The impossibility of compromise: some uniqueness properties of expected utility preferences

Summary. We focus on the following uniqueness property of expected utility preferences: Agreement of two preferences on one interior indifference class implies their equality. We show that, besides expected utility preferences under (objective) risk, this uniqueness property holds for subjective expected utility preferences in Anscombe-Aumann's (partially subjective) and Savage's (fully subjective) settings, while it does not hold for subjective expected utility preferences in settings without rich state spaces. Indeed, when it holds the uniqueness property is even stronger than described above, as it needs only agreement on binary acts. The extension of the uniqueness property to the subjective case is possible because beliefs in the mentioned settings are shown to satisfy an analogous property: If two decision makers agree on a likelihood indifference class, they must have identical beliefs. Received: November 15, 1999; revised version: December 29, 1999     

Half empty, half full and why we can agree to disagree forever

Aumann [Aumann R., 1976. Agreeing to disagree. Annals of Statisitics 4, 1236–1239] derives his famous we cannot agree to disagree result under the assumption that people are expected utility (=EU) decision makers. Motivated by empirical evidence against EU theory, we study the possibility of agreeing to disagree within the framework of Choquet expected utility (=CEU) theory which generalizes EU theory by allowing for ambiguous beliefs. As our first main contribution, we show that people may well agree to disagree if their Bayesian updating of ambiguous beliefs is psychologically biased in our sense. Remarkably, this finding holds regardless of whether people with identical priors apply the same psychologically biased Bayesian update rule or not. As our second main contribution, we develop a formal model of Bayesian learning under ambiguity. As a key feature of our approach the posterior subjective beliefs do, in general, not converge to “true” probabilities which is in line with psychological evidence against converging learning behavior. This finding thus formally establishes that CEU decision makers may even agree to disagree in the long-run despite the fact that they always received the same information.     

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     

Utility from anticipation and personal equilibrium

I develop a dynamic model of individual decisionmaking in which the agent derives utility from physical outcomes as well as from rational beliefs about physical outcomes (“anticipation”), and these two payoff components can interact. Beliefs and behavior are jointly determined in a personal equilibrium by the requirement that behavior given past beliefs must be consistent with those beliefs. I explore three phenomena made possible by utility from anticipation, and prove that if the decisionmaker’s behavior is distinguishable from a person’s who cares only about physical outcomes, she must exhibit at least one of these phenomena. First, the decisionmaker can be prone to self-fulfilling expectations. Second, she might be time-inconsistent even if her preferences in all periods are identical. Third, she might exhibit informational preferences, where these preferences are intimately connected to her attitudes toward disappointments. Applications of the framework to reference-dependent preferences, impulsive behaviors, and emotionally difficult choices are discussed.     

Proper belief revision and equilibrium in dynamic games

We present a theory of rationality in dynamic games in which players, during the course of the game, may revise their beliefs about the opponents’ utility functions. The theory is based upon the following three principles: (1) the players’ initial beliefs about the opponents’ utilities should agree on some profile u of utility functions, (2) every player should believe, at each of his information sets, that his opponents are carrying out optimal strategies and (3) a player at information set h should not change his belief about an opponent's ranking of strategies a and b if both a and b could have led to h. Scenarios with these properties are called preference conjecture equilibria for the profile u of utility functions. We show that every normal form proper equilibrium for u induces a preference conjecture equilibrium for u, thus implying existence of preference conjecture equilibrium.     

Status,Risk-Taking and Intertemporal Substitution in an Endogenous Growth Model

This paper examines intertemporal risk-taking in a stochastically growing economy with externalities in human capital accumulation where agents have preferences for social status. In order to isolate the effects of status concerns on long-run expected growth, the analysis is embedded in a non-expected utility setting, which disentangles the effects from risk aversion and intertemporal substitution. We examine the interaction between status desire and risk, risk aversion and intertemporal substitution. The externalities generated by the status game are able to correct the allocative distortions from the knowledge spillovers.Acknowledgement The author would like to thank an anonymous referee for his valuable comments.     

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.     

Belief and higher-order belief in the centipede games: An experimental investigation

This paper experimentally explores the epistemic conditions behind people's non-equilibrium behaviour in the centipede games. We propose a novel design of laboratory experiment to elicit people's first- and second-order beliefs regarding their opponents' choices and beliefs. The measured beliefs, together with the choice data, help us to estimate people's level of rationality, belief of rationality and second-order belief of rationality. To examine how these epistemic variables are affected by the social-efficiency property of the classic increasing-sum centipede game, we revisit the constant-sum centipede and compare the measured epistemic conditions from the constant-sum with those from the classic centipede. We find that people's non-backward induction behaviour may be attributed to the diffusion of beliefs and higher-order beliefs in the increasing-sum centipede. We consider a behavioural model in which people's preferences for social efficiency are incorporated into the extended utility maximization problem. Our analytical and estimation results indicate that the presence of efficiency-oriented players and people's belief towards the uncertain portion of such type of players may play a part in the non-backward-induction outcomes in experimental centipede games.     

