Notes on Supermodularity and Increasing Differences. in Expected Utility

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1 Notes on Supermodularity and Increasing Differences in Expected Utility Alejandro Francetich Department of Decision Sciences and IGIER Bocconi University, Italy March 7, 204 Abstract Many choice-theoretic and game-theoretic applications in Economics invoke some form of supermodularity or increasing differences for objective functions defined on lattices. These notes provide axiomatic foundations for these properties on expectedutility representations of preferences over lotteries. Keywords: Expected utility; supermodularity; quasi-supermodularity; increasing differences; single crossing JEL Classification Numbers: D0, D8 This is an updated working-paper version of the paper published in Economics Letters 2 (203) There is a typo on page 208 of the published version paper: On line 5 of the proof of Theorem 2, the second weak inequality should be a strict inequality. Via Röntgen, 2036 Milan, Italy. address: alejandro.francetich@unibocconi.it These notes are based on Chapter of my dissertation, submitted to the Graduate School of Business at Stanford University. I am grateful to David Kreps, Federico Echenique, Yossi Feinberg, Marco Li Calzi, Paul Milgrom, John Quah, Andrzej Skrzypacz, Christopher Tyson, and an anonymous referee, for helpful comments and suggestions to improve these notes. I also thank Rohan Dutta and Peter Troyan for carefully reading previous drafts. Any remaining errors and omissions are all mine.

2 Introduction These notes revisit the axiomatic foundations of the properties of supermodularity and of increasing differences for expected-utility representations of preferences defined over lotteries. While the first of these properties relates to a single preference relation and the second one involves a family of preferences, mathematically, they are closely related: A supermodular function on a product lattice has increasing differences. 2 Supermodular Expected Utility 2. Lattices and supermodularity Let (X, X ) be lattice, and denote the join and meet operations by and, respectively. For any f : X R, if f (x x ) + f (x x ) f (x) + f (x ) for any x, x X, then f is supermodular. Following Li Calzi (990) and Milgrom and Shannon (994), f is quasi-supermodular if, for any x, x X, we have that f (x) f (x x ) implies f (x x ) f (x ), with the corresponding implication for strict inequality. In words, for any x, x X, if meeting x with x downgrades x, then joining x and x upgrades x, according to f. In other words, if the value of x under f is strictly higher than the value under f of x x, then f cannot attain a higher value at x than at x x. Supermodularity is a cardinal property, while quasi-supermodularity is an ordinal implication of supermodularity. Any non-decreasing function satisfies the weak part of quasi-supermodularity, and any strictly increasing function is quasi-supermodular. 2.2 Mixture spaces and Borel probability measures Let T be a topology on X such that (X, T ) is a T space; that is, a space in which all singletons are closed sets. Denote by B(X) the Borel σ-field on X, and let Δ(X) denote the space of Borel probability measures on X. Finally, let Δ 0 (X) Δ(X) be the subset of simple Borel probability measures on X, that is, the probability measures in Δ(X) that have finite support. In particular, for any x X, the point mass concentrated at x, δ x, is an element of Δ 0 (X). A pair (Z, ), where Z is a set and is an operation : [0, ] Z Z Z, is a mixture space (Fishburn, 982) if: For all z, z Z, (, z, z ) = z; For all z, z Z and for all α [0, ], (α, z, z ) = ( α, z, z); 2

3 For all z, z Z and for all α, β [0, ], (α, (β, z, z ), z ) = (αβ, z, z ). Both Δ(X) and Δ 0 (X), coupled with the operation of taking convex combinations of probability measures, (α, μ, μ ) = αμ + ( α)μ, form mixture spaces. Henceforth, will denote this specific mixture operation. Let D Δ(X) be a subset of probability measures that contains all simple probability measures and that forms a mixture space on its own; that is to say, Δ 0 (X) D and (D, ) is a mixture space. A function f : D R is linear in if for all μ, μ D and all α [0, ], f ( (α, μ, μ )) = α f (μ) + ( α) f (μ ). 2.3 Preferences over lotteries and supermodular expected utility Let D D be a complete preorder on D. Since D contains all point masses, induces a complete preorder on X, X, given by x X x if δ x δ x for any x, x X. The asymmetric and symmetric parts of X, denoted by X and X, respectively, are the relations induced by the asymmetric and symmetric parts of. In choice-theoretic and game-theoretic applications, supermodularity is imposed on Bernoulli utility functions; these represent X. However, the primitive preference is that over lotteries, namely. Hence, the relevant link is the link between supermodularity of representations of X and properties of. In the Mixture Space Theorem (Herstein and Milnor, 953), the following three axioms are imposed on : (a) is a complete preorder; (b) For all μ, μ, μ D and for all α (0, ): μ μ implies (α, μ, μ ) (α, μ, μ ); (c) For all μ, μ, μ D such that μ μ μ, there exists some α, β (0, ) such that (α, μ, μ ) μ (β, μ, μ ). Axiom (a) is a necessary assumption for to admit a numerical representation. Axiom (b) is an independence assumption, stating that the presence of a third lottery μ does not change the ranking of μ, μ when mixed with equal weight. Finally, axiom (c) is a continuity or Archimedean axiom. Following Kreps (203), this last axiom rules out the existence of supergood or superbad lotteries: No matter how high μ is ranked by the agent, for some mixture, μ is still strictly preferred to this mixture of μ and μ. Similarly for μ : No matter how low it is ranked, μ is still strictly worse than some mixture of μ and μ. 3

