Weak Assumption and Iterative Admissibility

Transcription

1 Weak Assumption and Iterative Admissiility Chih-Chun Yang Institute of Economics, Academia Sinica, Taipei 115, Taiwan January 2014 Astract Brandenurger, Friedenerg, and Keisler [Econometrica , ] show that rationality and common assumption of rationality RCAR is impossile in a complete and continuous type structure. We show, y inroducing an alternative notion of weak assumption, that rationality and common weak assumption of rationality RCWAR is possile in a complete and continuous type structure. This possiility result provides an epistemic characterization for iterative admissiility. Keywords: Iterative admissiility, weak assumption, common weak assumption of rationality. This paper is ased on the first chaper of the author s thesis, University of Rochester, I am indeted to my adviser, Paulo Barelli, for his valuale comments and encouragement. I thank Adam Brandenurger, Yi-Chun Chen, Amanda Friedenerg, Tai-Wei Hu, Takashi Kunimoto, Xiao Luo, Romans Pancs, Marciano Siniscalchi, Satoru Takahashi, Gáor Virág, Yongchao Zhang and participants in seminars at Academia Sinica, National Taiwan University, National University of Singapore, Shanghai University of Finance and Economics, University of Rochester, the 2010 Workshop on Epistemic Game Theory Stony Brook, and 2012 SAET conference. Financial support from the National Science Council of Taiwan is gratefully acknowledged. Fax: cyang16@econ.sinica.edu.tw.

2 1 Introduction Samuelson 1992 points out that there is a tension etween admissiility which requires the inclusion all possile strategies in a elief and knowledge/elief aout admissiility which requires the exclusion of some strategies from consideration. Brandenurger, Friedenerg and Keisler 2008a henceforth, BFK provide an epistemic analysis for iterative admissiility henceforth IA y using lexicographic proaility systems LPS s. BFK define the notion of assumption, a counterpart of knowledge, in the model constructed y LPS s. BFK show that rationality and common assumption of rationality RCAR is equivalent to the solution concept of self-admissile set, a counterpart of the rationalizale set. In particular, the IA set is a self-admissile set. However, if a player is not indifferent in every outcome, RCAR is empty in a complete and continuous type structure. BFK s impossiility result implies that RCAR fails to provide epistemic foundations of iterative admissiility. That is, if one keeps the continuity assumption, then one has to drop completeness to otain a nonempty RCAR. But then one has to provide the exact epistemic reason for the particular form of incompleteness. Keisler and Lee 2011 prove that in a finite normal form game, there is a complete ut not continuous lexicographic structure where a state elongs to RCAR if and only if the strategy part of the state is iteratively admissile. In particular, Keisler and Lee 2011 construct the type space as a Cantor space {0, 1} N, the set of all infinite sequences of 0 s and 1 s with the product topology. Their work suggests that the continuity assumption, coupled with full-support LPS s, yields a strong notion of cautious ehavior. Barelli and Galanis 2013 adapt a different approach. They propose a new notion of rationality, namely event-rationality, in a standard universal type structure. They characterize IA as the outcome of event-rationality and common validated elief of eventrationality RCvBER. The purpose of this paper is to investigate IA and RCAR y modifying the notion of assumption in a complete and continuous type structure. BFK use two axioms to define the notion of assumption: Nontriviality and Strict Determination. By keeping Strict Determination and modifying Nontriviality, we propose an alternative notion of weak assumption. The modified Nontriviality axiom comes from the idea of the consistency check in BFK This modification is designed to accommodate the consistency issue y requiring weaker conditions such that we can avoid the impossiility result: 1

3 Instead of using an open set, we opt to use a strategy cylinder as a significant event. We define rationality and common weak assumption of rationality RCWAR and prove that RCWAR is nonempty in a complete and continuous type structure. The main result of this paper is the outcome equivalence etween RCWAR and IA. Hence, we provide an epistemic foundation for iterative admissiility. The rest of this paper is organized as follows. Section 2 reviews BFK s result and defines weak assumption. Section 3 formulates the notion of RCWAR and shows the main result. The proofs of minor results are relegated to the Appendix. 2 The Model We follow the model in BFK. Let S a, S, π a, π e a two-player finite strategic form game, where S a, S are the strategy sets and π a, π are the payoff functions. Extend the payoff functions to mixed strategies in the usual way. Let X Y S a S. A strategy s a X is admissile with respect to X Y if there is no mixed strategy σ a satisfying σ a X = 1 such that for each s Y, π a σ a, s π a s a, s and for some s Y, π a σ a, s > π a s a, s. Iterative Admissiility IA: For i = a,, let S i 0 = S i and define inductively S i n+1 = { s a S i n : s i is admissile with respect to S a n S n}. A strategy s i n=1s i n s n=1s n is called iteratively admissile. Let M Ω e the space of Borel proaility measures on a separale metric space Ω. Let N Ω e the set of all finite sequences of Borel proaility measures on Ω. Let µ, µ N Ω. If the lengths of µ and µ are different, then the distance is 1. Otherwise, the distance etween µ and µ is the maximum of the Prohorov distances etween µ i and µ i for all i. Write supp µ i for the support of µ i M Ω. Say µ = µ 0,..., µ n 1 N Ω is a lexicographic proaility system LPS if µ satisfies mutual singularity: There are Borel sets U i in Ω for each i = 0,..., n 1 such that µ i U i = 1 and µ i U j = 0 if j i. Say µ N Ω is a full-support sequence if Ω is the union of the supports of µ i. We write L Ω for the set of LPS s and L + Ω for the set of full-support LPS s. 2

