Revisiting independence and stochastic dominance for compound lotteries

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1 Revisiting independence and stochastic dominance for compound lotteries Alexander Zimper Working Paper Number 97 Department of Economics and Econometrics, University of Johannesburg

2 Revisiting independence and stochastic dominance for compound lotteries Alexander Zimper y October, 2008 Abstract We establish mathematical equivalence between independence of irrelevant alternatives and monotonicity with respect to rst order stochastic dominance. This formal equivalence result between the two principles is obtained under two key conditions. Firstly, for all m 2 N, each principle is de ned on the domain of compound lotteries with compoundness level m. Secondly, the standard concept of reduction of compound lotteries applies. JEL Classi cation Numbers: D8. I thank Jürgen Eichberger, Martin Hellwig, Ullrich Schmidt, Uzi Segal and two anonymous referees for helpful comments and suggestions. Financial support by ERSA (Economic Research South Africa) is gratefully acknowledged. y Department of Economics and Econometrics, University of Johannesburg, PO Box 524, Auckland Park, 2006, South Africa. zimper@bigfoot.com

3 Introduction In the theory of decision making under risk independence of irrelevant alternatives serves as key assumption for deriving von Neumann and Morgenstern s (947) representation of preferences over lotteries by an expected utility functional (cf. Fishburn 988). However, since many real-life decision makers persistently violate independence of irrelevant alternatives (e.g., Allais 979), alternative models of decision making under risk (for surveys on this vast literature see, e.g., Schmidt 2004; Starmer 2000; Sugden 2004) typically weaken the independence assumption while they additionally impose monotonicity with respect to rst order stochastic dominance. Whereas the normative appeal of independence of irrelevant alternatives is controversially discussed in the literature, researchers commonly agree upon that monotonicity with respect to rst order stochastic dominance must be obeyed by rational decision makers. As our main nding we establish mathematical equivalence of these two - seemingly di erent - principles under two conditions. As a rst condition we assume that each principle is de ned, for all m 2 N, on the domain of compound lotteries with compoundness level m. As a second condition we assume that the standard principle of reduction of compound lotteries applies without restriction. Referring to the development of descriptive decision theories as a reaction to the Allais paradoxes, Duncan Luce (2000) writes: Some theorists [:::] have to a degree abandoned independence, but have continued to devise theories on the assumption that the lotteries are well modeled as random variables, and so they accept the (automatic) reduction of compound gambles. Others of us have interpreted the body of evidence as favoring consequence monotonicity and as making dubious at a descriptive level the probability reduction principle built into the random variable notation [:::]. (p. 47) This note s equivalence result once more emphasizes that the reduction principle for compound lotteries is not merely an innocuous assumption but rather plays an important (and typically hidden) role in the normative interpretation of decision theoretic axioms. 2 Notation and preliminaries Let L 0 denote some nite set of deterministic outcomes and iteratively de ne for all m 2 N: ( ) nx L m = ( ; s ; :::; n ; s n ) j ; :::; n 2 R + ; i = and 9j 2 f; :::; ng s.t. s j 2 L m. i=

4 We interpret the elements of L m as compound lotteries of level m, that is, there is at least one compound lottery of level m in the support of a compound lottery in L m. According to this interpretation, the set L = S m2n Lm collects all compound lotteries without any upper bound for their compoundness level m. As standard notational conventions we introduce order-irrelevance of entries, i.e., (::; ; s; :::; ; t; ::) = (::; ; t; :::; ; s; ::) and distribution of weights for identical entries, i.e., (::; ; s; :::; ; s; ::) = (::; ( + ) ; s; ::). De nition. The preference relation on L satis es complete ordering if and only if the following three assumptions are ful lled: (i) There exists an asymmetric, non-re exive, and transitive binary relation over compound lotteries in L. (ii) s t if and only if not s t and not t s. (iii) s t if and only if s t or s t. De nition. The preference relation on L satis es reduction of compound lotteries if and only if, for all m 2 N, the following two assumptions are ful lled (where k k denotes the real number resulting from the multiplication of k with k, k 2 f; :::; ng): (i) If ( ; ( ; s ; :::; n ; s n ) ; :::) 2 L m+ ( ; s ; :::; n ; s n ; :::) 2 L m then ( ; ( ; s ; :::; n ; s n ) ; :::) (( ; s ; :::; n ; s n ) ; :::). (ii) If (; s) 2 L m+ and s 2 L m then (; s) s. 3 Result Consider a compound lottery s = ( ; s ; :::; n ; s n ) 2 L m+ such that s k+ s k for all k 2 f; :::; n g. The cumulative distribution function of s with respect to, F [s] : L m [0; ], is de ned as

