Purification and roulette wheels

Transcription

1 Economic Theory manuscript No. (will be inserted by the editor) Purification and roulette wheels Michael Greinecker Konrad Podczeck Received: date / Accepted: date Abstract We use concepts introduced by Aumann more than thirty years ago to throw new light on purification in games with extremely dispersed private information. We show that one can embed payoff-irrelevant randomization devices in the private information of players and use these randomization devices to implement mixed strategies as deterministic functions of the private information. This approach gives rise to very short and intuitive proofs for a number of purification results that previously required sophisticated methods from functional analysis or nonstandard analysis. We use our methods to prove the first general purification theorem for games with private information in which a player s payoffs can depend in arbitrary ways on events in the private information of other players and in which we allow for shared information in a general way. Keywords purification games with incomplete information roulette wheels JEL Classification C72 1 Introduction In this paper we explore a new road to purification results. In the early days of game theory, mixed strategies have been interpreted as deliberate acts of randomization based on randomization devices. Robert Aumann took the step to explicitly model We are grateful to Rabeè Tourky and Nicholas Yannelis. Discussions with them on existence of purestrategy equilibria and the relation between randomization and decomposability techniques inspired this research. M. Greinecker Institut für Wirtschaftstheorie, Universität Innsbruck, SOWI-Gebäude, Universitätsstraße 15, A-6020, Austria michael.greinecker@uibk.ac.at K. Podczeck Institut für Volkswirtschaftslehre, Universität Wien, Oskar-Morgenstern-Platz 1, A-1010 Wien, Austria konrad.podczeck@univie.ac.at

2 2 Michael Greinecker, Konrad Podczeck such randomization devices. In Aumann (1964), he used explicit randomization devices to bypass measurability problems with mixing over measurable functions, and in Aumann (1974) he allows a player to condition on payoff-irrelevant information to study the role of subjectivity and correlation. If this information is private and includes an atomless algebra of events independent of the private information of other players, a secret roulette wheel in the terminology of Aumann, the player can use these events to implement all mixed strategies as deterministic functions of the private information. We use these ideas to provide elementary and intuitive proofs of recent purification results, and even generalize them. Moreover, we use our method to give a very general purification theorem for games with incomplete information. This latter result is the first purification result for games in which payoffs may depend in an arbitrary way on players types and in which general forms of public information are allowed for. Purification has a long history in game theory. Bayesian decision theory based on expect utility implies that players have no incentives to randomize, because mixed optimal choices must be mixtures over optimal deterministic choices. Applied game theorists have therefore often expressed a preference for equilibria in pure strategies. But the convexifying effect of randomization often guarantees that an equilibrium in mixed strategies exists. Purification theorems allow a researcher to construct pure strategy equilibria out of mixed strategy equilibria. The seminal papers 1 in this category are Dvoretzky, Wald, and Wolfowitz (1950), Dvoretzky et al (1951), Radner and Rosenthal (1982), Milgrom and Weber (1985), Balder (2008), and Khan and Rath (2009). These results are based on players having private information that gives rise to an atomless distribution and action spaces being finite or countably infinite. Mixed strategy equilibria can be shown to exist in great generality, 2 so this approach is quite powerful. These earlier purification results have never been extended to general compact metric action spaces. Indeed, an example in Khan, Rath, and Sun (1999) shows that this cannot be done when private information is assumed to be merely atomless. The first positive result to overcome this problem was given in Loeb and Sun (2006), where the authors used nonstandard analysis to extend the approach of Dvoretzky- Wald-Wolfowitz to atomless Loeb spaces and general compact metric spaces. They then applied this result to obtain a strong purification result for games with incomplete information in which players have general compact metric action spaces and private information is modeled by a Loeb probability space. It was subsequently shown in Podczeck (2009) that one can replace Loeb spaces by a much larger class of probability spaces, called super-atomless in that paper, and that theses spaces can be actually characterized as those spaces for which the abstract purification result holds. Mathematically, Podczeck used extreme point methods as introduced in Lindenstrauss (1966). Another proof of this more general purification result was subsequently given in Loeb and Sun (2009), which used the result in Loeb and Sun (2006) and methods developed in Hoover and Keisler (1984). A further strengthening of 1 We do not discuss purification based on perturbing the game as in Harsanyi (1973) and Govindan, Reny, and Robson (2003). See Morris (2008) for a comparison of these approaches. 2 See for example Milgrom and Weber (1985) and Balder (1988).

