Representation and aggregation of preferences under uncertainty

We axiomatize in the Anscombe-Aumann setting a wide class of preferences called rank-dependent additive preferences that includes most known models of decision under uncertainty as well as state dependent versions of these models. We prove that aggregation is possible and necessarily linear if and only if (society's) preferences are uncertainty neutral. The latter means that society cannot have a non-neutral attitude toward uncertainty on a subclass of acts. A corollary to our theorem is that it is not possible to aggregate multiple prior agents, even when they all have the same set of priors. A number of ways to restore the possibility of aggregation are then discussed.     

Interim efficient allocations under uncertainty

This paper considers an exchange economy under uncertainty with asymmetric information. Uncertainty is represented by multiple priors and posteriors of agents who have either Bewley's incomplete preferences or Gilboa-Schmeidler's maximin expected utility preferences. The main results characterize interim efficient allocations under uncertainty; that is, they provide conditions on the sets of posteriors, thus implicitly on the way how agents update the sets of priors, for non-existence of a trade which makes all agents better off at any realization of private information. For agents with the incomplete preferences, the condition is necessary and sufficient, but for agents with the maximin expected utility preferences, the condition is sufficient only. A couple of necessary conditions for the latter case are provided.     

Sequentially consistent rules of choice under complete uncertainty

This work analyzes the problem of individual choice under complete uncertainty. In this context, each alternative action consists of a set of different possible outcomes with no associated probability distribution. The work examines and defines a class of rules such that: (a) the evaluation of sets (actions) follows a certain procedural pattern; and (b) an assumption of sequential contraction consistency, which is an adaptation of Sen's α condition, is satisfied. In this framework, some results of characterization show that several well-known rules for comparing sets of outcomes can be reinterpreted in procedural terms.     

The rationality and efficacy of decisions under uncertainty and the value of an experiment

Summary A decision maker faces a known prior distribution over payoff relevant states. We compare the expected utility of this individual under two scenarios. In the first, the decision maker makes a choice without further information. In the second, the decision maker has access to an experiment before choosing an action. However, the decision maker does not know the true joint distribution over states and messages. The value of the experiment as measured by the difference in the two utility levels can be negative as well as positive. We give a condition which is necessary and sufficient for the experiment to be valuable in our sense, for any decision problem.An earlier version of this paper was circulated under the title Noisy Bayes Updating and the Value of Information. We have gained from the comments of Stephen Coate, John Geanakoplos, Larry Samuelson, Timothy Van Zandt and seminar participants at Harvard Business School, Princeton, Boston University, the international conference on game theory at Stony Brook 1992 and the Winter meeting of the Econometric Society at Anaheim 1993. The first author received support for this project from NSF grant #SES-9308515 and a University of Pennsylvania Research Foundation Grant.     

A note on free entry under uncertainty: The role of asymmetric information

In a model of competing managerial firms I show that the equilibrium number of firms decreases with uncertainty if entry is relatively more costly than monitoring. The result adds to the earlier contributions and is consistent with the available evidence.     

Intertemporal equilibria with Knightian uncertainty

We study a dynamic and infinite-dimensional model with incomplete multiple prior preferences. In interior efficient allocations, agents share a common risk-adjusted prior and subjective interest rate. Interior efficient allocations and equilibria coincide with those of economies with subjective expected utility and priors from the agents? multiple prior sets. A specific model with neither risk nor uncertainty at the aggregate level is considered. Risk is always fully insured. For small levels of ambiguity, there exists an equilibrium with inertia where agents also insure fully against Knightian uncertainty. When the level of ambiguity exceeds a critical threshold, full insurance no longer prevails and there exist equilibria with inertia where agents do not insure against uncertainty at all. We also show that equilibria with inertia are indeterminate.     

