A simple axiomatization and constructive representation proof for choquet expected utility

>P>Summary. We provide a set of simple and intuitive set of axioms that allow for a direct and constructive proof of the Choquet Expected Utility representation for decision making under uncertainty. Received: October 29, 2002; revised version: November 13, 2002 RID="*" ID="*" We thank Matthew Ryan for very useful comments and suggestions on related work and for encouraging us to write this note. Correspondence to: S. Grant     

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     

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     

Subjective ambiguity, expected utility and Choquet expected utility

Summary. Using the Savage set up, this paper provides a simple axiomatization of the Choquet Expected Utility model where the capacity is an inner measure. Two attractive features of the model are its specificity and the transparency of its axioms. The key axiom states that the decision-maker uses unambiguous acts to approximate ambiguous ones. In addition, the notion of ‘ambiguity’ is subjective and derived from preferences. Received: March 23, 2000; revised version: April 24, 2001     

金融经济领域中的不确定性研究综述

本文综述了Choquet容度和Choquet积分在刻画金融经济领域中不确定性上的应用,包含非期望效用、资产定价、保险理论、投资消费储蓄理论和宏观经济等五个方面,同时给出不确定性领域的进一步研究方向:动态相容性、理性、效用的合理定义以及非期望效用的推广等。     

Background risk in generalized expected utility theory

Summary. In this paper, it is shown that, for a wide range of risk-averse generalized expected utility preferences, independent risks are complementary, contrary to the results for expected utility preferences satisfying conditions such as proper and standard risk aversion. Received: August 10, 2001; revised version: June 18, 2002 RID="*" ID="*"I thank Simon Grant and an anonymous referee for helpful comments and criticism. This research was supported by an Australian Research Council Senior Fellowship and Australian Research Council Large Grant A79800678.     

More on parametric characterizations of risk aversion and prudence

Summary. This note provides an alternative proof for the equivalence of decreasing absolute prudence (DAP) in the expected utility framework and in a two-parametric approach where utility is a function of the mean and the standard deviation. In addition, we elucidate that the equivalence of DAP and the concavity of utility as a function of mean and variance, which was shown to hold for normally distributed stochastics in Lajeri and Nielsen [4], cannot be generalized. Received: November 27, 2000; revised version: November 26, 2001 Correspondence to: T. Eichner     

Parametric characterizations of risk aversion and prudence

Summary. Our first main result says that whether one decision maker is more risk averse than another can be determined from their attitudes toward a given two-parameter family of risks. When all risks belong to this family, risk aversion can be compared even when initial wealth is random. Our second main result solves a long-standing problem in mean-variance analysis: what is the interpretation of the concavity of utility as a function of mean and variance? We show that in the case of normal distributions, this utility function is concave if and only if the agent has decreasing prudence. Received: July 29, 1996; revised: October 2, 1998     

Intertemporal substitution, risk aversion and ambiguity aversion

Summary. This paper axiomatizes a form of recursive utility on consumption processes that permits a role for ambiguity as well as risk. The model has two prominent special cases: (i) the recursive model of risk preference due to Kreps and Porteus [18]; and (ii) an intertemporal version of multiple-priors utility due to Epstein and Schneider [8]. The generalization presented here permits a three-way separation of intertemporal substitution, risk aversion and ambiguity aversion.Received: 5 August 2003, Revised: 12 March 2004, JEL Classification Numbers: D80, D81, D90.I am grateful to Larry Epstein for his guidance and invaluable advice, and to a referee for helpful comments and suggestions.     

Uncertainty and entry deterrence

Summary. We study a model where capacity installation by an incumbent firm serves to deter others from entering the industry. We argue that uncertainty about demand or costs forces the incumbent to choose a higher capacity level than it would under certainty. This higher level diminishes the attractiveness of deterrence (Proposition 1) and, therefore, the range of parameter values for which deterrence occurs (Proposition 2). Received: July 10, 1997; revised version: November 21, 1997     

Eliciting the core of a supermodular capacity

Summary. The paper utilizes duality theory to derive an exact representation of the core of a supermodular capacity for finite-state-space Choquet expected utility preferences. Using the dual representation we develop an algorithm that uses information on willingness to pay and willingness to sell to elicit a supermodular capacity in a finite number of iterations.Received: 21 February 2003, Revised: 26 May 2004, JEL Classification Numbers: D81. Correspondence to: Robert G. ChambersThe authors thank J. Quiggin and an anonymous referee for comments that improved the paper.     

