RISK AND UNCERTAINTY AVERSION WITH MULTIDIMENSIONAL CONSEQUENCES

Abstract

1. INTRODUCTION In a preceding paper (Montesano, 1999b) a systematic set of definitions on risk and uncertainty aversion was introduced with regard to unidimensional lotteries and acts. Taking into account only the preference preordering over the set of all possible lotteries and acts represented by the certainty equivalent function, many propositions were introduced and demostrated on global and local risk and uncertainty aversion, comparative risk and uncer­tainty aversion, and aversion to increasing risk and un­certainty. In this paper the preceding analysis is extended to the multidimensional case, i.e., to lotteries and acts whose consequences are points of a /:-dimensional Euclidean space, which can be interpreted as bundles with k com­modities. The analysis takes into consideration finite lot­teries and acts, i.e., the number of the possible states of nature is finite. No specific model for the decision-maker preferences is assumed, but only the existence of a regu­lar preference system represented by an ordinal utility function. Subsequently some specific models are intro­duced (the Expected Utility model, the Rank Dependent Expected Utility model, and the Choquet Expected Util­ity model), in order to extend some results already exam­ined (Montesano, 1999a) for the unidimensional case. In Section 2 the Decision Making Under Risk (DMUR) and Uncertainty (DMUU) situations are intro­duced, together with the main assumptions and some definitions. In particular, mean preserving spreads are also introduced for the multidimensional case, while their current definition only applies to the unidimen­sional case. Section 3 examines the DMUR situation. Aversion to risk and to increasing risk are introduced and some propositions are demonstrated which relate these aversions to some characteristics of the preference sys­tem. Unlike the unidimensional case, the notion of com­parative risk aversion (as well as that of comparative un­certainty aversion) is unattractive, since it requires that the preferences of two decision-makers coincide over the degenerate lotteries set. This is an overly compelling as­sumption, which discourages further analysis. Section 4 concerns the DMUU situation, where both risk and un­certainty matter. Aversion to them is distinctly introduc­ed and some propositions on these aversions are pro­posed. Although not all the propositions found for the un­idimensional case can be extended to the multidimen­sional case, many of them surprisingly can, providing us with tools for investigating real decision-makers' charac­teristics. 2. SOME INTRODUCTORY DEFINITIONS AND ASSUMPTIONS A Decision Making Under Uncertainty (DMUU) situ­ation is represented by a quadruple <5; Q; F; X} where S is a set of states of nature, II is an algebra on 5, X is a set of consequences and F is the set of possible acts (i.e., functions from S to X). In this paper 5 is finite, 0 = (the empty set included), and X is a compact and convex sub­set of the Euclidean /:-dimensional space IR^. A Decision Making Under Risk (DMUR) situation is represented by a quintuple (^S; O; F; X; p}, where j9 is an exogenous probability: p E P, where P is the set of all possible probability distributions over S = {s^, ..., s,^}, i.e., P = {p(s)^0 for J = 1, ..., m : EJL, pis) = 1} or P = {p(e,)^ 0 for every e. e 2^ : p(0) = 0, p(S) = 1 , and p(ei n e^) + p(e^ u e^) = pie^) + p{e^) for every pair ^., e^ e 2*^}, so that Z^ j p{ei) = 1 if {ei)'i^, is a partition of S. The set Q of lotteries on X is the set of probability measures on X induced through the acts by some probability, i.e., q = (f, p), where f e F and p e P, md Q = F X P.

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