Violations of betweenness and choice shifts in groups

In decision theory, the betweenness axiom postulates that a decision maker who chooses an alternative A over another alternative B must also choose any probability mixture of A and B over B itself and can never choose a probability mixture of A and B over A itself. The betweenness axiom is a weaker version of the independence axiom of expected utility theory. Numerous empirical studies documented systematic violations of the betweenness axiom in revealed individual choice under uncertainty. This paper shows that these systematic violations can be linked to another behavioral regularity—choice shifts in a group decision making. Choice shifts are observed if an individual faces the same decision problem but makes a different choice when deciding alone and in a group.     

The two faces of independence: betweenness and homotheticity

This paper shows that expected utility belongs to the intersection of models satisfying betweenness and a homotheticity condition for risky choice. Betweenness models can accommodate variable risk attitudes, originally highlighted by the Allais paradox, by restricting indifference curves to be linear while allowing non-parallelism. Homotheticity, in contrast, restricts indifference curves to be parallel while permitting non-linearities, such as those highlighted by inverse-S probability weighting. Data from an experiment indicate that approximately 2/3s of participants satisfied homotheticity. Of this group, about half satisfied the expected utility hypothesis. The other half violated expected utility and therefore betweenness. These results indicate that homothetic preferences, and in particular those violating betweenness, can be empirically relevant for decision making under risk.

     

The preservation of multivariate comparative statics in nonexpected utility theory

This article investigates the preservation of multivariate expected utility comparative statics for “smooth” nonexpected utility representations. Specifically, we answer the following question: if an expected utility comparative statics property depends only on preferences over sure prospects, then when will a nonexpected utility maximizer with identical sure preferences also satisfy that property? We demonstrate that the effects of increased risk aversion are preserved under the “Almost Degenerate Independence” axiom, but that those of distribution changes of exogenous risks are not preserved under stringent assumptions. Hence, nonexpected utility comparative statics may diverge from expected utility, even for “first-order” properties—those whose effect is determinable from restrictions on “local” utility functions.     

Ambiguity aversion under maximum-likelihood updating

Maximum-likelihood updating (MLU) is a well-known approach for extending static ambiguity sensitive preferences to dynamic set-ups. This paper develops an example in which MLU induces an ambiguity averse maxmin expected utility (MEU) decision-maker to (1) prefer a bet on an ambiguous over a risky urn and (2) be more willing to bet on the ambiguous urn compared to an (ambiguity neutral) subjective expected utility (SEU) decision-maker. This is challenging, since prior to observing (symmetric) draws from the urns, the MEU decision-maker (in line with the usual notion of ambiguity aversion) actually preferred the risky over the ambiguous bet and was less willing to bet on the ambiguous urn than the SEU decision-maker. The identified switch in betting preferences is not due to a violation of dynamic consistency or consequentialism. Rather, it results from MLU’s selection of extreme priors, causing a violation of the stability of set inclusion over the course of the updating process.     

Sequential decision making without independence: a new conceptual approach

This paper presents a critical reflection on dynamic consistency as commonly used in economics and decision theory, and on the difficulty to test it experimentally. It distinguishes between the uses of the term dynamic consistency in order to characterize two different properties: the first accounts for the neutrality of individual preferences towards the timing of resolution of uncertainty whereas the second guarantees that a strategy chosen at the beginning of a sequential decision problem is immune to any reevaluation and will effectively be implemented from then on in the decision problem. Although these two properties are equivalent under expected utility (EU), this is not the case under non-EU. Building on the possible characteristics of individual dynamic preferences under risk, this paper proposes a conceptual categorization, that is experimentally testable, of possible sequential decision making behaviors of non-EU maximizers.     

Valuing future cash flows with non separable discount factors and non additive subjective measures: conditional Choquet capacities on time and on uncertainty

We consider future cash flows that are contingent both on dates in time and on uncertain states. The decision maker (DM) values the cash flows according to its decision criterion: Here, the payoffs’ expectation with respect to a capacity measure. The subjective measure grasps the DM’s behaviour in front of the future, in the spirit of de Finetti’s (1930) and of Yaari’s (1987) Dual Theory in the case of risk. Decomposition of the criterion into two criteria that represent the DM’s preferences on uncertain payoffs and time contingent payoffs are derived from Ghirardato’s (1997) results. Conditional Choquet integrals are defined by dynamic consistency (DC) requirements and conditional capacities are derived, under some conditions on information. In contrast with other models referring to DC, ours does not collapse into a linear one because it violates a weak version of consequentialism.     

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.     

Expanding state space and extension of beliefs

This article provides a simple decision theoretic model in which elements of the world successively enter the decision maker??s scope and the state space expands over time, which is intended to be the closest correspondence to the standard subjective expected utility theory. We propose a dynamic consistency condition that after any expansion of the scope, the preference ranking should remain unchanged over acts to which the expansion is irrelevant. Together with other natural axioms, it characterizes a model in which the decision maker??s belief extends over time in order that the marginal distribution of the new belief induced over the old state space coincides with the old belief. It is extended to encompass both expansion of scope and learning events, and we characterize the model with an additional property that the decision maker??s belief updating follows Bayes?? rule when she learns events.     