3 Liapounoff s theorem in Banach spaces 3 these results, based on machinery in Sun (2006) and Podczeck (2010), was given in Wang and Zhang (2012), showing that continuity and compactness assumptions required in earlier results can be dispensed with. Wang and Zhang thereby showed that purification is a purely measure theoretic problem. Along these lines, we present two abstract purification results in Section 3 below (Theorems 2 and 3). However, our proofs of these results substantially differ from the proofs found in the papers mentioned above, and are much simpler. In fact, our proofs are a straightforward application of the idea that a roulette wheel can be used to reproduce random distributions in an abstract setting (Theorem 1 in Section 3). In Section 4, we apply our approach to obtain a purification theorem for games with private information, our Theorem 4. In contrast to all previous purification theorems, a player s payoff may depend in arbitrary ways on the states of the world. We do not require a player s utility function to be measurable with respect to her private information or, in the language of Bayesian games, to depend only on her own type. We also allow for players to receive arbitrary Borel measurable signals about the private information of other players. We set our purification result in the framework of Aumann (1974), augmented by state dependent utility. This framework incorporates Bayesian games as a special case. 3 As will become clear, this level of generality is possible by identifying roulette wheels in the private information of players that players can use to mimic mixed strategies. The same idea will illuminate the saturation principle (for probability spaces) in Hoover and Keisler (1984) and we show that it can be seen as a purification principle in its own right in Section 5. In Section 6, we relate our work to a recent paper by He and Sun (2014) on games with incomplete information and to the problem of equilibrium existence in games with a continuum of players. Finally, in Section 7, we discuss the economic relevance of using super-atomless probability spaces. Closing the introduction, we remark that the classical purification results in Dvoretzky et al (1950), Dvoretzky et al (1951), Radner and Rosenthal (1982), Milgrom and Weber (1985), Balder (2008), and Khan and Rath (2009) do not depend on players being able to condition on payoff-irrelevant information. Thus our arguments cannot be applied to these settings directly. 2 Preliminaries This section contains several mathematical facts and introduces some terminology. These facts are largely well known, we collect them here for ease of reference. We start by introducing some notation and terminology. If U, V, and W are sets, and f : U V and g: U W are functions, then ( f,g) denotes the parallel product 3 A far reaching generalization of Aumann s framework can be found in Grant, Meneghel, and Tourky (2013), where the authors are able to prove the existence of pure strategy equilibria by employing a novel fixed-point theorem from Meneghel and Tourky (2013), in which convexity assumptions are replaced by a decomposability assumption from nonlinear analysis. They obtain pure strategy equilibria in Bayesian games directly without purifying mixed strategy equilibria. It is not clear to us whether a general purification result would hold in their framework, as their assumptions are not directly comparable to ours.

4 4 Michael Greinecker, Konrad Podczeck of f and g; thus ( f,g)(u) = ( f (u),g(u)) for each u U. By ι X we denote the identity on a set X. If Ω is a set and F 2 Ω, we denote by σ(f ) the smallest σ-algebra under set inclusion that contains F and call it the σ-algebra generated by F. A σ-algebra Σ on Ω is countably generated if there is a countable family C such that Σ = σ(c ). If Σ i i I is a family of sub-σ-algebras, we let i I Σ i = σ ( ) i I Σ i. If f is a function whose codomain is a measurable space (X,X ), the σ-algebra generated by f is the σ-algebra { f 1 (H) : H X } on the domain of f. It is the smallest σ-algebra that makes f measurable. A probability space is called super-atomless 4 if there is no non-negligible measurable set on which the subspace measure induces a measure algebra which is completely generated by a countable sub-algebra. The canonical example is the fair coinflipping measure on {0,1} κ with κ uncountable. But it is worth pointing out that the underlying space might be quite small. As a matter of fact, every atomless Borel probability measure on a Polish space can be extended to a super-atomless probability measure, as shown in Podczeck (2009, Appendix). We elaborate on this in Section 7. Maharam s representation theorem on the classification of finite measure spaces modulo null sets, given in Maharam (1942), can be used to provide a useful characterization of super-atomless probability spaces: A probability space is super-atomless if and only if there exists an uncountable family of independent random variables on it with uniform distribution on [0,1] (cf. Lemma 5). We denote the Borel σ-algebra of [0,1] by B, and the restriction of Lebesgue measure to B by λ. For a general topological space X, the Borel-σ-algebra of X is denoted by B(X). If (X,X ) is a measurable space, M (X) denotes the set of probability measures on (X,X ), endowed with the smallest σ-algebra such that for every B X the evaluation function given by ν ν(b) is measurable. If X is a topological space, then M (X) denotes the set of Borel probability measures on X, endowed with the σ-algebra defined in the previous paragraph, substituting B(X) for X. A topological space X is called Souslin if it is Hausdorff and if there is a continuous surjection from a Polish space onto X. Thus any Polish space is a Souslin space. If X is a Souslin space, then B(X) is countably generated. It is not hard to see that this implies that for a Souslin space X, the σ-algebra on M (X) defined above is countably generated. We shall make much use of distributions of random measures. Let (Ω,Σ, µ) be a probability space, and (X,X ) a measurable space. Let f : Ω M (X) be measurable. We define the distribution µ f of f by µ f (B) = f ( )(B) dµ Ω for all B X. We shall frequently identify a measurable function with values in X with a measurevalued function whose values are Dirac-measures. It is readily verified that, under 4 Terminology varies. See footnote 4 in Wang and Zhang (2012) for an overview over the various concepts that are equivalent to what we call super-atomless.

