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.     

A SMOOTH AMBIGUITY MODEL OF THE COMPETITIVE FIRM

This paper examines the optimal production decision of the competitive firm under price uncertainty when the firm's preferences exhibit smooth ambiguity aversion. Ambiguity is modeled by a second‐order probability distribution that captures the firm's uncertainty about which of the subjective beliefs govern the price risk. Ambiguity preferences are modeled by the (second‐order) expectation of a concave transformation of the (first‐order) expected utility of profit conditional on each plausible subjective distribution of the price risk. Within this framework, we derive necessary and sufficient conditions under which the ambiguity‐averse firm optimally produces less in response either to the introduction of ambiguity or to greater ambiguity aversion when ambiguity prevails. In the case that the price risk is binary, we show that ambiguity and greater ambiguity aversion always adversely affect the firm's production decision.     

Collusion and discrimination in organizations

In a principal-multi-agent setting we investigate how optimal contracts should be modified under relative performance evaluation when agents collude. Agents may write side-contracts, which are not contingent on their effort choices but indirectly control them through side-transfers. We show that the optimal collusion-proof contract is to introduce a “discriminatory policy” in the sense that the wage schemes offered to agents depend on their identities even if they are identical with respect to productive abilities. Such discriminatory wage schemes explain the organizational strategy of “divide and conquer” as an optimal response to collusion.     

Auctions with uncertain numbers of bidders

We investigate bidders’ and seller's responses to ambiguity about the number of bidders in the first price auction (FPA) and the second price auction (SPA) with independent private valuations. We model ambiguity aversion using the maxmin expected utility model. We find that bidders prefer the number of bidders to be revealed in the FPA, are indifferent between revealing and concealing in the SPA, and prefer the SPA to the FPA. If bidders are more pessimistic than the seller then the seller prefers to conceal the number of bidders in the FPA, and prefers the FPA to the SPA.     

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.     

Observational equivalence and nonequivalence of subjective and robust mean–variance preferences

We identify the conditions where robust mean–variance preferences, which capture ambiguity aversion, are observationally nonequivalent to subjective mean–variance preferences. Conversely, we also provide an example showing that observational equivalence holds regardless of the degree of ambiguity aversion.     

A dynamic mechanism and surplus extraction under ambiguity

We study the question of auction design in an IPV setting characterized by ambiguity. We assume that the preferences of agents exhibit ambiguity aversion; in particular, they are represented by the epsilon-contamination model. We show that a simple variation of a discrete Dutch auction can extract almost all surplus. This contrasts with optimal auctions under IPV without ambiguity as well as with optimal static auctions with ambiguity—in all of these, types other than the lowest participating type obtain a positive surplus. An important point of departure is that the modified Dutch mechanism is dynamic rather than static, establishing that under ambiguity aversion—even when the setting is IPV in all other respects—a dynamic mechanism can have additional bite over its static counterparts. A further general insight is that the standard revelation principle does not automatically extend to environments not characterized by subjective expected utility.     

The optimality of nominal contracts

Summary. This paper presents a model in which agents choose to use money as a medium of exchange, a means of payment, and a unit of account. The paper defines conditions under which nominal contracts, promising future payment of a fixed number of units of fiat money, prove to be the optimal contract form in the presence of either relative or aggregate price risk. When relative prices are random, nominal contracts are optimal if individuals have ex ante similar preferences over future consumption. When the aggregate price level is random, whether from shocks to the money supply or aggregate output, nominal contracts (perhaps coupled with equity contracts) lead to optimal risk-sharing if individuals have the same degree of relative risk aversion. Finally, nominal contracts may be optimal if the repayment of contracts is subject to a binding cash-in-advance constraint. In this case, a contingent contract increases the risk of holding excessive cash balances. Received: March 29, 1996; revised version: February 25, 1997     

Recursive smooth ambiguity preferences

This paper axiomatizes an intertemporal version of the Smooth Ambiguity decision model developed in [P. Klibanoff, M. Marinacci, S. Mukerji, A smooth model of decision making under ambiguity, Econometrica 73 (6) (2005) 1849-1892]. A key feature of the model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective beliefs, and ambiguity attitude, a characteristic of the decision maker's tastes. In applications one may thus specify/vary these two characteristics independent of each other, thereby facilitating richer comparative statics and modeling flexibility than possible under other models which accommodate ambiguity sensitive preferences. Another key feature is that the preferences are dynamically consistent and have a recursive representation. Therefore techniques of dynamic programming can be applied when using this model.     

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.     

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.     

Interim efficiency with MEU-preferences

Recently Kajii and Ui (2009) [17] proposed to characterize interim efficient allocations in an exchange economy under asymmetric information when uncertainty is represented by multiple posteriors. When agents have Bewley's incomplete preferences, Kajii and Ui (2009) [17] proposed a necessary and sufficient condition on the set of posteriors. However, when agents have Gilboa-Schmeidler's MaxMin expected utility preferences, they only propose a sufficient condition. The objective of this paper is to complete Kajii and Ui's work by proposing a necessary and sufficient condition for interim efficiency for various models of ambiguity aversion and in particular MaxMin expected utility. Our proof is based on a direct application of some results proposed by Rigotti, Shannon and Stralecki (2008) [24].     

