Countable Additivity and the Foundations of Bayesian Statistics

At a very fundamental level an individual (or a computer) can process only a finite amount of information in a finite time. We can therefore model the possibilities facing such an observer by a tree with only finitely many arcs leaving each node. There is a natural field of events associated with this tree, and we show that any finitely additive probability measure on this field will also be countably additive. Hence when considering the foundations of Bayesian statistics we may as well assume countable additivity over a σ-field of events.     

Statistical decisions under ambiguity

This article provides unified axiomatic foundations for the most common optimality criteria in statistical decision theory. It considers a decision maker who faces a number of possible models of the world (possibly corresponding to true parameter values). Every model generates objective probabilities, and von Neumann–Morgenstern expected utility applies where these obtain, but no probabilities of models are given. This is the classic problem captured by Wald’s (Statistical decision functions, 1950) device of risk functions. In an Anscombe–Aumann environment, I characterize Bayesianism (as a backdrop), the statistical minimax principle, the Hurwicz criterion, minimax regret, and the “Pareto” preference ordering that rationalizes admissibility. Two interesting findings are that c-independence is not crucial in characterizing the minimax principle and that the axiom which picks minimax regret over maximin utility is von Neumann–Morgenstern independence.     

The Likelihood Method for Decision under Uncertainty

This paper introduces the likelihood method for decision under uncertainty. The method allows the quantitative determination of subjective beliefs or decision weights without invoking additional separability conditions, and generalizes the Savage–de Finetti betting method. It is applied to a number of popular models for decision under uncertainty. In each case, preference foundations result from the requirement that no inconsistencies are to be revealed by the version of the likelihood method appropriate for the model considered. A unified treatment of subjective decision weights results for most of the decision models popular today. Savage’s derivation of subjective expected utility can now be generalized and simplified. In addition to the intuitive and empirical contributions of the likelihood method, we provide a number of technical contributions: We generalize Savage’s nonatomiticy condition (“P6”) and his assumption of (sigma) algebras of events, while fully maintaining his flexibility regarding the outcome set. Derivations of Choquet expected utility and probabilistic sophistication are generalized and simplified similarly. The likelihood method also reveals a common intuition underlying many other conditions for uncertainty, such as definitions of ambiguity aversion and pessimism.     

What are the minimal requirements of rational choice? Arguments from the sequential-decision setting

There are at least two plausible generalisations of subjective expected utility (SEU) theory: cumulative prospect theory (which relaxes the independence axiom) and Levi’s decision theory (which relaxes at least ordering). These theories call for a re-assessment of the minimal requirements of rational choice. Here, I consider how an analysis of sequential decision making contributes to this assessment. I criticise Hammond’s (Economica 44(176):337–350, 1977; Econ Philos 4:292–297, 1988a; Risk, decision and rationality, 1988b; Theory Decis 25:25–78, 1988c) ‘consequentialist’ argument for the SEU preference axioms, but go on to formulate a related diachronic-Dutch-book-style’ argument that better achieves Hammond’s aims. Some deny the importance of Dutch-book sure losses, however, in which case, Seidenfeld’s (Econ Philos 4:267–290, 1988a) argument that distinguishes between theories that relax independence and those that relax ordering is relevant. I unravel Seidenfeld’s argument in light of the various criticisms of it and show that the crux of the argument is somewhat different and much more persuasive than what others have taken it to be; the critical issue is the modelling of future choices between ‘indifferent’ decision-tree branches in the sequential setting. Finally, I consider how Seidenfeld’s conclusions might nonetheless be resisted.     

Eliciting beliefs

We develop an algorithm that can be used to approximate a decisionmaker’s beliefs for a class of preference structures that includes, among others, α-maximin expected utility preferences, Choquet expected utility preferences, and, more generally, constant additive preferences. For both exact and statistical approximation, we demonstrate convergence in an appropriate sense to the true belief structure.     

Competence effects for choices involving gains and losses

We investigate how choices for uncertain gain and loss prospects are affected by the decision maker’s perceived level of knowledge about the underlying domain of uncertainty. Specifically, we test whether Heath and Tversky’s (J Risk Uncertain 4:5–28, 1991) competence hypothesis extends from gains to losses. We predict that the commonly-observed preference for high knowledge over low knowledge prospects for gains reverses for losses. We employ an empirical setup in which participants make hypothetical choices between gain or loss prospects in which the outcome depends on whether a high or low knowledge event occurs. We infer decision weighting functions for high and low knowledge events from choices using a representative agent preference model. For gains, we replicate the results of Kilka and Weber (Manage Sci 47:1712–1726, 2001), finding that decision makers are more attracted to choices that they feel more knowledgeable about. However, for losses, we find limited support for our extension of the competence effect.     

