The same but different: an empirical investigation of the reducibility principle

According to classical utility theory the valuation of any lottery should depend only on its outcomes and their respective probabilities, and should not be affected by the nature, complexity and structure of the chance mechanism. Previous research has documented systematic violations of this principle. For example, most subjects prefer lotteries in which the payoffs are contingent on the joint occurrence of multiple (high probability) events to simple lotteries, and lotteries in which the earlier stages offer higher probabilities than the later stages. We review the various violations of this principle and suggest a classification into two major types associated with misunderstanding of chance mechanisms and attitude towards the chance mechanism and process. In the present study 40 subjects were presented with 30 pairs of binary gambles. In any given pair the lotteries had identical outcomes and equal ‘reduced’ probabilities (and thus equal expected values). However, the chance mechanisms varied along a variety of factors such as the size of the sample space, the number of stages, temporal ordering, order of probabilities, their transparency and time constraints. Half the subjects saw lotteries involving gains and the other half considered only losses. After choosing one lottery in each pair, the subjects were asked to explain and justify their choices. The findings revealed systematic violations of the reducibility principle: subjects displayed a preference for lotteries with larger sample spaces, and for lotteries that allow quicker resolutions in the earlier stages. A clear distinction between some patterns of preferences in the gains and loss domains was revealed. In gambles involving gains subjects preferred to have the highest probability on the first stage (and the lowest probability on the last stage), but displayed the opposite preferences for losses. A content analysis of the subjects' stated reasons for their choices identified eight major categories. The most frequently invoked were hope, fun, simplicity, stress and time. Copyright © 2001 John Wiley & Sons, Ltd.     

Testing Credit and Blame Attributions as Explanation for Choices under Ambiguity

People make decisions every day based on their estimates of the chances of various events occurring. By their very nature, these subjective estimates are less certain, and thus more ambiguous, than objective probabilities such as those associated with roulette wheels or lotteries. In two experimental studies, we investigate choices between bets on the decision maker′s knowledge versus bets on a lottery with the same chance of winning. In Study 1, we partially replicate the Heath and Tversky (1991) finding that subjects have an increasing preference for betting on their own knowledge over the matched-chance lottery as probability increases. We also discover a significant minority of subjects who exhibit the opposite pattern and prefer the lottery as probability increases. In addition, we explicitly test the subjects′ attributions about the credit they would feel for a win and the blame they would feel for a loss, and these attributions do provide some explanation for their choice behavior. In Study 2, we manipulate knowledge and accountability, in an attempt to heighten the attributions of credit and blame, and we replicate the findings.     

Order Preservation of PEST Determined Certainty Equivalents: A Test Based on Lotteries with a Fixed Preference Strength

A major assumption of many utility theories is that preferences over lotteries are monotonic in consequences. Under additional reasonable conditions, this means that the choice certainty equivalent (CE) of a lottery should be substitutable for the lottery, but some empirical evidence has suggested otherwise. So several reasonable conditions need to be studied. They are monotonicity of consequences, order preservation of CEs, and bias in estimates of CEs. This paper addresses the second assumption. To do this an algorithm is devised that generates pairs of lotteries with exactly the same preference probability of choice. This permits one to average over both subjects and lotteries. With a sample size of 102, there was no evidence that order preservation of CEs is violated.     

The Presence of Odd Pricing in the Texas State Lottery1

We examined 443 drawings from the Texas state lottery from 1992 to 1997 to determine whether odd pricing effects would be found in multimillion-dollar lotteries. Specifically, we predicted that many more lottery tickets than expected would be purchased when the jackpot amount was $10 million than when it was $9 million. Three findings emerged. First, the higher the jackpot, the more tickets were purchased. Second, odd pricing effects were found at $10 million such that approximately 670,000 more tickets were purchased than expected. Third, controlling for the jackpot amount, there were approximately 620,000 fewer tickets purchased for each drawing per year. This final finding provides evidence that people are becoming less interested in multimillion-dollar lotteries.     

