Subjective expected utility theory revisited: A reductio ad absurdum paradox

A paradox is posed and analyzed in which people reverse their preferences for information on probabilities versus prizes once the range of the unknown probabilities is sufficiently narrowed. This reversal is shown to be incompatible with both objective expected utility (EU) as well as subjective versions in which the same probability transformation applies to all random variables. Experimental data are presented showing that the reversals occur with small, medium and large payoffs.The present paradox is compared with those of Allais and Ellsberg, and found to differ in substantive ways. It raises further questions about the normative status of expected utility theory, especially its treatment of probability and value. The paradox specifically calls into question EU's substitution and compound probability axioms.     

Violations of dominance in pricing judgments

The dominance principle states that the judged price of gamble A should be equal to or greater than the judged price of gamble B whenever A's outcomes are equal to or better than the corresponding outcomes of B, holding everything else constant. Subjects often violate the dominance principle by assigning a higher price to a gamble with some probability of winning a positive amount, Y, otherwise zero, than to a superior gamble with the same chances of winning Y, otherwise winning X. Violations also occur with losses. Results are consistent with a configural-weight theory in which the decision weight for each outcome depends on the rank of the outcome with respect to the other outcomes in the lottery and the value of the outcome (zero vs. nonzero).The authors thank Shi-jie Chang, Duncan Luce, and Lisa Ordóñez for comments on an earlier draft. This research was supported by a National Science Foundation grant to the first author (BNS-8451368). Requests for reprints should be sent to Barbara Mellers, Dept. of Psychology, University of California, Berkeley, CA 94720.     

Risk Aversion when Gains are Likely and Unlikely: Evidence from a Natural Experiment with Large Stakes

In the television show Deal or No Deal a contestant is endowed with a sealed box, which potentially contains a large monetary prize. In the course of the show the contestant learns more information about the distribution of possible monetary prizes inside her box. Consider two groups of contestants, who learned that the chances of their boxes containing a large prize are 20% and 80% correspondingly. Contestants in both groups receive qualitatively similar price offers for selling the content of their boxes. If contestants are less risk averse when facing unlikely gains, the price offer is likely to be more frequently rejected in the first group than in the second group. However, the fraction of rejections is virtually identical across two groups. Thus, contestants appear to have identical risk attitudes over (large) gains of low and high probability.      

Dopamine and risk choices in different domains: Findings among serious tournament bridge players

We explore how risk-taking in the card game contract bridge, and in a financial gamble, correlate with variation in the dopamine receptor D4 gene (DRD4) among serious tournament bridge players. In bridge risk-taking, we find significant interactions between genetic predisposition and skill. Among men with the 7-repeat allele of DRD4, namely 7R + men, those with more bridge skill take more good risks and fewer bad risks, while the opposite is found for less-expert 7R + men. Conversely, skill does not predict risk-taking among men without the 7R + allele. Consistent with some prior studies, we also find that 7R + men take more risk in the financial gamble. We find no relationship between 7R + and either risk measure among our female subjects. Our results suggest that the dopamine system plays an important role in individual differences in risk-taking among men, and is the first to distinguish between advantageous and disadvantageous risk-taking.     

Reanalysis of the Chechile-Cooke Experiment: Correcting for Mismatched Gambles

Chechile and Cooke (1997) experimentally tested a broad class of utility models subsumed under the Miyamoto (1988, 1992) generic utility theory. The Chechile and Cooke study required participants to match on each trial, a fully specified reference gamble to a partially specified comparison gamble by adjusting the probability of a win on the comparison gamble. The Chechile and Cooke experiment, however, contained a subset of trials which were intrinsically unmatchable. In such cases, the participants could only give an extreme probability (either 0 or 1). In this paper, those extreme trials were omitted and the results from the experiment reanalyzed. Despite the mismatch problem, the conclusions of the Chechile and Cooke experiment were again supported. For nine implementations of generic utility there is model failure due to the systematic variation of a parameter that should be a constant.     

The law of large numbers and the attractiveness of compound gambles

A compound of many independent replicas of a gamble may be chosen by a risk averter even if the single gamble would be rejected given any initial wealth level. Samuelson has shown that such choices are impossible if the expected utility theory of preference is adopted. But they are consistent with more general theories of preference. Consequently, the intuition suggested by the law of large numbers can be supported.     