A tractable method to measure utility and loss aversion under prospect theory

This paper provides an efficient method to measure utility under prospect theory. Our method minimizes both the number of elicitations required to measure utility and the cognitive burden for subjects, being based on the elicitation of certainty equivalents for two-outcome prospects. We applied our method in an experiment and were able to replicate the main findings on prospect theory, suggesting that our method measures what it is intended to. Our data confirmed empirically that risk seeking and concave utility can coincide under prospect theory. Utility did not depend on the probability used in the elicitation, which offers support for the validity of prospect theory.

Rethinking Risk Attitude: Aspiration as Pure Risk

There exists no completely satisfactory theory of risk attitude in current normative decision theories. Existing notions confound attitudes to pure risk with unrelated psychological factors such as strength of preference for certain outcomes, and probability weighting. In addition traditional measures of risk attitude frequently cannot be applied to non-numerical consequences, and are not psychologically intuitive. I develop Pure Risk theory which resolves these problems – it is consistent with existing normative theories, and both internalises and generalises the intuitive notion of risk being related to the probability of not achieving one’s aspirations. Existing models which ignore pure risk attitudes may be misspecified, and effects hitherto modelled as loss aversion or utility curvature may be due instead to Pure Risk attitudes.     

Perspectives on a Pair of Envelopes

The two envelopes problem has generated a significant number of publications (I have benefitted from reading many of them, only some of which I cite; see the epilogue for a historical note). Part of my purpose here is to provide a review of previous results (with somewhat simpler demonstrations). In addition, I hope to clear up what I see as some misconceptions concerning the problem. Within a countably additive probability framework, the problem illustrates a breakdown of dominance with respect to infinite partitions in circumstances of infinite expected utility. Within a probability framework that is only finitely additive, there are failures of dominance with respect to infinite partitions in circumstances of bounded utility with finitely many consequences (see the epilogue).     

Multiple motives of pro-social behavior: evidence from the solidarity game

The article analyses experimental “solidarity games” with two benefactors and one beneficiary. Depending on their motive for giving—e.g., warm glow, altruism, or guilt—the benefactors’ response functions are either constant, decreasing, or increasing. If motives interact, or if envy is a concern, then more complex (unimodal) shapes may emerge. Controlling for random utility perturbations, we determine which and how many motives affect individual decision making. The main findings are that the motives of about 75% of the subjects can be identified fairly sharply, that all of the motives discussed in the literature co-exist in the population, and that for any given individual no more than two motives (out of six motives considered overall) are identified. We conclude that a unifying motive for solidarity cannot be derived even when we allow for individually heterogeneous parameterization: different subjects give for different reasons and all existing social preference theories are partially correct.     

Elementary proof that mean–variance implies quadratic utility

An extensive literature overlapping economics, statistical decision theory and finance, contrasts expected utility [EU] with the more recent framework of mean–variance (MV). A basic proposition is that MV follows from EU under the assumption of quadratic utility. A less recognized proposition, first raised by Markowitz, is that MV is fully justified under EU, if and only if utility is quadratic. The existing proof of this proposition relies on an assumption from EU, described here as “Buridan’s axiom” after the French philosopher’s fable of the ass that starved out of indifference between two bales of hay. To satisfy this axiom, MV must represent not only “pure” strategies, but also their probability mixtures, as points in the (σ, μ) plane. Markowitz and others have argued that probability mixtures are represented sufficiently by (σ, μ) only under quadratic utility, and hence that MV, interpreted as a mathematical re-expression of EU, implies quadratic utility. We prove a stronger form of this theorem, not involving or contradicting Buridan’s axiom, nor any more fundamental axiom of utility theory.     

A Method for Eliciting Utilities and its Application to Collective Choice

Designing a mechanism that provides a direct incentive for an individual to report her utility function over several alternatives is a difficult task. A framework for such mechanism design is the following: an individual (a decision maker) is faced with an optimization problem (e.g., maximization of expected utility), and a mechanism designer observes the decision maker’s action. The mechanism does reveal the individual’s utility truthfully if the mechanism designer, having observed the decision maker’s action, infers the decision maker’s utilities over several alternatives. This paper studies an example of such a mechanism and discusses its application to the problem of optimal social choice. Under certain simplifying assumptions about individuals’ utility functions and about how voters choose their voting strategies, this mechanism selects the alternative that maximizes Harsanyi’s social utility function and is Pareto-efficient.     

Ambiguity,pessimism, and rational religious choice

Using a subclass of the α-maximin expected-utility preference model, in which the decision maker’s degree of ambiguity and degree of pessimism are each parameterized, we present a theory of religious choice in the Pascalian decision theory tradition, one that can resolve dilemmas, address the “many Gods objection,” and address the ambiguity inherent in religious choice. Parameterizing both the degree of ambiguity and the degree of pessimism allows one to examine how the two interact to impact choice, which is useful regardless of the application. Applying this model to religious choice is a move beyond subjective expected-utility theory, allowing us to show that a change in either the degree of ambiguity or the degree of pessimism can lead a decision maker to “convert” from one religion to another.     