5 F [s] (x) = 0 for all x 2 L m such that s x, F [s] (x) = j kx j for all x 2 L m such that s k+ x s k, j= F [s] (x) = for all x 2 L m such that x s n. We say s strictly dominates t with respect to rst-order stochastic dominance, denoted s F t, if and only if F [s] (x) F [t] (x) for all x 2 L m, and F [s] (x) < F [t] (x) for some x 2 L m. Analogously, s F t if and only if F [s] (x) = F [t] (x) for all x 2 L m. De nition. The preference relation on L satis es monotonicity with respect to rst-order stochastic dominance with support on L m if and only if, for all s; t 2 L m+, s F t implies s t, and s F t implies s t. De nition. The preference relation on L satis es independence of irrelevant alternatives with support on L m if and only if, for all compound lotteries s; t; r 2 L m, s t ) (; s; ( ) ; r) (; t; ( ) ; r) s t ) (; s; ( ) ; r) (; t; ( ) ; r) Proposition. Suppose that the preference relation on L satis es complete ordering and reduction of compound lotteries. Then the following two statements are equivalent: (i) For all m 2 N, satis es independence of irrelevant alternatives with support on L m. (ii) For all m 2 N, satis es monotonicity with respect to rst order stochastic dominance with support on L m. 4 Proof At rst we introduce an alternative dominance relation, denoted E, for compound lotteries and prove - by a lemma - equivalence between E and F.

6 De nition. For two compound lotteries s; t 2 L m+ such that s = ( ; s ; 2 ; s 2 ; :::; n ; s n ) t = ( ; t ; 2 ; t 2 ; :::; n ; t n ) we write s E t if and only if s k t k for all k 2 f; :::; ng. Furthermore, s E t, if and only if s k t k for all k 2 f; :::; ng. Remark. Observe that the dominance relation E is equivalent to so-called consequence monotonicity (De nition in Luce 2000) whenever transitivity holds and the decision maker s uncertainty about the occurrence of events is resolved by an additive probability measure. Lemma: Suppose that is a complete ordering. For any pair of compound lotteries ( ; s ; :::; l ; s l ) 2 L m+ (a) ; t ; :::; q ; t q 2 L m+ (b) there exists some pair of compound lotteries ( ; s 0 ; :::; n ; s 0 n) 2 L m+ (2a) ( ; t 0 ; :::; n ; t 0 n) 2 L m+ (2b) where s 0 j+ s 0 j and t 0 j+ t 0 j, for j 2 f; :::; n g, such that ( ; s ; :::; l ; s l ) = ( ; s 0 ; :::; n ; s 0 n) (3a) ; t ; :::; q ; t q = ( ; t 0 ; :::; n ; t 0 n) (3b) and if and only if ( ; s ; :::; l ; s l ) F ; t ; :::; q ; t q ( ; s 0 ; :::; n ; s 0 n) E ( ; t 0 ; :::; n ; t 0 n). Proof of the Lemma Step : For any given compound lotteries (a) and (b) we construct two compound lotteries (2a) and (2b) such that equations (3a) and (3b) are satis ed. Note that, by the notational convention order-irrelevance of entries, we can - without loss of generality - assume that s j+ s j and t j+ t j, for j 2 f; :::; n g.

7 st iteration step. Without loss of generality suppose and observe that there exists a unique number i such that i X ix + k < k. Construct now two lotteries s ; t 2 L m+ such that Xi Xi s = :::; k ; s i +; i + k ; s i +; ::: Xi t = :::; k ; t ; 2 ; t 2 ; ::: and observe that the rst i + probability weights in s and in t are identical. 2nd iteration step. Focus now on the lotteries s ; t and start with the (i + 2)nd entry to determine whether Xi i + k 2 or Xi i + k > 2. Then proceed analogously to the st iteration step. The above procedure determines at the kth iteration step a unique number i k such that the rst i k + probability weights for entries in s k coincide with the rst i k + probability weights for entries in t k. Moreover, by the notational convention distribution of weights for identical entries, we have, for all k, s k = ( ; s ; :::; l ; s l ) t k = ; t ; :::; q ; t q. By the above procedure, any two compound lotteries s; t are transformed after a nite number of iteration steps, say M, into notationally equivalent compound lotteries s M ; t M. Moreover, since each iteration step generates the same number of entries with identical probability weights for all entry pairs, s M and t M must share the same number of entries. Step 2: In the light of step, for arbitrary ( ; s ; :::; l ; s l ) 2 L m+ ; t ; :::; q ; t q 2 L m+,

