Ambiguity in a Two-Country World

Irasema Alonso

The correlation between consumption levels in different countries is much lower than what is suggested by models of efficient risk sharing with common beliefs. Relatedly, observed asset portfolios of consumers in different countries suggest a "bias" toward home-country securities, even for countries where financial markets are quite well developed. This paper examines a mechanism that can generate these observations by considering preferences that allow ambiguity aversion of the sort illustrated by the Ellsberg Paradox. A key assumption is that the home consumer is more ambiguous about the process generating uncertainty in the foreign country than about that in the home country. This permits formalization of a statement like "I don't hold foreign stocks because I don't know much about them." The specific context here is that of a two-country dynamic general equilibrium model. It is shown that the model generates low consumption correlations, higher output correlations, biased financial portfolios, and biased real investment flows.

Sharing Beliefs: Between Agreeing and Disagreeing

Antoine Billot

In an exchange economy with no aggregate uncertainty and Bayesian agents, Pareto optimal allocations provide full insurance if and only if the agents have a common prior. It is hard to explain why there is relatively so little betting taking place. One is led to ask: when are full insurance allocations optimal for uncertainty averse agents? It turns out that commonality of beliefs, appropriately defined, is key again. Specifically, consider agents who are uncertainty averse and who maximize the minimal expected utility according to a set of possible priors. Pareto optimal allocations provide full insurance if and only if the agents share at least one prior. In the proof of this result, we develop a separation theorem among a number "n" of convex sets, that might be of independent interest.

"Coarse Contingencies"

Larry Epstein

Consider an agent who must choose an action today under uncertainty about the consequence of any chosen action but without having in mind a complete list of all the con tangencies that could influence outcomes. She conceives of some relevant (subjective) contingencies or states of the world but she is aware that these contingencies are coarse — they leave out some details that may affect outcomes. Though she may not be able to describe these finer details, she is aware that they exist and this may affect her behavior. How does one model such an agent?

The standard Savage framework, based on a primitive state space, is inappropriate for two reasons. First, in the Savage model each state is a complete description of the world — it determines a unique outcome for any chosen action. Second, even if we knew how to model a "coarse or incomplete state" and we redefined the Savage state space accordingly, the resulting approach would still be unsatisfactory if, as in Savage, the state space is adopted as a primitive. In that case, the state space is presumed observable by the modeler, an assumption that is all the more problematic when states are coarse. Ideally, the agent’s conceptualization of the future should be subjective — it should be derived from preference.

Kreps (1979, 1992) and Dekel, Lipman and Rustichini (2001) have rendered the state space subjective, thus addressing the second concern. However, we argue in this paper that their models do not capture coarse perceptions. We focus primarily on the model of Dekel, Lipman and Rutichini. They describe (p. 893) the agent they are modeling: "…she sees some relevant considerations, but knows there may be others that she cannot specify. For simplicity, we assume henceforth that the agent conceives of only one situation, ‘something happens,’ but knows that her conceptualization is incomplete." Though they frequently refer to "unforeseen contingencies", it seems that, at least in part, they have in mind what we prefer to call "coarse contingencies." Later (pp. 919-20), they describe what is needed for a critique of their model: "… just as Ellsberg identified the role of the sure-thing-principle in precluding uncertainty-averse behavior, we believe that one must find a concrete example of behavior that is a sensible response to unforeseen contingencies but that is precluded by our axioms. An important direction for further research is to see if there is such an Ellsbergian example for this setting and, if so, to explore relaxations of our axioms." This is the direction we pursue here.

Can Case-Based Decision Makers Survive in a Financial Market?

Ani Guerdjikova

The paper explores whether case-based decision makers (CBDM) can survive in an asset market and influence asset prices in the presence of expected utility maximizers. Conditions are identified, under which the CBDM retain a positive mass with probability one. CBDM can cause predictability of asset returns, high volatility and bubbles. It is found that the expected utility maximizers can disappear from the market for a finite period of time, if the mispricing of the risky asset caused by the CBDM aggravates too much. Only expected utility maximizers with a logarithmic utility function are able to drive the CBDM out of the market for all parameter values.

Extensive Robustness Revisited

Ehud Kalai

In "Large Robust Games" (Econometrica, forthcoming), Kalai shows that inone-simultaneous-move Baysian games with many semi-anonymous players all the equilibria are (approximately) extensively robust. This means that the equilibria survive even if the one-simultaneous-move assumption is relaxed to allow for many variations. These include sequential moves with repeated revisions of choices, information leakage, commitments, cheap talk and more. In addition to its own interest, the above result has the following implications:

1. A stronger purification result than Schmeidler's about the equilibria of Normal form games with a continuum of anonymous players,
2. Ex-post stability property of games with many players, and
3. Strong rational expectations property for market games with independent types.

The current paper shows that in the same class of games studied above oneobtains a significantly stronger, yet simpler to describe, version of the extensive robustness property.

Edi Karni

This paper extends the analysis of incentive schemes, designed to mitigate the welfare loss associated with moral hazard, to the case in which the principal and the agent are maxmin expected utility players. Invoking the parametrized distribution approach to agency theory, the paper examines the axiomatic foundations of the principal's and agent's choice behaviors that are representable by the maximization of the minimum expected utility over action-dependent sets of priors. In the context of this model, the paper also discusses some implications of ambiguity aversion for the design of optimal incentive schemes.