Mean-dispersion preferences and constant absolute uncertainty aversion

We axiomatize, in an Anscombe–Aumann framework, the class of preferences that admit a representation of the form V(f)=μ−ρ(d)$V(f)=\mu -\rho (d)$, where μ is the mean utility of the act f with respect to a given probability, d   is the vector of state-by-state utility deviations from the mean, and ρ(d)$\rho (d)$ is a measure of (aversion to) dispersion that corresponds to an uncertainty premium. The key feature of these mean-dispersion   preferences is that they exhibit constant absolute uncertainty aversion. This class includes many well-known models of preferences from the literature on ambiguity. We show what properties of the dispersion function ρ(⋅)$\rho (\cdot )$ correspond to known models, to probabilistic sophistication, and to some new notions of uncertainty aversion.     

Precaution under mixed uncertainty: Implications for environmental management

A model of pollution control subject to two types of uncertainty is presented. First, the natural decay of the pollution stock follows stochastic dynamics that drives a diffusion pollution process (“stochastic uncertainty”). Moreover, the damage coefficient which determines the amount of damage corresponding to each pollution stock can undergo an abrupt increase at some uncertain future time (“event uncertainty”). The model admits an explicit and simple dynamic characterization of the optimal emission rule and the ensuing pollution process. When only one type of uncertainty is permitted (by setting either the variance of the stochastic process or the hazard rate for the damage increase to zero) it acts to promote the intuitive response of precaution. However, allowing the two types to interact gives rise to a non-monotonic behavior, whereby increasing the stochastic variance first enhances, then diminishes the response to the hazard. The results confirm and expand recent findings based on discrete-time formulations.     

Introduction to model uncertainty and robustness

This article introduces the symposium on model uncertainty and robustness.     

A revealed-preference implication of weighted utility decisions under uncertainty

Summary. In a novel formulation of revealed preference analysis, Green and Osband [8] show that for expected-utility maximizers, acts partition the state-simplex into linear polyhedral blocks. The question naturally arises whether this characterization distinguishes expected utility theory from non-expected utility theories. This paper investigates the weighted utility theory of Chew [2] and shows that the corresponding partition is systematically different from the expected utility theory: the boundaries of the partition blocks are quadratic rather than linear. This result contains useful empirical contents. Received: May 15, 1995; revised version: November 22, 1996     

Small worlds: Modeling attitudes toward sources of uncertainty

We introduce the concept of a conditional small world event domain—an extension of Savage's [The Foundations of Statistics, Wiley, New York, 1954] notion of a ‘small world’—as a self-contained collection of comparable events. Under weak behavioral conditions we demonstrate probabilistic sophistication in any small world event domain without relying on monotonicity or continuity. Probabilistic sophistication within, though not necessarily across, small worlds provides a foundation for modeling a decision maker that has source-dependent risk attitudes. This also helps formalize the idea of source preference and suggests an interpretation of ambiguity aversion, often associated with Ellsberg-type behavior, in terms of comparative risk aversion across small worlds.     

Knightian uncertainty and moral hazard

This paper presents a principal-agent model in which the agent has imprecise beliefs. We model this situation formally by assuming the agent?s preferences are incomplete as in Bewley (1986) [2]. In this setting, incentives must be robust to Knightian uncertainty. We study the implications of robustness for the form of the resulting optimal contracts. We give conditions under which there is a unique optimal contract, and show that it must have a simple flat payment plus bonus structure. That is, output levels are divided into two sets, and the optimal contract pays the same wage for all output levels in each set. We derive this result for the case in which the agent?s utility function is linear and then show it also holds if this utility function has some limited curvature.     

Search and Knightian uncertainty

Suppose that “uncertainty” about labor market conditions has increased. Does this change induce an unemployed worker to search longer, or shorter? This paper shows that the answer is drastically different depending on whether an increase in “uncertainty” is an increase in risk or that in true uncertainty in the sense of Frank Knight. We show in a general framework that, while an increase in risk (the mean-preserving spread of the wage distribution that the worker thinks she faces) increases the reservation wage, an increase in Knightian uncertainty (a decrease in her confidence about the wage distribution) reduces it.     

