Maxmin Expected Utility Model Research Articles

Rationalisable belief selection

Considering a decision maker with beliefs from a probability–possibility set representing objective information about uncertainty, this study investigates the behavioural meanings of the assumption that the belief selection correspondence of the Max–min Expected Utility model can be rationalised. It can provide an axiomatic foundation for a class of models in which the decision maker exhibits a preference over probability measures that remains consistent across different probability–possibility sets and generate confirmatory bias.

Strength of preference over complementary pairs axiomatizes alpha-MEU preferences

Preferences over acts have an α-Maxmin Expected Utility (α-MEU) representation if they can be represented by the α-mixture of the worst and the best expected utility over a set of priors. The case α=1 is the Maxmin Expected Utility (MEU) model characterized in Gilboa and Schmeidler (1989).This paper provides the first axiomatic characterization of the α-MEU model in the Anscombe-Aumann framework for all α∈[0,1]﹨{12}. My first axiom is a weakening of Uncertainty Aversion, the second axiom is a strengthening of Obvious Dominance. Both axioms rely on the concept of complementary pairs (Siniscalchi, 2009). I show that the two parameters of the model are uniquely set identified with a one-to-one monotone relationship. The set of α-values that allow a representation have a clear behavioral interpretation and can be elicited from choice data. These results clarify the behavioral implications of the model, which is crucial for decision-theoretic interpretations, applications, and testing/falsification.

Maxmin expected utility in Savage's framework

We axiomatize the maxmin expected utility model in Savage's original framework, which does not require a rich set of consequences nor objective probabilities. The key conditions in our result reformulate Gilboa–Schmeidler's (1989) Uncertainty Aversion and Certainty Independence for Savage's acts. Also, we propose a definition for comparative ambiguity aversion in this context.

A comment on the axiomatics of the Maxmin Expected Utility model

A comment on the axiomatics of the Maxmin Expected Utility model

Random ambiguity

We introduce a model of random ambiguity aversion. Choice is stochastic due to unobserved shocks to both information and ambiguity aversion. This is modeled as a random set of beliefs in the maxmin expected utility model of Gilboa and Schmeidler (1989). We characterize the model and show that the distribution of ambiguity aversion can be uniquely identified from binary choices. A novel stochastic order on random sets is introduced that characterizes greater uncertainty aversion under stochastic choice. If the set of priors is the Aumann expectation of the random set, then choices satisfy dynamic consistency. This corresponds to an agent who knows the distribution of signals but is uncertain about how to interpret signal realizations. More broadly, the analysis of stochastic properties of random ambiguity attitudes provides a theoretical foundation for the study of other random nonlinear utility models.

Auction Design with Ambiguity: Optimality of the First-Price and All-Pay Auctions

We study the optimal auction design problem when bidders' preferences follow the maxmin expected utility model. We suppose that each bidder's set of priors consists of beliefs close to the seller's belief, where closeness is defined by a divergence. For a given allocation rule, we identify a class of optimal transfer candidates, named the win-lose dependent transfers, with the following property: each type of bidder's transfer conditional on winning or losing is independent of the competitor's type report. Our result reduces the infinite-dimensional optimal transfer problem to a two-dimensional optimization problem. By solving the reduced problem, we find that: (i) among efficient mechanisms with no premiums for losers, the first-price auction is optimal; and, (ii) among efficient winner-favored mechanisms where each bidder pays smaller amounts when she wins than loses: the all-pay auction is optimal. Under a simplifying assumption, these two auctions remain optimal under the endogenous allocation rule.