The sure thing principle and the value of information

This paper examines the relationship between Savage's sure thing principle and the value of information. We present two classes of results. First, we show that, under a consequentialist axiom, the sure-thing principle is neither sufficient nor necessary for perfect information to be always desirable: specifically, under consequentialism, the sure thing principle is not implied by the condition that perfect information is always valuable; moreover, the joint imposition of the sure thing principle, consequentialism and either one of two state independence axioms does not imply that perfect information is always desirable. Second, we demonstrate that, under consequentialism, the sure thing principle is necessary for a nonnegative value of possibly imperfect information (though of course the principle is still not sufficient). One implication of these results is that the sure thing principle, under consequentialism, plays a somewhat different role in ensuring dynamic consistency in decision making under uncertainty than does the independence axiom in decision making under risk.     

Even Risk-Averters may Love Risk

A decision maker bets on the outcomes of a sequence of coin-tossings. At the beginning of the game the decision maker can choose one of two coins to play the game. This initial choice is irreversible. The coins can be biased and the player is uncertain about the nature of one (or possibly both) coin(s). If the player is an expected-utility maximizer, her choice of the coin will depend on different elements: the nature of the game (namely, whether she can observe the outcomes of the previous tosses before making her next decision), her utility function, the prior distribution on the bias of the coin. We will show that even a risk averter might optimally choose a riskier coin when learning is allowed. We will express most of our results in the language of stochastic orderings, allowing comparisons that are valid for large classes of utility functions.     

The likelihood of various stock market return distributions, part 1: Principles of inference

This is the first of two articles which apply certain principles of inference to a practical, financial question. The present article argues and cites arguments which contend that decision making should be Bayesian, that classical (R. A. Fisher, Neyman-Pearson) inference can be highly misleading for Bayesians as can the use of diffuse priors, and that Bayesian statisticians should show remote clients with a variety of priors how a sample implies shifts in their beliefs. We also consider practical implications of the fact that human decision makers and their statisticians cannot fully emulate Savage's rational decision maker.     

A Rule For Updating Ambiguous Beliefs

When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on E ^c, obey axioms which lead to maxmin expected utility representation with multiple priors, and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule.     

Process fairness,outcome fairness,and dynamic consistency: Experimental evidence for risk and ambiguity

Literature on fairness preferences distinguishes between outcome fairness, concerning the final allocation of payoffs, and process fairness, concerning the expected allocation of payoffs. It is not obvious, however, whether process fairness can consistently be implemented. Once uncertainty is resolved and outcomes are determined, the ex-ante procedurally fair decision maker may become consequentialist ex-post, and reconsider her choice on the basis of the observed outcomes. We present experimental evidence on dynamic consistency of social preferences under both known risk and ambiguity. A significant share of people subscribe to process fairness both before and after the resolution of uncertainty.     

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.     

A Theory of Rational Choice under Ignorance

This paper contributes to a theory of rational choice for decision-makers with incomplete preferences due to partial ignorance, whose beliefs are representable as sets of acceptable priors. We focus on the limiting case of `Complete Ignorance' which can be viewed as reduced form of the general case of partial ignorance. Rationality is conceptualized in terms of a `Principle of Preference-Basedness', according to which rational choice should be isomorphic to asserted preference. The main result characterizes axiomatically a new choice-rule called `Simultaneous Expected Utility Maximization'. It can be interpreted as agreement in a bargaining game (Kalai-Smorodinsky solution) whose players correspond to the (extremal) `acceptable priors' among which the decision maker has suspended judgment. An essential but non-standard feature of Simultaneous Expected Utility choices is their dependence on the entire choice set. This is justified by the conception of optimality as compromise rather than as superiority in pairwise comparisons.     

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.     

An experimental evaluation of the descriptive validity of lottery-dependent utility theory

This article compares the performance of the expected utility (EU) and lottery-dependent expected utility (LDEU) models in predicting the actual choices of experimental subjects among risky options. In the process, we present two approaches for calibrating the LDEU model for an individual decision maker. The results indicate that while LDEU exhibits a higher potential for correctly predicting choice, the version of the model calibrated by indifference judgments does not outperform EU. We suggest a functional form for the parametric functions that defines the LDEU model, and discuss ways in which this function can be incorporated into choice-based assessment approaches to improve predictions.This research was supported in part by the Business Associates Fund at the Fuqua School of Business, Duke University.     

On the consistency of choice

Consistency of choice is a fundamental and recurring theme in decision theory, social choice theory, behavioral economics, and psychological sciences. The purpose of this paper is to study the consistency of choice independent of the particular decision model at hand. Consistency is viewed as an inherently logical concept that is fundamentally void of connotation and is thus disentangled from traditional rationality or consistency conditions imposed on decision models. The proposed formalization of consistency takes two forms: internal consistency, which refers to the property that a choice model does not generate contradictory statements; and semantic consistency, which refers to the idea that a theory’s predictions are valid with respect to some observed data. In addressing semantic consistency, the relationship between theory and data is analyzed in terms of so-called duality mappings, which allow a passage between the two universes in a way that consistency is preserved. The formalization of consistency concepts relies on adapting the revealed preference theory to the context-dependent setting. The paper concludes by discussing the implications of the proposed framework and how it relates to classical revealed preference theory and other formalizations of the relationship between the theory and reality of choice.     

Indirect methods for valuing changes in environmental risks with nonexpected utility preferences

Theoretical models for estimating individuals' values for sure improvements in environmental quality are well developed. These models can be classified as being based on averting behavior, hedonic prices, or weak complementarity. Some of these models have also been applied to the task of valuing changes in risk based on expected utility theory. This article provides a systematic development of these models for changes in either the probability or the magnitude of an uncertain event and shows that the derived expressions for individual marginal willingness to pay can be generalized to nonexpected utility preferences as long as the index of preferences is continuous, convex, and twice differentiable.