A variational model of preference under uncertainty

A familiar example devised by Daniel Ellsberg to highlight the effects of event ambiguity on preferences is transformed to separate aleatory uncertainty (chance) from epistemic uncertainty. The transformation leads to a lottery acts model whose states involve epistemic uncertainty; aleatory uncertainty enters into the statedependent lotteries. The model proposes von Neumann-Morgenstern utility for lotteries, additive subjective probability for states, and the use of across-states standard deviation weighted by a coefficient of aversion to variability to account for departures from Anscombe-Aumann subjective expected utility. Properties of the model are investigated and a partial axiomatization is provided.     

The Independence Postulate,Hypothetical and Called-off Acts: a further reply to Rabinowicz

The Independence postulate links current preferences between called-off acts with current preferences between constant acts. Under the assumption that the chance-events used in compound von Neumann-Morgenstern lotteries are value-neutral, current preferences between these constant acts are linked to current preferences between hypothetical acts, conditioned by those chance events. Under an assumption of stability of preferences over time, current preferences between these hypothetical acts are linked to future preferences between what are then and there constant acts. Here, I show that a failure of Independence with respect to current preferences leads to an inconsistency in sequential decisions. Two called-off acts are constructed such that each is admissible in the same sequential decision and yet one is strictly preferred to the other. This responds to a question regarding admissibility posed by Rabinowicz ([2000] Preference stability and substitution of indifferents: A rejoinder to Seidenfeld, Theory and Decision 48: 311–318 [this issue]).     

A simplified axiomatic approach to ambiguity aversion

This paper takes the Anscombe–Aumann framework with horse and roulette lotteries, and applies the Savage axioms to the horse lotteries and the von Neumann–Morgenstern axioms to the roulette lotteries. The resulting representation of preferences yields a subjective probability measure over states and two utility functions, one governing risk attitudes and one governing ambiguity attitudes. The model is able to accommodate the Ellsberg paradox and preferences for reductions in ambiguity.     

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.     

Weighted averaging,Jeffrey conditioning and invariance

Jeffrey conditioning tells an agent how to update her priors so as to grant a given probability to a particular event. Weighted averaging tells an agent how to update her priors on the basis of testimonial evidence, by changing to a weighted arithmetic mean of her priors and another agent’s priors. We show that, in their respective settings, these two seemingly so different updating rules are axiomatized by essentially the same invariance condition. As a by-product, this sheds new light on the question how weighted averaging should be extended to deal with cases when the other agent reveals only parts of her probability distribution. The combination of weighted averaging (for the events whose probability the other agent reveals) and Jeffrey conditioning (for the events whose probability the other agent does not reveal) is a comprehensive updating rule to deal with such cases, which is again axiomatized by invariance under embedding. We conclude that, even though one may dislike Jeffrey conditioning or weighted averaging, the two make a natural pair when a policy for partial testimonial evidence is needed.     

Signed orders in linear and nonlinear utility theory

Self-reflecting signed orders on a set A and its anti-set A ^* were introduced previously as a way to account for negative as well as positive feelings about the inclusion of items in A in potential subsets of choice. The present paper extends the notion of signed orders to lotteries on A A ^*, describes reflection axioms for the lottery context, and shows how these axioms simplify utility representations for preference between lotteries. The simplified representations are then used to guide procedures for extending preferences from A A ^* and its lotteries to preferences between subsets of items.     

Stochastic Choice and Consistency in Decision Making Under Risk: An Experimental Study

This paper reports the results of an experiment designed to uncover the stochastic structure of individual preferences over lotteries. Unlike previous experiments, which have presented subjects with pair-wise choices between lotteries, our design allowed subjects to choose between two lotteries or (virtually) any convex combination of the two lotteries. We interpret the mixtures of lotteries chosen by subjects as a measure of the stochastic structure of choice. We test between two alternative interpretations of stochastic choice: the random utility interpretation and the deterministic preferences interpretation. The main findings of the experiment are that the typical subject prefers mixtures of lotteries rather than the extremes of a linear lottery choice set. The distribution of choices does not change between a first and second asking of the same question. We argue that this provides support for the deterministic preferences interpretation over the random utility interpretation of stochastic choice. As a subsidiary result, we find a small proportion of subjects make choices that violate transitivity, but the level of intransitive choice falls significantly over time.     

On the Representation of Beliefs by Probabilities

This paper explores two axiomatic structures of subjective expected utility assuming a finite state-space and state-dependent, connected, topological outcome-spaces. Building on the work of Karni and Schmeidler (1981) the analytical framework includes, in addition to the preference relation on acts, introspective preferences on hypothetical lotteries that are linked to the preference relation on acts by consistency axioms. The two models accommodate state-dependent preferences and yield subjective probabilities that correctly represent the decision-maker's beliefs. State-independent preferences are a special case.     