Ambiguity and Probabilistic Information

Experiments detecting ambiguity aversion often rely on the assumption that probabilities are exogenously given for some uncertain events. However, the canonical models that accommodate ambiguity into economic theory, such as the maxmin expected utility (MEU) and Choquet expected utility (CEU) models, are purely subjective. These models do not specify how subjects could incorporate exogenous probabilities into decisions. We study two approaches for embedding exogenous probabilities in the context of the thought experiments suggested by Mark Machina. We show that Machina’s choice behavior entails fundamentally different consequences for the ambiguity models mentioned; although it violates the CEU model, it is consistent with the MEU model. For the latter model, Machina’s experiments can test whether individuals adhere to expected utility for prospects whose consequences occur with the exogenously given probabilities. This paper was accepted by Manel Baucells, decision analysis.

Climate policy: How to deal with ambiguity?

We study the impact of ambiguity and ambiguity attitudes on optimal adaptation and mitigation decisions when the future environmental quality is ambiguous and the decision maker’s (DM) preferences are represented by the $$\alpha $$-Maxmin Expected Utility model. We show that ambiguity aversion plays a significant role in designing an optimal climate policy that is different from risk aversion. We also focus on the induced effects of changes in ambiguity, captured by the arrival of additional information. We state that a change in the informational structure may trigger more efforts of both mitigation and adaptation depending on both the DM’s attitude toward ambiguity and her environmental preferences.

On the falsifiability and learnability of decision theories

We study the degree of falsifiability of theories of choice. A theory is easy to falsify if relatively small data sets are enough to guarantee that the theory can be falsified: the Vapnik–Chervonenkis (VC) dimension of a theory is the largest sample size for which the theory is “never falsifiable.” VC dimension is motivated strategically. We consider a model with a strategic proponent of a theory and a skeptical consumer, or user, of theories. The former presents experimental evidence in favor of the theory; the latter may doubt whether the experiment could ever have falsified the theory.We focus on decision‐making under uncertainty, considering the central models of expected utility, Choquet expected utility, and max–min expected utility models. We show that expected utility has VC dimension that grows linearly with the number of states, while that of Choquet expected utility grows exponentially. The max–min expected utility model has infinite VC dimension when there are at least three states of the world. In consequence, expected utility is easily falsified, while the more flexible Choquet and max–min expected utility are hard to falsify. Finally, as VC dimension and statistical estimation are related, we study the implications of our results for machine learning approaches to preference recovery.

Optimal Insurance under Maxmin Expected Utility

We examine a problem of demand for insurance indemnification, when the insured is sensitive to ambiguity and behaves according to the Maxmin-Expected Utility model of Gilboa and Schmeidler (1989), whereas the insurer is a (risk-averse or risk-neutral) Expected-Utility maximizer. We characterize optimal indemnity functions both with and without the customary ex ante no-sabotage requirement on feasible indemnities, and for both concave and linear utility functions for the two agents. This allows us to provide a unifying framework in which we examine the effects of the no-sabotage condition, marginal utility of wealth, belief heterogeneity, as well as ambiguity (multiplicity of priors) on the structure of optimal indemnity functions. In particular, we show how the singularity in beliefs leads to an optimal indemnity function that involves full insurance on an event to which the insurer assigns zero probability, while the decision maker assigns a positive probability. We examine several illustrative examples, and we provide numerical studies for the case of a Wasserstein and a Renyi ambiguity set.

Risk, ambiguity, and Giffen assets

We provide necessary and sufficient conditions for the demand for financial assets to satisfy the law of demand (prices and quantity demanded move in opposite directions) when preferences take the expected utility, multiplier preferences, Choquet expected utility, and the maxmin expected utility forms. For the first three models the law of demand holds when the variation in the coefficient of relative risk aversion stays within a specified bound. Ensuring the law of demand holds in the maxmin expected utility model requires constant relative risk aversion.