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.     

Greeks and games: Forerunners of modern game theory

Modern game theory is connected with its ancient Greek originals in three respects.

Strategy-proof risk sharing

We consider risk sharing problems with a single good and a finite number of states. Agents have a common prior and their preferences are represented in the expected utility form and are risk averse. We study efficient and individually rational risk sharing rules satisfying strategy-proofness, the requirement that no one can ever benefit by misrepresenting his preference. When aggregate certainty holds, we show that “fixed price selections” from Walrasian correspondence are the only rules satisfying efficiency, individual rationality, and strategy-proofness. However, when aggregate uncertainty holds, we show that there exists no rule satisfying the three requirements. Moreover, in the two agents case, we show that dictatorial rules are the only efficient and strategy-proof rules. Dropping the common prior assumption in the model, we show that this assumption is necessary and sufficient for the existence of rules satisfying the three main requirements in the two agents and aggregate certainty case.     

Efficient allocations under ambiguity

Important implications of the expected utility hypothesis and risk aversion are that if agents have the same probability belief, then consumption plans in every efficient allocation of resources under uncertainty are comonotone with the aggregate endowment, and if their beliefs are concordant, then the consumption plans are measurable with respect to the aggregate endowment. We study these two properties of efficient allocations for models of preferences that exhibit ambiguity aversion using the concept of conditional beliefs, which we introduce in this paper. We provide characterizations of such conditional beliefs for the standard models of preferences used in applications.     

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     

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.     

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.     

Growth trends, cyclical fluctuations, and welfare with non-expected utility preferences

Using standard time-separable, discounted expected utility (EU) models, Lucas (Models of Business Cycles, 1987) and others argue that growth trends have a crucial, and cyclical fluctuations a trivial, effect on individuals' utility. These conclusions are drastically altered if, instead, non-EU preferences are assumed.     

Certainty Independence and the Separation of Utility and Beliefs

Economists often operate under an implicit assumption that the tastes of a decision maker are quite stable, while his beliefs change with the availability of new information. We show that for a general class of preferences, a separation of a key component of tastes, the utility function, from the other components of the representation is possible only if the decision maker's preferences satisfy a mild but not completely innocuous condition, called ‘certainty independence’. We also outline the axiomatic characterization of the preferences that obtain such separation, which are a subset of the biseparable preferences.     

Uncertainty and the Voluntary Provision of a Pure Public Good in a Two‐Moment Decision Model

In this paper, we explore the potential benefits of uncertainty that may arise in a two‐moment model of the voluntary provision of a pure public good. We find that an increase in a given contributor i’s risk associated with the aggregate contribution level of the other contributors (i.e., an increase in social uncertainty) induces that contributor to increase his own contribution level if and only if the uncertainty's incremental effect on the expected value of his net marginal utility is negative. Contributor i’s welfare likewise increases when a closely related condition is met, namely that the uncertainty's marginal effect on his expected marginal utility value of the public good exceeds its countervailing effect on the numeraire. Further, the corresponding aggregate contribution to the public good increases in the presence of free‐riding if and only if the incremental effect of contributor i’s contribution on the aggregate expected value of all other contributors’ net marginal utilities is small‐enough positive. We derive similar conditions for the case of private uncertainty, where the increase in contributor i’s risk is associated with his own marginal valuation of the public good. A simple example illustrates these conceptual results. Numerical analysis demonstrates that an increase in private uncertainty can have a nonmonotonic impact on contributor i’s welfare.     

Nash equilibrium without mutual knowledge of rationality

Summary. In a Nash equilibrium, players' rationality is mutual knowledge. However, both intuition and experimental evidence suggest that players do not know for sure the rationality of opponents. This paper proposes a new equilibrium concept, cautious equilibrium, that generalizes Nash equilibrium in terms of preferences in two person strategic games. In a cautious equilibrium, players do not necessarily know the rationality of opponents, but they view rationality as infinitely more likely than irrationality. For suitable models of preference, cautious equilibrium predicts that a player might take a “cautious” strategy that is not a best response in any Nash equilibrium. Received: January 28, 1998; revised version October 2, 1998