The framing of nothing and the psychology of choice

Journal of Risk and Uncertainty - Zero outcomes are inconsequential in most models of choice. However, when disclosing zero outcomes they must be designated. It has been shown that a gamble is...     

It is Whether You Win or Lose: The Importance of the Overall Probabilities of Winning or Losing in Risky Choice

Imagine that you own a five-outcome gamble with the following payoffs and probabilities: ($100, .20; $50, .20; Imagine that you own a five-outcome gamble with the following payoffs and probabilities: ($100, .20; $50, .20; $0, .20; –$25, .20; –$50, .20). What happens when the opportunity to improve such a gamble is provided by a manipulation that adds value to one outcome versus another outcome, particularly when the opportunity to add value to one outcome versus another outcome changes the overall probability of a gain or the overall probability of a loss? Such a choice provides a simple test of the expected utility model (EU), original prospect theory (OPT), and cumulative prospect theory (CPT). A study of risky choices involving 375 respondents indicates that respondents were most sensitive to changes in outcome values that either increased the overall probability of a strict gain or decreased the overall probability of a strict loss. These results indicate more support for OPT rather than CPT and EU under various assumptions about the shape of the utility and value and weighting functions. Most importantly, the main difference between the various expectation models of risky choice occurs for outcomes near the reference value. A second study of risky choice involving 151 respondents again demonstrated the sensitivity of subjects to reducing the probability of a strict loss even at the cost of reduced expected value. Consequently, we argue that theories of how people choose among gambles that involve three or more consequences with both gains and losses need to include measures of the overall probabilities of a gain and of a loss.JEL Classification  D81     

Expected utility without utility

This paper advances an interpretation of Von Neumann-Morgenstern's expected utility model for preferences over lotteries which does not require the notion of a cardinal utility over prizes and can be phrased entirely in the language of probability. According to it, the expected utility of a lottery can be read as the probability that this lottery outperforms another given independent lottery. The implications of this interpretation for some topics and models in decision theory are considered.     

The St. Petersburg gamble and risk

Pursuing a line of thought initiated by Maurice Allais (1979), I consider whether the mean-risk method of decision making introduced by Harry Markowitz (1959) and other resolves Karl Menger's (1934) version of the St. Petersburg paradox. I provide a conditional answer to this question. I demonstrate that given certain plausible assumption about attitudes toward risk, a certain plausible development of the mean-risk method does resolve the paradox. My chief premiss is roughly that in the St. Petersburg gamble the small chances for large prizes create big risks.     

Endowment effects? “Even” with half a million on the table!

In the television show Deal or No Deal, a contestant is endowed with a sealed box containing a monetary prize between one cent and half a million euros. In the course of the show, the contestant is offered to exchange her box for another sealed box with the same distribution of possible monetary prizes inside. This offers a unique natural experiment for studying endowment effects under high monetary incentives. We find evidence of only a weak endowment effect when contestants exchange their box for another box with the same distribution of possible prizes.     

On a lottery pricing anomaly: Time tells the tale

This article identifies a lottery pricing anomaly, which I call the “r=x anomaly,” that is present in past pricing experiments—namely, a tendency for subjects to announce that their minimum selling price for some binary lottery is the greater of the two lottery prizes. The study shows that the anomaly is inconsistent with two theoretical explanations for another well-known pricing anomaly (preference reversal) and experimentally replicates these inconsistencies. The new experiment also measures the time subjects spend making their pricing decisions. These decision-time measurements suggest that ther=x anomaly may be a decision-cost effect.     

Ellsberg paradox: Ambiguity and complexity aversions compared

We present a simple model where preferences with complexity aversion, rather than ambiguity aversion, resolve the Ellsberg paradox. We test our theory using laboratory experiments where subjects choose among lotteries that “range” from a simple risky lottery, through risky but more complex lotteries, to one similar to Ellsberg’s ambiguity urn. Our model ranks lotteries according to their complexity and makes different—at times contrasting—predictions than most models of ambiguity in response to manipulations of prizes. The results support that complexity aversion preferences play an important and separate role from beliefs with ambiguity aversion in explaining behavior under uncertainty.     

