Agreement theorem for neo-additive beliefs

In this paper, we extend Aumann’s (Ann Stat 4:1236–1239, 1976) probabilistic agreement theorem to situations in which agents’ prior beliefs are represented by a common neo-additive capacity. In particular, we characterize the family of updating rules for neo-additive capacities, which are necessary and sufficient for the impossibility of “agreeing to disagree” on the values of posterior capacities as well as on the values of posterior Choquet expectations for binary acts. Furthermore, we show that generalizations of this result to more general acts are impossible.     

Comparing three ways to update Choquet beliefs

We analyze three rules for updating neo-additive capacities. Only for Generalized Bayesian Updating is relative optimism the same for both updated and unconditional capacities. For updates of the other two, either the updated capacity is fully optimistic or fully pessimistic.     

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.     

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).     

Preference for safety under the Choquet model: in search of a characterization

Victor prefers safety more than Ursula if whenever Ursula prefers a constant to an uncertain act, so does Victor. This paradigm, whose expected utility (EU) version is Arrow and Pratt’s more risk aversion concept, will be studied in the Choquet expected utility (CEU) model. Necessary condition Pointwise inequality between a function of the utility functions and another of the capacities is necessary and sufficient for the preference by Victor of safety over a dichotomous act whenever such is the preference of Ursula. However, increased preference for safety versus dichotomous acts does not imply preference by Victor of safety over a general act whenever such is the preference of Ursula. A counterexample will be provided, via the casino theory of Dubins and Savage. Sufficient condition Separation of the two functions by some convex function is sufficient for Victor to prefer safety more than Ursula, over general acts. Furthermore, a condition on the capacities will be presented for simplicity seeking, the preference by Victor over any act for some dichotomous act that leaves Ursula indifferent. This condition is met in particular if Victor’s capacity is a convex function of Ursula’s capacity. For these cases, the pointwise inequality (necessary) condition is a characterization of greater preference for safety, extending the Arrow–Pratt notion from EU to CEU and rank-dependent utility (RDU). These inequalities preserve the flavor of the “more pessimism than greediness” characterization of monotone risk aversion by Chateauneuf, Cohen and Meilijson in the RDU model and its extension by Grant and Quiggin to CEU. Preferences between safety and dichotomous acts are at the core of the biseparable preferences model of Ghirardato and Marinacci.     

Piecewise additivity for non-expected utility

A model of choice under purely subjective uncertainty, Piecewise Additive Choquet Expected utility, is introduced. PACE utility allows for optimism and pessimism simultaneously, but represents a minimal departure from expected utility. It can be seen as a continuous version of NEO-expected utility (Chateauneuf et al. in J Econ Theory 137:538–567, 2007) and, as such, is especially suited for applications with rich state spaces. The main theorem provides a preference foundation for PACE utility in the Savage framework of purely subjective uncertainty with an arbitrary outcome set.     

Subjective probabilities on “small” domains

The Savagian choice-theoretic construction of subjective probability does not apply to preferences, like those in the Ellsberg Paradox, that reflect a distinction between risk and ambiguity. We formulate two representation results—one for expected utility, the other for probabilistic sophistication—that derive subjective probabilities but only on a “small” domain of risky events. Risky events can be either specified exogenously or in terms of choice behavior; in the latter case, both the values and the domain of probability are subjective. The analysis identifies a mathematical structure—called a mosaic—that is intuitive for both exogenous and behavioral specifications of risky events. This structure is weaker than an algebra or even a λ-system.     

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     

Subjective evaluations of n-person games

Using an axiomatization of subjective expected utility due to Fishburn, we characterize a class of utility functions over a set of n-person games in characteristic-function form. A probabilistic value is defined as the expectation of some player's marginal contribution with respect to some probability measure on the set of coalitions of other players. We decribe conditions under which a utility function on the set of n-person games is a probabilistic value; we prove as well an analogous result for simple games. We present additional axioms that characterize the semivalues and, in turn, the Shapley and Banzhaf values.     