5 Liapounoff s theorem in Banach spaces 5 this identification, the distribution as defined above coincides with the usual notion of distribution of a measurable function. Let (Ω,Σ, µ) be a probability space, and let X i,x i i I, I = {1,...,n}, be a finite family of measurable spaces. Endow i I X i with the product-σ-algebra i I X i. If ( f 1,..., f n ) is a family of measurable functions f i : Ω M (X i ), we define the distribution or joint distribution µ ( f1,..., f n ) of this family to be the distribution of the function i f i : Ω M ( i I X i ) given by ( i f i )(ω) = i f i (ω), where i f i (ω) is the usual product measure of the measures f i (ω), i I, on i I X i. A monotone class argument shows that this function is again measurable, so its distribution on ( i I X i, i I X i ) is well-defined. If all fi are deterministic measurable functions, our notion of joint distribution coincides with the usual one for random variables. If we look at the joint distribution of an ordinary random variable and a measure valued random variable, such as the µ (ιω, f ) in the following lemma, the ordinary random variable is identified with a measure-valued random variable with Dirac measures as values. Lemma 1 Let (Ω,Σ, µ) be a probability space, (X,X ) a measurable space, and ϕ : Ω X R a Σ X - measurable function. Suppose that for some integrable function ρ : Ω R +, sup x X ϕ(ω,x) ρ(ω) for almost all ω Ω. Then given any measurable function f : Ω M (X), ϕ(ω,x)dµ (ιω, f )(ω,x) = ϕ(ω,x)d f (ω)(x)dµ(ω). Ω X In particular, if g: Ω X is measurable, then ϕ(ω,x)dµ (ιω,g)(ω,x) = Ω X Ω X Ω ϕ(ω,g(ω))dµ(ω). Proof The statement concerning g is a special case of that concerning f, identifying g with the map ω δ g(ω), where δ g(ω) denotes Dirac measure at g(ω). Now as for f, we have Ω X ϕ(ω,x) dµ (ιω, f )(ω,x) Ω X ρ(ω)dµ (ιω, f )(ω,x) = ρ(ω)dµ(ω) <, Ω so the claim follows from the generalized Fubini theorem for random measures, see Bogachev (2007, Theorem ). Lemma 2 Let (Ω,Σ, µ) be a probability space, let (X,X ) and (Y,Y ) be measurable spaces, and let f : Ω X and g : Ω Y be independent measurable functions. Then µ ( f,g) = µ f µ g. Proof It suffices to show that µ ( f,g) and µ f µ g agree on all measurable rectangles A B in X Y. Now, µ ( ( f,g) 1 (A B) ) = µ ( f 1 (A) g 1 (B) ) = µ ( f 1 (A) ) µ ( g 1 (B) ) = µ f (A)µ g (B).

6 6 Michael Greinecker, Konrad Podczeck The following lemma is fundamental for our constructions. It says that a random probability measure can be seen as the distribution of a random function. According to Rustichini (1993), the result dates back to work of Skorokhod in the 1950s. Lemma 3 Let (Ω,Σ) be a measurable space, X a Souslin space, and p: Ω M (X) a measurable map. Then there exists a Σ B-measurable map h: Ω [0,1] X such that for all ω Ω and all B B(X), p(ω) ( B ) { } = λ r [0,1] : h(ω,r) B Proof Bogachev (2007, Proposition ), or Aumann (1964, Lemma F). Lemma 4 Let (X,X ) and (Y,Y ) be measurable spaces and let A be a countably generated sub-σ-algebra of X Y. Then there exists a countably generated sub-σalgebra C of X such that A C Y. Proof We use the easily proved and well known fact that if a set is in the σ-algebra generated by some family, it is already in the σ-algebra generated by a countable sub-family. Let G X Y be a countable family generating A. By the fact just stated, for each G G we can find a countable family R G of measurable rectangles in X Y such that G is in the σ-algebra generated by R G. Let π X be the projection of X Y onto X. Then {π X (R) : R R G,G G } is countable, and we can take C to be the σ-algebra generated by this family. Lemma 5 Let (Ω,Σ, µ) be a probability space. Then: (a) µ is atomless if and only if there is an infinite independent family E i i I in Σ with µ(e i ) = 1/2 for each i I. (b) µ is super-atomless if and only if there is an uncountable independent family E i i I in Σ with µ(e i ) = 1/2 for each i I. Proof (a) follows from the fact that a probability space (Ω,Σ, µ) is atomless if and only if there is a map h: Ω {0,1} N whose distribution is the fair coin-flipping measure. For (b), see Podczeck (2010, Remark 1 and Lemma 2). If (Ω,Σ, µ) is a probability space and Σ Σ is a sub-σ-algebra, we say that µ is atomless on Σ if the restriction µ Σ is atomless; similarly for super-atomless. Lemma 6 Let (Ω,Σ, µ) be a probability space, and Σ, Σ 1 two sub-σ-algebra of Σ. Suppose Σ 1 is countably generated and that µ Σ is super-atomless. Then there is a countably generated sub-σ-algebra Σ 2 Σ which is independent of Σ 1 and such that µ Σ 2 is atomless. Proof By Lemma 5(b) there is an uncountable independent family E i i I in Σ with µ(e i ) = 1/2 for each i I. Now by Fremlin (2010, Theorem 272Q), there is a countable H I such that the σ-algebras Σ 1 and σ ( {E i : i I\H} ) are independent. As I is uncountable, there is a countable infinite set J I\H. Let Σ 2 = σ ( {E i : i J} ). In view of Lemma 5(a), Σ 2 is as required.