8 ( ; s ; :::; m ; s m ) F ; t ; :::; q ; t q if and only if ( ; s 0 ; :::; n ; s 0 n) F ( ; t 0 ; :::; n ; t 0 n). (4) Thus, for proving the lemma, it remains to be shown that (4) is satis ed if and only if ( ; s 0 ; :::; n ; s 0 n) E ( ; t 0 ; :::; n ; t 0 n). (5) Rewriting (4), gives, for all x 2 L m, F [ ; s 0 ; :::; n ; s 0 n] (x) F [ ; t 0 ; :::; n ; t 0 n] (x) where F [ ; s 0 ; :::; n ; s 0 n] (x) = F [ ; t 0 ; :::; n ; t 0 n] (x) = jx k such that s 0 j+ x s 0 j jx k such that t 0 j+ x t 0 j. Our claim is trivially proved if, for all j 2 f; :::; ng, s j t j. Thus, assume that (5) is satis ed and denote by s 0 j; t 0 j, for j 2 f; :::; ng, the rst entries such that s 0 j t 0 j whereas s 0 k t0 k for all k < j. Observe that, for all r with t0 j x, F [ ; s 0 ; :::; n ; s 0 n] (x) = < X j k j X k + j F [ ; t 0 ; :::; n ; t 0 n] (x) Repeating this argument for the remaining r 2 L m shows that (4) is satis ed as well. Assume now that (4) is satis ed and denote by r 2 L m the compound lottery such that for all x with r x, F [ ; s 0 ; :::; n ; s 0 n] (x) = F [ ; t 0 ; :::; n ; t 0 n] (x) whereas F [ ; s 0 ; :::; n ; s 0 n] (r ) < F [ ; t 0 ; :::; n ; t 0 n] (r ) Obviously, s 0 j t 0 j for all j 2 f; :::; ng such that r s 0 j and r t 0 j. Contrary to (5), presume that t 0 j+ s 0 j+. But then jx F [ ; s 0 ; :::; n ; s 0 n] s 0 j+ = k + j+ > jx k = F [ ; t 0 ; :::; n ; t 0 n] s 0 j+

9 A contradiction to (4). Repeating this argument nally proves our claim. Proof of the Proposition. Restricted to compound lotteries s; t 2 L m+ with two entries only, ME (=monotonicity with respect to E ) with support on L m obviously coincides with the de nition of independence with support on L m. As a consequence, we obtain that, for arbitrary m 2 N, ME with support on L m implies independence with support on L m. In view of the lemma we can therefore prove the proposition by showing that, for arbitrary m 2 N, independence with support on L m+ implies ME with support on L m. in L m. We prove this claim by induction over the number of entries in the compound lotteries At rst we show that ME with support on L m is satis ed for all compound lotteries s; t 2 L m+ having exactly one entry in their support. Suppose this claim is false. Then there are compound lotteries s = (; s 0 ) 2 L m+ t = (; t 0 ) 2 L m+ such that s 0 t 0 whereas t s. By reduction of compound lotteries, s s 0 and t t 0, implying, by transitivity, s t s. A contradiction to the ordering assumption. Assume now that ME with support on L m is satis ed for all compound lotteries s; t 2 L m+ with k entries. We prove: if independence with support on L m+ is satis ed, then ME with support on L m must be satis ed for all compound lotteries s; t 2 L m+ with k + entries. Suppose on the contrary that ME with support on L m is violated for some compound lotteries s; t 2 L m+ with k + entries, i.e., there are compound lotteries s = ; s 0 ; :::; k+ ; s 0 k+ 2 L m+ t = ; t 0 ; :::; k+ ; t 0 k+ 2 L m+ such that s j t j for all j 2 f; :::; k + g whereas t s. By reduction of compound lotteries, s s 0 and t t 0 whereby s 0 = k+ ; k+ ; s 0 ; :::; t 0 = k+ ; ; t 0 ; :::; k k+ ( k ) ; t0 k k ( k ) ; s0 k ; k+ ; s 0 k+ 2 L m+2 ; k+ ; t 0 k+ 2 L m+2. Moreover, by the induction assumption ; s 0 ; :::; k k+ ( k ) ; s0 k ; t 0 ; :::; k k+ ( k ) ; t0 k

10 since s j t j for all j 2 f; :::; kg. Because of s k+ t k+, independence with support on L m+ implies s 0 t 0. By transitivity, s t s, contradicting the ordering assumption.

11 References Allais, M. (979) The Foundation of a Positive Theory of Choice Involving Risk and a Criticism of the Postulates and Axioms of the American School. Part II in Allais, M., and O. Hagen [eds.], Expected Utility Hypotheses and the Allais Paradox, D. Reidel: Dordrecht etc. Fishburn, P. C. (988) Nonlinear Preferences and Utility Theory under Risk, The John Hopkins University Press: Baltimore, London. Luce, R.D. (2000) Utility of Gains and Losses: Measurement-Theoretical and Experimental Approaches, Lawrence Erlbaum Associates, Publishers, New Jersey. von Neuman, J., and O. Morgenstern (947) Theory of Games and Economic Behavior, Princeton University Press, 2nd edn., Princeton. Schmidt, U. (2004) Alternatives to Expected Utility: Some Formal Theories, in: P.J. Hammond, S. Barberá, and C. Seidl [eds.], Handbook of Utility Theory Vol. II, Kluwer, Boston, chapter 5. Starmer, C. (2000) Developments in Non-Expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk, Journal of Economic Literature 38, Sugden, R. (2004) Alternatives to Expected Utility: Foundation, in: P.J. Hammond, S. Barberá, and C. Seidl [eds.], Handbook of Utility Theory Vol. II, Kluwer, Boston, chapter 4.

Economics Bulletin, 2012, Vol. 32 No. 1 pp Introduction. 2. The preliminaries

Economics Bulletin, 2012, Vol. 32 No. 1 pp Introduction. 2. The preliminaries 1. Introduction In this paper we reconsider the problem of axiomatizing scoring rules. Early results on this problem are due to Smith (1973) and Young (1975). They characterized social welfare and social