Higher-order generalizations of Arrow-Pratt and Ross risk aversion: A comparative statics approach

We analyze comparative risk aversion in a new way, through a comparative statics problem in which, for a cost, agents can shift from an initial probability distribution toward a preferred distribution. The Ross characterization arises when the original distribution is riskier than the preferred distribution and the cost is monetary, and the Arrow-Pratt characterization arises when the original distribution differs from the preferred distribution by a simple mean-preserving spread and the cost is a utility cost. Higher-order increases in risk lead to higher-order generalizations, and the comparative statics method yields a unified approach to the problem of comparative risk attitudes.     

Scientific Ambiguity and Climate Policy

Economic evaluation of climate policy traditionally treats uncertainty by appealing to expected utility theory. Yet our knowledge of the impacts of climate policy may not be of sufficient quality to be described by unique probabilistic beliefs. In such circumstances, it has been argued that the axioms of expected utility theory may not be the correct standard of rationality. By contrast, several axiomatic frameworks have recently been proposed that account for ambiguous knowledge. In this paper, we apply static and dynamic versions of a smooth ambiguity model to climate mitigation policy. We obtain a general result on the comparative statics of optimal abatement and ambiguity aversion, and then extend our analysis to a more realistic, dynamic setting, where we introduce scientific ambiguity into the well-known DICE model of the climate-economy system. For policy-relevant exogenous mitigation policies, we show that the value of emissions abatement increases as ambiguity aversion increases, and that this ‘ambiguity premium’ can in some plausible cases be very large. In these cases the effect of ambiguity aversion on welfare is comparable to that of other much studied welfare parameters. Thus ambiguity aversion may be an important neglected aspect of climate change economics, and seems likely to provide another argument for strong abatement policy.     

The dual theory of the smooth ambiguity model

This paper studies the “dual” theory of the smooth ambiguity model introduced by Klibanoff et al. (Econometrica 73:1849–1892, 2005). Unlike the original model, we characterize attitudes toward ambiguity captured by second-order probabilities. First, we give a set of axioms to derive a dual representation of the smooth ambiguity model. Second, we present a characterization of ambiguity aversion. Last, as an application of our dual model to a portfolio problem, we conduct comparative static predictions which give sufficient conditions to guarantee that an increase in smooth ambiguity aversion decreases the optimal portfolio.     

A theory of subjective compound lotteries

We develop a Savage-type model of choice under uncertainty in which agents identify uncertain prospects with subjective compound lotteries. Our theory permits issue preference; that is, agents may not be indifferent among gambles that yield the same probability distribution if they depend on different issues. Hence, we establish subjective foundations for the Anscombe-Aumann framework and other models with two different types of probabilities. We define second-order risk as risk that resolves in the first stage of the compound lottery and show that uncertainty aversion implies aversion to second-order risk which implies issue preference and behavior consistent with the Ellsberg paradox.     

A note on comparative downside risk aversion

We provide comparative global conditions for downside risk aversion, which are similar to the ones studied by Ross for risk aversion. We define a coefficient of downside risk aversion, and study its local properties.     

Dragon Slaying with Ambiguity: Theory and Experiments

This paper studies the impact of ambiguity in the best‐shot and weakest‐link models of public good provision. The models are first analyzed theoretically. Then, we conduct experiments to study how ambiguity affects behavior in these games. We test whether subjects' perception of ambiguity differs between a local opponent and a foreign one. We find that an ambiguity‐safe strategy is often chosen by subjects. This is compatible with the hypothesis that ambiguity aversion influences behavior in games. Subjects tend to choose contributions above (respectively, below) the Nash equilibrium in the best‐shot (respectively, weakest‐link) model.     

Sealed bid auctions with ambiguity: Theory and experiments

This study presents a laboratory experiment of the first and second price sealed bid auctions with independent private values, where the distribution of bidder valuations may be unknown. In our experimental setting, in first price auctions, bids are lower with the presence of ambiguity. This result is consistent with ambiguity loving in a model that allows for different ambiguity attitudes. We also find that the first price auction generates significantly higher revenue than the second price auction with and without ambiguity.     

INCOMPLETE INFORMATION GAMES WITH MULTIPLE PRIORS*

We present a model of incomplete information games, where each player is endowed with a set of priors. Upon arrival of private information, it is assumed that each player “updates” his set of priors to a set of posterior beliefs, and then evaluates his actions by the most pessimistic posterior beliefs. So each player's preferences may exhibit aversion to ambiguity or uncertainty. We define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players’ views on uncertainty about the environment affect the strategic outcomes.