A solution to a problem raised in Luce and Marley (2005)

Luce and Marley [2005. Ranked additive utility representations of gambles: Old and new axiomatizations. Journal of Risk and Uncertainty, 30, 21-62] examined various relations between mathematical forms for the utility of joint receipt ⊕ of gambles and for the utility of uncertain gambles. Their assumptions lead to a bisymmetry functional equation which, when the gambles are ranked, is defined on a restricted domain. Maksa [1999. Solution of generalized bisymmetry type equations without surjectivity. Aequationes Mathematicae, 57, 50-74] solved the general case and Kocsis [2007. A bisymmetry equation on restricted domain. Aequationes Mathematicae, 73, 280-284] presents the solution for the ranked case. The latter solution allows us to solve open problem 5 in Luce and Marley (2005) by showing that the assumptions of their Theorem 19 for an order-preserving ranked additive utility (RAU) representation U imply that U is a ranked weighted utility (RWU) representation that is additive over ⊕.     

Intuitions about declining marginal utility

In two studies, subjects judged the desirability of distributions of life expectancy or money. Their judgments showed declining marginal utility. That is, they were less sensitive to changes at the high end of each scale. Subjects also made utility ratings of the outcomes of individuals. And they made ratings of the distributions when these were described in terms of utility ratings rather than goods (years or dollars). The judgments of utility ratings showed equivalent declining marginal utility, even though they were based on utilities that themselves declined marginally. People extend their intuition about declining marginal utility to utility itself, as if utility had utility that declined marginally. In one experiment, a similar result was found with gambles: people are risk averse for utility as well as for money. We argue that this is an overextension of a reasonable heuristic and that this heuristic may account for one classic objection to utilitarian distributions. Copyright © 2001 John Wiley & Sons, Ltd.     

Functional equations arising in a theory of rank dependence and homogeneous joint receipts

This paper focuses on a class of utility representations of uncertain alternatives with two possible consequences (binary gambles) when they are linked via a distributivity property called segregation to an operation of joint receipt, which may be non-commutative. The assumption that the gambling structure and the joint receipt operation both have homogeneous representations that are order preserving leads to a functional equation that has too many solutions to be useful for characterizing a reasonably specific utility representation. A plausible restriction on the form of the utility of gambles leads to the functional equation

     

Utility of gambling under p(olynomial)-additive joint receipt and segregation or duplex decomposition

This article continues our exploration of the utility of gambling, but here under the assumption of a non-additive but p(olynomial)-additive representation of joint receipt. That assumption causes the utility of gambling term to have a multiplicative impact, rather than just an additive one, on what amounts to ordinary weighted utility. Assuming the two rational recursions known as branching and upper gamble decomposition, we investigate separately the rational property of segregation and the non-rational one of duplex decomposition. Under segregation, we show that each pair of disjoint events either exhibits weight complementarity or both support no utility of gambling, a property called UofG-singular. We develop representations for both cases. The former representation is a simple weighted utility with each term multiplied by a function of the event underlying that branch. The latter representation is an ordinary rank-dependent utility with Choquet weights but with no utility of gambling. Under duplex decomposition, we show that weights have the intuitively unacceptable property that there is essentially no dependence upon the events, making duplex decomposition, in this context, of little behavioral interest.     

Empirical underidentification in estimating random utility models: The role of choice sets and standardizations

A standard approach to distinguishing people’s risk preferences is to estimate a random utility model using a power utility function to characterize the preferences and a logit function to capture choice consistency. We demonstrate that with often-used choice situations, this model suffers from empirical underidentification, meaning that parameters cannot be estimated precisely. With simulations of estimation accuracy and Kullback–Leibler divergence measures we examined factors that potentially mitigate this problem. First, using a choice set that guarantees a switch in the utility order between two risky gambles in the range of plausible values leads to higher estimation accuracy than randomly created choice sets or the purpose-built choice sets common in the literature. Second, parameter estimates are regularly correlated, which contributes to empirical underidentification. Examining standardizations of the utility scale, we show that they mitigate this correlation and additionally improve the estimation accuracy for choice consistency. Yet, they can have detrimental effects on the estimation accuracy of risk preference. Finally, we also show how repeated versus distinct choice sets and an increase in observations affect estimation accuracy. Together, these results should help researchers make informed design choices to estimate parameters in the random utility model more precisely.     