Ecological-economic viability as a criterion of strong sustainability under uncertainty

Strong sustainability, according to the common definition, requires that different natural and economic capital stocks be maintained as physical quantities separately. Yet, in a world of uncertainty this cannot be guaranteed. To therefore define strong sustainability under uncertainty in an operational manner we propose to use the concept of viability. Viability means that the different components and functions of a dynamic, stochastic system at any time remain in a domain where the future existence of these components and functions is guaranteed with sufficiently high probability. We develop a unifying and general ecological-economic concept of viability that encompasses the traditional ecological and economic notions of viability as special cases. It provides an operational criterion of strong sustainability under a mild form of uncertainty and for medium spatial and temporal scales. We illustrate this concept and demonstrate its usefulness by applying it to livestock grazing management in semi-arid rangelands.     

Preemption in a real option game with a first mover advantage and player-specific uncertainty

In this paper a two-player real option game with a first-mover advantage is analyzed, where payoffs are driven by a player-specific stochastic state variable. It is shown that there exists an equilibrium which has qualitatively different properties from those in standard real option games driven by common stochastic shocks. The properties of the equilibrium are four-fold: (i) preemption does not necessarily occur, (ii) if preemption takes place, the rent-equalization property holds, (iii) for almost all sample paths it is clear ex-ante which player invests first, and (iv) it is possible that both players invest simultaneously, even if that is not optimal. It is argued from simulations that real option games with a common one-dimensional shock do not provide a good approximation for games with player-specific uncertainty, even if these are highly correlated.     

Irreversible investment and Knightian uncertainty

When firms make a decision about irreversible investment, they may not have complete confidence about their perceived probability measure describing future uncertainty. They may think other probability measures perturbed from the original one are also possible. Such uncertainty, characterized by not a single probability measure but a set of probability measures, is called “Knightian uncertainty.” The effect of Knightian uncertainty on the value of irreversible investment opportunity is shown to be drastically different from that of traditional uncertainty in the form of risk. Specifically, an increase in Knightian uncertainty decreases the value of investment opportunity while an increase in risk increases it.     

Optimal consumption and the timing of the resolution of uncertainty

As is well-known, consumers want to accumulate precautionary savings in the face of income risks when their marginal utility is convex (prudence). In this paper, we explore the effect of the timing of the resolution of income uncertainty on savings. An agent faces uncertainty about his income at date t+2. What is the effect of being informed that the uncertainty will be resolved at date t+1 on the consumption at date t? We show that the effect is positive, if and only if, marginal utility is convex (prudence), when either the risk free rate is equal to the rate of pure preference for the present, or when the utility function is HARA. The intuition is that an early resolution of uncertainty allows for time-diversifying the risk. It therefore plays a role similar to a reduction of the income risk, whose effect on savings is negative under prudence.     

Product differentiation and location decisions under demand uncertainty

We investigate Hotelling's duopoly game of location-then-price choices with quadratic transportation costs and uniformly distributed consumers under the assumption that firms are uncertain about the exact location of demand. We characterize the unique equilibrium and the socially optimal locations. Contrary to the individual-level random utility models, location uncertainty is a differentiation force. In equilibrium, increases in the variance of the uncertainty lead to greater differentiation, higher expected equilibrium prices and profits, and a greater welfare loss.     

Sealed bid auctions with uncertainty averse bidders

Summary. Traditional analysis of auctions assumes that each bidder's beliefs about opponents' valuations are represented by a probability measure. Motivated by experimental findings such as the Ellsberg Paradox, this paper examines the consequences of relaxing this assumption in the first and second price sealed bid auctions with independent private values. The multiple priors model of Gilboa and Schmeidler [Journal of Mathematical Economics, 18 (1989), 141–153] is adopted specifically to represent the bidders' (and the auctioneer's) preferences. The unique equilibrium bidding strategy in the first price auction is derived. Moreover, under an interesting parametric specialization of the model, it is shown that the first price auction Pareto dominates the second price auction. Received: December 15, 1995; revised version: February 19, 1997     

Allocation effects of uncertainty on resources in Japan

This paper provides evidence of a negative association between macroeconomic uncertainty and the cross sectional dispersion of investment rate for a panel of Japanese manufacturing firms. We show that an increase in uncertainty leads to the narrowing of the cross section dispersion of investment rate and vice versa. This finding suggests that firms’ fixed capital investment behavior becomes more homogeneous in times of increased uncertainty.