SSB Utility Research Articles

Non-binary choice in a non-deterministic model

We here consider a new move in the direction of Nehring’s approach to non-binary maximization [Econ. Lett. 50 (1996) 337; Econ. Lett. 63 (1999) 29] so that mixtures spaces [Econometrica 21 (1953) 291] can be incorporated naturally. The set of conditions that we shall set forth include the SSB model naturally, and permit to ensure that in every set spanned by a finite number of alternatives there is forcefully one that must be chosen, thus generalizing a celebrated result of SSB utility theory [cf. Econometrica 56 (1988) 1221].

Totally convex preferences for gambles

Numerical representations that admit total convexity of preferences include Chew’s weighted linear utility, Dekel’s implicit representation, and Fishburn’s SSB utility. This paper shows that existence of a universally indifferent gamble and acyclicity of strict preferences have significant implications for general axiomatizations of those representations.

Linear aggregation of SSB utility functionals

A necessary and sufficient condition for linear aggregation of SSB utility functionals is presented. Harsanyi's social aggregation theorem for von Neumann–Morgenstern utility functions is shown to be a corollary to this result. Two generalizations of Fishburn and Gehrlein's conditional linear aggregation theorem for SSB utility functionals are also established.

The Selection of Preferences Through Imitation

The paper presents a model in which a population of agents repeatedly play games against nature; the rules of behaviour followed are revised over time through a process of imitation. For binary decisions, imitation selects rules consistent with a preference relation of the kind proposed by SSB utility theory and regret theory. In general, this preference relation need not satisfy either independence or transitivity; we state conditions on imitation necessary for it to do so. For decisions over three or more options, the long-run tendency is for options that are maximally preferred in terms of SSB preferences to be chosen. If no maximally preferred option exists, the process of imitation may not converge.

Bilinear utility and a threshold structure for nontransitive preferences

This paper develops a generalization of Fishburn's SSB utility theory. Let P be a set of probability measures. A new representational form yields a bilinear functional u on P× P for which u( p, q)> 0 if and only if p is preferred to q, where u( p, p)≤0, and u( q, p)<0 whenever u( p, q)> 0. Since u is not necessarily skew-symmetric, i.e., u( p, q)≠− u( q,), it provides imprecise preference discriminability. It is also discussed that u is related to SSB utility with a threshold structure.

Risk attitudes for nonlinear measurable utility

This paper concerns SSB utility theory in the monetary context. It introduces a measure of risk aversion and establishes necessary and sufficient conditions for comparative risk aversion. Nonseparable decompositions of an SSB utility function that have the form Φ(x, y)=w(x)w(y)f(x−y) are examined. Risk properties such as constant risk aversion and a delta property are shown to give specific functional forms for an SSB utility function.

Transitivity in the small and in the large for states-additive SSB utilities

Starting with induced state-conditional preferences for an additive and potentially state-dependent SSB utility model, we investigate the effects of transitive preferences on various super-sets of a set of available lottery acts. Sufficient richness of the act set forces transitive preferences to be fully linear; failing such richness, transitive but nonlinear preferences may obtain.

Multicriteria efficiency with arbitrary finite sets and cyclic preferences

It is shown that when X is an arbitrary finite subset of an n-factor product set and preference relations on each factor or criterion are assumed only to be asymmetric, efficient (undominated) points always exist in the set P of probability distributions on X when the preference relations are extended to probability distributions on the factors according to SSB utility theory

Transitivity is equivalent to independence for states-additive SSB utilities

The states-additive SSB utility model is a lottery-acts generalization of Savage's subjective-probability expected utility model for decision under uncertainty that presumes additive event probabilities but does not assume that preferences are transitive or satisfy traditional independence axioms. Given the states-additive SSB model, we prove that the additional axiom of transitive indifference is equivalent to the Herstein-Milnor independence axiom, provided that some event has probability strictly between 0 and 1.

