Rationalizability and the savage axioms

Summary. Suppose there is a finite set of acts defined on a finite state space and a decision maker chooses an act from the set. In this setting, the subjective expected utility model is observationally indistinguishable from all models of preference that satisfy Savage's axiom P3. The result has implications also for rationalizability in strategic games. Received: September 18, 1998; revised version: January 29, 1999     

When is there state independence?

Whether a preference relation can be represented using state-independent utilities as opposed to state-dependent utilities may depend on which acts count as constant acts. This observation underlies an extension of Savage's expected utility theory to the state-dependent case that was proposed in this journal by Edi Karni. His result contains a condition requiring the existence of a set of acts which can play the role of constant acts and support a representation involving a state-independent utility function. This paper contains necessary and sufficient conditions on the preference relation for such a set of acts to exist. Results are obtained both for the Savage and the Anscombe and Aumann frameworks. Among the corollaries are representation theorems for state-dependent utilities. Relationships to Karni's work and extensions of the results are discussed.     

On the uniqueness of subjective probabilities

Summary The purpose of this paper is twofold: First, within the framework of Savage (1954), we suggest axiomatic foundations for the representation of event-dependent preference relations over acts. This representation has the form of expectation of event-debendent utility with respect to non-unique subjective probabilities on the set of states. Second, we give an economic-theoretic motivation for selecting a unique probability distribution as an appropriate concept of subjective probabilities. However, unlike in Savage's theory, this notion of subjective probabilities does not necessarily represent the decisions-maker's belief regarding the likelihood of events.Our approach involves a departure from Savage's postulate P4, which guarantees the completeness of Savage's likelihood relation on the set of all events. Instead, we assume the existence of a finite partition of the set of states, {S ₁,...S _n}, such that, for events within each element of this partition P4 is satisfied. This weakening of Savage's axioms suffices for the existence of an expected event-dependent utility representation, but not for the uniqueness of the subjective probabilities.In many economic problems involving decision-making under uncertainty the existence of a unique probability is presumed and, in fact, is essential for the statement of the result. An example is Arrow's (1965) finding that all risk averse decision-makers will invest in a risky asset provided its expected rate of return exceeds that of an alternative risk-free asset. We show that a unique probability distribution can be chosen so as to render such results meaningful. Namely, any risk averse decision-maker will hold a positive position in the risky asset if and only if its expected rate of return with respect to the chosen probability exceeds that of the riskless asset.The research described in this paper began while the authors visited the Mathematisches Forschungsinstitut Oberwolfach, Germany. It was carried out in part while the second author visited the Santa Fe Institute in Sante Fe, New Mexico, USA and the Autonomous University of Barcelona, Spain. The second author would also like to acknowledge the financial support by NSF grant 911873.     

Bayesian beliefs with stochastic monotonicity: An extension of Machina and Schmeidler

Machina and Schmeidler show that the probabilistic sophistication can be obtained in an Anscombe–Aumann setting without imposing expected utility by maintaining stochastic monotonicity and adding a new axiom loosely analogous to Savage's P4. This analogous axiom, however, is very strong. In this note, we obtain probabilistic sophistication using a weaker (and more natural) analog of Savage's P4. Stochastic monotonicity is sufficient to bridge the gap, where Anscombe and Aumman use independence twice, we use stochastic monotonicity twice.     

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.     

Preference for flexibility in infinite horizon problems

Summary. We consider the extension of the classical problem of preference for flexibility to many periods. Preferences are defined over sets of infinite paths of choices. The main result provides a set of axioms on preferences that yield an additive representation over a subjective state space. This space is the set of preferences over choice today and feasible set tomorrow. The main new axiom, stochastic dominance, is a stronger form of the assumption of monotonicity. Received: September 11 2000; revised version: December 18, 2001     

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     

Expected utility theory without the completeness axiom

We study the problem of obtaining an expected utility representation for a potentially incomplete preference relation over lotteries by means of a set of von Neumann-Morgenstern utility functions. It is shown that, when the prize space is a compact metric space, a preference relation admits such a multi-utility representation provided that it satisfies the standard axioms of expected utility theory. Moreover, the representing set of utilities is unique in a well-defined sense.     