Duplex decomposition and general segregation of lotteries of a gain and a loss: An empirical evaluation

We investigate the empirical validity of two hypotheses, duplex decomposition (DD) and general segregation (GS), regarding decomposition of a binary gamble of a gain and a loss into two unitary gambles in which one consequence of each gamble is no change from the status quo. Four binary lotteries (money gambles with specified probabilities) and four decomposed lotteries designed to test GS and four decomposed lotteries to test DD were constructed, and certainty equivalents (CEs) were estimated for each lottery. Respondents’ indifference between a binary lottery and a decomposed lottery was determined by evaluating the equality between the CE of a mixed binary lottery and the CE of the corresponding decomposed lottery. Given the variability of estimates of CEs and the lack of a clear statistical definition for the equality between two CEs, we applied several criteria: we counted responses where the difference between two CEs was either ±2% (the strictest criterion), ±4%, ±6%, or ±8% (the most lenient criterion) of the range of lottery outcomes. The results showed that under the strictest criterion, GS held for 25% of the responses and DD for 22%. Under the most lenient criterion, GS held for 56% of the responses and DD for 52%. Depending upon the criterion used, between 39 and 75% of the responses were consistent with at least one of the hypotheses. Several methodological problems in determining the indifference between two lotteries are discussed.     

Prosocial behavior in families: Moderators of resource sharing

Archival studies of within-family prosocial behavior (Judge & Hrdy, 1992; Smith, Kish, & Crawford, 1987) have shown that inheritance patterns support Hamilton’s (1964) theory of inclusive fitness, such that people will more of their estates to relatives of closer genetic relatedness. In a survey-based simulation of these studies, students allocated one of three lotteries to their blood relatives. As expected, participants allocated greater proportions of their lotteries to relatives of greater genetic relatedness. This effect became stronger with decreasing lottery amounts. Relatives of certain relatedness were favored over relatives of uncertain relatedness (via paternal uncertainty), and this effect was stronger among those with less money to allocate. Male participants tended to favor their young female relatives over their young male relatives when given the smallest lottery, but this sex preference reversed when male participants were given the largest lottery. In general, scarcer resources led to more evolutionarily conservative investment strategies.     

Choosing Between Lotteries: Remarkable Coordination Without Communication

The current research examines tacit coordination behavior in a lottery selection task. Two hundred participants in each of three experiments and 100 in a fourth choose to participate in one of two lotteries, where one lottery has a larger prize than the other. Independent of variations in the complexity of the mechanism of prize allocation, the prize amounts, and whether the lottery is the participant's first or second choice, we typically find that the percentage of participants who choose the high versus low‐prize lotteries does not significantly differ from the equilibrium predictions. This coordination is achieved without communication or experience. We additionally find that participants with an analytical thinking style and a risk‐averse tendency are more likely to choose the low‐prize lottery over the high‐prize lottery. This tendency seems to be stable across choices. The pattern of our results suggests that to achieve tacit coordination, having a subset of individuals who attend to the choices of others is sufficient. Copyright © 2012 John Wiley & Sons, Ltd.     

Emotion,Decision, and Risk: Betting on Gambles versus Betting on People

We examined the effects of two emotions, fear and anger, on risk‐taking behavior in two types of tasks: Those in which uncertainty is generated by a randomizing device (“lottery risk”) and those in which it is generated by the uncertain behavior of another person (“person‐based risk”). Participants first completed a writing task to induce fear or anger. They then made choices either between lotteries (Experiment 1) or between actions in risky two‐person decisions (Experiments 2 and 3). The experiments involved substantial real‐money payoffs. Replicating earlier studies (which used hypothetical rewards), Experiment 1 showed that fearful participants were more risk‐averse than angry participants in lottery‐risk tasks. However—the key result of this study—fearful participants were substantially less risk‐averse than angry participants in a two‐person task involving person‐based risk (Experiment 2). Experiment 3 offered options and payoffs identical to those of Experiment 2 but with lottery‐type risk. Risk‐taking returned to the pattern of Experiment 1. The impact of incidental emotions on risk‐taking appears to be contingent on the class of uncertainty involved. For lottery risk, fear increased the frequency of risk‐averse choices and anger reduced it. The reverse pattern was found when uncertainty in the decision was person‐based. Further, the effect was specifically on differences in willingness to take risks rather than on differences in judgments of how much risk was present. The impact of different emotions on risk‐taking or risk‐avoiding behavior is thus contingent on the type, as well as the degree, of uncertainty the decision maker faces. Copyright © 2010 John Wiley & Sons, Ltd.     