Aggregation theory for SSB utility functionals

John Hansanyi's aggregation theorem for the weighted additive form u( x) = c 1 u 1( x) + … + c n u n ( x) + k of linear functionals u, u 1,…, u n on a mixture set X serves as the point of departure for an investigation of similar additive forms for skew-symmetric bilinear functionals φ, φ 1,…, φ n on X × X. The paper considers φ( x, y) = w 1( y) φ 1( x, y) + … + w n ( y) φ n ( x, y) and its constant-weights specialization. Conditions relating the φ i to φ that are necessary for the weighted form are discussed. A detailed examination of the n = 2 case reveals situations in which the weighting functionals w 1 and w 2 must be effectively constant.

Some implications of a more general form of regret theory

Regret theory entails the possibility of non-transitive pairwise choices. It therefore raises questions about how individuals choose from sets of more than two actions, especially when there exists a subset of pairs of actions over which preferences cycle. A generalization of regret theory is suggested and is compared and contrasted with Fishburn's generalization of SSB utility theory. It is also shown that under this generalization an individual with non-transitive pairwise preferences with not be caught in a never-ending cycle and is not vulnerable to being “money-pumped” into bankruptcy.

State-dependent SSB utility

Axioms for preference on lottery acts are shown to imply a utility model that is additive over states and represents within-state preferences by skew-symmetric bilinear functionals specific to each state. Additional axioms lead to unique subjective probabilities for the states.

Noncooperative games and nontransitive preferences

A proof exactly analogous to Nash's proof for the existence of equilibria in finite noncooperative games with von Neumann-Morgenstern utilities shows that such games have Nash equilibria when preferences satisfy the weaker conditions of ‘SSB utility theory’. An example illustrates the dual roles of mixed strategies in the SSB game context, namely to disguise a player's actual strategy choice and to resolve the potential intrapersonal dilemma of cyclic preferences among pure strategies.

SSB utility theory and decision-making under uncertainty

In the few decades since the expected utility theories of von Neumann and Morgenstern, Ramsey, and Savage first gained wide acceptance, an impressive body of evidence has appeared which indicates that certain of their principles of coherent and consistent preferences are systematically violated by many people. This has encouraged the development of more general theories of decision-making under risk and under uncertainty that weaken earlier axioms to accommodate observed violations of those axioms. One of the generalizations of the von Neumann-Morgenstern theory weakens their transitivity and independence axioms to obtain a notion of ‘nontransitive measurable utility’ in which preferences are represented by the positive part of a skew-symmetric bilinear functional on pairs of lotteries or risky decisions. The present paper applies the axioms of this ‘SSB utility theory’ to preferences between lottery acts in a Savage-type states-of-the-world formulation of decision-making under uncertainty. It also examines the effects of within-state and between-state consistency principles on the basic representation of preferences provided by this application. A dominant concern of the paper is the need for more flexible foundational theories of decision-making under uncertainty. It is hoped that the present generalizations of the Ramsey-Savage theory will contribute positively to this concern.

Dominance in SSB utility theory

SSB utility theory represents preferences between probability measures by the positive part of a skew-symmetric bilinear functional φ defined on pairs of measures. Three facets of the theory are examined. First, dominance axioms are used to extend φ to an integral form. Second, the maximizing behavior of φ on subspaces of measures is investigated. Third, aspects of stochastic dominance are explored in the SSB setting.

SSB Utility theory: an economic perspective

Abstract SSB utility theory for decision-making under risk is a generalization of the expected utility theory of von Neumann and Morgenstern that preserves much of their structure for the quantitative representation of preferences, including continuity, convexity, and monotonicity. but drops their transitivity and independence axioms. It represents preferences by a skew-symmetric bilinear functional ϕ that is unique up to multiplication by a positive constant and hasϕ( p,q)>0 if and only if risky prospect p is preferred to risky prospect q. This generalization has far-reaching consequences. It is compatible with the phenomena of preference cycles, preference reversals, and systematic violations of independence: provides a viable theory of choice based on maximally preferred alternatives; supports a rationale for social choice lotteries that is untroubled by cyclical majorities and is consistent with majority choice and Pareto optimality; offers a much more general theory of risk attitudes; retains intact the central propositions of stochastic dominance; and suggests a new approach to decision making in the face of uncertainty.