Ambiguity Made Precise: A Comparative Foundation

The theory of subjective expected utility has been recently extended to allow ambiguity to matter for choice. We propose a notion of absolute ambiguity aversion by building on a notion of comparative ambiguity aversion. We characterize it for a preference model which encompasses some of the most popular models in the literature. We next build on these ideas to provide a definition of unambiguous act and event and show the characterization of the latter. As an illustration, we consider the classical Ellsberg 3-color urn problem and find that the notions developed in the paper provide intuitive answers. 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     

Cumulative prospect theory and the St. Petersburg paradox

Summary. We find that in cumulative prospect theory (CPT) with a concave value function in gains, a lottery with finite expected value may have infinite subjective value. This problem does not occur in expected utility theory. The paradox occurs in particular in the setting and the parameter regime studied by Tversky and Kahneman [15] and in subsequent works. We characterize situations in CPT where the problem can be resolved. In particular, we define a class of admissible probability distributions and admissible parameter regimes for the weighting- and value functions for which finiteness of the subjective value can be proved. Alternatively, we suggest a new weighting function for CPT which guarantees finite subjective value for all lotteries with finite expected value, independent of the choice of the value function. Some of these results have already been found independently by Blavatskyy [4] in the context of discrete lotteries.Received: 14 October 2004, Revised: 6 May 2005, JEL Classification Numbers: C91, D81.We thank Pavlo Blavatskyy and Thorsten Hens for their helpful remarks regarding our paper. Moreover, we thank the referee for his constructive suggestions. This research was supported by the University Research Priority Program “Finance and Financial Markets” a research instrument of the University of Zürich.     

The subjective expected utility hypothesis and revealed preference

Summary A choice behavior of a decision maker is said to satisfy the subjective expected utility hypothesis if there exist a utility and a subjective probability such that a decision maker chooses acts with the highest expected utility. We present a revealed preference characterization of choice behavior that is consistent with the subjective expected utility hypothesis. Our result applies to any state space and compact sets of prizes and observations (either finite or infinite).     

Decision Theory without Logical Omniscience: Toward an Axiomatic Framework for Bounded Rationality

I propose modelling boundedly rational agents as agents who are not logically omniscient—that is, who do not know all logical or mathematical implications of what they know. I show how a subjective state space can be derived as part of a subjective expected utility representation of the agent's preferences. The representation exists under very weak conditions. The representation uses the familiar language of probability, utility, and states of the world in the hope that this makes this model of bounded rationality easier to use in applications.     

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.     

Ambiguity Without a State Space

Many decisions involve both imprecise probabilities and intractable states of the world. Objective expected utility assumes unambiguous probabilities; subjective expected utility assumes a completely specified state space. This paper analyses a third domain of preference: sets of consequential lotteries. Using this domain, we develop a theory of objective ambiguity without explicit reference to any state space. We characterize a representation that integrates a non-linear transformation of first-order expected utility with respect to a second-order measure. The concavity of the transformation and the weighting of the measure capture ambiguity aversion. We propose a definition for comparative ambiguity aversion.     

Testable implications of subjective expected utility theory

I show that the predictive content of the hypothesis of subjective expected utility maximization critically depends on what the analyst knows about the details of the problem a particular decision maker faces. When the analyst does not know anything about the agent's payoffs or beliefs and can only observe the sequence of actions taken by the decision maker any arbitrary sequence of actions can be implemented as the choice of an agent that solves some intertemporal utility maximization problem under uncertainty.     

Effects of instantaneous groundwater availability on irrigated agriculture and implications for aquifer management

Groundwater is an important input for agricultural production in many parts of the world. Aquifer depletion has been shown to affect the rate that groundwater can be extracted from an aquifer. In this paper, we develop an analytical framework that accounts explicitly for the effects of limited instantaneous groundwater extraction rate (well capacity) on a producer's irrigation decisions. We show that limited well capacities can affect the producer’s groundwater use and profit. We draw three important insights from these findings. First, we demonstrate that the price elasticity of demand for groundwater is higher for lower well capacities. Second, farmers’ irrigation decisions are non-monotonic with respect to well capacity and climate conditions. Under a drier climate, producers with greater well capacities increase their groundwater use, and producers with lower well capacities reduce their water use. Third, through numerical analysis, we show that considering spatial heterogeneity in well capacities is important for estimating the cost-effectiveness and distributional impacts of groundwater management policies. Our results shed new light on the importance of extraction capacity for groundwater management policies and the potential impacts of climate change on agricultural production.     

Justifiable preferences

We characterize preferences over acts that can be represented by a utility function and a multiple-prior, such that an act f is preferred to act g if there is a prior under which the expected utility induced by f is higher than that induced by g. These preferences are referred to as justifiable preferences. We further introduce a generalized model of ambiguity that involves a collection of multiple-priors, namely, multiple multiple-priors and incorporate Bewley?s Knightian model in justifiability: f is preferred to g if, according to at least one set of priors, f is unanimously preferred to g.