On the existence of expected multi-utility representations

We prove the following facts related to the expected multi-utility representation of an affine preorder: If the prize space is not compact and if the lottery set consists of all probabilities on the prize space, standard independence and continuity axioms do not guarantee the existence of a (continuous) representation. If the prize space is σ-compact and lotteries have compact support, a representation exists. When the preorder in question is bounded, this result extends to the set of lotteries that consists of all probabilities on the prize space. For the case of monetary lotteries, the boundedness assumption in this last result can be dropped, provided that the preference relation at hand is monotone and risk-averse.     

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.     

Attitude toward imprecise information

This paper presents an axiomatic model of decision making under uncertainty which incorporates objective but imprecise information. Information is assumed to take the form of a probability–possibility set, that is, a set P of probability measures on the state space. The decision maker is told that the true probability law lies in P and is assumed to rank pairs of the form (P,f) where f is an act mapping states into outcomes. The key representation result delivers maxmin expected utility (MEU) where the min operator ranges over a set of probability priors—just as in the MEU representation result of Gilboa and Schmeidler [Maxmin expected utility with a non-unique prior, J. Math. Econ. 18 (1989) 141–153]. However, unlike the MEU representation, the representation here also delivers a mapping, , which links the probability–possibility set, describing the available information, to the set of revealed priors. The mapping is shown to represent the decision maker's attitude to imprecise information: under our axioms, the set of representation priors is constituted as a selection from the probability–possibility set. This allows both expected utility when the selected set is a singleton and extreme pessimism when the selected set is the same as the probability–possibility set, i.e., is the identity mapping. We define a notion of comparative imprecision aversion and show it is characterized by inclusion of the sets of revealed probability distributions, irrespective of the utility functions that capture risk attitude. We also identify an explicit attitude toward imprecision that underlies usual hedging axioms. Finally, we characterize, under extra axioms, a more specific functional form, in which the set of selected probability distributions is obtained by (i) solving for the “mean value” of the probability–possibility set, and (ii) shrinking the probability–possibility set toward the mean value to a degree determined by preferences.     

Spanning,valuation and options

Summary We model the space of marketed assets as a Riesz space of commoditics. In this setting two altenative characterizations are given of the space of continuous options on a bounded asset,s, with limited liability. The first characterization represents every continuous option ons as the uniform limit of portfolios of calls ons. The second characterization represents an option as a continuous sum (or integral) of Arrow-Debreu securities, with respect tos. The pricing implications of these representations are explored. In particular, the Breeden-Littzenberger pricing formula is shown to be a direct consequence of the integral representation theorem.Research supported in part by NSF Grant SES83-19611     

Regret aversion and opportunity dependence

This paper provides an axiomatic model of decision making under uncertainty in which the decision maker is driven by anticipated ex post regrets. Our model allows both regret aversion and likelihood judgement over states to coexist. Also, we characterize two special cases, minimax regret with multiple priors that generalizes Savage's minimax regret, and a smooth model of regret aversion.     

A new integral for capacities

A new integral for capacities is introduced and characterized. It differs from the Choquet integral on non-convex capacities. The main feature of the new integral is concavity, which might be interpreted as uncertainty aversion. The integral is extended to fuzzy capacities, which assign subjective expected values to random variables (e.g., portfolios) and may assign subjective probability only to a partial set of events. An equivalence between the minimum over sets of additive capacities (not necessarily probability distributions) and the integral w.r.t. fuzzy capacities is demonstrated. The extension to fuzzy capacities enables one to calculate the integral also in cases where the information available is limited to a few events. I wish to thank Eran Hanany, David Schmeidler, Eilon Solan and especially Yaron Azrieli and the anonymous referee of Economic Theory for their helpful comments.     