4 The Mixture Space Theorem states that a binary relation on D satisfies these three axioms if and only if there exists a real-valued function u on D representing that is linear in and unique up to positive affine transformations. If D = Δ 0 (X), the von Neumann and Morgenstern Theorem (von Neumann and Morgenstern, 953) establishes the existence of a real-valued function U on X that is also unique up to positive affine transformations and such that u(μ) = Udμ. The result extends to the case D = Δ(X) if there exists a metric d on X such that (X, d) is separable, and if is continuous in the topology of weak convergence. The function U in the von Neumann and Morgenstern Theorem represents X, as U(x) = u(δ x ). Hence, the problem is to establish a link between properties of and supermodularity of U. The obvious link is given by the following axiom, (S): ( ) ( ) Axiom (S). For all x, x X, 2, δ x x, δ x x 2, δ x, δ x. Axiom (S) states that, for any two outcomes, the mixture between the highest and the lowest of the two (under X ) is weakly preferred to the mixture between the outcomes. If we think of X as the product of two lattices, the axiom can be read as saying that a mixture between all-high or all-low coordinates is weakly preferred to a lottery between elements that feature both high and low coordinates. Thus, it can be read as an axiom about complementarity across dimensions. Theorem. Let (X, X ) be a lattice and let be a binary relation on Δ 0 (X). Then, satisfies axioms (a), (b), (c), and (S), if and only if there exists a supermodular real-valued function U : X R such that u : Δ 0 (X) R given by u(μ) = x supp(μ) U(x)μ({x}) represents. Moreover, U is unique up to positive affine transformations. The mixture specified by axiom (S) is crucial in the proof of Theorem. For other mixtures, quasi-supermodularity follows instead. Consider the following weaker version of axiom (S), (qs): Axiom (qs). For all x, x X, there exists some α (0, ) such that (α, δ x x, δ x x ) (α, δ x, δ x ). Axiom (qs) states that, for any two outcomes, there exists a (strict) mixture between the highest and the lowest of the two that is weakly preferred to the same mixture 4

5 between the outcomes themselves. However, this mixture may be different that /2, and it may depend on the choice of x, x X. Theorem 2. Let (X, X ) be a lattice and let be a binary relation on Δ 0 (X). Then, satisfies axioms (a), (b), (c), and (qs), if and only if there exists a quasi-supermodular real-valued function U : X R such that u : Δ 0 (X) R given by u(μ) = x supp(μ) U(x)μ({x}) represents. Moreover, U is unique up to positive affine transformations. 3 Increasing Differences in Expected Utility In this section, (X, X ) is a poset (not necessarily a lattice), and R Θ := { θ : θ Θ } is an indexed family of complete preorders on D Δ(X). The index set Θ is also endowed with a partial order, denoted by Θ. Following Milgrom and Shannon (994), a function F : X Θ R satisfies the single-crossing property if, for each x, x X and each θ, θ Θ such that x > X x and θ > Θ θ, F(x, θ) F(x, θ) implies F(x, θ ) F(x, θ ), and F(x, θ) > F(x, θ) implies F(x, θ ) > F(x, θ ); if we have F(x, θ ) F(x, θ ) F(x, θ) F(x, θ), then F has increasing differences. Just as with supermodularity and quasi-supermodularity, the property of increasing differences is a cardinal property, and it has the single-crossing property as an ordinal implication. As in Section 2, the relevant link is the link between increasing differences of representations of preferences over outcomes, θ X, and properties of the corresponding preferences over lotteries, θ, for each θ Θ. To simplify the analysis, I will maintain the following assumption: Condition. There exist some x 0, x X such that δ x θ δ x 0 for all θ Θ. Condition means that there are two outcomes x 0, x X such that all preference relations in R Θ agree that the lottery that pays x with certainty is strictly preferred to the one that pays x 0 with certainty. Under this condition, the desired link is given by the following axiom, which will be called axiom (ɛ): If the mixture in axiom (qs) is uniform across x, then representations will satisfy the following property, weaker than supermodularity but stronger than quasi-supermodularity. A function f : X R defined on a lattice (X, X ) is α-supermodular if there exists some α [0, ] such that, for all x, x X, α f (x x ) + ( α) f (x x ) max{α f (x) + ( α) f (x ), α f (x ) + ( α) f (x)}. 5