A Simple Axiomatization of Binary Rank-Dependent Utility of Gains (Losses)

For binary gambles composed only of gains (losses) relative to a status quo, the rank-dependent utility model with a representation that is dense in intervals is shown to be equivalent to ten elementary properties plus event commutativity and a gamble partition assumption. The proof reduces to a (difficult) functional equation that has been solved by Aczél, Maksa, and Páles (in press).     

VIOLATIONS OF BRANCH INDEPENDENCE IN JUDGMENTS OF THE VALUE OF GAMBLES

Abstract —Branch independence is weaker than Savage's "sure thing" principle. It requires that judgments of gambles with a common outcome produced by the same probability-event must not reverse order when that common outcome is changed. Subjects judged 168 gambles from viewpoints of both buyer (highest buying price) and in both viewpoints. Violations also changed systematically between viewpoints, consistent with the theory that viewpoint affects configural weighting but not the utility function. Violations of branch independence were opposite those predicted by the model of cumulative prospect theory. The middle of three equally likely outcomes received the most weight in the seller's viewpoint. In the buyer's, lower outcomes received greater weights. In both viewpoints, the ratio of weights of the middle outcome to the highest outcome exceeded the ratio of weights of the lowest outcome to the middle outcome.     

Trichotomous preferences for gambles

Let G_X be the set of all gambles on a set X of decision outcomes. Trichotomous preferences divide G_X into three disjoint subsets, viz., the set P⁺ of preferable gambles, the set P⁰ of indifferent gambles, and the set P^- of unpreferable gambles. Those preference comparisons are made against the status quo, i.e., the do-nothing alternative. This paper presents and discusses dyadic representations of such trichotomous preferences and identifies necessary and sufficient axioms for the existence of those representations.     

A multiplicative model of the utility of survival duration and health quality

Survival duration and health quality are fundamentally important aspects of health. A utility model for survival duration and health quality is a model of the subjective value of these attributes. We investigate the hypothesis that the utility (subjective value) of survival duration and health quality is determined by a multiplicative model. According to this model, there are separate subjective scales for the utility of survival duration and health quality. If F(Y) equals the utility of surviving Y years, and G(Q) equals the utility of living in health state Q, then the multiplicative model proposes that F(Y)G(Q) equals the utility of surviving Y years in health state Q. This model provides a simple explanation for several intuitively compelling relationships. First, the distinction between better-than-death and worse-than-death health states corresponds to the assignment of positive or negative utilities to different health states. Second, a zero duration of survival removes any reason to prefer one health state over any other, just as multiplying the utility of health quality by zero eliminates differences between the utilities of different health states. Third, the subjective difference between Y years in pain and Y years free from pain increases as Y increases as if the difference in utility between pain and no pain were being multiplied by the utility of surviving Y years. A critical prediction of the multiplicative model is the hypothesis that preferences between gambles for health outcomes satisfy a property called utility independence. Individual analyses revealed that most subjects satisfy utility independence, thereby supporting the multiplicative utility model. Some subjects appear to violate a fundamental assumption of utility theory: They appear to violate the assumption that a single utility scale represents both the ordinal preference relations between certain outcomes and the subjective averaging that underlies the utility of gambles. The violation is inferred from an inconsistency between preferences for multiattribute outcomes when they are viewed as certain outcomes and when they are viewed as the outcomes of gambles.     

We are more selfish than we think: The endowment effect and reward processing within the human medial-frontal cortex

Perceived ownership has been shown to impact a variety of cognitive processes: attention, memory, and—more recently—reward processing. In the present experiment we examined whether or not perceived ownership would interact with the construct of value—the relative worth of an object. Participants completed a simple gambling game in which they gambled either for themselves or for another while electroencephalographic data were recorded. In a key manipulation, gambles for oneself or for another were for either small or large rewards. We tested the hypothesis that value affects the neural response to self-gamble outcomes, but not other-gamble outcomes. Our experimental data revealed that while participants learned the correct response option for both self and other gambles, the reward positivity evoked by wins was impacted by value only when gambling for oneself. Importantly, our findings provide additional evidence for a self-ownership bias in cognitive processing and further demonstrate the insensitivity of the medial-frontal reward system to gambles for another.     