7 Liapounoff s theorem in Banach spaces 7 3 Purification in an abstract setting Purification theorems in the tradition of Dvoretzky-Wald-Wolfowitz show that a family of functions evaluated with respect to a deterministic function agrees with the evaluation with respect to a given random measure. By Lemma 1, this can be reduced to showing that the induced measures on a product space agree on a σ-algebra that makes all these functions measurable. If there is a σ-algebra which is independent of the former and on which µ is atomless, we can use it as a roulette wheel. It is a randomization device that can be used to make the implicit randomization in a random measure explicit by using Lemma 3. Theorem 1 Let (Ω,Σ, µ) be a probability space and let Σ 1 and Σ 2 be independent sub-σ-algebras of Σ such that µ is atomless on Σ 2. Let X be a Souslin space, and f : Ω M (X) a Σ 1 -measurable function. Then there is a measurable map g : Ω X such that the restriction of µ (ιω,g) to Σ 1 B(X) coincides with the restriction of µ (ιω, f ) to Σ 1 B(X). Proof Choose a Σ 2 -measurable map w : Ω [0,1] with µ w = λ, as is possible because µ is atomless on Σ 2, and let h : Ω [0,1] X be a Σ 1 B-measurable map chosen according to Lemma 3. Now let g = h (ι Ω,w). To see that (ι Ω,g) has the desired distribution, it suffices to consider a rectangle A B Σ 1 B(X). By Lemma 2, µ (ιω,w) Σ 1 B = µ Σ 1 λ. Thus, using Fubini s theorem, A f (ω)(b)dµ(ω) = λ ( h(ω, ) 1 (B) ) dµ(ω) A = µ Σ 1 λ ( (A [0,1]) h 1 (B) ) = µ (ιω,w) Σ 1 B ( (A [0,1]) h 1 (B) ) = µ ( (ι Ω,w) 1( (A [0,1]) h 1 (B) )) = µ ( (ι Ω,w) 1 (A [0,1]) (ι Ω,w) 1 (h 1 (B)) ) = µ(a g 1 (B)) = µ (ιω,g)(a B). As a corollary, we get a generalization of the purification results in Podczeck (2009) and Loeb and Sun (2009): Theorem 2 Let (Ω,Σ, µ) be a probability space, let X be a Souslin space, and let f : Ω M (X) be a measurable function. Let J be a countable set, and for each j J let ϕ j : Ω X R be a jointly measurable mapping such that for some integrable function ρ j : Ω R +, sup x X ϕ j (ω,x) ρ j (ω) for almost all ω Ω. Suppose µ is super-atomless. Then there is a function g: Ω X such that (a) g is measurable; (b) Ω X ϕ j(ω,x)d f (ω)(x)dµ(ω) = Ω ϕ j(ω,g(ω))dµ(ω) for all j J.

8 8 Michael Greinecker, Konrad Podczeck Proof The σ-algebra on Ω X generated by the countable family ϕ j j J of realvalued functions is countably generated. It follows from Lemma 4 that Σ has a countably generated sub-σ-algebra Σ 1 such that all these functions are Σ 1 B(X)- measurable. By Lemma 6, there exists a sub-σ-algebra Σ 2 that is independent of Σ 1 and on which µ is atomless. The claim now follows from Theorem 1 in conjunction with Lemma 1. As was pointed out in Wang and Zhang (2012), in a context like that of Theorem 2 there must actually exist uncountably many purifications. This may also be seen from our proof: If Σ i i I is any countable and independent family of sub-σ-algebras of Σ such that each Σ i is countably generated and independent of Σ 1, and such that µ is atomless on each Σ i, then Lemma 6 applied to Σ 1 i I Σ i gives another countably generated sub-σ-algebra of Σ, say Σ +, such that Σ + is independent of Σ 1 and each Σ i, and such that µ Σ + is again atomless. Now such a Σ + leads to another function w as used to construct g in the proof of Theorem 1, and leads thus to another purification in the context of Theorem 2. It is sometimes useful to have purifications that satisfy additional constraints. For example, actions available to a player in a Bayesian game might be type-dependent and then we want only purified strategies that are actually feasible. The following theorem allows for such constraints. 5 In this theorem, G Γ denotes the graph of the correspondence Γ. Theorem 3 Let (Ω,Σ, µ) be a probability space, let X be a Souslin space, and let Γ : Ω 2 X be a correspondence which admits a measurable selection and such that G Γ Σ B(X). Let f : Ω M (X) be a measurable map such that µ (ιω, f )(G Γ ) = 1. Endow G Γ with the subspace-σ-algebra defined from Σ B(X). Let J be a countable set, and for each j J let ϕ j : Γ R be a measurable map such that for some integrable ρ j : Ω R +, sup x Γ (ω) ϕ j (ω,x) ρ j (ω) for almost all ω Ω. Suppose µ is super-atomless. Then there is a g: Ω X such that (a) g is a measurable selection of Γ ; (b) Ω X ϕ j(ω,x)d f (ω)(x)dµ(ω) = Ω ϕ j(ω,g(ω))dµ(ω) for all j J. Remark 1 Sufficient conditions for the existence of a measurable selection as required in this theorem are for example that (Ω,Σ, µ) is complete and Γ is nonemptyvalued (Castaing and Valadier 1977, Theorem III.22), or that X is Polish and that Γ is a measurable correspondence with closed and nonempty values (Castaing and Valadier 1977, Theorem III.8). Proof of Theorem 3 For all j J, extend φ j to all of Ω X, setting φ j (ω,x) = 0 for (ω,x) / Γ. We can now use Theorem 2 to obtain a measurable function g : Ω X such that (b) and in addition Ω X 1 GΓ dµ (ιω, g) = 1 GΓ dµ (ιω, f ) Ω X 5 This result generalizes Theorem 15 in Carmona and Podczeck (2013), which is used there for purifying mixed equilibria in games with a continuum of players.