6 Axiom (ɛ). ( For all x, ) x X such ( that x > ) X x, for all ( θ, θ Θ such ) that( θ > Θ θ, and) for all ɛ 0, +ɛ, δ x, δ x 0 θ +ɛ, δ x, δ x implies +ɛ, δ x, δ x 0 θ +ɛ, δ x, δ x. Axiom (ɛ) states that if any mixture of the higher of two elements with x 0 is weakly preferred under some preference relation to the corresponding mixture between the lower of the two and x, then the same ranking applies to all preference relations identified by higher indices. Thus, it can be read as an axiom about risk comparisons across preferences in the family. Theorem 3. Let R Θ be an indexed family of binary relations on Δ 0 (X). Then, the relations in R Θ satisfy axioms (a), (b), (c), and (ɛ) if and only if there exists a real-valued function U : X Θ R with increasing differences such that, for every θ Θ, u(, θ) : Δ 0 (X) R given by u(μ, θ) = x supp(μ) U(x, θ)μ({x}) represents θ. Moreover, for each θ Θ, U(, θ) is unique up to positive affine transformations. If the implication in (ɛ) can only be guaranteed for some rather than for all ɛ 0, and if the ɛ s on each side of the implication may be different, then we get the single-crossing property instead. The weaker axiom that captures this ordinal property will be called axiom (ɛ ): Axiom (ɛ ). For all x, x X such ( that x > ) X x and ( for all θ, θ ) Θ such that θ > Θ θ, if there exists some ɛ 0 such that +ɛ, δ x, δ x 0 θ +ɛ, δ x, δ x, then there exists some ɛ 0 ( ) ( ) such that +ɛ, δ x, δ x 0 θ +ɛ, δ x, δ x. Axiom (ɛ ) states that if some mixture of the higher of two elements with x 0 is weakly preferred under some preference relation to the corresponding mixture between the lower of the two and x, then some other mixtures are similarly weakly preferred under preferences with higher indices. Theorem 4. Let R Θ be an indexed family of binary relations on Δ 0 (X). Then, the relations in R Θ satisfy axioms (a), (b), (c), and (ɛ ) if and only if there exists a real-valued function U : X Θ R with the single-crossing property such that, for every θ Θ, u(, θ) : Δ 0 (X) R given by u(μ, θ) = x supp(μ) U(x, θ)μ({x}) represents θ. Moreover, for each θ Θ, U(, θ) is unique up to positive affine transformations. 6

7 A Proofs Proof of Theorem. Assume that are represented by u(μ) = x supp(μ) U(x)μ({x}) for some real-valued supermodular function U. Then, u is linear in. That satisfies axioms (a), (b), and (c), follows from the Mixture Space Theorem. Take any x, x X. Using linearity in of u and supermodularity of U, ( ( )) u 2, δ x x, δ x x = U(x x ) + U(x x ) 2 U(x) + U(x ) 2 ( ( )) = u 2, δ x, δ x. Thus, Axiom (S) follows. Conversely, assume that preferences satisfy axioms (a), (b), (c), and (S). The von Neumann and Morgenstern Theorem produces a function U : X R such that u(μ) = x supp(μ) U(x)μ({x}). Clearly, U represents X. Supermodularity of U is a simple consequence of (S) and linearity of u: ( ( )) ( ( )) u(δ x x ) + u(δ x x ) = 2u 2, δ x x, δ x x 2u 2, δ x, δ x = u(δ x ) + u(δ x ), and thus U(x x ) + U(x x ) = u(δ x x ) + u(δ x x ) u(δ x ) + u(δ x ) = U(x) + U(x ) for any two x, x X, as desired. The statement about uniqueness up to positive affine transformations follows from the von Neumann and Morgenstern Theorem. Proof of Theorem 2. Assume that are represented by u(μ) = x supp(μ) U(x)μ({x}) for some quasi-supermodular real-valued function U. As before, axioms (a), (b), and (c), are consequences of the Mixture Space Theorem. Take any x, x X. Without loss of generality, assume that U(x) U(x ). If U(x) > U(x x ), by quasi-supermodularity, we have U(x x ) > U(x ). Define h : [0, ] R as h(α) = ( α)[u(x x ) U(x )] + α[u(x x ) U(x)]; this function is linear and satisfies h(0) > 0 and h() < 0. Thus, there exists some α (0, ) close enough to 0 such that h(α ) > 0, which implies u ( (α, δ x x, δ x x )) > u ( (α, δ x, δ x )). If U(x) = U(x x ), quasi-supermodularity implies U(x x ) U(x ); thus, U(x x ) + U(x x ) U(x ) + U(x x ) = U(x ) + U(x). In this case, we can take α = 2. Finally, consider the case U(x) < U(x x ). If U(x x ) U(x α ), for any α (0, ), we have α (U(x x ) U(x )) + U(x x ) > U(x); rearranging terms yields the desired ranking. If U(x x ) < U(x ), we can still ɛ find some ɛ > 0 such that U(x) < U(x x U(x x ) ɛ. Let α := ) U(x ) ɛ + ; the same U(x x ) U(x ) inequality as before follows. Conversely, assume that preferences satisfy axioms (a), (b), (c), and (qs). Let U be as in the proof of Theorem. Take any two x, x X. By (qs), there exists some α (0, ) such that αu(x x ) + ( α)u(x x ) αu(x) + ( α)u(x ), 7