VIOLATIONS OF MONOTONICITY AND CONTEXTUAL EFFECTS IN CHOICE-BASED CERTAINTY EQUIVALENTS

Abstract —This article investigates choices between gambles and amounts of money to explore two issues in decision making First, in recent studies, judgments of the values of gambles violated monotonicity (dominance), yet choices between the same gambles satisfied monotonicity, producing reversals of preference This experiment tested whether certainty equivalents based on choices between gambles and money would violate monotonicity Results indicated that these choices violated monotonicity in the same way as had judgments Second, this experiment investigated whether the certainty equivalent of a gamble would depend on the distribution of amounts offered for comparison It was found that certainty equivalents based on choices depended on the context m the same fashion as psychophysical comparisons Apparently, paradoxes of behavioral decision making are not eliminated by using choices instead of judgments to investigate human preferences     

An empirical assessment of the form of utility functions

Utility functions, which relate subjective value to physical attributes of experience, are fundamental to most decision theories. Seven experiments were conducted to test predictions of the most widely assumed mathematical forms of utility (power, log, and negative exponential), and a function proposed by Rachlin (1992). For pairs of gambles for real monetary gains, undergraduate and nonstudent subjects either reported an equalizing amount for 1 outcome that made the gambles subjectively equal or chose between gambles where the amounts were varied across trial, which allowed the equalizing amount to be estimated from their pattern of choices. Using a novel method that eliminates several limitations of previous research, I manipulated the outcomes across trials such that each type of utility function predicted a linear relationship between the equalizing amounts and the amounts of the other outcomes, and made point predictions for either the slope or intercept of that relationship. In a meta-analysis across experiments, systematic departures from the point predictions were observed for each type of utility function. Thus, the data imply that despite their historical importance and incorporation in many psychological and economic decision theories, the most widely assumed models of utility are incorrect.     

Utility-uncertainty trade-off structures

A qualitative axiomatization of a generalization of expected utility theory is given in which the expected value of simple gambles is not necessarily the product of subjective probability and utility. Representation and uniqueness theorems for these generalized structures are derived for both Archimedean and nonarchimedean cases. It is also shown that a simple condition called distributivity is necessary and sufficient in the case of simple gambles for one of these generalized expected utility structures to have simultaneously an additive subjective probability function and a multiplicative combining rule for expected values.     

Expectation theories of decision making for duplex gambles

Complete tests of subjectively expected utility (SEU), subjectively expected value (SEV), expected utility (EU) and expected value (EV) theories were made for duplex gambles without measuring subjective probability or subjective utility. All gambles were hypothetical and offered on booklets. The duplex gambles consisted of winning gambles, which offered a chance to win a certain amount of money or to break even; and losing gambles, which offered a chance to lose a certain amount of money or break even.The results indicated that SEU, SEV and EU theories could not account for the strategies of 33%, 53% and 86% of the Ss respectively in the losing form of gambles, while EV theory accounted for 78% of the behavior of Ss.In the winning form of gambles, SEU, SEV and EU theory held for 77%, 65%, and 54% of the Ss respectively, while EV theory held for only 40% of the Ss. Suggestions for further research were made.     

RATIONAL VERSUS PLAUSIBLE ACCOUNTING EQUIVALENCES IN PREFERENCE JUDGMENTS

Subjective expected utility (SEU) embodies four distinct principles of rational behavior. Although all have been called into some question empirically, the least plausible and least studied is the property that formally equivalent gambles are treated as indifferent in preference. The paper describes some results that arise when this property is sharply weakened and to some degree replaced by alternative rational and not-so-rational-assumptions. The resulting utility representations, like SEU, are weighted averages of the utilities of consequences, but with the weights dependent on more than the underlying chance event. In rank-dependent cases, which arise from a restricted assumption about formally equivalent gambles, the weights depend on the rank position of the corresponding consequence. In ank-and sign-dependent models, they depend both on the rank position of the consequence associated to the event and on whether it is a gain or a loss. The theory giving rise to the latter involves an additional primitive, namely, joint receipt of gambles, in terms of which new rational and irrational assumptions are invoked. The result generalizes prospect theory to gambles with more than a single gain and a single loss.     