Auctions with Uncertain Numbers of Bidders

Dan Levin

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.

Dense Types

Stephen Morris

A strategic topology on higher order belief types says that two types are close if they behave in similar ways in similar situations. An upper (lower) strategic topology on types is the finest topology generating upper (lower) hemicontinuity of strategic outcomes. We show that the upper strategic topology is equivalent to the product topology and the lower strategic topology is strictly finer. Nonetheless, the set of "finite types" (types describable by finite type space) are dense in the lower strategic topology.

Search and Knightian Uncertainty

Kiyohiko Nishimura and Hiroyuki Ozaki

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.

Optimal Auctions with Ambiguity

Emre Ozdenoren

A crucial assumption in the optimal auction literature is that each bidder’s valuation is known to be drawn from a single unique distribution. In this paper we study the optimal auction problem allowing for ambiguity about the distribution of valuations and where the agents may display ambiguity aversion (modeled using the maxmin expected utility model of Gilboa and Schmeidler (1989). When the bidders are ambiguity averse with all types having the same set of priors (and the seller is ambiguity neutral) we show that (i) the optimal incentive compatible and individually rational mechanism must be such that for each type of bidder the minimum expected utility is attained by using the seller’s prior; (ii) an auction that provides full insurance to all types of bidders is always in the set of optimal auctions; (iii) and in particular, when the bidders’ set of priors is the epsilon-contamination of the seller’s prior the unique optimal auction provides full insurance to bidders of all types; (iv) in general, neither the first nor the second price auction is optimal (even with suitably chosen reserve prices). When the seller is ambiguity averse (and the bidders are ambiguity neutral) the optimal auction involves the seller being perfectly insured.

Next we look at the case where different types may have different sets of priors. In this case, we show that if lower types are more ambiguity averse than higher types, full surplus extraction is not possible. Furthermore, the impossibility of full surplus extraction result is robust to small perturbations of bidders’ sets of priors.

Consumption Commitments and Preferences for Risk

Andrew Postlewaite

We examine an economy in which the cost of consuming some goods can be reduced by making commitments to consumption levels independent of the state. For example, it is cheaper to produce housing services via owner-occupied than rented housing, but the transactions costs associated with the former prompt relatively inflexible housing consumption paths. We show that consumption commitments can cause risk-neutral consumers to care about risk, creating incentives to both insure risks and bunch uninsured risks together. For example, workers may prefer to avoid wage risk while bearing an unemployment risk that is concentrated in as few states as possible.

Coping with Imprecise Information

Jean-Marc Tallon

We provide a model of decision making under uncertainty in which the decision maker reacts to imprecision of the available data. Data is represented by a set of probability distributions. We axiomatize a decision criterion of the maxmin expected utility type, in which the revealed set of priors explicitly depends on the available data. We then characterize notions of comparative aversion to imprecision of the data as well as traditional notions of risk aversion. Interestingly, the study of comparative aversion to imprecision can be done independently of the utility function, which embeds risk attitudes. We also give a more specific result, in which the functional representing the decision maker's preferences is the convex combination of the minimum expected utility with respect to the available data and expected utility with respect to a subjective probability distribution, interpreted as a reference prior. This particular form is shown to be equivalent to some form of constant aversion to imprecision. We finally provide examples of applications first to unanimity rankings of imprecision and risk and then to optimal risk sharing arrangements.

Two-Fund Separation under Homogeneous Ambiguity

Kats Wakai

This paper derives two-fund separation for an economy with agents of multiple-priors utility: Given homogeneous sets of priors and linear risk tolerance felicity functions, an interior equilibrium is Pareto efficient and all agents hold a combination of the riskless security and the market portfolio (i.e., aggregate endowment). This result permits the construction of a representative agent whose preferences follow the multiple-priors model with a utilitarian von-Neumann–Morgenstern utility function.

Non-Additive Probability Theory in Insurance

Peter Wakker

Schmeidler’s discovery of a sensible theoretical manner to derive decisions from non-Bayesian beliefs has led to a flurry of applications, such as a theoretical correction of prospect theory and, more importantly, an extension of this theory to the prevailing context of unknown probabilities ("ambiguity"). Whereas the views on risk and ambiguity attitude of the great majority of economists are still based on expected utility, considerable refinements and improvements are possible through Schmeidler's model, after which for instance the coexistence of gambling and insurance is no longer a paradox but can be explained naturally. This study reports on a consultancy job for an insurance company where risk and ambiguity attitudes of clients were crucial factors, and where the modern views on those attitudes were instrumental in understanding data, and in the policy recommendations derived there from.

A remarkable finding in this field study was considerable risk seeking. This would be inexplicable under classical expected utility, but can be explained by means of the modern models. More remarkable is the extensive ambiguity seeking that was found.  Most of the empirical support for ambiguity aversion derives from experimental studies in laboratories with artificial stimuli such as Ellsberg’s urns with proportions of balls deliberately kept secret to the subjects. Our study dealt with natural decisions in a field study. In natural environments we commonly deal with unknown probabilities, and we have no special aversion to them.