Caregiver depression and child behaviour problems: A longitudinal mixed effects approach

Caregiver mental health has a powerful influence on child well‐being. Both maternal depression and paternal depression predict behaviour problems in young children. However, little is known about the nature of this link over time. The present study leveraged a large sample of at‐risk families to examine the impact of maternal and paternal depression on trajectories of child behaviour from age 3 to 15. Data came from the At‐Home substudy of the Fragile Families and Child Well‐Being Study, which observed a cohort of children born 1998–2000 in 20 large American cities (N = 1,090). Multiple imputation by chained equations addressed missing data to maximize available information without biasing models. Linear mixed effects models tested the association between caregiver depression and child behaviour over 12 years; interaction models tested whether this association changed as children transitioned into adolescence. Results showed caregiver depression is related with increases in both child internalizing and externalizing problems, and these relationships strengthened over time. Findings inform risk trajectories of distinct behavioural problems, shedding light on the development of child psychopathology and highlighting opportunities for intervention.     

The Micro and Macro Dimensions of Gambling in the United States

A national probability sample of 1736 respondents and a Nevada State probability sample of 296 respondents were surveyed during the summer of 1975 to determine the extent of gambling activity in the United States, to estimate the amount of government revenue that could result from various changes in the gambling laws, and to predict the social consequences of these changes. While there were large variations among regions and demographic groups, a majority of all adult Americans gamble. Wagers on commercial games amounted to $22.4 billion in 1974 of which $17 billion was wagered legally and approximately $5 billion illegally.     

On Elements of Chance

One aspect of the utility of gambling may evidence itself in failures of idempotence, i.e., when all chance outcomes give rise to the same consequence the `gamble' may not be indifferent to its common consequence. Under the assumption of segregation, such gambles can be expressed as the joint receipt of the common consequence and what we call `an element of chance', namely, the same gamble with the common consequence replaced by the status quo. Generalizing, any gamble is indifferent to the joint receipt of its element of chance and a certain consequence, which is called the `kernel equivalent' of the gamble. Under idempotence, the kernel equivalent equals the certainty equivalent. Conditions are reported (Theorem 4) that are sufficient for the kernel equivalents to have the kind of utility representation first discussed by Luce and Fishburn (1991), including being idempotent. This utility representation of the kernel equivalents together with the derived form of utility over joint receipts yields a utility representation of the original structure. Possible forms for the utility of an element of chance are developed.     

Stubbornness,Power, and Equilibrium Selection in Repeated Games with Multiple Equilibria

Axelord’s [(1970), Conflict of Interest, Markham Publishers, Chicago] index of conflict in 2 × 2 games with two pure strategy equilibria has the property that a reduction in the cost of holding out corresponds to an increase in conflict. This article takes the opposite view, arguing that if losing becomes less costly, a player is less likely to gamble to win, which means that conflict will be less frequent. This approach leads to a new power index and a new measure of stubbornness, both anchored in strategic reasoning. The win probability defined as power constitutes an equilibrium refinement which differs from Harsanyi and Selten’s [(1988), A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge] refinement. In contrast, Axelrod’s approach focuses on preferences regarding divergences from imaginary outmost rewards that cannot be obtained jointly. The player who is less powerful in an asymmetric one-shot game becomes more powerful in the repeated game, provided he or she values the future sufficiently more than the opponent. This contrasts with the view that repetition induces cooperation, but conforms with the expectation that a more patient player receives a larger share of the pie.      

The Social Composition of China's Private Entrepreneurs: Class and Cohort Differences*

On the basis of survey data on Chinese private enterprises over the years, we try to respond to the classic subject of “the social composition of private entrepreneurs (siyingqiyezhu 私营企业主).” In nearly forty years of development, the overall composition of private entrepreneurs has undergone major changes. The group contains a growing proportion of people who have a market background and higher education and are non-political CPCs. On further classifying the occupational mobility of private entrepreneurs into categories such as “xiahai 下海” (jump into business), “gaizhi 改制” (restructuring), “kuajie 跨界” (crossover), “tiaoban 跳板” (springboard) and “caogen 草 根” (grassroots), we find there are significant differences in the occupational mobility of entrepreneurs in large, medium, and small enterprises in terms of what they did before they founded their businesses. In particular, entrepreneurs in large enterprises are more likely to have “jumped into business” from inside the government system or after restructuring, while most small and medium entrepreneurs develop outside the system. Multiple regression and coefficient clustering analysis shows that education level and political status have varying effects on the occupational mobility of entrepreneurs in terms of class and cohort.     

Objective Belief Functions as Induced Measures

Given a belief function ν on the set of all subsets of prizes, how should ν values be understood as a decision alternative? This paper presents and characterizes an induced-measure interpretation of belief functions.