Discounting by Probabilistic Waiting

In everyday life, many probabilistic situations may be characterized as probabilistic waiting. A gambler, for example, bets repeatedly at the racetrack, the casino, or the card table. The gambler may not win on the first try, but if a gamble is repeated enough times, a win is almost certain to occur eventually. If repeated gambles are structured as strings of losses ending in a win (probabilistic waiting) and the amount won is discounted by the delay caused by the series of losses, then strings with many losses will be discounted more than those with fewer losses, thereby causing subjective value of the series of gambles as a whole to increase. The current study used the opposite effect that amount has on the degree of delay and probability discounting as a marker to determine whether people evaluate situations involving probabilistic waiting as they evaluate situations involving delayed outcomes or as situations involving probabilistic outcomes. We find that the more likely a probabilistic waiting situation is to end in reward (e.g., a gamble is repeated indefinitely until reward is obtained), the more that situation conforms to delay discounting; the less likely a probabilistic waiting situation is to end in reward (e.g., a fixed, small number of gambles), the more that situation conforms to probability discounting. We argue that the former situation is applicable to pathological gambling, and that people with steep delay discount functions would therefore be more likely to have gambling problems. Copyright © 2015 John Wiley & Sons, Ltd.     

A Representation Theorem for Frequently Irrational Agents

The standard representation theorem for expected utility theory tells us that if a subject’s preferences conform to certain axioms, then she can be represented as maximising her expected utility given a particular set of credences and utilities—and, moreover, that having those credences and utilities is the only way that she could be maximising her expected utility (given her preferences). However, the kinds of agents these theorems seem apt to tell us anything about are highly idealised, being (amongst other things) always probabilistically coherent with infinitely precise degrees of belief and full knowledge of all a priori truths. Ordinary subjects do not look very rational when compared to the kinds of agents usually talked about in decision theory. In this paper, I will develop an expected utility representation theorem aimed at the representation of those who are neither probabilistically coherent, logically omniscient, nor expected utility maximisers across the board—that is, agents who are frequently irrational. The agents in question may be deductively fallible, have incoherent credences, limited representational capacities, and fail to maximise expected utility for all but a limited class of gambles.     

The Disjunction Effect: Does It Exist for Two-Step Gambles?

One of the basic axioms of the rational theory of decision under uncertainty is Savage's (1954) Sure Thing Principle. It states that if Prospect x is preferred to Prospect y knowing that Event A occurred, and if x is preferred to y knowing that A did not occur, then x should also be preferred to y when it is not known whether A occurred. Tversky and Shafir (1992) claim to have demonstrated a violation of this principle in two-step gambles, which is termed a disjunction effect. The present article evaluates the replicability of the disjunction effect for two-step gambles. The findings show that people do not violate the sure thing principle in repeated gambles. The validity of alleged violations in other paradigms is discussed.     

Simplified approaches for multicriteria decision making under uncertainty

The problem of uncertainty in the outcome of decisions has tended to be glossed over by many MCDM methods, with the exception of multiattribute utility theory (MAUT). MAUT does, however, require quite complicated preference elicitations and knowledge of the full multivariate distribution of outcomes. Results from a series of simulation studies indicate that the preference orderings of MAUT are only minimally changed when using a simple additive aggregation of marginal utilities, especially in relation to the natural imprecisions inherent in preference elicitation. It is shown in the simulations that by far the most critical aspect of multicriteria decision analysis under uncertainty is not the form of aggregation, but the correct elicitation of marginal utilities which properly represent decision maker preferences over gambles. We relate the results obtained here to other results on the approximation of distributions by three- or five-point discrete distributions, and suggest that the use of deterministic MCDM methods of any form (not necessarily value function techniques), applied to an extended formulation in which each criterion measure is repeated for three or five ‘scenarios’, can be justified.