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Mathematics and Financial Economics

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07.09.2020

Preferences over rich sets of random variables: on the incompatibility of convexity and semicontinuity in measure

verfasst von: Alexander Zimper, Hirbod Assa

Erschienen in: Mathematics and Financial Economics | Ausgabe 2/2021

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Abstract

This paper considers a decision maker whose preferences are locally upper- or/and lower-semicontinuous 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.

Vorheriger Artikel Equilibrium effects of intraday order-splitting benchmarks

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Of course, continuity concepts are not directly testable through finitely many observations of choice behavior. Violations of convexity alone—as, e.g., expressed by the popularity of the value-at-risk criterion—cannot prove that these decision makers’ preferences are continuous in measure. We come back to this point when we discuss the behavioral meaning of continuity.

For a general class of utility representations with convex strictly better sets—including the variational preferences of Maccheroni et al. [27]—see Cerreira-Vioglio et al. [10].

For references to the economic literature which uses subjective belief data from the HRS or/and the SCF, see, e.g., Groneck, Ludwig, and Zimper [25].

A connected topological space is characterized by the property that only the empty set and the universal set are simultaneously open and closed (cf. Chapter 1.11.1 in Bourbaki [9]). Whereas some of the spaces considered in this paper are connected (e.g., the space of all random variables) others are not (e.g., the ‘small’ rich set of our Example 1 below).

Our preferred interpretation is that the decision maker is ‘aware’ of the random variables in L. At this point, we do not even require the decision maker to have complete preferences over all random variables in L.

The converse statement is, in general, not true. A violation of lower-semicontinuity of U at some $Y\in L$ implies that the strictly better set $S^{*}\left( c\right) =\left\{ Z\in L\mid c<U\left( Z\right) \right\} $ cannot be open for some $c\in \mathbb {R}$ (cf., Theorem 1, p. 76 in Berge 1996). However, we do not always have that $c=U\left( X\right) $ for some $X\in \mathcal {F}$.

The distribution $\mu _{Z}$ of random variable Z is the probability measure on $\left( \mathbb {R},\mathcal {B}\left( \mathbb {R} \right) \right) $ such that

$$\begin{aligned} \mu _{Z}\left( A\right) =\mu \left( \left\{ \omega \in \Omega \mid Z\left( \omega \right) \in A\right\} \right) \text { for all }A\in \mathcal {B}\left( \mathbb {R}\right) \text {.} \end{aligned}$$

Point (i) is explained in detail in the present paper. For point (ii) see our analysis in Assa and Zimper [5] and references therein. For example, Savage’s [29] subjective expected utility axiomatization requires a bounded (Bernoulli) utility function as he considers complete preferences over the set of all random variables. Compare Wakker [37] who writes:

Ever since, the extension of Savage’s theorem to unbounded utility has been an open question, and with that the question ”what is wrong with Savage’s axioms?”. $\left[ \ldots \right] $ I think that ”what is wrong with Savage’s axioms”, is primarily his requirement of completeness of the preference relation on the set of all (alternatives=) acts $\left[ \ldots \right] $. (p. 448)

To see the formal relationship between the CEU- and the mutliple priors representation of ambiguity aversion, observe that for a convex $\nu $

$$\begin{aligned} \int _{\Omega }^{Choquet}u\left( Z\right) d\nu =\min _{\pi \in \mathcal {P} }\int _{\Omega }u\left( Z\right) d\pi \end{aligned}$$

where $\mathcal {P}$ is defined as the core of $\nu $.

Coherent risk measures, defined on some vector space $L\subseteq L^{0}$, have to satisfy positive homogeneity and subadditivity which implies convexity.

Note that lower-semicontinuity of U becomes, by (8), upper-semicontinuity of $\rho $.

In Assa and Zimper (2018) we had incorrectly claimed that value-at-risk preferences are continuous in measure without the qualifying condition (10).

We write $\mathrm {VaR}_{\alpha }\left( Z\right) $ whenever (10) holds for Z.

Our formal definition of the average value-at-risk also appears in the literature as the definition of the “conditional value-at-risk” or of the “expected shortfall”. We follow here Föllmer and Schied ([21], p. 233) who argue that the notion of the average over the interval $\left( 0,\beta \right) $ is more precise as it clarifies that the conditional distribution in question is the uniform distribution.

To see this, note that all converging sequences $\left\{ Y_{n}\right\} \rightarrow Y$ which determine whether the strictly better set at X is open are, by (12), of the form

$$\begin{aligned} Y_{n}=\alpha \left( n\right) Y+\left( 1-\alpha \left( n\right) \right) Z^{\prime } \end{aligned}$$

such that $\lim _{n\rightarrow \infty }\alpha \left( n\right) =1$.