9 Liapounoff s theorem in Banach spaces 9 holds. By hypothesis, 1 GΓ dµ (ιω, f ) = µ (ιω, f )(Γ ) = 1, and it follows that there is N Σ with µ(n) = 0 such that g(ω) Γ (ω) for all ω Ω\N. Let s : Ω X be a measurable selection of Γ. Define g : Ω X by { g(ω) if ω Ω\N g(ω) = s(ω) if ω N. Then g is a measurable selection of Γ. Since g = g almost surely, the integrals in (b) will be unchanged if we replace g by g. 4 Purification in games with private information In this section, we prove a general purification theorem for games with private information. Our model is a slight modification of the extremely elegant framework in Aumann (1974). The main difference is that we allow for state dependent preferences and assume that all players have the same prior. 6 There is a finite set N of players and a probability space (Ω,Σ, µ) of states of the world with µ being the common prior of all players. For each player i N, there is an action space A i which is a Souslin space. Let A = i N A i. Each player i N has a bounded and measurable utility function u i : Ω A R. 7 The information of player i N is given by a sub-σ-algebra I i Σ. The idea is that nature picks a state according to µ and then every player is informed about the events in her σ-algebra that contain the state. Players receive private information about the state and then choose their actions. A mixed strategy of i N is a I i -measurable function m i : Ω M (A i ). A pure strategy of i N is a I i -measurable function p i : Ω A i. Clearly, we may treat pure strategies as degenerate mixed strategies. We assume the following: (A) There are families P i i N and S i i N of sub-σ-algebras of Σ such that: (i) I i = P i S i for each i N. (ii) For each i N, S i is independent of j i S j. (iii) µ S i is super-atomless for each i N. (iv) P i is countably generated for each i N. We think of S i as the secret private information of player i, and of P i as the part of her information other players may be informed about. But S i may not be completely secret. It is not possible to infer anything about S i from the pooled secret information of other players j i S i, but we allow for any dependence between S i and the P j. In particular, every countable family of Borel measurable numerical signals a player receives about the information of another player is admissible. In Loeb and Sun (2006) and Khan and Zhang (2012) it is required that all private information is mutually independent conditional on a countably valued random variable. We 6 For equilibrium existence results in the framework of states of nature instead of the formulation in terms of types, see Yannelis and Rustichini (1991). 7 We actually never use boundedness, but it ensures that expected utility is well defined. Clearly, weaker assumptions would do.