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     

Risk aversion in the Talmud

Summary. Evidence is adduced that the sages of the ancient Babylonian Talmud, as well as some of the medieval commentators thereon, were well aware of sophisticated concepts of modern theories of risk-bearing. Received: April 10, 2002; revised version: May 7, 2002 RID="*" ID="*"Presented at the Institute for Mathematical Studies in the Social Sciences-Economics, Stanford University, August 4, 1981. Subsequent to that presentation, the author's attention was drawn to an article by Zvi Ilani, “Models in the Economics of Uncertainty: The Cost of Concluding a Conditional Contract, according to the Talmud and the Halachic Literature,” Iyunim Bekalkala (Investigations in Economics), The Israel Association for Economics, Jerusalem, Nissan 5740 (April 1980), 246–261 (in Hebrew). Inter alia, Ilani treats the Talmudic passage that forms the subject of this paper, and provides a fairly comprehensive review of the medieval commentaries thereon; undoubtedly, he was the first to recognize in print the relevance of this passage to modern economic theories of uncertainty. It is not clear, though, whether or not his understanding of the passage agrees with ours. The current paper appeared in January 2002 in the Research Bulletin Series of the Research Center on Jewish Law and Economics, Department of Economics, Bar Ilan University.     

Risk aversion, moral hazard, and the principal's loss

Summary. In their seminal paper on the principal-agent model with moral hazard, Grossman and Hart (1983) show that if the agent's utility function is , then the loss to the principal from being unable to observe the agent's action is increasing in the agent's degree of absolute risk aversion. Their proof is restricted to the case where the number of observable outcomes is equal to two, and it uses an argument that is specific to that case. In this note, we provide an alternative proof that generalizes their result to any (finite) number of outcomes. Received: March 21, 2001; revised version: June 21, 2001     

Comparing risk aversion in a probability triangle

An agent's acceptance set consists of the probability distributions preferred to the status quo. One agent is more risk averse than another if the more risk averse agent's acceptance set is a proper subset of the less risk averse agent's acceptance set. An agent's odds premium expresses the odds in favor of winning the largest cash prize in a lottery over the best and worst alternatives that is indifferent to the the agent's initial wealth. Comparisons of two agents odds premia completely characterizes the risk aversion relations between them when facing lotteries in a probability triangle. The result applies to expected utility and some non-expected utility theories. Received: December 30, 1998; revised version: February 10, 1999     

More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model

Summary. This paper studies monotone risk aversion, the aversion to monotone, mean-preserving increase in risk (Quiggin [21]), in the Rank Dependent Expected Utility (RDEU) model. This model replaces expected utility by another functional, characterized by two functions, a utility function u in conjunction with a probability-perception function f. Monotone mean-preserving increases in risk are closely related to the notion of comparative dispersion introduced by Bickel and Lehmann [3,4] in Non-parametric Statistics. We present a characterization of the pairs (u,f) of monotone risk averse decision makers, based on an index of greediness G _u of the utility function u and an index of pessimism P _f of the probability perception function f: the decision maker is monotone risk averse if and only if . The index of greediness (non-concavity) of u is the supremum of taken over . The index of pessimism of f is the infimum of taken over 0 < v < 1. Thus, , with G _u = 1 iff u is concave. If then , i.e., f is majorized by the identity function. Since P _f = 1 for Expected Utility maximizers, forces u to be concave in this case; thus, the characterization of risk aversion as is a direct generalization from EU to RDEU. A novel element is that concavity of u is not necessary. In fact, u must be concave only if P _f = 1.Received: 10 April 2001, Revised: 18 November 2003, JEL Classification Numbers: D81. Correspondence to: Michéle CohenAlain Chateauneuf, Michéle Cohen, Isaac Meilijson: We are most grateful to Mark Machina, Peter Wakker and two anonymous referees for very helpful suggestions and comments.