Expected discounted utility

Standard axioms of additively separable utility for choice over time and classic axioms of expected utility theory for choice under risk yield a generalized expected additively separable utility representation of risk-time preferences over probability distributions over sure streams of intertemporal outcomes. A dual approach is to use the analogues of the same axioms in a reversed order to obtain a generalized additively separable expected utility representation of time–risk preferences over intertemporal streams of probability distributions over sure outcomes. The paper proposes an additional axiom, which is called risk-time reversal, for obtaining a special case of the two representations—expected discounted utility. The axiom of risk-time reversal postulates that if a risky lottery over streams of sure intertemporal outcomes and an intertemporal stream of risky lotteries yield the same probability distribution of possible outcomes in every point in time then a decision-maker is indifferent between the two. This axiom is similar to assumption 2 “reversal of order in compound lotteries” in Anscombe and Aumann (Ann Math Stat 34(1):199–205, 1963, p. 201).

     

Experimental evidence on valuation with multiple priors

Popular models for decision making under ambiguity assume that people use not one but multiple priors. This paper is a first attempt to experimentally elicit the min and the max of multiple priors directly. In an ambiguous scenario we measure a participant’s single prior, her min and max of multiple priors, and the valuation of an ambiguous asset with the same underlying states as the ambiguous scenario. We use the min and the max of multiple priors to directly test two popular multiple priors models: the maxmin model and the α maxmin model. We find more support for the α maxmin model: although people put about twice the weight on the minimum of multiple priors, they also consider the maximum. Furthermore, we indirectly elicit confidence weights over the whole set of multiple priors and test two additional models: variational preferences and the smooth model of ambiguity. Two particular versions of the variational preferences model explain less than the α maxmin but more than the maxmin model. Overall, the smooth model of ambiguity performs best among all models tested.     

Even-chance lotteries in social choice theory

This paper discusses aspects of the theory of social choice when a nonempty choice set is to be determined for each situation, which consists of a feasible set of alternatives and a preference order for each voter on the set of nonempty subsets of alternatives. The individual preference assumptions include ordering properties and averaging conditions, the latter of which are motivated by the interpretation that subset A is preferred to subset B if and only if the individual prefers an even-chance lottery over the basic alternatives in A to an even-chance lottery over the basic alternatives in B. Corresponding to this interpretation, a choice set with two or more alternatives is resolved by an even-chance lottery over these alternatives. Thus, from the traditional no-lottery social choice theory viewpoint, ties are resolved by even-chance lotteries on the tied alternatives. Compared to the approach which allows all lotteries to compete along with the basic alternatives, the present approach is a contraction which allows only even-chance lotteries.After discussing individual preference axioms, the paper examines Pareto optimality for nonempty subsets of a feasible set in a social choice context with n voters. Aspects of simple-majority comparisons in the even-chance context follow, including an analysis of single-peaked preferences. The paper concludes with an Arrowian type impossibility theorem that is designed for the even-chance setting.     

On the (Sample) Condorcet Efficiency of Majority Rule: An alternative view of majority cycles and social homogeneity

The Condorcet efficiency of a social choice procedure is usually defined as the probability that this procedure coincides with the majority winner (or majority ordering) in random samples, given a majority winner exists (or given the majority ordering is transitive). Consequently, it is in effect a conditional probability that two sample statistics coincide, given certain side conditions. We raise a different issue of Condorcet efficiencies: What is the probability that a social choice procedure applied to a sample matches with the majority preferences of the population from which the sample was drawn? We investigate the canonical case where the sample statistic is itself also majority rule and the samples are drawn from real world distributions gathered from national election surveys in Germany, France, and the United States. We relate the results to the existing literature on majority cycles and social homogeneity. We find that these samples rarely display majority cycles, whereas the probability that a sample misrepresents the majority preferences of the underlying population varies dramatically and always exceeds the probability that the sample displays cyclic majority preferences. Social homogeneity plays a fundamental role in the type of Condorcet efficiency investigated here.

     

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.     

Comparative probability for conditional events: A new look through coherence

We study, from the standpoint of coherence, comparative probabilities on an arbitrary familyE of conditional events. Given a binary relation ·, coherence conditions on · are related to de Finetti's coherent betting system: we consider their connections to the usual properties of comparative probability and to the possibility of numerical representations of ·. In this context, the numerical reference frame is that of de Finetti's coherent subjective conditional probability, which is not introduced (as in Kolmogoroff's approach) through a ratio between probability measures.Another relevant feature of our approach is that the family & need not have any particular algebraic structure, so that the ordering can be initially given for a few conditional events of interest and then possibly extended by a step-by-step procedure, preserving coherence.     