Precision of probability information and prominence of outcomes: A description and evaluation of decisions under uncertainty

Information about event probability upon which decisions depend may be more or less precise. The first section of this paper reports three experiments that investigated the relationship between this type of imprecision and the prominence that outcomes obtain in decisions. Participants had to rank order sets of six lotteries according to attractiveness. While the lotteries’ values were always precisely known precision of information about lottery chances varied. These experiments showed that increasing ambiguity tied decisions closer to lottery values. The second section shows that modeling participants’ decisions with the contingent weighting model suggests that this outcome prominence effect was not necessarily caused by any change in the respective weighting of probability and outcome information, but that it had probably occurred for purely mathematical reasons. The third part of this paper explores, by means of a computer simulation, (i) which weighting strategy is optimal when probabilities are imprecise and (ii) how participants’ decision behavior compared to a simple, but better adapted strategy. It shows that the weighting of probability information should not change with decreasing precision and it implies that participants’ performance suffered most from a lack of strategic consequence. Implications for decision making policy in general are discussed.     

Receiving Two Consequences: Tests of Monotonicity and Scale Invariance

The present study investigated two behavioral properties of decision making. Participants were presented two alternatives, each of which yielded two consequences jointly (joint receipt, JR). Monotonicity of JR states that choices remain invariant when the same lottery is adjoined to both of the original alternatives. The data failed to reject monotonicity of JR. Scale invariance states that choices remain invariant under proportionate (scale) changes of the lottery consequences. This property was not rejected for three of four classes of JR lotteries having a common consequence, but it was rejected for two of three classes of binary lotteries and for two of the four classes of general JR lotteries. We conclude that scale invariance does not hold in general. The implication of these findings for utility theory, testing its fundamental assumptions, and determining the functional form of utility is discussed. Copyright 2000 Academic Press.     

Subjective relative income and lottery ticket purchases

Despite a return of only $.53 on the dollar, state lotteries are extremely popular, especially among the poor, who play the most but can least afford to play. In two experiments conducted with low‐income participants, we examine how implicit comparisons with other income classes increase low‐income individuals' desire to play the lottery. In Experiment 1, participants were more likely to purchase lottery tickets when they were primed to perceive that their own income was low relative to an implicit standard. In Experiment 2, participants purchased more tickets when they considered situations in which rich people or poor people receive advantages, implicitly highlighting the fact that everyone has an equal chance of winning the lottery. Copyright © 2007 John Wiley & Sons, Ltd.     

Delay-discounting probabilistic rewards: Rates decrease as amounts increase

The independence of delay-discounting rate and monetary reward size was tested by offering subjects (N = 621) a series of choices between immediate rewards and larger, delayed rewards. In contrast to previous studies, in which hypothetical rewards have typically been employed, subjects in the present study were entered into a lottery in which they had a chance of actually receiving one of their choices. The delayed rewards were grouped into small ($30–$35), medium ($55–$65), and large amounts ($70–$85). Using a novel parameter estimation procedure, we estimated discounting rates for all three reward sizes for each subject on the basis of his/her pattern of choices. The data indicated that the discounting rate is a decreasing function of the size of the delayed reward (p < .0001), whether hyperbolic or exponential discounting functions are assumed. In addition, a reliable gender difference was found (p = .005), with males discounting at higher rates than females, on average.     

