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.     

Dynamic Choice Mechanisms

A dynamic choice mechanism (e.g. sophisticated choice) is a rule transforming a sequence of orderings into actual choices. The dynamic choice literature analysing such rules is restricted to the case of strict preferences. The present paper allows also for weak preferences. This generalisation is necessary since in dynamic utility models decision makers are typically endowed with orderings which can be represented by continuous utility functions. Such a representation, however, requires the orderings to be a weak ordering. In the presence of weak orderings the classical choice mechanisms have to be re-examined. This paper develops and analyses new choice mechanisms and relates them to the notion of a subgame-perfect equilibrium.     

Metapreferences and the reasons for stability in social choice: Thoughts on broadening and clarifying the debate

The standard model of collective choice looks at aggregation procedures which take individual preferences as existing for a specified set of alternatives. In this paper we propose that actors also have preferences for rules of choice or characteristics of choice processes (e.g., the perceived fairness of procedures or the popularity of outcomes) rather than simply for alternatives (outcomes) themselves. We argue that the positing of the existence of meta-preferences can illuminate a number of areas of choice theory. Here we focus on one such area: the problem of “too much” stability in majority rule decision making - a stability which belies the standard theoretical results on the generic instability of majority rule processes. We also show that discussion of the prevalence of stability in collective decision making needs to be clarified because there are at least six distinct types of stability which are sometimes confounded in the literature.     

Choice under risk and the security factor: An axiomatic model

The particular attention paid by decision makers to the security level ensured by each decision under risk, which is responsible for the certainty effect, can be taken into account by weakening the independence and continuity axioms of expected utility theory. In the resulting model, preferences depend on: (i) the security level, (ii) the expected utility, offered by each decision. Choices are partially determined by security level comparison and completed by the maximization of a function, which express the existing tradeoffs between expected utility and security level, and is, at a given security level, an affine function of the expected utility. In the model, risk neutrality at a given security level implies risk aversion.     

Invariant multiattribute utility functions

We present a method to characterize the preferences of a decision maker in decisions with multiple attributes. The approach modifies the outcomes of a multivariate lottery with a multivariate transformation and observes the change in the decision maker’s certain equivalent. If the certain equivalent follows this multivariate transformation, we refer to this situation as multiattribute transformation invariance, and we derive the functional form of the utility function. We then show that any additive or multiplicative utility function that is formed of continuous and strictly monotonic utility functions of the individual attributes must satisfy transformation invariance with a multivariate transformation. This result provides a new interpretation for multiattribute utility functions with mutual utility independence as well as a necessary and sufficient condition that must be satisfied when assuming these widely used functional forms. We work through several examples to illustrate the approach.     

Stochastic dominance in multicriterion analysis under risk

Traditionally, in the literature on the modelling of decision aids one notes the propensity to treat expected utility models and outranking relation models as rivals. It may be possible, however, to benefit from the use of both approaches in a risky decision context. Stochastic dominance conditions can be used to establish, for each criterion, the preferences of a decision maker and to characterise them by a concave or convex utility function.Two levels of complexity in preference elicitation, designated as clear and unclear, are distinguished. Only in the case of unclear preferences is it potentially interesting to attempt to estimate the value function of the decision maker, thus obtaining his (her) preferences with a reduced number of questions. The number of questions that must be asked of the decision maker depends upon the level of the concordance threshold that he(she) requires in the construction of the outranking relations using the ELECTRE method.     

A new test of convexity–concavity of discount function

Discounted utility theory and its generalizations (e.g., quasihyperbolic discounting, generalized hyperbolic discounting) use discount functions for weighting utilities of outcomes received in different time periods. We propose a new simple test of convexity–concavity of discount function. This test can be used with any utility function (which can be linear or not) and any preferences over risky lotteries (expected utility theory or not). The data from a controlled laboratory experiment show that about one third of experimental subjects reveal a concave discount function and another one third of subjects reveal a convex discount function (for delays up to two month).

     

Dynamic Choice and NonExpected Utility

This paper explores how some widely studied classes of nonexpected utility models could be used in dynamic choice situations. A new "sequential consistency" condition is introduced for single-stage and multi-stage decision problems. Sequential consistency requires that if a decision maker has committed to a family of models (e.g., the multiple priors family, the rank-dependent family, or the betweenness family) then he use the same family throughout. Conditions are presented under which dynamic consistency, consequentialism, and sequential consistency can be simultaneously preserved for a nonexpected utility maximizer. An important class of applications concerns cases where the exact sequence of decisions and events, and thus the dynamic structure of the decision problem, is relevant to the decision maker. It is shown that for the multiple priors model, dynamic consistency, consequentialism, and sequential consistency can all be preserved. The result removes the argument that nonexpected utility models cannot be consistently used in dynamic choice situations. Rank-dependent and betweenness models can only be used in a restrictive manner, where deviation from expected utility is allowed in at most one stage.     

What can multiple price lists really tell us about risk preferences?

Multiple price lists have emerged as a simple and popular method for eliciting risk preferences. Despite their popularity, a key downside of multiple price lists has not been widely recognized — namely that the approach is unlikely to generate sufficient information to accurately identify different dimensions of risk preferences. The most popular theories of decision making under risk posit that preferences for risk are driven by a combination of two factors: the curvature of the utility function and the extent to which probabilities are weighted non-linearly. In this paper, we show that the widely used multiple price list introduced by Holt and Laury (The American Economic Review 92(5) 1644–1655 2002) is likely more accurate at eliciting the shape of the probability weighting function, and we construct a different multiple price list that is likely more accurate at eliciting the shape of the utility function. We show that by combining information from different multiple price lists, greater predictive performance can be achieved.     

