Utility theory and the Bayesian paradigm

In this paper, a problem for utility theory - that it would have an agent who was compelled to play Russian Roulette with one revolver or another, to pay as much to have a six-shooter with four bullets relieved of one bullet before playing with it, as he would be willing to pay to have a six-shooter with two bullets emptied - is reviewed. A less demanding Bayesian theory is described, that would have an agent maximize expected values of possible total consequence of his actions. And utility theory is located within that theory as valid for agents who satisfy certain formal conditions, that is, for agents who are, in terms of that more general theory, indifferent to certain dimensions of risk. Raiffa- and Savage-style arguments for its more general validity are then resisted. Addenda are concerned with implications for game theory, and relations between utilities and values.     

Security level,potential level,expected utility: A three-criteria decision model under risk

Among the violations of expected utility (E.U.) theory which have been observed by experimenters, the violations of its independence axiom is, by far, the most common. It seems that, in many cases, these inconsistencies can be ascribed to the desire for security - called the security factor by L. Lopes (1986) - which makes people attach special importance to the worst outcomes of risky decisions as well as to the sole outcomes of riskless decisions (certainty effect). J.-Y. Jaffray (1988) has proposed a model which generalizes E.U. theory by taking into account this factor and is then able to account for certain violations. However, especially in experiments on choice involving prospective losses, violations of the von Neumann-Morgenstern independence axiom cannot be explained by the security factor alone and have to be partially ascribed to the potential factor (L. Lopes, 1986) which reflects heightened attention to the best outcomes of decisions, especially when the best outcome is the status quo. In this paper, we construct an axiomatic model for subjects taking into account simultaneously or alternatively the security factor and the potential factor. For this, as in Jaffray's model, it has been necessary to weaken not only the standard independence axiom but also the continuity axiom and, in the same time, to reinforce the dominance axiom. In the resulting model, choices are partially determined by the mere comparison of the (security level, potential level) (i.e. the (worst outcome, best outcome)) pairs offered, and completed by the maximization of an affine function of the expected utility, the coefficients of which depend on both the security level and potential level.In this model, a decision maker who (i) has constant marginal utility for money, (ii) is sensitive to the security factor alone in the domain of gains, (iii) is sensitive to the potential factor alone in the domain of losses, behaves as a risk averter for gains and a risk seeker for losses.     

When an Event Makes a Difference

For (S, Σ) a measurable space, let and be convex, weak^* closed sets of probability measures on Σ. We show that if ∪ satisfies the Lyapunov property , then there exists a set A ∈ Σ such that min_μ1∈ μ₁(A) > max_μ2 ∈ (A). We give applications to Maxmin Expected Utility (MEU) and to the core of a lower probability.     

Testing Descriptive Utility Theories: Violations of Stochastic Dominance and Cumulative Independence

Choices between gambles show systematic violations of stochastic dominance. For example, most people choose ($6, .05; $91, .03; $99, .92) over ($6, .02; $8, .03; $99, .95), violating dominance. Choices also violate two cumulative independence conditions: (1) If S = (z, r; x, p; y, q) R = (z, r; x, p; y, q) then S = (x, r; y, p + q) R = (x, r + p; y, q). (2) If S = (x, p; y, q; z, r) R = (x, p; y, q; z, r) then S = (x, p + q; y, r) R = (x, p; y, q + r), where 0 < z < x < x < y < y < y < z.Violations contradict any utility theory satisfying transivity, outcome monotonicity, coalescing, and comonotonic independence. Because rank-and sign-dependent utility theories, including cumulative prospect theory (CPT), satisfy these properties, they cannot explain these results.However, the configural weight model of Birnbaum and McIntosh (1996) predicted the observed violations of stochastic dominance, cumulative independence, and branch independence. This model assumes the utility of a gamble is a weighted average of outcomes\' utilities, where each configural weight is a function of the rank order of the outcome\'s value among distinct values and that outcome\'s probability. The configural weight, TAX model with the same number of parameters as CPT fit the data of most individuals better than the model of CPT.     

