two options on each of five decision pages. On each decision page you will choose between a different pair of options. The example below shows two options that are similar to ones on decision pages.
In Option A you receive either $3 or $10. Your payoff is determined by drawing one ball from a bingo cage containing 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 4 is drawn then you receive $3. If a ball with number 5 to 20 is drawn then you receive $10.
In Option B you receive either $5 or $7 or $8. Your payoff is determined by drawing one ball from a bingo cage that contains 20 balls numbered 1, 2, 3, …, 20. If a ball with number 1 to 3 is drawn then you receive $5. If a ball with number 4 to 7 is drawn then you receive $7. If a ball with numbers 8 to 20 is drawn then you receive $8.

Making Choices
Please make your choice on each of the five decision pages by clicking on Option A or Option B

Payoffs
After you make a decision on each of the five decision pages, all your chosen options will be played as follows. One numbered ball will be drawn from a bingo cage that contains balls numbered 1, 2, 3,…,20. The ball drawn determines your payoff from the option you chose on all five decision pages.
Your total payoff is one fifth of the sum of your payoffs from all five decision pages; all payoffs are determined by the one ball drawn.