Pareto efficiency in multiple referendum

We consider situations of multiple referendum: finitely many yes-or-no issues have to be socially assessed from a set of approval ballots, where voters approve as many issues as they want. Each approval ballot is extended to a complete preorder over the set of outcomes by means of a preference extension. We characterize, under a mild richness condition, the largest domain of top-consistent and separable preference extensions for which issue-wise majority voting is Pareto efficient, i.e., always yields out a Pareto-optimal outcome. Top-consistency means that voters’ ballots are their unique most preferred outcome. It appears that the size of this domain becomes negligible relative to the size of the full domain as the number of issues increases.     

Discount-neutral utility models for denumerable time streams

This paper formulates and axiomatizes utility models for denumerable time streams that make no commitment in regard to discounting future outcomes. The models address decision under certainty and decision under risk. Independence assumptions in both contexts lead to additive or multiplicative utilities over time periods that allow unambiguous comparisons of the relative importance of different periods. The models accommodate all patterns of future valuation. This discount-neutral feature is attained by restricting preference comparisons to outcome streams or probability distributions on outcome streams that differ in at most a finite number of periods.     

Risk-adjusted martingales and the design of “indifference” gambles

In the probability literature, a martingale is often referred to as a “fair game.” A martingale investment is a stochastic sequence of wealth levels, whose expected value at any future stage is equal to the investor’s current wealth. In decision theory, a risk neutral investor would therefore be indifferent between holding on to a martingale investment, and receiving its payoff at any future stage, or giving it up and maintaining his current wealth. But a risk-averse decision maker would not be indifferent between a martingale investment and his current wealth level, since he values uncertain deals less than their mean. A risk seeking decision maker, on the other hand, would readily accept a martingale investment in exchange for his current wealth, and would repeat this investment any number of times. These ideas lead us to introduce the notion of a “risk-adjusted martingale”; a stochastic sequence of wealth levels that a rational decision maker with any attitude toward risk would value constantly with time, and would be indifferent between receiving its pay-off at any future stage, or giving it up and maintaining his current wealth level. We show how to construct such risk-adjusted investments for any decision maker with a continuous monotonic utility function. The fundamental result we derive is that a pay-off structure of an investment (i) is a risk-adjusted martingale and (ii) can be represented by a lattice if and only if the pay-off functions are invariant transformations of the given utility function.     

Options traders exhibit subadditive decision weights

Professional options traders priced risky prospects as well as uncertain prospects whose outcomes depended on future values of various stocks. The prices of the risky prospects coincided with their expected value, but the prices of the uncertain prospects violated expected utility theory. An event had greater impact on prices when it turned an impossibility into a possibility or a possibility into a certainty than when it merely made a possibility more or less likely, as predicted by prospect theory. This phenomenon is attributed to the subadditivity of judged probabilities.     

On the Application of Multiattribute Utility Theory to Models of Choice

Ellsberg (The Quarterly Journal of Economics 75, 643–669 (1961); Risk, Ambiguity and Decision, Garland Publishing (2001)) argued that uncertainty is not reducible to risk. At the center of Ellsberg’s argument lies a thought experiment that has come to be known as the three-color example. It has been observed that a significant number of sophisticated decision makers violate the requirements of subjective expected utility theory when they are confronted with Ellsberg’s three-color example. More generally, such decision makers are in conflict with either the ordering assumption or the independence assumption of subjective expected utility theory. While a clear majority of the theoretical responses to these violations have advocated maintaining ordering while relaxing independence, a persistent minority has advocated abandoning the ordering assumption. The purpose of this paper is to consider a similar dilemma that exists within the context of multiattribute models, where it arises by considering indeterminacy in the weighting of attributes rather than indeterminacy in the determination of probabilities as in Ellsberg’s example.      

Valuing Others’ Information under Imperfect Expectations

Sometimes we believe that others receive harmful information. However, Marschak’s value of information framework always assigns non-negative value under expected utility: it starts from the decision maker’s beliefs – and one can never anticipate information’s harmfulness for oneself. The impact of decision makers’ capabilities to process information and of their expectations remains hidden behind the individual and subjective perspective Marschak’s framework assumes. By introducing a second decision maker as a point of reference, this paper introduces a way for evaluating others’ information from a cross-individual, imperfect expectations perspective for agents maximising expected utility. We define the cross-value of information that can become negative – then the information is “harmful” from a cross-individual perspective – and we define (mutual) cost of limited information processing capabilities and imperfect expectations as an opportunity cost from this same point of reference. The simple relationship between these two expected utility-based concepts and Marschak’s framework is shown, and we discuss evaluating short-term reactions of stock market prices to new information as an important domain of valuing others’ information.      

Taste uncertainty and status quo effects in consumer choice

We use reference-dependent expected utility theory to develop a model of status quo effects in consumer choice. We hypothesise that, when making their decisions, individuals are uncertain about the utility that will be yielded by their consumption experiences in different ‘taste states’ of the world. If individuals have asymmetric attitudes to gains and losses of utility, the model entails acyclic reference-dependent preferences over consumption bundles. The model explains why status quo effects may vary substantially from one decision context to another and why some such effects may decay as individuals gain market experience.