Rational preference: Decision theory as a theory of practical rationality

In general, the technical apparatus of decision theory is well developed. It has loads of theorems, and they can be proved from axioms. Many of the theorems are interesting, and useful both from a philosophical and a practical perspective. But decision theory does not have a well agreed upon interpretation. Its technical terms, in particular, utility and preference do not have a single clear and uncontroversial meaning.How to interpret these terms depends, of course, on what purposes in pursuit of which one wants to put decision theory to use. One might want to use it as a model of economic decision-making, in order to predict the behavior of corporations or of the stock market. In that case, it might be useful to interpret the technical term utility as meaning money profit. Decision theory would then be an empirical theory. I want to look into the question of what utility could mean, if we want decision theory to function as a theory of practical rationality. I want to know whether it makes good sense to think of practical rationality as fully or even partly accounted for by decision theory. I shall lay my cards on the table: I hope it does make good sense to think of it that way. For, I think, if Humeans are right about practical rationality, then decision theory must play a very large part in their account. And I think Humeanism has very strong attractions.     

Coherent decision analysis with inseparable probabilities and utilities

This article explores the extent to which a decision maker's probabilities can be measured separately from his/her utilities by observing his/her acceptance of small monetary gambles. Only a partial separation is achieved: the acceptable gambles are partitioned into a set of belief gambles, which reveals probabilities distorted by marginal utilities for money, and a set of preference gambles, which reveals utilities reciprocally distorted by marginal utilities for money. However, the information in these gambles still enables us to solve the decision maker's problem: his/her utility-maximizing decision is the one that avoids arbitrage (i.e., incoherence or Dutch books).     

A foundation of Bayesian statistics (How to deal with fears)

A rational statistical decision maker whose preferences satisfy Savage's axioms will minimize a Bayesian risk function: the expectation with respect to a revealed (or subjective) probability distribution of a loss (or negative utility) function over the consequences of the statistical decision problem. However, the nice expected utility form of the Bayesian risk criterion is nothing but a representation of special preferences. The subjective probability is defined together with the utility (or loss) function and it is not possible, in general, to use a given loss function - say a quadratic loss - and to elicit independently a subjective distribution.I construct the Bayesian risk criterion with a set of five axioms, each with a simple mathematical implication. This construction clearly shows that the subjective probability that is revealed by a decider's preferences is nothing but a (Radon) measure equivalent to a linear functional (the criterion). The functions on which the criterion operates are expected utilities in the von Neumann-Morgenstern sense. It then becomes clear that the subjective distribution cannot be eliciteda priori, independently of the utility function on consequences.However, if one considers a statistical decision problem by itself, losses, defined by a given loss function, become the consequences of the decisions. It can be imagined that experienced statisticians are used to dealing with different losses and are able to compare them (i.e. have preferences, or fears over a set of possible losses). Using suitable axioms over these preferences, one can represent them by a (linear) criterion: this criterion is the expectation of losses with respect to a (revealed) distribution. It must be noted that such a distribution is a measure and need not be a probability distribution.     

The incoherence of agreeing to disagree

The agreeing-to-disagree theorem of Aumann and the no-expected-gain-from-trade theorem of Milgrom and Stokey are reformulated under an operational definition of Bayesian rationality. Common knowledge of beliefs and preferences is achieved through transactions in a contingent claims market, and mutual expectations of Bayesian rationality are defined by the condition of joint coherence,i.e., the collective avoidance of arbitrage opportunities. The existence of a common prior distribution and the impossibility of agreeing to disagree follow from the joint coherence requirement, but the prior must be interpreted as a risk-neutral distribution: a product of probabilities and marginal utilities for money. The failure of heterogenous information to create disagreements or incentives to trade is shown to be an artifact of overlooking the potential role of trade in constructing the initial state of common knowledge.     

Games,goals, and bounded rationality

A generalization of the standard n-person game is presented, with flexible information requirements suitable for players constrained by certain types of bounded rationality. Strategies (complete contingency plans) are replaced by policies, i.e., endmean pairs of goals and controls (partial contingency plans), which results in naturally disconnected player choice sets. Well-known existence theorems for pure strategy Nash equilibrium and bargaining solutions are generalized to policy games by modifying connectedness (convexity) requirements.     

