Preferences over rich sets of random variables: on the incompatibility of convexity and semicontinuity in measure
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Preferences over rich sets of random variables: on the incompatibility of convexity and semicontinuity in measure Alexander Zimper1
· Hirbod Assa2
Received: 24 May 2020 / Accepted: 30 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper considers a decision maker whose preferences are locally upper- or/and lowersemicontinuous in measure. We introduce the notion of a rich set which encompasses any standard vector space of random variables but also much smaller sets containing only random variables with at most two different outcomes in their support. Whenever preferences are complete on a rich set of random variables, lower- (resp. upper-) semicontinuity in measure becomes incompatible with convexity of strictly better (resp. worse) sets. We discuss implications for utility representations and risk-measures. In particular, we show that the value-at-risk criterion violates convexity exactly because it is lower-semicontinuous in measure. Keywords Continuous preferences · Utility representations · Convex risk measures · Value-at-risk Mathematics Subject Classification D81
1 Introduction We consider a decision maker whose preferences over random variables are locally upperor/and lower-semicontinuous in the topology of convergence in measure μ. On the one hand, convergence in measure stands for a plausible description of how some decision makers might perceive similarity of random variables. More precisely, we show that this topology
We are grateful to comments and suggestions from Frank Riedel, Alex Ludwig, an anonymous referee as well as from seminar participants at the Universities of Bielefeld, Frankfurt, Padova, Udine, and the ETH Zürich.
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Alexander Zimper [email protected] Hirbod Assa [email protected]
1
Department of Economics, University of Pretoria, Private Bag X20, Hatfield 0028, South Africa
2
Institute for Financial and Actuarial Mathematics and Institute for Risk and Uncertainty, University of Liverpool, Center for Doctoral Training, Chadwick Building, G62, Liverpool, UK
123
Mathematics and Financial Economics
can be generated by the k-truncated expectation metric E μ (|X − Y | ∧ k)
(1)
where the truncation value k > 0 cuts off any large differences between the random variables X and Y . This topology therefore describes decision makers who care about extremely bad and extremely good events but who also tend to ignore extreme differences in the magnitude of such events. On the other hand, convergence in measure turns out to be particularly interesting because a decision maker whose preferences are lower-semicontinuous in measure is bound to violate in specific choice situations the convexity of strictly better sets. Convexity of strictly better sets, however, is central to standard characterizations of global risk/uncertainty/ambiguity aversion, cf. Cerreira-Vioglio et al. ([10], p. 1276): Convexity reflects a basic negative attitude of decision makers toward the presence of uncertainty in their choices, an attitude arguably shared by most
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