Agner, E., Loewenstein, G.: Behavioral Economics. In: Mäki, Uskali (ed.) Handbook of the Philosophy of Science: Philosophy of Economics Chapter 13, pp. 641–690. Elsevier, Amsterdam (2012)

Aliprantis, D.C., Border, K.: Infinite Dimensional Analysis, 2nd edn. Springer, Berlin (2006)MATH

Artzner, P., Delbaen, F., Eber, J., Heath, D.: Thinking coherently. Risk 10, 68–71 (1997)

Artzner, P., Delbaen, F., Eber, J., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)MathSciNetCrossRef

Assa, H., Zimper, A.: Preferences over all random variables: incompatibility of convexity and continuity. J. Math. Econ. 75, 71–83 (2018)MathSciNetCrossRef

Azevedo, E.M., Gottlieb, D.: Risk-neutral firms can extract unbounded profits from consumers with prospect theory preferences. J. Econ. Theory 147, 1291–1299 (2012)MathSciNetCrossRef

Berge, C.: Topological Spaces. Dover Publications, New York. (An “unabridged and unaltered republication” of the original English publication from 1963.) (1997)

Billingsley, P.: Probability and Measure. Wiley, New York (1995)MATH

Bourbaki, N.: Elements of Mathematics. General Topology. Chapters 1–4. Springer, Berlin (1989)

10.

Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Uncertainty averse preferences. J. Econ. Theory 146, 1275–1330 (2011)MathSciNetCrossRef

11.

Chateauneuf, A., Cohen, M., Meilijson, I.: More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model. Econ. Theory 25, 649–667 (2005)MathSciNetCrossRef

12.

Chew, S., Karni, E., Safra, Z.: Risk aversion in the theory of expected utility with rank dependent preferences. J. Econ. Theory 42, 370–381 (1987)CrossRef

13.

Day, M.M.: The spaces $L^{p}$ with $0{\le }p{\le }1$. Bull. Am. Math. Soc. 46, 816–823 (1940)

14.

Danan, E., Gajdos, T., Tallon, J.-M.: Harsanyi’s aggregation theorem with incomplete preferences. Am. Econ. J. Microecon. 7, 61–69 (2015)CrossRef

15.

Dekel, E.: Asset demands without the independence axiom. Econometrica 57, 163–169 (1989)MathSciNetCrossRef

16.

Delbaen, F.: Coherent risk measures on general probability spaces. In: Advances in Finance and Stochastics, pp. 1-37. Springer, Berlin (2002)

17.

Delbaen, F.: Risk measures for non-integrable random variables. Math. Finance 19, 329–333 (2009)MathSciNetCrossRef

18.

Eilenberg, S.: Ordered topological spaces. Am. J. Math. 63, 39–45 (1941)MathSciNetCrossRef

19.

Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1970)CrossRef

20.

Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 6, 429–447 (2002)MathSciNetCrossRef

21.

Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, 4th edn. Walter de Gruyter GmbH, Berlin (2016)

22.

Gerasimou, G.: On Continuity of Incomplete Preferences. Social Choice and Welfare 41, 157–167 (2013)MathSciNetCrossRef

23.

Gilboa, I.: Expected utility with purely subjective non-additive probabilities. J. Math. Econ. 16, 65–88 (1987)CrossRef

24.

Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique priors. J. Math. Econ. 18, 141–153 (1989)CrossRef

25.

Groneck, M., Ludwig, A., Zimper, A.: A life-cycle model with ambiguous survival beliefs. J. Econ. Theory 162, 137–180 (2016)MathSciNetCrossRef

26.

Kahn, M.A., Uyanık, M.: Topological connectedness and behavioral assumptions on preferences: a two-way relationship. Economic Theory (forthcoming) (2019) https://doi.org/10.1007/s00199-019-01206-7

27.

Maccheroni, F., Marinacci, M., Rustichini, A.: Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74, 1447–1498 (2006)MathSciNetCrossRef

28.

Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)MATH

29.

Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954)MATH

30.

Samuelson, P.A.: A note on the pure theory of consumer’s behaviour. Economica 5, 61–71 (1938)CrossRef

31.

Schmeidler, D.: A condition for the completeness of partial preference relations. Econometrica 39, 403–404 (1971)MathSciNetCrossRef

32.

Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)MathSciNetCrossRef

33.

Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)MathSciNetCrossRef

34.

Sen, A.: Quasi-transitivity, rational choice and collective decisions. Rev. Econ. Stud. 36, 381–393 (1969)CrossRef

35.

Sonnenschein, H.: The relationship between transitive preference and the structure of the choice space. Econometrica 33, 624–634 (1965)CrossRef

36.

Wakker, P.P.: The algebraic versus the topological approach to additive representations. J. Math. Psychol. 32, 421–435 (1988)MathSciNetCrossRef

37.

Wakker, P.P.: Unbounded utility for Savage’s “Foundations of Statistics” and other models. Math. Oper. Res. 18, 446–485 (1993)

38.

Wakker, P.P.: Prospect Theory for Risk and Ambiguity. Cambridge University Press, Oxford (2010)CrossRef

Titel: Preferences over rich sets of random variables: on the incompatibility of convexity and semicontinuity in measure
verfasst von: Alexander Zimper
Hirbod Assa
Publikationsdatum: 07.09.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Mathematics and Financial Economics / Ausgabe 2/2021
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-020-00280-z

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