Preference aggregation under uncertainty: Savage vs. Pareto

Following Mongin [J. Econ. Theory 66 (1995) 313; J. Math. Econ. 29 (1998) 331], we study social aggregation of subjective expected utility preferences in a Savage framework. We argue that each of Savage's P3 and P4 are incompatible with the strong Pareto property. A representation theorem for social preferences satisfying Pareto indifference and conforming to the state-dependent expected utility model is provided.     

What do uncertainty-averse decision-makers believe?

Summary. This paper introduces the concept of firm belief, which is proposed as a new epistemic model for a wide class of preferences. In particular, firm beliefs are shown to have the following desirable properties: (i) they are derived from preferences according to a plausible rule of epistemic inference; (ii) they satisfy standard logical properties; and (iii) tractable representations of firm belief are available for all (suitably continuous) biseparable preferences [13, 14], including the Choquet expected utility [30] and maxmin expected utility [16] classes. We also use firm belief to construct a generalization of Nash equilibrium for (two-player) normal form games. Received: December 14, 1999; revised version: February 26, 2001     

Epistemic Conditions for Equilibrium in Beliefs without Independence

Aumann and Brandenburger [Econometrica63(1995), 1161–1180.] provide sufficient conditions on the knowledge of the players in a game for their beliefs to constitute a Nash equilibrium. They assume, among other things, mutual knowledge of rationality. By rationality of a player, it is meant that the action chosen by him maximizes his expected utility, given his beliefs. There is, however, no need to restrict the notion of rationality to expected utility maximization. This paper shows that their result can be generalized to the case where players' preferences over uncertain outcomes belong to a large class of non-expected utility preferences.Journal of Economic LiteratureClassification Numbers: C72, D81.     

Foundations of Bayesian theory

This paper states necessary and sufficient conditions for the existence, uniqueness, and updating according to Bayes’ rule, of subjective probabilities representing individuals’ beliefs. The approach is preference based, and the result is an axiomatic subjective expected utility model of Bayesian decision making under uncertainty with state-dependent preferences. The theory provides foundations for the existence of prior probabilities representing decision makers’ beliefs about the likely realization of events and for the updating of these probabilities according to Bayes’ rule.     

Second-order ambiguous beliefs

This paper axiomatizes models of second-order ambiguous beliefs in the original domain of preferences of Anscombe and Aumann (Ann Math Stat 34:199–205, 1963) by weakening the first-stage independence postulate. The models we propose include the second-order subjective expected utility (SOSEU) of Seo (Econometrica 77:1575–1605, 2009) as a particular case. We characterize the intersection of our models of second-order ambiguity with the canonical models of (first-order) ambiguity aversion and provide a further generalization of SOSEU by relaxing the completeness axiom.     

Range convexity and ambiguity averse preferences

We show that range convexity of beliefs, a `technical' condition that appears naturally in axiomatizations of preferences in a Savage-like framework, imposes some unexpected restrictions when modelling ambiguity averse preferences. That is, when it is added to a mild condition, range convexity makes the preferences collapse to subjective expected utility as soon as they satisfy structural conditions that are typically used to characterize ambiguity aversion. Received: February 25, 2000; revised version: April 17, 2000     

Trembles in extensive games with ambiguity averse players

We introduce and analyze three definitions of equilibrium for finite extensive games with imperfect information and ambiguity averse players. In a setting where players’ preferences are represented by maxmin expected utility, as characterized in Gilboa and Schmeidler (J Math Econ 18(2):141–153, 1989), our definitions capture the intuition that players may consider the possibility of slight arbitrary mistakes. This generalizes the idea leading to trembling-hand perfect equilibrium as introduced in Selten (Int J Game Theory 4(1):25–55, 1975), by allowing for ambiguous trembles characterized by sets of distributions. We prove existence for two of our equilibrium notions and relate our definitions to standard equilibrium concepts with expected utility maximizing players. Our analysis shows that ambiguity aversion can lead to behavioral implications that are distinct from those attained under expected utility maximization, even if ambiguous beliefs only arise from the possibility of slight mistakes in the implementation of unambiguous strategies.     