8 or α[u(x x ) U(x)] ( α)[u(x ) U(x x )], and so U(x ) > U(x x ) implies U(x x ) > U(x). Again, the last statement in the proposition follows from the von Neumann and Morgenstern Theorem. Lemma. A function F : X Θ R satisfies increasing differences if and only if, for all x, x X such that x > X x and θ, θ Θ such that θ > Θ θ: F(x, θ) F(x, θ) + ɛ implies F(x, θ ) F(x, θ ) + ɛ for all ɛ 0. Proof. If F satisfies increasing differences, for any x, x X such that x > X x and any θ, θ Θ such that θ > Θ θ, for any ɛ 0, F(x, θ ) F(x, θ ) F(x, θ) F(x, θ) ɛ. Conversely, assume that there exists some x 0 > X, x 0 X and θ 0, θ 0 Θ such that F(x 0, θ 0 ) F(x 0, θ 0 ) < F(x 0, θ 0) F(x 0, θ 0 ). Then, there exists some ɛ 0 > 0 such that F(x 0, θ 0 ) F(x 0, θ 0 ) < ɛ 0 < F(x 0, θ 0) F(x 0, θ 0 ). Proof of Theorem 3. The only portion of the theorem that remains to be shown corresponds to axiom (ɛ). Given some U : X Θ R, consider the family of preferences with expected-utility representation induced by U(, θ) : θ Θ. We can normalize these representations so that U(x 0, θ) = 0 and U(x, θ) = for each θ Θ. 2 Take x, x X such that x > X x, and θ, θ Θ such that θ > Θ θ. For ɛ 0, assume that +ɛ U(x, θ) = ( ( ) ) ( ( ) ) u +ɛ, δ x, δ x 0, θ u +ɛ, δ x, δ x, θ = +ɛ ɛ U(x, θ) + +ɛ. If U has increasing differences, U(x, θ ) U(x, θ ) U(x, θ) U(x, θ) = ɛ; the implication in axiom (ɛ) follows. Conversely, under axiom (ɛ), we get +ɛ U(x, θ ) +ɛ U(x, θ ) + +ɛ ɛ, or U(x, θ ) U(x, θ ) + ɛ; the result follows by Lemma. Proof of Theorem 4. Take x, x X such that x > X x, and θ, θ ( Θ( such that θ ) > ) Θ θ. Assume that there exists some ɛ 0 such that +ɛ U(x, θ) = u +ɛ, δ x, δ x 0, θ ( ( ) ) u +ɛ, δ x, δ x, θ = +ɛ ɛ U(x, θ) + +ɛ. Then, U(x, θ) U(x, θ) ɛ. If U has the single-crossing ( ( property, ) then ɛ := U(x, θ ) U(x, θ ) 0, and rearranging terms yields +ɛ, δ x, δ x 0, θ ) ( ( ) u +ɛ, δ x, δ x, θ ). Conversely, if ɛ := U(x, θ) U(x, θ) 0, we can find some ɛ 0 such that U(x, θ ) U(x, θ ) ɛ 0. Thus, U has the single-crossing property. References Fishburn, P. (982). The Foundations of Expected Utility. D. Reidel Publishing Company. 2 For each θ Θ, take Ũ(, θ) := U(,θ) U(x0,θ) U(x,θ) U(x 0,θ). 8

9 Herstein, I.N. and Milnor, J. (953). An axiomatic approach to measurable utility. Econometrica 2(2): Kreps, D. (203). Microeconomic Foundations I: Choice and Competitive Markets. Princeton University Press. Li Calzi, M. (990). Generalized symmetric supermodular functions. University. Mimeo, Stanford Milgrom, P. and Shannon, C. (994). Monotone comparative statics. Econometrica 62(): von Neumann, J. and Morgenstern, O. (953). Theory of Games and Behaviour. Princeton University Press. 9