Majority rule and general decision rules

For ann-member body making dichotomous decisions, decision rules can be formulated as simple games. Of all possible decision rules, majority rule maximizes the average reponsiveness of the body to individual preferences. It is the unique decision rule which does this whenn is odd, but not whenn is even. Other conditions limit the choice of decision rules whenn is even.     

Preferences over consumption and status

Experimental evidence suggests that individual consumption has not only personal value but also enters the social part of the utility. Existing models of social preferences make ad hoc parametric assumptions about the nature of this duality. This creates a problem of experimental identification of preferences since without such assumptions it is impossible to distinguish whether consumption or social concerns are driving the behavior. Given observed choice, the Axiomatic model of preferences in this article makes it possible to unambiguously determine personal and social utility without any assumptions about their relationship. The unique separation can be achieved only if the individual choices in different subgroups of other people are available. Preferences over consumption and status are used as an example to demonstrate how the utility is constructed. The model shows what kind of information about choice is needed to empirically determine the nature of social preferences without making restrictive assumptions. This can help to estimate whether personal consumption or social value is more important in economic decisions.     

Evaluating Time Streams of Income: Discounting What?

For decisions whose consequences accrue over time, there are several possible techniques to compute total utility. One is to discount utilities of future consequences at some appropriate rate. The second is to discount per-period certainty equivalents. And the third is to compute net present values (NPVs) of various possible streams and to then apply the utility function to these net present values. We find that the best approach is to first compute NPVs of various possible income streams and then take the utility of such NPVs. We show the drawbacks of other alternative models of evaluating income streams. The article discusses the advantages of the power and logarithmic forms in the modeling of time preference. These are the only forms for which utility of income and utility of consumption are strategically equivalent. Further, these forms permit the flexibility in the choice of a time period (e.g., monthly or quarterly) without modifying the utility function, thus simplifying analysis.     

Indecisiveness aversion and preference for commitment

We present an axiomatic model of preferences over menus that is motivated by three assumptions. First, the decision maker is uncertain ex ante (i.e., at the time of choosing a menu) about her ex post (i.e., at the time of choosing an option within her chosen menu) preferences over options, and she anticipates that this subjective uncertainty will not resolve before the ex post stage. Second, she is averse to ex post indecisiveness (i.e., to having to choose between options that she cannot rank with certainty). Third, when evaluating a menu she discards options that are dominated (i.e., inferior to another option whatever her ex post preferences may be) and restricts attention to the undominated ones. Under these assumptions, the decision maker has a preference for commitment in the sense of preferring menus with fewer undominated alternatives. We derive a representation in which the decision maker’s uncertainty about her ex post preferences is captured by means of a subjective state space, which in turn determines which options are undominated in a given menu, and in which the decision maker fears, whenever indecisive, to choose an option that will turn out to be the worst (undominated) one according to the realization of her ex post preferences.     

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.     

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.     

Reference Wealth Effects in Sequential Choice

It is argued that in order to accommodate experimentally-observed choice patterns, it is not enough to model the utility function as being dependent on changes from a reference wealth point. Instead, individuals should be modeled as treating decisions as part of an identifiable sequence of decisions, and utility should be a function of reference wealth, income so far from the sequence, and payoffs from the current decision. The three-argument utility function allows for risk aversion over gains and risk seeking over losses for the first choice in the sequence, and for the house money and break-even effects in later decisions.     

Expected utility with ambiguous probabilities and ‘irrational’ parameters

Savage's expected utility theory has been criticized for various reasons. Two of these are: (1) Subjective probabilities may be more or less ambiguous; this may affect the preference order. Savage's theory rules out any such effect. (2) Savage's theory rules out any influence of certain parameters of a probability distribution on the preference order. Such parameters are fractiles or pairs of fractiles, e.g., which have gained widespread attention as risk measures.This paper generalizes expected utility theory to include effects of ambiguity and those parameters on the preference order. Basic to this generalization is the classic paper by Herstein/Milnor.     

A note on decisions under uncertainty: the impact of the choice of the welfare measure

The paper addresses the question, how policy decisions under uncertainty depend on the underlying welfare concept. We study three different welfare measures: The first is directly based on the ex ante (expected) utility of a representative consumer whereas the second relies on an ex ante and the third on an ex post valuation of policy changes compared to the status quo. We show that decisions based on these measures coincide if and only if risk-neutral expected utility maximization is applied. Differences between the decisions are analyzed for both, risk-averse expected utility maximization and the MaxiMin criterion. For risk-averse decision makers, differences between the first and the second concept arise if the absolute risk-aversion of the decision maker is not constant in income. For risk-aversion and the MaxiMin criterion, the effort levels to provide a public good based on an optimization of ex post utility changes exceed those based on the first or second concept. Implications for environmental policy decisions based on the concepts of abatement costs and benefits from abatement are discussed.     

A theorem for Bayesian group decisions

This paper presents a natural extension of Bayesian decision theory from the domain of individual decisions to the domain of group decisions. We assume that each group member accepts the assumptions of subjective expected utility theory with respect to the alternatives from which they must choose, but we do not assume, a priori, that the group as a whole accepts those assumptions. Instead, we impose a multiattribute utility independence condition on the preferences of the group with respect to the expected utilities of its actions as appraised by its members. The result is that the expected utility of an alternative for the group is a weighted average of the expected utilities of that alternative for its members. The weights must be determined collectively by the group. Pareto optimality is not assumed, though the result is consistent with Pareto optimality.