10 10 Michael Greinecker, Konrad Podczeck made our independence assumption on the S i in unconditional form because there is no sensible notion of a random variable we should condition on. We make no assumptions on the utility functions, they can depend in arbitrary ways on both actions and states. In contrast, in Loeb and Sun (2006) and Khan and Zhang (2012) it is required that payoffs depend only on private information and some countably valued random variable. The reason why we can work with very weak independence assumptions is the following Lemma. Intuitively, it says that if µ is super-atomless on a σ-algebra independent of another σ-algebra, then a large number of events in the former σ-algebra remain independent of the latter after conditioning on an arbitrary countably generated σ-algebra. Lemma 7 Let (Ω,Σ, µ) be a probability space, and F 1, F 2, P sub-σ-algebras of Σ. Suppose that F 1 and F 2 are independent, that µ F 2 is super-atomless, and that P is countably generated. Then there is a sub-σ-algebra F 3 F 2 such that F 3 and F 1 P are independent and such that µ F 3 is super-atomless Proof ) By Lemma 5 there is an uncountable ( independent ) family E i i I in F 2 with µ(e i = 1/2 for each i I. Let F = σ Ei i I,F 1. Let C P be a countable algebra generating P, and for each C C let h C be a conditional expectation of C on F. Let H F be a countably generated sub-σ-algebra such that each h C is H -measurable. There is a countable J I such that H σ ( ) E i i J,F 1. Set H = I\J, let F3 = σ ( ) E i i H, and let F4 = σ ( ) E i i J,F 1. Observe that F3 and F 4 are independent (use Fremlin 2010, 272F and 272K). In particular, therefore, G h = µ(g) Ω h whenever G F 3 and h is F 4 -measurable. Pick any G F 3, F F 1 and C C. By the last sentence of the previous paragraph, as both H F 4 and F 1 F 4, we have h C = 1 F h C = µ(g) 1 F h C = µ(g)µ(f C) G F G where the last equality holds because F F and h C is a conditional expectation of C on F. On the other hand, by this property of h C, as also G F F, h C = µ(g F C). G F It follows that each G F 3 is independent of the intersection of any two elements of F 1 C. Now by a monotone class argument, it follows that each G F 3 is independent of every element of σ ( F 1 C ), i.e., F 3 and σ ( F 1 C ) are independent. Finally, just note that σ ( F 1 C ) = F 1 P. Before we can state and prove our main theorem, we introduce another Lemma, which is a variant of Theorem 1. The proof can be found in an appendix. Lemma 8 Let (Ω,Σ, µ) be a probability space, let Σ 1, Σ 2, I be sub-σ-algebras of Σ, let X, Y be Souslin spaces, and let f x : Ω M (X) a I -measurable function. Suppose that Σ 1 and Σ 2 are independent, that Σ 2 I, and that µ Σ 2 is atomless. Then there is a I -measurable map g x : Ω X such that whenever f y : Ω M (Y ) is Σ 1 -measurable, then µ (ιω, f y,g x ) and µ (ιω, f y, f x ) agree on Σ 1 B(Y ) B(X). Ω

11 Liapounoff s theorem in Banach spaces 11 Our purification theorem shows that a player s mixed strategy can be replaced by a pure strategy that depends in exactly the same way as the given mixed strategy on all events that determine payoffs, action distributions, and shared information. Strategically, they are essentially indistinguishable: Theorem 4 (a) There is a countably generated sub-σ-algebra Σ of Σ such that u i is Σ B(A)-measurable for all i N and i P i Σ. (b) For any player i N and any mixed strategy m i of i, there is a pure strategy p i such that given any profile m j j i of mixed strategies of the other players, the joint distributions of ( ι Ω, m j j N ) and ( ιω, p i,m j j i ) agree on Σ B(A). Proof By Lemma 4, for each player i N there is a countably generated σ-algebra Σ i Σ such that u i is Σ i B(A)-measurable. Set Σ = i N Σ i i P i, so that Σ is as required in (a). Fix any player i and any mixed strategy m i of i. Let Σ 1 = Σ j is j. By Lemma 7 and condition (A), there is a σ-algebra Σ 2 I i which is independent of Σ 1 and such that µ Σ 2 is atomless. Note that any strategy m j of any player j i must be Σ 1 -measurable. The claim now follows from Lemma 8, substituting I i for I, m i for f x, and j i m j for f y, noting that B(A) = j I B(A j ), as all the spaces A j are Souslin. We have formulated Theorem 4 in terms of a single player to make clear that mixed and pure best replies materially coincide in our framework, but it is of course straightforward to purify strategy profiles. By purifying strategies in a profile one player at a time, it follows from Theorem 4 by a straightforward induction argument that every mixed strategy profile can be replaced by a pure strategy profile that gives rise to the same joint distribution over realized payoffs and action profiles, even when some players deviate from the profile. In particular, every Nash equilibrium can be purified. Now theorem 4 guarantees that not only the distributions of action profiles and the distributions over realized payoffs coincide for the purified and the original strategies, but also the joint distributions of action profiles and realized payoffs of these two strategies. Our notion of purification is therefore stronger than the notion of strong purification introduced in Khan et al (2006). The remarkable part of our proof is that it depends on player i being able to condition on events in a σ-algebra that is completely payoff-irrelevant 8, independent of the private information of everyone else, and rich enough that µ is atomless on it. In short, she is using a secret roulette wheel. It is worth pointing out that this does not hold when private information is merely assumed to be atomless and action spaces are countable. In that case, u i might well generate I i B(A). 8 Fu has drawn attention to differences between information that is payoff-relevant and other forms of information, and used this to obtain a purification result in the classical finite-action setting in Fu (2008). He takes the categorization of forms of information to be basic. In our view, payoff-relevance should be derived from the structure of the payoff-functions as we do here.