Risk seeking with diminishing marginal utility in a non-expected utility model

The present work takes place in the framework of a non-expected utility model under risk: the RDEU theory (Rank Dependent Expected Utility, first initiated by Quiggin under the denomination of Anticipated Utility), where the decision maker's behavior is characterized by two functionsu andf. Our first result gives a condition under which the functionu characterizes the decision maker's attitude towards wealth. Then, defining a decision maker as risk averter (respectively risk seeker) when he always prefers to any random variable its expected value (weak definition of risk aversion), the second result states that a decision maker who has an increasing marginal utility of wealth (a convex functionu) can be risk averse, if his functionf issufficiently below his functionu, hence if he is sufficientlypessimistic. Obviously, he can also be risk seeking with a diminishing marginal utility of wealth. This result is noteworthy because with a stronger definition of risk aversion/risk seeking, based on mean-preserving spreads, Chew, Karni, and Safra have shown that the only way to be risk averse (in their sense) in RDEU theory is to have, simultaneously, a concave functionu and a convex functionf.     

Ordinal utility models of decision making under uncertainty

This paper studies two models of rational behavior under uncertainty whose predictions are invariant under ordinal transformations of utility. The quantile utility model assumes that the agent maximizes some quantile of the distribution of utility. The utility mass model assumes maximization of the probability of obtaining an outcome whose utility is higher than some fixed critical value. Both models satisfy weak stochastic dominance. Lexicographic refinements satisfy strong dominance.The study of these utility models suggests a significant generalization of traditional ideas of riskiness and risk preference. We define one action to be riskier than another if the utility distribution of the latter crosses that of the former from below. The single crossing property is equivalent to a minmax spread of a random variable. With relative risk defined by the single crossing criterion, the risk preference of a quantile utility maximizer increases with the utility distribution quantile that he maximizes. The risk preference of a utility mass maximizer increases with his critical utility value.     

Utilities for distributive justice

This paper falls within the field of Distributive Justice and (as the title indicates) addresses itself specifically to the meshing problem. Briefly stated, the meshing problem is the difficulty encountered when one tries to aggregate the two parameters of beneficence and equity in a way that results in determining which of two or more alternative utility distributions is most just. A solution to this problem, in the form of a formal welfare measure, is presented in the paper. This formula incorporates the notions of equity and beneficence (which are defined earlier by the author) and weighs them against each other to compute a numerical value which represents the degree of justice a given distribution possesses. This value can in turn be used comparatively to select which utility scheme, of those being considered, is best.Three fundamental adequacy requirements, which any acceptable welfare measuring method must satisfy, are presented and subsequently demonstrated to be formally deducible as theorems of the author's system. A practical application of the method is then considered as well as a comparison of it with Nicholas Rescher's method (found in his book, Distributive Justice). The conclusion reached is that Rescher's system is unacceptable, since it computes counter-intuitive results. Objections to the author's welfare measure are considered and answered. Finally, a suggestion for expanding the system to cover cases it was not originally designed to handle (i.e. situations where two alternative utility distributions vary with regard to the number of individuals they contain) is made. The conclusion reached at the close of the paper is that an acceptable solution to the meshing problem has been established.I would like to gratefully acknowledge the assistance of Michael Tooley whose positive suggestions and critical comments were invaluable in the writting of this paper.     

Separating marginal utility and probabilistic risk aversion

This paper is motivated by the search for one cardinal utility for decisions under risk, welfare evaluations, and other contexts. This cardinal utility should have meaningprior to risk, with risk depending on cardinal utility, not the other way around. The rank-dependent utility model can reconcile such a view on utility with the position that risk attitude consists of more than marginal utility, by providing a separate risk component: a probabilistic risk attitude towards probability mixtures of lotteries, modeled through a transformation for cumulative probabilities. While this separation of risk attitude into two independent components is the characteristic feature of rank-dependent utility, it had not yet been axiomatized. Doing that is the purpose of this paper. Therefore, in the second part, the paper extends Yaari's axiomatization to nonlinear utility, and provides separate axiomatizations for increasing/decreasing marginal utility and for optimistic/pessimistic probability transformations. This is generalized to interpersonal comparability. It is also shown that two elementary and often-discussed properties — quasi-convexity (aversion) of preferences with respect to probability mixtures, and convexity (pessimism) of the probability transformation — are equivalent.     