‘Expected utility / subjective probability’ analysis without the sure-thing principle or probabilistic sophistication

Summary. The basic analytical concepts, tools and results of the classical expected utility/subjective probability model of risk preferences and beliefs under subjective uncertainty can be extended to general event-smooth preferences over subjective acts that do not necessarily satisfy either of the key behavioral assumptions of the classical model, namely the Sure-Thing Principle or the Hypothesis of Probabilistic Sophistication. This is accomplished by a technique analogous to that used by Machina (1982) and others to generalize expected utility analysis under objective uncertainty, combined with an event-theoretic approach to the classical model and the use of a special class of subjective events, acts and mixtures that exhibit almost-objective like properties. The classical expected utility/subjective probability characterizations of outcome monotonicity, outcome derivatives, probabilistic sophistication, comparative and relative subjective likelihood, and comparative risk aversion are all globally robustified to general event-smooth preferences over subjective acts.Received: 4 May 2004, Revised: 4 October 2004, JEL Classification Numbers: D81.This paper presents a considerably improved version of the concept of event-differentiability from Machina (1992). An alternative definition has been independently developed by Epstein (1999) in his analysis of the concept of uncertainty aversion. I am grateful to Kenneth Arrow, Mark Durst, Jürgen Eichberger, Daniel Ellsberg, Clive Granger, Simon Grant, Edi Karni, Peter Klibanoff, David Kreps, Duncan Luce, Robert Nau, Uzi Segal, Peter Wakker, Joel Watson and especially Larry Epstein, Ted Groves and Joel Sobel for helpful discussions and comments. This material is based upon work supported by the National Science Foundation under Grants No. 9209012 and 9870894.     

Subjective Expected Utility Theory with State-Dependent Preferences

This paper extends Savage′s subjective expected utility theory to include state-dependent preferences. The dependence of the decision maker′s preferences over consequences on the states of nature is represented by state-specific mappings of the set of consequences onto itself. Within this framework Savage′s postulates are reformulated and it is shown that there exist subjective expected utility representations of the preference relation over acts with unique, nonatomic, probability measure on the algebra of all events, and a state-dependent utility function over the set of consequences. Journal of Economic Literature Classification Number: D81.     

Diversification, convex preferences and non-empty core in the Choquet expected utility model

Summary. We show, in the Choquet expected utility model, that preference for diversification, that is, convex preferences, is equivalent to a concave utility index and a convex capacity. We then introduce a weaker notion of diversification, namely “sure diversification.” We show that this implies that the core of the capacity is non-empty. The converse holds under concavity of the utility index, which is itself equivalent to the notion of comonotone diversification, that we introduce. In an Anscombe-Aumann setting, preference for diversification is equivalent to convexity of the capacity and preference for sure diversification is equivalent to non-empty core. In the expected utility model, all these notions of diversification are equivalent and are represented by the concavity of the utility index. Received: July 27, 1999; revised version: November 7, 2000     

Proper scoring rules for general decision models

If a decision maker whose behavior conforms to the max-min expected utility model is faced with a scoring rule for a subjective expected utility decision maker, she will always announce a probability belonging to her set of priors; moreover, for any prior in the set, there is a scoring rule inducing the agent to announce that prior. We also show that on the domain of Choquet expected utility preferences with risk neutral lottery evaluation and totally monotone capacities, proper scoring rules do not exist. This implies the non-existence of proper scoring rules for any larger class of preferences (CEU with convex capacities, multiple priors).     

Probabilistic sophistication and variational preferences

This paper shows that in the class of variational preferences the notion of probabilistic sophistication is equivalent to expected utility as long as there exists at least one event such that the independence axiom holds for bets on that event. This extends the result of Marinacci (2002) [13] and provides a novel interpretation of his result.     

Fixed endpoint optimal control

Summary. Properness has been introduced in the expected utility framework and it recently has been transfered to mean-variance utility functions. Here, we show that properness implies the slope of the mean-standard deviation indifference curve being convex in the standard deviation. This indifference curve property allows us to characterize the comparative static effects of changing the background risk and dependency structure in a simple portfolio choice model.Received: 8 December 2003, Revised: 16 February 2004, JEL Classification Numbers: D81.     

Completing the State Space with Subjective States

We define an opportunity act as a mapping from an exogenously given objective state space to a set of lotteries over prizes, and consider preferences over opportunity acts. We allow the preferences to be possibly uncertainty averse. Our main theorem provides an axiomatization of the maxmin expected utility model. In the theorem we construct subjective states to complete the objective state space. As in E. Dekel et al. (Econometrica, in press), we obtain a unique subjective state space. We also allow for preference for flexibility in some of the subjective states and commitment in others. Journal of Economic Literature Classification Number: D81.     