12 12 Michael Greinecker, Konrad Podczeck 5 Saturation revisited The following saturation principle was introduced in Hoover and Keisler (1984) and used as a basis for proving a purification result in Loeb and Sun (2009): A probability space (Ω,Σ, µ) is saturated if for any two Polish spaces X and Y, any probability measure ν M (X Y ), and any random variable φ : Ω X with distribution equal to the X-marginal of ν, there exists a random variable ψ : Ω Y such that (φ,ψ) has distribution ν. It was shown implicitly in Hoover and Keisler (1984) and more explicitly in Fajardo and Keisler (2002, Theorem B.7) that a probability space is saturated if and only if it is super-atomless. 9 Actually, saturation can be seen as a form of automatic purification. We first relate the saturation principle, strengthened to Souslin spaces, to independent randomization by employing Theorem 1, in order to show what happens in the background when saturation is employed as a black box. Closely related to the material in this section is Lemma 4 in Hoover and Keisler (1984), which is proven using methods from Maharam (1942). Proposition 1 Let (Ω,Σ, µ) be a probability space, let X and Y be Souslin spaces, let ν M (X Y ) and let φ : Ω X be a random variable with distribution equal to the X-marginal ν X of ν. Suppose there exists a sub-σ-algebra Σ 2 of Σ on which µ is atomless and such that Σ 2 is independent of the σ-algebra generated by φ. Then there exists a random variable ψ : Ω Y such that (φ,ψ) has distribution ν. Proof Since regular conditional probabilities are guaranteed to exist in Souslin spaces (Bogachev 2007, Corollary ) and the product of Souslin spaces is again a Souslin space (Bogachev 2007, Lemma 6.6.5(iii)) there exists a measurable function r : X M (Y ) such that (ι X,r) has distribution ν when we endow X with the marginal measure ν X. Let Σ 1 be the σ-algebra generated by φ. Let f = r φ. We can now take ψ to be the g guaranteed to exist by Theorem 1. Then the distribution of (ι Ω,ψ) restricted to Σ 1 B(Y ) coincides with the distribution of (ι Ω,r φ) restricted to Σ 1 B(Y ). Since φ is Σ 1 -measurable, the distribution of (ι Ω,φ,ψ) restricted to Σ 1 B(X) B(Y ) coincides with the distribution of (ι Ω,φ,r φ) restricted to Σ 1 B(X) B(Y ). For let A B C be a rectangle in Σ 1 B(X) B(Y ). We have µ (ιω,φ,ψ)(a B C) = µ (ιω,ψ)( A φ 1 (B) C ) = µ (ιω,r φ)( A φ 1 (B) C ) = µ (ιω,φ,r φ)(a B C). As B(X) B(Y ) equals B(X Y ), (φ,ψ) has distribution ν. The following proposition shows that joint applications of Theorem 1 and Lemma 6 could be based on saturation directly. We therefore think of saturation as a form of automatic purification. Proposition 2 Let (Ω,Σ, µ) be a saturated probability space, Σ 1 a countably generated sub-σ-algebra, X a Souslin space and f : Ω M (X) be Σ 1 - measurable. Then there exists a function g : Ω X such that the restriction of µ (ιω,g) to Σ 1 B(X) coincides with the restriction of µ (ιω, f ) to Σ 1 B(X). 9 The more involved notion of saturation for adapted stochastic processes introduced in Hoover and Keisler (1984) gives rise to a much more restricted class of probability spaces.

13 Liapounoff s theorem in Banach spaces 13 Proof As Σ 1 is countably generated, there is a measurable map φ : Ω {0,1} N such that Σ 1 is the σ-algebra generated by φ. Let ν be the distribution of (φ, f ). By saturation, there is a measurable function g : Ω X such that (φ,g) has distribution ν. But this implies that the restriction of µ (ιω,g) to Σ 1 B(X) coincides with the restriction of µ (ιω, f ) to Σ 1 B(X). Indeed, let A Σ 1 and B B(X). Because φ generates Σ 1, there is a Borel set C B ( {0,1} N) such that A = φ 1 (C). Now µ (ιω,g)(a B) = µ(φ 1 (C) g 1 (B)) = ν(c B). As ν(c B) = µ (φ, f ) (C B) = µ (ιω, f )( φ 1 (C) B ) = µ (ιω, f )(A B), the result follows. 6 The scope of our approach An approach to purification that is mathematically close to ours is followed in He and Sun (2014). That paper gives a sufficient condition for purification in Bayesian games with general compact metrizable action spaces, but payoffs are allowed to depend on only a player s own type, there is no public information, and players type distributions are required to be independent. The condition used is that paper is that the measure on the private information is "relatively diffuse" with respect to a countably generated σ-algebra with respect to which a player s payoff-function is measurable. This relative diffuseness condition can be shown to be equivalent to there being an independent sub-σ-algebra on which the measure of private information is atomless; see He, Sun, and Sun (2013, Proposition 1). The actual purification is then based on a result from Hoover and Keisler (1984), which is analogous to our Proposition 2. The resulting form of purification is essentially equivalent to our form of purification, but defined in terms of certain conditional expectations. Even though it is not the focus of this paper, our approach has an obvious application to the theory of large games. In particular, it illuminates the relation between the first proofs of the existence of pure strategy Nash equilibrium for games with a super-atomless space of players and general compact metrizable action spaces in the individualistic framework of Schmeidler (1973). Existence has already been proven for distributional equilibria in Mas-Colell (1984), where equilibria are certain joint distributions on actions and payoffs. The result of Mas-Colell has been used in both Carmona and Podczeck (2009) and Keisler and Sun (2009) to prove the existence of a pure strategy Nash equilibrium in the individualistic framework. Taking a disintegration of a distributional equilibrium gives immediately rise to a mixed strategy equilibrium in the individualistic framework. Carmona and Podczeck purify such a mixed strategy equilibrium using the super-nonatomicity of the space of players. Keisler and Sun on the other hand apply the saturation property directly to the distributional equilibrium to obtain an equilibrium in pure strategies. But Proposition 1 and its proof show that these approaches are essentially the same. For large games, an essentially necessary and sufficient condition for the existence of a pure strategy Nash equilibrium in the individualistic framework has been established in He, Sun, and Sun (2013). This condition requires that there exists a sub-σ-algebra in the space of players that is independent of that generated by the characteristics of the players. Since this amounts exactly to our condition for purifi-