A new approach to the rational expectations equilibrium: existence, optimality and incentive compatibility

Rational expectations equilibrium seeks a proper treatment of behavior under private information by assuming that the information revealed by prices is taken into account by consumers in their decisions. Typically agents are supposed to maximize a conditional expectation of state-dependent utility function and to consume the same bundles in indistiguishable states [see Allen (Econometrica 49(5):1173–1199, 1981), Radner (Econometrica 47(3):655–678, 1979)]. A problem with this model is that a rational expectations equilibrium may not exist even under very restrictive assumptions, may not be efficient, may not be incentive compatible, and may not be implementable as a perfect Bayesian equilibrium (Glycopantis et al. in Econ Theory 26(4):765–791, 2005). We introduce a notion of rational expectations equilibrium with two main features: agents may consume different bundles in indistinguishable states and ambiguity is allowed in individuals’ preferences. We show that such an equilibrium exists universally and not only generically without freezing a particular preferences representation. Moreover, if we particularize the preferences to a specific form of the maxmin expected utility model introduced in Gilboa and Schmeidler (J Math Econ 18(2):141–153, 1989), then we are able to prove efficiency and incentive compatibility. These properties do not hold for the traditional (Bayesian) Rational Expectation Equilibrium.

Allais, Ellsberg, and preferences for hedging

Two of the most well-known regularities of preferences under risk and uncertainty are ambiguity aversion and the Allais paradox. We study the behavior of an agent who can display both tendencies at the same time. We introduce a novel notion of preference for hedging that applies to both objective lotteries and uncertain acts, and captures both aversion to ambiguity and attraction towards certainty in objective lotteries. We show that this axiom, together with other standard axioms, is equivalent to two representations that generalize the MaxMin Expected Utility model of Gilboa and Schmeidler (1989). In both representations the agent evaluates ambiguity using multiple priors, but does not use Expected Utility to evaluate objective lotteries. In the rst representation, lotteries are evaluated by distorting probabilities as in the Rank Dependent Utility model, but using the worst from a set of such distortions. In the second, equivalent, representation the agent treats objective lotteries as ‘ambiguous objects,’ and uses a set of priors to evaluate them. We show that a preference for hedging is not sucient to guarantee an Ellsberg-like behavior if the agent violates Expected

Allais, Ellsberg, and Preferences for Hedging

We study the relation between ambiguity aversion and the Allais paradox. To this end, we introduce a novel de nition of hedging which applies to objective lotteries as well as to uncertain acts, and we use it to de ne a novel axiom that captures a preference for hedging which generalizes the one of Schmeidler (1989). We argue how this generalized axiom captures both aversion to ambiguity, and attraction towards certainty for objective lotteries. We show that this axiom, together with other standard ones, is equivalent to to two representations both of which generalize the MaxMin Expected Utility model of Gilboa and Schmeidler (1989). In both, the agent reacts to ambiguity using multiple priors, but does not use expected utility to evaluate objective lotteries. In our rst representation, the agent treats objective lotteries as `ambiguous objects,' and use a set of priors to evaluate them. In the second, equivalent representation, lotteries are evaluated by distorting probabilities as in the Rank-Dependent Utility model, but using the worst from a set of such distortions. Finally, we show how a preference for hedging is not sucient to guarantee an Ellsberg-like behavior if the agent violate expected utility for objective lotteries. We then provide an axiom that guarantees that this is the case, and nd an associated representation in which the agent rst maps acts to an objective lottery using the worst of the priors in a set; then evaluates this lottery using the worst distortion from a set of concave Rank-Dependent Utility functionals.

Aggregating Tastes, Beliefs, and Attitudes Under Uncertainty

We provide possibility results on the aggregation of beliefs and tastes for Monotone, Bernoullian and Archimedian preferences of Cerreia-Vioglio, Ghirardato, Maccheroni, Marinacci, and Siniscalchi (2011). We propose a new axiom, Unambiguous Pareto Dominance, which requires that if the unambiguous part of individuals’ preferences over a pair of acts agree, then society should follow them. We characterize the resulting social preferences and show that it is enough that individuals share a prior to allow non dictatorial aggregation. A further weakening of this axiom on common-taste acts, where cardinal preferences are identical, is also characterized. It gives rise to a set of relevant priors at the social level that can be any subset of the convex hull of the individuals’ sets of relevant priors. We then apply these general results to the Maxmin Expected Utility model, the Choquet Expected Utility model and the Smooth Ambiguity model. We end with a characterization of the aggregation of ambiguity attitudes.

