Massimiliano Amarante

Ambiguous events and maxmin expected utility

Abstract We study the properties associated to various definitions of ambiguity [L.G. Epstein, J. Zhang, Subjective probabilities on subjectively unambiguous events, Econometrica 69 (2001) 265–306; P. Ghirardato et al., Differentiating ambiguity and ambiguity attitude, J. Econ. Theory 118 (2004) 133–173; K. Nehring, Capacities and probabilistic beliefs: a precarious coexistence, Math. Soc. Sci. 38 (1999) 197–213; J. Zhang, Subjective, ambiguity, expected utility and Choquet expected utility, Econ. Theory 20 (2002) 159–181] in the context of Maximin Expected Utility (MEU). We show that each definition of unambiguous events produces certain restrictions on the set of priors, and completely characterize each definition in terms of the properties it imposes on the MEU functional. We apply our results to two open problems. First, in the context of MEU, we show the existence of a fundamental incompatibility between the axiom of “Small unambiguous event continuity” (Epstein and Zhang, 2001) and the notions of unambiguous event due to Zhang (2002) and Epstein and Zhang (2001). Second, we show that, in the context of MEU, the classes of unambiguous events according to either Zhang (2002) or Epstein and Zhang (2001) are always λ -systems. Finally, we reconsider the various definitions in light of our findings, and identify some new objects (Z-filters and EZ-filters) corresponding to properties which, while neglected in the current literature, seem relevant to us.

Ambiguity on the insurer's side: the demand for insurance

Empirical evidence suggests that ambiguity is prevalent in insurance pricing and underwriting, and that often insurers tend to exhibit more ambiguity than the insured individuals (e.g., [23]). Motivated by these findings, we consider a problem of demand for insurance indemnity schedules, where the insurer has ambiguous beliefs about the realizations of the insurable loss, whereas the insured is an expected-utility maximizer. We show that if the ambiguous beliefs of the insurer satisfy a property of compatibility with the non-ambiguous beliefs of the insured, then there exist optimal monotonic indemnity schedules. By virtue of monotonicity, no ex-post moral hazard issues arise at our solutions (e.g., [25]). In addition, in the case where the insurer is either ambiguity-seeking or ambiguity-averse, we show that the problem of determining the optimal indemnity schedule reduces to that of solving an auxiliary problem that is simpler than the original one in that it does not involve ambiguity. Finally, under additional assumptions, we give an explicit characterization of the optimal indemnity schedule for the insured, and we show how our results naturally extend the classical result of Arrow [5] on the optimality of the deductible indemnity schedule.

Mm-OWA: A Generalization of OWA Operators

We characterize those operators that satisfy the properties of monotonicity, permutation invariance, positive homogeneity, and translation invariance. As these operators do not necessarily satisfy comonotonic additivity, their class is larger than that of ordered weighted averaging (OWA) operators. We give a representation theorem for these operators, which shows, nonetheless, that this more general class can be constructed directly from that of OWA operators. In addition, we characterize the special classes consisting of operators that are either subadditive or superadditive. We suggest applications to the evaluation of complex systems.

Contracting on Ambiguous Prospects

We study contracting problems where one party perceives ambiguity about the relevant contingencies. We show that the party who perceives ambiguity has to observe only the revenue/loss generated by the prospect object of negotiation, but not the underlying state. We, then, introduce a novel condition (vigilance), which extends the popular monotone likelihood ratio property to settings featuring ambiguity. Under vigilance, optimal contracts are monotonic and, thus, produce the right incentives in the presence of both concealed information and hidden actions. Our result holds irrespectively of the party’s attitude towards ambiguity. Sharper results obtain in the case of global ambiguity‐loving behaviour.

Extraction of the surplus in standard auctions

Cremer and McLean [1] and McAfee and Reny [3] showed that, in “nearly all auctions”, the seller can offer a mechanism that obtains full rent extraction. The mechanism designed by Cremer and McLean differs from standard procedures in several respects. Notably, (a) bidders are required to buy “lotteries” in order to partecipate at the auction; (b) bidders are treated non-anonymously, in the sense that different bidders are given different strategic options; (c) the bidders’ payments to the seller depend not only on the bids but also on some other “actions” taken by the bidders. In this paper, I keep (a) and (b), and impose that — like in the “standard model of auctions” — the bidders’ payments to the seller depend on the bids alone. I find that the full-surplus extraction result no longer holds: There are “open sets” of auctions where the full extraction is not possible.

Sandwich theorems for set functions: An application of the Lehrer–Teper integral

Abstract We show that a new integral, recently introduced by Lehrer and Teper (2008) [11] , provides all the tools to determine the necessary and sufficient conditions for the existence of an additive set function, m, sandwiched between two arbitrary set functions γ and ζ: γ ≤ m ≤ ζ .