14 14 Michael Greinecker, Konrad Podczeck cation in Theorem 1, the problem in establishing existence is really just a problem of going from mixed to pure strategy equilibria. 7 What do super-atomless probability spaces represent? There are two justifications for the assumption that the private information is superatomless or, in large games, that the space of players is super-atomless. One is philosophical, and one pragmatic. Let us assume that a player receives private signals lying in the unit interval [0,1]. Now, for which subsets of the unit interval do we want to be able to assign a probability? A common view is that one ideally wants to assign a probability to all subsets and merely refrains from doing so for technical reasons. However, it follows from an old theorem of Ulam that under the continuum hypothesis every probability measure defined on all subsets of the unit interval must be supported on countably many atoms; see Fremlin (2006, Corollary 419G). So it is consistent with the usual axioms of set theory that there is no atomless probability measure defined on all subsets of the unit interval. But one can discuss the set theoretic implications of the assumption that an atomless probability measure defined on all subsets of the unit interval exists. In that case, it follows from an extremely deep result of Gitik and Shelah that such a probability space must be super-atomless, see Fremlin (2008, 543E). In this respect, super-atomless measures represent the idea that all subsets of the unit interval are measurable in a better way than, say, Lebesgue measure. Since every Borel probability measure on a Polish space can be extended to a super-atomless probability measure, see Podczeck (2009, Appendix), we merely allow more sets to be measurable when working with super-atomless probability spaces. There is also a pragmatic justification for using super-atomless probability spaces. If action spaces are finite, we can take atomless private information to be very finegrained information relative to the size of the action space. Under strong independence assumptions, purification is possible in this setting as the classic purification results show. If we allow for compact metric spaces as models of a very large number of not too diverse actions, we should represent relatively fine grained information so that purification is still possible. This is exactly the case when information is represented by a super-atomless probability space. Similarly, in large applied games, as in many economic models in general, it is often assumed that players face individual uncertainty that is sufficiently diluted in the aggregate so that an exact law of large numbers holds. It follows from Podczeck (2010, Theorem 3) that formalizing this idea coherently requires the space of players to be super-atomless. The general point is that many properties that we are familiar with from the finite case, or that are intuitively reasonable, hold only when working with super-atomless probability spaces. 8 Conclusion The intuition behind our results is very simple: A super-atomless probability space contains a very large number of nontrivial independent events, but only a small num-

15 Liapounoff s theorem in Banach spaces 15 ber of events are important in determining the payoffs being realized and the actions being played. So a lot of events serve no relevant function and can be used as randomization devices. This rich supply of independent events makes purification with super-atomless spaces qualitatively different from purification with arbitrary atomless spaces of private information and finite or countably infinite action spaces. It is the rich supply of independent events in a super-atomless probability space that allows for a lot of constructions not available for general atomless probability spaces. One of the most striking consequences of this richness for economic applications is the existence of nontrivial jointly measurable processes that allow for enough independence to guarantee an exact law of large numbers as provided in Sun (2006) to hold. Such processes were shown to exist in Sun (2006) by nonstandard methods. General existence results were given in Podczeck (2010), where explicit use of the rich independence structure of super-atomless probability spaces is made. Aumann took the crucial step of incorporating the randomization devices, usually assumed to be working in the background, into the description of a game. This gives a new perspective on the general theme of purification: Purification is about the identification of randomization devices within a player s private information. Superatomless probability spaces are simply the easiest case, the point itself is general. Appendix Proof of Lemma 8 As in the proof of Theorem 1, choose a Σ 2 -measurable map w: Ω [0,1] so that µ w = λ, and by Lemma 3, choose a I B measurable h: Ω [0,1] X so that f x (ω)(b x ) = λ(h(ω, ) 1 (B x )) for each B x B(X) and each ω Ω. Let g x = h (ι Ω,w) and note that g x is I -measurable, as Σ 2 I. Now pick any Σ 1 -measurable function f y : Ω M (Y ). Note that by Lemma 2, µ (ιω, f y,w) Σ 1 B(Y ) B = ( µ (ιω, f y ) Σ 1 B(Y ) ) λ. Now given any rectangle A B y B x Σ 1 B(Y ) B(X), we can calculate as follows, where the sixth equality follows by Fubini s theorem, the seventh by the generalized version of Fubini s theorem, and where h: Ω Y [0,1] X is given

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