Uncommon Priors Require Origin Disputes

In standard belief models, priors are always common knowledge. This prevents such models from representing agents’ probabilistic beliefs about the origins of their priors. By embedding standard models in a larger standard model, however, pre-priors can describe such beliefs. When an agent’s prior and pre-prior are mutually consistent, he must believe that his prior would only have been different in situations where relevant event chances were different, but that variations in other agents’ priors are otherwise completely unrelated to which events are how likely. Due to this, Bayesians who agree enough about the origins of their priors must have the same priors.     

Conditional Desirability

Conditional attitudes are not the attitudes an agent is disposed to acquire in event of learning that a condition holds. Rather they are the components of agent's current attitudes that derive from the consideration they give to the possibility that the condition is true. Jeffrey's decision theory can be extended to include quantitative representation of the strength of these components. A conditional desirability measure for degrees of conditional desire is proposed and shown to imply that an agent's degrees of conditional belief are conditional probabilities. Rational conditional preference is axiomatised and by application of Bolker's representation theorem for rational preferences it is shown that conditional preference rankings determine the existence of probability and desirability measures that agree with them. It is then proven that every conditional desirability function agrees with an agent's conditional preferences and, under certain assumptions, every desirability function agreeing with an agent's conditional preferences is a conditional desirability function agreeing with her unconditional preferences.     

Preference and belief: Ambiguity and competence in choice under uncertainty

We investigate the relation between judgments of probability and preferences between bets. A series of experiments provides support for the competence hypothesis that people prefer betting on their own judgment over an equiprobable chance event when they consider themselves knowledgeable, but not otherwise. They even pay a significant premium to bet on their judgments. These data connot be explained by aversion to ambiguity, because judgmental probabilities are more ambiguous than chance events. We interpret the results in terms of the attribution of credit and blame. The possibility of inferring beliefs from preferences is questioned.¹     

The curse of hope

In K?szegi and Rabin’s (Q J Econ 1133–1165, 2006, Am Econ Rev 97:1047–1073, 2007) reference-dependent model of preferences, the chance of obtaining a better outcome can reduce an agent’s expected utility through an increase in the stochastic reference point. This means that individuals may prefer stochastically dominated lotteries. In this sense, hope, understood as a small probability of a better outcome, can be a curse. While K?szegi and Rabin focus on a linear specification of the utility function, we show that this effect occurs more broadly. Using fairly plausible assumptions and parameter values, we specify the conditions under which it occurs, as well as the type of lotteries in which this should be expected. We then show that while a simple subjective transformation of probability into weights of the reference point may in some cases mitigate the issue, in others, it can intensify it or even generate new ones. Finally, we extend the model by adding the individual’s current reference point (status quo) to the stochastic reference point. We show that this modification can reconcile K?szegi and Rabin’s model with the apparent empirical infrequency of stochastically dominated choices while maintaining its main qualitative results.     

Reporting probabilistic expectations with dynamic uncertainty about possible distributions

We study how experimental subjects report subjective probability distributions in the presence of ambiguity characterized by uncertainty over a fixed set of possible probability distributions generating future outcomes. Subjects observe draws from the true but unknown probability distribution generating outcomes at the beginning of each period of the experiment and state at selected periods a) the likelihoods that each probability distribution in the set is the true distribution, and b) the likelihoods of future outcomes. We estimate heterogeneity of rules used to update uncertainty about the true distribution and rules used to report distributions of future outcomes. We find that approximately 65% of subjects report distributions by properly weighing the possible distributions using their expressed uncertainty over them, while 22% of subjects report distributions close to the distribution they perceive as most likely. We find significant heterogeneity in how subjects update their expressed uncertainty. On average, subjects tend to overweigh the importance of their prior uncertainty relative to new information, leading to ambiguity that is substantially more persistent than would be predicted using Bayes’ rule. Counterfactual simulations suggest that this persistence will likely hold in settings not covered by our experiment.     

Ambiguity aversion in the small and in the large for weighted linear utility

The widely observed preference for lotteries involving precise rather than vague of ambiguous probabilities is called ambiguity aversion. Ambiguity aversion cannot be predicted or explained by conventional expected utility models. For the subjectively weighted linear utility (SWLU) model, we define both probability and payoff premiums for ambiguity, and introduce alocal ambiguity aversion function a(u) that is proportional to these ambiguity premiums for small uncertainties. We show that one individual's ambiguity premiums areglobally larger than another's if and only if hisa(u) function is everywhere larger. Ambiguity aversion has been observed to increase 1) when the mean probability of gain increases and 2) when the mean probability of loss decreases. We show that such behavior is equivalent toa(u) increasing in both the gain and loss domains. Increasing ambiguity aversion also explains the observed excess of sellers' over buyers' prices for insurance against an ambiguous probability of loss.