Utility of gambling I: entropy modified linear weighted utility (original) (raw)
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Utility of Gambling when Events are Valued: an Application of Inset Entropy
Theory and Decision, 2009
The present theory leads to a set of subjective weights such that the utility of an uncertain alternative (gamble) is partitioned into three terms involving those weights -a conventional subjectively weighted utility function over pure consequences, a subjectively weighted value function over events, and a subjectively weighted function of the subjective weights. Under several assumptions, this becomes one of several standard utility representations, plus a weighted value function over events, plus an entropy term of the weights. In the finitely additive case, the latter is the Shannon entropy; in all other cases it is entropy of degree not 1. The primary mathematical tool is the theory of inset entropy.
Utility of gambling II: risk, paradoxes, and data
Economic Theory, 2008
We specialize our results on entropy-modified representations of eventbased gambles to representations of probability-based gambles by assuming an implicit event structure underlying the probabilities, and adding assumptions linking the qualitative properties of the former and the latter. Under segregation and under duplex decomposition, we obtain numerical representations consisting of a linear weighted utility term plus a term corresponding to information-theoretical entropies. These representations accommodate the Allais paradox and most of the data due to Birnbaum and associates. A representation of mixed event-and probability-based gambles accommodates the Ellsberg paradox. We suggest possible extensions to handle the data not accommodated.
On the utility of gambling: extending the approach of Meginniss (1976)
Aequationes mathematicae, 2008
J. R. Meginniss modified expected utility to accommodate a concept of the utility of gambling that led to a representation composed of a utility expectation term plus an entropy of degree κ term. He imposed several apparently strong assumptions. One of these is that a number of unknown generating functions are identical. A second is that he assumed he was working with given probabilities. Here we follow his general framework but weaken considerably those assumptions. Our problem is reduced to solving some functional equations induced by gamble decomposition. From the solutions, we obtain the representation of the utility function. Further axiomatic restrictions are imposed that lead ultimately to Meginniss' earlier result. . 94A17, 91B16, 39B22.
2000
Ž. Ž. Miyamoto's 1988, 1992 generic utility theory GUT subsumes a broad class of bilinear utility models. Ž. Chechile and Cooke 1997 tested the GUT class of models and found model failure due to the systematic variation of a parameter that should be a positive constant across a range of contexts. In the current study, an improved experimental design is employed to evaluate utility theory. The current study provides further evidence against the GUT class of models for mixed gambles. Moreover, evidence is also provided to demonstrate individual behavior that is incompatible with a coherent bilinear utility theory of choice behavior in the context of mixed gambles with gains and losses.
Constraints on the representation of gambles in prospect theory
Journal of Mathematical Psychology, 1987
's ((1979) Prospect theory: An analysis of decision under risk, Econometrica, 47, 263-291) prospect theory postulates that the subjective value of a gamble is determined by different combination rules depending on the outcomes of the gamble. The theory distinguishes regular from irregular gambles, and in addition, distinguishes two different kinds of irregular gambles. The representations of value for these classes of gambles are formally analogous, but they differ in the way that gamble probabilities are integrated with the worth of gamble outcomes. The present work identifies properties of preference, restricted betweenness and equivalence under interchange, that play a central role in justifying the use of different combination rules for different classes of gambles in prospect theory. 'i" 1987 Academic Press, Inc.
Configural weighting in judgments of two-and four-outcome gambles
1998
This study tested branch independence, a key property distinguishing nonconfigural from configural theories of decision making. Sixty undergraduates judged buying and selling prices of 168 lotteries composed of 2 or 4 equally likely outcomes, (x, y, z, v). Branch independence requires that (x, y, z, v) is judged higher than (x', y', z, v) whenever (x, y, z', v') is judged higher than (x', y', z', v'). Different violations observed in different viewpoints are consistent with the theory that the utility function is independent of viewpoint and that only configural weights differ between viewpoints. Lower ranked outcomes have greater weights in the buyer's than in the seller's viewpoint. Sellers place more weight than buyers on higher ranked outcomes. In both viewpoints, violations of branch independence are contrary to the inverseS weighting function of cumulative prospect theory: Moderate outcomes receive more weight than adjacent extreme outcomes. This study tests a key distinction between a class of subjective expected utility (SEU) models (Savage, 1954) and configural weight models (Birnbaum & Beeghley, 1997; Bimbaum & Mclntosh, 1996). SEU theory and other nonconfigural models imply branch independence, a weaker form of Savage's "sure thing" principle. Branch independence requires that if two gambles have one or more common branches (the same outcome produced by the same event with the same known probability), then the preference order induced by other components of gambles will be independent of the value of the common outcome(s). Let (x, v, z) denote a gamble in which the outcomes x, y, and z are equally likely. Branch independence requires that (x, y, z) is judged better than (x', y\ z) if and only if (x, y, z') is judged better than (x\ y', z'}. Birnbaum and Mclntosh (1996) found that most judges prefer
Indexing Gamble Desirability by Extending Proportional Stochastic Dominance
RePEc: Research Papers in Economics, 2016
We characterise two new orders of desirability of gambles (risky assets) that are natural extensions of the stochastic dominance order to complete orders, based on choosing optimal proportions of gambles. These orders are represented by indices, which we term the S index and the G index, that are characterised axiomatically and by wealth and utility uniform dominance concepts. The S index can be viewed as a generalised Sharpe ratio, and the G index can be used for maximising the growth path of a portfolio.
Insurance: Mathematics and Economics, 1993
Choice is viewed as a derived, not a primitive, concept. Individual gambles are assigned subjective certainty equivalents (CE1); the choice set X has an associated reference level [RL(X)] based on the CEls of its members; the outcomes of each gamble are recoded as deviations from the RL(X); and new CEes are constructed. The gamble having the largest CE2 is chosen. The CEs are described by the rank-and sign-dependent theory of Luce (1992b). The concept of RL is studied axiomatically. The model predicts many behavioral anomalies and is tested with data sets of .
Violation of utility theory in unique and repeated gambles
Journal of Experimental Psychology: Learning, Memory, and Cognition, 1987
This article is concerned with a recent debate on the generality of utility theory. It has been argued by Lopes (198 i) that decisions regarding preferences between gambles are different for unique and repeated gambles. The present article provides empirical support for the need to distinguish between these two. It is proposed that violations of utility theory obtained under unique conditions (e.g., Kahneman & Tversky, 1979), cannot necessarily be generalized to repeated conditions. We would like to thank Baruch Fischhoff, Sarah Lichtenstein, Charles Lewis, and Charles Vlek for many valuable comments on previous drafts of this article.