Cognitive processes underlying lottery and sports gambling decisions: The role of stated probabilities and background knowledge

Two studies compared choice and underlying cognitive processes in equivalent decision tasks involving risk and uncertainty (lotteries versus sports gambles including displayed expert probability judgements). In sports gambles, background knowledge was triggered via information on team location, home or away. Otherwise, displayed risk information (stake, winnings, odds, and outcome probabilities) was controlled across gamble type. In a choice study, home win bets were chosen significantly more frequently than draws or away wins, compared to lottery equivalents. In a parallel study eliciting concurrent verbal protocols, participants made fewer evaluations of odds and probabilities, and more statements involving background knowledge in sports gambles. Furthermore, some sports gamble protocols indicated modifications of stated probabilities and decision strategies contingent on domain knowledge. It was concluded that stated probability revision and knowledge-based reasoning are key cognitive processes in sports gambling not normally applied in the lottery paradigms often employed in decision research.     

Consequences of regret aversion in real life: The case of the Dutch postcode lottery

Although ample research has shown that decisions may cause regret and that the anticipation of regret may influence decision-making, this previous research was largely limited to hypothetical choices with student participants. The current research replicates and extends these findings for real life lottery participation decisions in non-student samples. Four studies are reported in which two lotteries in the Netherlands, the Postcode Lottery and the National State Lottery, were compared. The State Lottery is a traditional lottery in which one has to buy a ticket with a number printed on it. In the Postcode Lottery, one’s postcode is the ticket number, and hence even if not participating one may still find out that one would have won had one played. As our research shows, this particular feedback that is present in the Postcode Lottery but absent in the State Lottery influences the level of anticipated post-decisional regret, and moderates the influence that anticipated regret has on lottery participation. Study 1, 100 street interviews, confirmed our expectations that the Postcode Lottery may elicit regret. Study 2 found under controlled conditions, that people anticipate more regret over not playing when there is feedback about the neighbors winning a prize in the Postcode Lottery than in the State Lottery. However, when this feedback is absent they anticipate equal amounts of regret over not playing. Study 3 replicated these findings for regret, while showing that the two lotteries do not differ with respect to envy and jealousy, emotions that might also be invoked in this context. Study 4 validated that, as we predicted, anticipations of post-decisional regret influence decisions to play the Postcode lottery, but not the State Lottery. These findings demonstrate the external and discriminant validity of anticipated regret for decision-making, and indicate its pragmatic relevance. The implications or recent developments in regret research are discussed.     

WEIGHTED LOTTERIES IN LIFE AND DEATH CASES

Faced with a choice between saving one stranger and saving a group of strangers, some people endorse weighted lotteries, which give a strictly greater chance of being saved to the group of strangers than the single stranger. In this paper I attempt to criticize this view. I first consider a particular version of the weighted lotteries, Frances Kamm's procedure of proportional chances, and point out two implausible implications of her proposal. Then, I consider weighted lotteries in general, and claim (1) that the correct thing to distribute is not the chance of being saved but the good of being saved, (2) that assigning some chance to the single stranger is not the only way to give a positive (and equal) respect to the people concerned, and (3) that the weighted lottery appears to be deceptive since it would show the respect to the single stranger in a negligible way.     

Testing the indifference between a binary lottery and its edited components using observed estimates of variability

Past studies have shown inconsistent conclusions on the empirical validity of the duplex decomposition assumption that Luce proposed (1997). The duplex decomposition assumption states that a person would feel indifferent between playing a mixed (gain and loss outcomes) lottery and playing two lotteries simultaneously or successively where one lottery is the gain part and the other lottery is the loss part of the original mixed lottery. The current study tested this assumption by investigating the equality between the certainty equivalent of a binary lottery, CE (BL), and the certainty equivalent of the duplex decomposed lottery, CE (DDL), while controlling for the response variability in the estimated CEs. The results from two experiments showed that for 69% of responses, the difference between the CE (BL) and CE (DDL) was smaller than or equal to the response variability in estimating CEs, implying that duplex decomposition cannot be rejected for a majority of responses. The size of response variability in estimated CEs was discussed.     