Information and Ambiguity: Toward a Foundation of Nonexpected Utility

The concept of ambiguity designates those situations where the information available to the decision maker is insufficient to form a probabilistic view of the world. Thus, it has provided the motivation for departing from the subjective expected utility (SEU) paradigm. Yet, the formalization of the concept is missing. This is a grave omission as it leaves nonexpected utility models hanging on shaky ground. In particular, it leaves unanswered basic questions such as the following: (1) Does ambiguity exist? (2) If so, which situations should be labeled as “ambiguous”? (3) Why should one depart from SEU in the presence of ambiguity? (4) If so, what kind of behavior should emerge in the presence of ambiguity? The present paper fills these gaps. Specifically, it identifies those information structures that are incompatible with SEU theory, and shows that their mathematical properties are the formal counterpart of the intuitive idea of insufficient information. These are used to give a formal definition of ambi...

Analogy in Decision Making

In the context of decision making under uncertainty, I formalize the concept of analogy: an analogy between two decision problems is a mapping that transforms one problem into the other while preserving the problem’s structure. After identifying the basic structure of a decision problem, I introduce the concepts of analogical reasoning operator and of analogical reasoning preference . The former maps the decision problem at hand into a family of decision problems, which are analogous to the problem under consideration. The latter is the result of aggregating the various analogies. I provide several representations (in decreasing order of generality) of the analogical reasoning operators. After introducing two mild assumptions on the aggregators of analogies, I characterize analogical reasoning (AR) preferences. I give several examples of AR preferences and of the associated aggregators. These include Gilboa-Schmeidler similarities, Choquet integrals, and quantiles. Finally, I show that the class of monotone continuous invariant biseparable (MCIB) preferences (which includes many popular models of decision making under uncertainty) has an important stability property: any MCIB preference is an AR preference; conversely, every AR preference that results from aggregating MCIB preferences is an MCIB preference.

Entrepreneurship, Ambiguity, and the Shape of Innovation Contracts

In Amarante, Ghossoub, and Phelps (AGP) [2], we proposed a model of innovation and entrepreneurship where the entrepreneur generates innovation, innovation generates Ambiguity for all economic agents except the entrepreneur, and the financier deals with this Ambiguity through bilateral contracts that we called innovation contracts. Under a requirement on the financier’s ambiguous beliefs, we showed the existence and monotonicity of optimal innovation contracts. Moreover, when the financier is ambiguity-loving in the sense of Schemeidler [26], we showed that the problem of contracting for innovation under Ambiguity can be reduced to a situation of non-ambiguous but heterogeneous Bayesian beliefs. This is important since the latter situations have been examined by Ghossoub [10, 12], and the solutions can be characterized in that case. In this paper, we consider a special case of the setting of AGP [2] which will allow us to fully characterize an optimal innovation contract, all the while maintaining a situation where the financier has ambiguous beliefs.

States, Models and Unitary Equivalence I: Representation Theorems and Analogical Reasoning

I show that virtually any model of decision making under uncertainty is associated to a certain structure. This contains three fundamental ingredients: (1) The domain of the acts; (2) Another set, which is called the set of models for the decision maker; and (3) The decision maker’s information about the set of models (an algebra of subsets of the set of models). A consequence of this finding is that that the decision maker’s choices can be viewed as the outcome of a two-stage process. First, the set of acts is mapped into a system of hypothetical bets on the set of models. Then, the latter are ranked by the decision maker. I show that this procedure can be thought of as describing a general form of analogical reasoning. I also observe that the appearance of two different sets implies that the decision maker is uncertain about two different objects and that he may receive information about any of them. In particular, information about the set of models affects the decision maker’s ranking of the available alternatives. In the sequel to this paper, I show that certain natural information structures lead to an inherent inability of assigning probabilities on the domain of the acts. In a formal sense, their properties describe the idea of Knightian Uncertainty. ∗I benefited from comments and suggestions from Larry Epstein, Paolo Ghirardato, Tzachi Gilboa, Fabio Maccheroni, Massimo Marinacci, Rich McLean, Ket Richter, Aldo Rustichini, Marco Scarsini, Paolo Siconolfi, Max Stinchcombe, William Thomson, Stan Wellisz and Bill Zame. I also wish to thank seminar participants at University of Iowa, RUD 2003, CalTech, University of Minnesota, University of Texas-Austin, Columbia University, University of Rochester, Rutgers University, University of Turin, Ohio State University, Columbia University Workshop on Multiple Priors. †Department of Economics, Columbia University, 1019 IAB, 420 W 118th street, NY, NY 10027. Email: ma734@columbia.edu.