On the continuity of correspondences on sets of measures with restricted marginals

Summary. Consider the set of probability measures on a product space with the property that all have the same marginal distributions on the coordinate spaces. This set may be viewed as a correspondence, when the marginal distributions are varied. Here, it is shown that this correspondence is continuous. Numerous problems in economics involve optimization over a space of measures where one or more marginal distributions is given. Thus, for this class of problem, Berge's theorem of the maximum is applicable: the set of optimizers is upper-hemicontinuous and the value of the optimal solution varies with the parameters (marginals) continuously. Received: April 23, 1997; revised version: January 16, 1998     

Representing equilibrium aggregates in aggregate games with applications to common agency

An aggregate game is a normal-form game with the property that each playerʼs payoff is a function of only his own strategy and an aggregate of the strategy profile of all players. Such games possess properties that can often yield simple characterizations of equilibrium aggregates without requiring that one solves for the equilibrium strategy profile. When payoffs have a quasi-linear structure and a degree of symmetry, we construct a self-generating maximization program over the space of aggregates with the property that the solution set corresponds to the set of equilibrium aggregates of the original n-player game. We illustrate the value of this approach in common-agency games where the playersʼ strategy space is an infinite-dimensional space of nonlinear contracts. We derive equilibrium existence and characterization theorems for both the adverse selection and moral hazard versions of these games.     

Peer Group Formation in an Adverse Selection Model

This paper develops an adverse selection model where peer group systems are shown to trigger lower interest rates and remove credit rationing in the case where borrowers are uninformed about their potential partners and ex post state verification (or auditing) by banks is costly. Peer group formation reduces interest rates due to a 'collateral effect', namely, cross subsidisation amongst borrowers acts as collateral behind a loan. By uncovering such a collateral effect, this paper shows that peer group systems can be viewed as an effective risk pooling mechanism, and thus enhance efficiency, not just in the full information set up.     

Conditional utility

Summary The motivation for this paper starts out with a decision situation under risk where the decision-maker has to choose among various lottery tickets. We will ask what happens to a person's lottery choice if he is given additional information in terms of probabilities on some states of nature which might affect his choice among lottery tickets. In other words, in evaluating his decision situation, a person should not only consider the probability of a certain prospect to be realized but also the problem how and to which extent some state(s) of nature modify the utility of this prospect. This problem has not been dealt with in Bernoullian utility theory.We state several conditions which are necessary and sufficient to treat conditional utility as Bernoullian utility. Then we show as a main result that it is possible to represent expected utility of decision acts (in Savage's terminology) by conditional expected utility of prospects which preserves well-known properties of expected utility with the exception of linearity.We give a potential application of the notion of conditional utility to the estimation of the value of information as a residual value of prior and posterior utility.     

Core of convex distortions of a probability

This paper characterizes the core of a differentiable convex distortion of a probability measure on a nonatomic space by identifying it with the set of densities which dominate the derivative of the distortion, for second order stochastic dominance. The densities that have the same distribution as the derivative of the distortion are the extreme points of the core. These results are applied to the differentiability of a Yaari's or Rank Dependent Expected utility function. The superdifferential of a Choquet integral at any point is fully characterized. Examples of use of these results in simple models where some agent is a RDEU maximizer are given.     

A dimensionally consistent aggregation framework for biophysical metrics

We develop a formal representation of the economy-ecosystem interaction problem by distinguishing between stock-flow, and fund-flux spaces (Georgescu-Roegen, 1971). We then define dimensionless quantities in both the cardinal stock-flow space and the ordinal fund-flux space. This leads to analytic definitions of natural capital and natural income in the fund-flux space. We show that a stock-fund representation of the economy-ecosystem interaction problem helps investigate aggregation properties of biophysical metrics. In particular, we show how a metric that is dimensionally consistent in the stock-flow space can have dimensional problems in the fund-flux space. Ecological footprint is used as an illustrative example. Finally, we argue that dimensionally consistent metrics are keys to further the development of biophysical assessments as a tool for practical environmental policy.