Underestimation of probabilities modifications: characterization and economic implications

The aim of this paper is to propose a behavioral characterization of individuals who underestimate probabilities modifications and to characterize this behavior in the standard preferences representation models under risk (expected utility, dual theory, rank dependent utility theory and MaxMin expected utility). Our main results are the following. Underreaction to probabilities modifications is in general independent from standard risk aversion and prudence. In models involving probabilities transformation functions, it is characterized by the slope of the probability transformation function. In the MaxMin expected utility model under risk, it is related to the weights of the maximal and minimal consequences in the preferences representation function. Considering a simple prevention decision, consisting in the reduction in the probability of a monetary loss, we show that individuals who underreact to probabilities modifications, invest less in prevention than individuals who objectively evaluate these modifications. Underreaction to probabilities modification is thus a possible explanation for low investment in prevention.

Maxmin expected utility with additivity on unambiguous events

This paper provides a model of beliefs representation in which ambiguity and unambiguity are endogenously distinguished in the maxmin expected utility model of Gilboa and Schmeidler (1989). Specifically, I first extend it by getting a representation of beliefs such that the probabilistic beliefs over each ambiguous event are represented by a nondegenerate interval, while the ones over each unambiguous event are represented by a number. I then suggest a behavioral definition of ambiguity. It provides a choice theoretical foundation for the Knightian distinction between ambiguity and unambiguity.

Allais, Ellsberg, and Preferences for Hedging

We study the relation between ambiguity aversion and the Allais paradox. To this end, we introduce a novel definition of hedging which applies to objective lotteries as well as to uncertain acts, and we use it to define a novel axiom that captures a preference for hedging which generalizes the one of Schmeidler (1989). We argue how this generalized axiom captures both aversion to ambiguity, and attraction towards certainty for objective lotteries. We show that this axiom, together with other standard ones, is equivalent to two representations both of which generalize the MaxMin Expected Utility model of Gilboa and Schmeidler (1989). In both, the agent reacts to ambiguity using multiple priors, but does not use expected utility to evaluate objective lotteries. In our first representation, the agent treats objective lotteries as 'ambiguous objects,' and use a set of priors to evaluate them. In the second, equivalent representation, lotteries are evaluated by distorting probabilities as in the Rank-Dependent Utility model, but using the worst from a set of such distortions. Finally, we show how a preference for hedging is not sufficient to guarantee an Ellsberg-like behavior if the agent violate expected utility for objective lotteries. We then provide an axiom that guarantees that this is the case, and find an associated representation in which the agent first maps acts to an objective lottery using the worst of the priors in a set; then evaluates this lottery using the worst distortion from a set of concave Rank-Dependent Utility functionals.

On the confidence preferences model

In this paper we study the model of decision under uncertainty consistent with confidence preferences. In that model, a decision maker held beliefs represented by a fuzzy set of priors and tastes captured by a standard affine utility index on consequences. First, we find some interesting properties concerning the well-known maxmin expected utility model, taking into account the point of view of the confidence preferences model. Further, we provide new examples of preferences that capture ambiguity-averse attitudes weaker than ambiguity attitudes featured by maxmin expected utility theory. Finally, we discuss the axiomatic foundations for the confidence preferences model with optimistic behavior.

Dynamically consistent updating of multiple prior beliefs – An algorithmic approach

This paper develops algorithms for dynamically consistent updating of ambiguous beliefs in the maxmin expected utility model of decision making under ambiguity. Dynamic consistency is the requirement that ex-ante contingent choices are respected by updated preferences. Such updating, in this context, implies dependence on the feasible set of payoff vectors available in the problem and/or on an ex-ante optimal act for the problem. Despite this complication, the algorithms are formulated concisely and are easy to implement, thus making dynamically consistent updating operational in the presence of ambiguity.