Contribution to a public good under subjective uncertainty

This paper examines how voluntary contributions to a public good are affected by the contributors' heterogeneity in beliefs about the uncertain impact of their contributions. It assumes that contributors have Savagian preferences that are represented by a two‐state‐dependent expected utility function and different beliefs about the benefit that will result from the sum of their contributions. We establish general comparative statics results regarding the effect of specific changes in the distribution of beliefs on the (unique) Nash equilibrium provision of the public good, under certain conditions imposed on the preferences. We specifically show that the equilibrium public good provision is increasing with respect to both first‐ and second‐order stochastic dominance changes in the distribution of beliefs. Hence, increasing the contributors' optimism about the uncertain benefit of their contributions increases aggregate public good provision, as does any homogenization of these beliefs around their mean.     

Resolving distributional conflicts between generations

Summary. We describe a new approach to the problem of resolving distributional conflicts between an infinite and countable number of generations. We impose conditions on the social preferences that capture the following idea: If preference (or indifference) holds between truncated paths for infinitely many truncating times, then preference (or indifference) holds also between the untruncated infinite paths. In this framework we use such conditions to (1) characterize different versions of leximin and utilitarianism by means of equity conditions well-known from the finite setting, and (2) illustrate the problem of combining Strong Pareto and impartiality in an intergenerational setting.Received: 8 May 2002, Revised: 12 June 2003, JEL Classification Numbers: D63, Q01.Correspondence to: Geir B. AsheimWe thank Kaushik Basu, Marc Fleurbaey, David Miller, Tapan Mitra, Lars-Gunnar Svensson, and an anonymous referee for helpful comments. Asheim gratefully acknowledges the hospitality of the Stanford University research initiative on the Environment, the Economy and Sustainable Welfare, and financial support from the Hewlett Foundation through this research initiative.     

Correlated Nash equilibrium

We modify the epistemic conditions for Nash equilibrium only to accommodate Gilboa and Schmeidler's [I. Gilboa, D. Schmeidler, Maxmin expected utility with nonunique prior, J. Math. Econ. 18 (1989) 141-153] maxmin expected utility preferences, and identify the equilibrium concept in n-player strategic games that characterizes the modified epistemic conditions. The epistemic characterization supports the equilibrium concept as a minimal generalization of Nash equilibrium, in the sense that it deviates from Nash equilibrium only in terms of players' attitude towards ambiguity. Consequently, comparing it with Nash equilibrium constitutes a ceteris paribus study of the effects of ambiguity on how a game is played. For example, with ambiguity, (beliefs about) action choices are in general correlated.     

A SMOOTH AMBIGUITY MODEL OF THE COMPETITIVE FIRM

This paper examines the optimal production decision of the competitive firm under price uncertainty when the firm's preferences exhibit smooth ambiguity aversion. Ambiguity is modeled by a second‐order probability distribution that captures the firm's uncertainty about which of the subjective beliefs govern the price risk. Ambiguity preferences are modeled by the (second‐order) expectation of a concave transformation of the (first‐order) expected utility of profit conditional on each plausible subjective distribution of the price risk. Within this framework, we derive necessary and sufficient conditions under which the ambiguity‐averse firm optimally produces less in response either to the introduction of ambiguity or to greater ambiguity aversion when ambiguity prevails. In the case that the price risk is binary, we show that ambiguity and greater ambiguity aversion always adversely affect the firm's production decision.     

Non-Expected Utility, Saving and Portfolios

Despite increased stockholding opportunities, standard expected-utility models overpredict household participation and stock holdings. It has been suggested that departures from expected utility could resolve both puzzles. We investigate three measurable departures: (i) Kreps-Porteus preferences, (ii) Yaari's Dual Theory, and (iii) Quiggin's Rank-dependent Utility. Improvements tend to occur in predicted portfolio composition rather than participation. They are limited under (i), questionable under (ii), and more sizeable under (iii). Contrary to priors in the literature, improvements under (iii) do not result from solutions at kinks of indifference curves. We conclude that stockholding puzzles are unlikely to be resolved through preferences alone.     

Coalitional Strategy-Proof House Allocation

We consider house allocation without endowments. We show that there is a unique maximal domain including all strict preferences on which efficiency and coalitional strategy-proofness are compatible. A preference relation belongs to the unique maximal domain if it is a strict descending ranking of houses to a certain house and indifference holds over it and the remaining houses. We also show that on this domain mixed dictator-pairwise-exchange rules are the only rules satisfying efficiency and coalitional strategy-proofness. Journal of Economic Literature Classification Numbers: C78, D63, D71.