Finite expected multi-utility representation
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Finite expected multi‑utility representation Dino Borie1 Received: 13 November 2019 / Accepted: 18 June 2020 © Society for the Advancement of Economic Theory 2020
Abstract I study the problem of obtaining a finite expected multi-utility representation for an incomplete preference relation over lotteries. It is shown that, when the prize space is a compact metric space, a preference relation admits such a multi-utility representation provided that it satisfies the standard axioms of expected utility theory and an additional necessary and sufficient axiom. Keywords Incomplete preference relations · Expected utility · Multi-utility · Finite representation JEL Classification D11 · D81
1 Introduction Incomplete preferences structures have a wide scope of application. In particular, many authors have made a formal connection between incomplete preferences and multi-objective decision making, and others have studied choice models in which an agent may not be able to compare some alternatives due to the presence of uncertainty. Several authors have considered preference models under risk and uncertainty in which incompleteness of individual preferences plays a key role.1 1 The formal connection between multi-objective decision making and incomplete preferences are studied by Seidenfeld et al. (1995), Ok (2002), and Dubra et al. (2004), among others. In uncertainty, a person may find it impossible to compare the desirability of two acts because, for some reason, she cannot formulate a “precise guess” about the likelihood of the states of the world. This sort of an incompleteness of preferences is what Bewley (2002) captures in his well known model of Knightian uncertainty. Several models of decision-making use incomplete preferences, derived from revealed preferences or introduce them. For instance, the “unambiguous preference” defined by Ghirardato et al. (2004). Gilboa et al. (2010) model the objective rationality as an incomplete preference relation.
Supported by Labex MME-DII (ANR11-LBX-0023-01). * Dino Borie dino.borie@u‑cergy.fr 1
THEMA, Université de Cergy-Pontoise, Cergy‑Pontoise Cedex, France
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By now, there is a sizable literature on multi-utility representations of preference relations. A multi-utility representation for a preorder becomes increasingly tractable as one chooses the representing set of utility functions “parsimoniously” so as to use as few utility functions as possible in the representation. In other words, the structure of preference relations that is represented by a set of utility functions is more amenable to analysis (say, from the perspective of optimization) if this set is small. For this reason, it is of great interest to determine when one can choose this set to be finite. The only result we know so far is given by Ok (2002). The author gives only a sufficient condition to obtain a finite utility representation of a semicontinuous order-separable preorder on an arbitrary topological space. Evren and Ok (2011) extend Ok’s result by dropping the o
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