Empirical evidence of two-attribute utility dependence on probability

We investigate utility dependence on probability using a new methodology that examines how indifference statements vary with the probability of obtaining times and costs of individual trips. Of 127 subjects, 8 supplied 3 (out of 3) sets of indifference statements consistent with probability independence. Those subjects with 2 or more sets of indifference statements violating probability independence exhibited a systematic dependence, in that knowing the direction of a subject's violation in one set of indifference statements would increase the likelihood of his or her violating other sets of indifference statements in the same direction. Data show that this systematic violation of dependence should not be attributed to artifacts of the experiment.     

A formulation of the determinism hypothesis

The author tries to formulate what a determinist believes to be true. The formulation is based on some concepts defined in a systems-theoretical manner, mainly on the concept of an experiment over the sets A ^m (a set of m-tuples of input values) and B ⁿ (a set of n-tuples of output values) in the time interval (t ₁, ..., t _k ) (symbolically E[t ₁,..., t _k , A ^m , B ⁿ ]), on the concept of a behavior of the system S ^m,n (=(A ^m , B ⁿ )) on the basis of the experiment E[t ₁, ..., t _k , A ^m , B ⁿ ] and, indeed, on the concept of deterministic behavior .... The resulting formulation of the deterministic hypothesis shows that this hypothesis expresses a belief that we always could find some hidden parameters.     

Towards a more precise decision framework

The terms negative utility of gambling and risk aversion conflate three things:

How a certain internal consistency entails the expected utility dogma

If I am coherent, in the sense that I can always replace any subset of outcomes by their certainty equivalent (occurring with the sum of their probabilities), then I must act according to the dogma of maximizing an Exp {U}, ruling out Machina [1982], Allais [1952], and Ysidro [1950] functionals.     

Approaches to cardinal utility

In a previous article (see [3]) a system of axioms is proposed stating conditions which are necessary and sufficient to determine a cardinal utility function on any set, finite or infinite, of outcomes X. The present paper discusses and interprets the meaning of those axioms, and compares this new approach to cardinal utility with the utility differences approach proposed by Alt and Frisch, among others, and with the expected utility approach of von-Neuman and Morgenstern. The notion of repetition of the same choice situation is presented and its interpretation discussed. It is then argued that this notion leads naturally to the system of axioms presented in On Cardinal Utility. It is also argued that this notion must be used if we want to have a more clear understanding of the meaning of the axioms proposed by Alt and Frisch. Finally, it is remarked that since uncertainty is not present in the new approach, it is free of the paradoxes that have plagued the expected utility hypothesis.     

Rational choice and economic behavior

This paper discusses several concepts that can be used to provide a foundation for a unified, theory of rational, economic behavior. First, decision-making is defined to be a process that takes place with reference to both subjective and objective time, that distinguishes between plans and actions, between information and states and that explicitly incorporates the collection and processing of information. This conception of decision making is then related to several important aspects of behavioral economics, the dependence of values on experience, the use of behavioral rules, the occurrence of multiple goals and environmental feedback.Our conclusions are (1) the non-transitivity of observed or revealed preferences is a characteristic of learning and hence is to be expected of rational decision-makers; (2) the learning of values through experience suggests the sensibleness of short time horizons and the making of choices according to flexible utility; (3) certain rules of thumb used to allow for risk are closely related to principles of Safety-First and can also be based directly on the hypothesis that the feeling of risk (the probability of disaster) is identified with extreme departures from recently executed decisions. (4) The maximization of a hierarchy of goals, or of a lexicographical utility function, is closely related to the search for feasibility and the practice of satisficing. (5) When the dim perception of environmental feedback and the effect of learning on values are acknowledged the intertemporal optimality of planned decision trajectories is seen to be a characteristic of subjective not objective time. This explains why decision making is so often best characterized by rolling plans. In short, we find that economic man - like any other - is an existential being whose plans are based on hopes and fears and whose every act involves a leap of faith.This paper is based on a talk presented at the Conference, New Beginnings in Economics, Akron, Ohio, March 15, 1969. Work on this paper was supported by a grant from the National Science Foundation.     

Metatickles and the dynamics of deliberation

The traditional or orthodox decision rule of maximizing conditional expected utility has recently come under attack by critics who advance alternative causal decision theories. The traditional theory has, however, been defended. And these defenses have in turn been criticized. Here, I examine two objections to such defenses and advance a theory about the dynamics of deliberation (a diachronic theory about the process of deliberation) within the framework of which both objections to the defenses of the traditional theory fail.