Good fortune but bad luck? Predictable versus unpredictable gains and losses in risky choice

This study examines the effects of two different types of good and bad experiences on risk‐taking preferences: fortune and luck. We define fortune as a relatively stable positive or negative context within which choices are made and luck as a more unpredictable series of better or worse outcomes. With the use of a lottery‐based paradigm, fortune was operationalized as a preponderance of all‐gain or all‐loss two‐outcome option pairs within a larger set of mixed‐outcome control lotteries. Luck was operationalized as the experienced frequency of better versus worse outcomes when playing the lotteries. We predicted that fortune and luck would lead to opposite risk‐taking tendencies within control lotteries. An assimilation effect of fortune was predicted, with risk‐averse preferences for control lotteries when surrounded by good fortune and risk‐seeking preferences when surrounded by bad fortune. In contrast, we expected that high rates of success with good luck would lead to risk‐seeking preferences, whereas low rates of success with bad luck would yield risk‐averse preferences. Our predictions for fortune were confirmed; however, there was no evidence of any effect on risk taking based on experiencing good versus bad luck. Moreover, we observed a striking disconnect between impressions of the experience and risk‐taking behavior. Both identification and attributions of luck and fortune were highly correlated with the number of gain outcomes that participants experienced but were uncorrelated with risk taking. We review these surprising findings considering several prominent theories of risk‐taking behavior, particularly drawing attention to the differential roles of predecisional and postdecisional information in choice.     

Prospects behind bars: Analyzing decisions under risk in a prison population

Criminal activity often involves considerable risks. It is therefore not surprising that criminals have been speculated to differ from noncriminals in risk attitude. Yet, few data exist to support this assumption. Moreover, the psychological underpinnings of differences in risk attitude are currently little understood. We presented prisoners and controls with sets of risky decision tasks and modeled their responses using cumulative prospect theory (CPT). The two groups showed several differences. Prisoners were more risk seeking than nonprisoners in lotteries involving losses, but they were less risk seeking in lotteries involving high-probability gains. Bestfitting CPT parameters indicated a reduced sensitivity to outcomes, for both gains and losses, and a stronger loss aversion among prisoners. In addition, prisoners showed a diminished sensitivity to the probability of gains. Our results contribute to a better understanding of prisoners’ risk attitudes and the underlying mechanisms that distinguish prisoners from nonprisoners and may thus help improve interventions designed to prevent crime.     

Attitude bolstering when behavior is inconsistent with central attitudes

An experiment was conducted to study the effects of behaving inconsistently with a central attitude. In a theoretical account, it is suggested that subjects who score high on a feminism scale and who fail to solve a sex-role problem and thus demonstrate sexist thinking will be motivated to adopt feminist (nonsexist) behavior more than subjects who are either lower in feminism or who don't fail such a problem. Since the attitude is central, attitude bolstering rather than change in the direction of the behavior should be the preferred mode of inconsistency reduction for such subjects. Bolstering, in the form of positive affirmative action decisions, was generally demonstrated by subjects who failed the sex-role problem. This effect was more pronounced for those highest in feminism. No such effects on affirmative action decisions were observed for subjects who didn't fail the sex-role problem. Results support the view that threats to one's self-image caused by one's own counter-attitudinal actions lead to attempts to reestablish those threatened values. A conceptual analysis of the conditions under which bolstering rather than attitude change in the direction of the behavior should be expected is presented.     

Natural experiments and the educational context: The environment and effects of an alternative inner-city public school on adolescents

The 1-year impact of attending a public alternative high school on two cohorts of adolescents who gained entrance to the school through a lottery was studied. Adolescents who had applied to the school but were not selected in the lottery served as a control group. The nature of the alternative high school environment is described, and the outcome of this natural experiment defined in terms of reactions to school, attitude change, and student achievement. In general, the alternative school positively affected student satisfaction with a variety of aspects of school life and induced some positive change in interracial attitudes while not harming student achievement. Race effects were found in the areas of achievement and interracial attitudes. Implications for the study of natural experiments in general and the specific data in particular are drawn.