4 An S a, S -ased type structure is a structure S a, S, T a, T, λ a, λ, where the type spaces T a and T are nonempty Polish spaces and λ a, λ are Borel measurale functions: λ a : T a N S T and λ : T N S a T a. A type structure S a, S, T a, T, λ a, λ is complete if L + S T range λ a and L + S a T a range λ. It is continuous if λ a and λ are continuous. Each type of a player is associated with a sequence of proaility measures on the opponent s strategies and types. Let the notion L represent the lexicographic order. That is, for each a 1,..., a n, 1,..., n R n, a 1,..., a n L 1,..., n iff whenever a i < i, there is j < i such that a i > i. A strategy s a is optimal under µ = µ 0,..., µ n 1 N S T if π a s a, marg S µ i s n 1 L π a r a, marg S µ i s n 1 for all r a S a. A strategy-type pair s a, t a is rational if s a is optimal under λ a t a and λ a t a is a full-support LPS. Let R a e the set of rational strategy-type pairs s a, t a. The full support requirement says that no event is thought completely impossile y a rational player. A rational player must e so cautious that all his opponent strategies, eliefs over strategies and higher order eliefs are taken into account. This definition of rationality is designed to meet the condition of admissiility: It is well known that a strategy in X S a is admissile with respect to X Y S a S if and only if it is a est response in X to a elief with full support in Y. 2.1 BFK s result BFK define the notion of assumption, a counterpart of knowledge, in the model constructed y LPS s. Let A e the set of measurale functions from Ω to [0, 1]. For each 3

5 µ = µ 0,..., µ n 1 L + Ω, define a preference µ on A as follows. For each x, y A, define x µ y if x i ω dµ i ω ω Ω n 1 L ω Ω n 1 y i ω dµ i ω. They use two axioms on µ to define the notion of assumption. Let e a complete, transitive, reflexive inary relation, satisfying Independence axiom, on A. For each x, y A and E Ω, define x E, y Ω\E A if xe, y Ω\E = x on E and x E, y Ω\E = y on Ω\E. Let E Ω. An event E 0 is a part of E if there is an open set U Ω such E 0 = E U. An event E is assumed under if E is Borel and satisfies the following conditions: i Nontriviality: E is nonempty and for each part of E, E 0, there are acts x, y A such that x E 0 y; ii Strict Determination: For each act x, y A, x E y implies x y. Strict Determination says that the consequences in the event E are determining for. Nontriviality says that for each significant event U, an open suset of Ω, if U E, then U E is not Savage-null. Nontriviality is designed to accommodate the consistency check in BFK 2008: If a player steps into the analyst s shoes, then his elief and ehavior should e consistent with the analyst s prediction. BFK define that an event E is infinitely more likely than F under a full-support LPS µ. Say an event E is assumed under µ if E is infinitely more likely than Ω\E under µ. BFK show that an event E is assumed under µ if and only if E is assumed under µ. Define A a E {t a T a : E is assumed under λ a t a }. Define A E analogously. Let R a 1 R a and R 1 R. For each n 1, define R a n+1 and R n+1 inductively y R a n+1 R a n [ S a A a R n] and R n+1 R n [ S A R a n ]. If s a, s, t a, t n=1r a n n=1r n, then we say there is Rationality and Common Assumption of Rationality RCAR at this state. BFK show that for each integer n, proj S ar a n proj S R n = S a n S n. 4

6 Unfortunately, RCAR is empty in a natural class of games. Fix a complete and continuous type structure S a, S, T a, T, λ a, λ. If player a is not indifferent in every outcome, then there is no state at which there is RCAR. In fact, n=1r a n n=1r n =. 2.2 Weak Assumption Let Ω = S T. For each s S, let [s] {s} T. We define the notion of weak assumption as follows. Definition 1. An event E Ω is weakly assumed under if E is Borel and satisfies the following conditions: i Nontriviality : E is nonempty and for each s proj S E, there are acts x, y A such that x E [s] y; ii Strict Determination: For each act x, y A, x E y implies x y. We keep BFK s Strict Determination. Our Nontriviality is also designed to accommodate the consistency check. Our approach is ased on the idea that if the analyst s prediction, in particular, on the strategy space, is availale to a player, then the player s elief and ehavior should e consistent with this prediction. What if Ann steps into the analyst s shoes? Specifically, let us now focus on the strategies that can e played i.e., on the space S and imagine that the analyst s prediction is availale to Ann.... There, ecause Ann is confident of her prediction {L, C}, she must e more confident that Bo plays in accordance with the prediction and, in particular, plays C than that Bo violates his prediction and plays R. BFK 2008, p.1-2 Nontriviality is then modified to reflect the idea that the player is more confident of the nonempty intersection of the prediction and a strategy cylinder than its complement: Instead of using a nonempty open set, we opt to use a strategy cylinder as a significant event. Nontriviality says that for each significant event U, a strategy cylinder, if U E, then U E is not Savage-null. This modification helps us to avoid the impossiility result. 5

7 We conclude this section y the characterization of weak assumption. Lemma 1. Let µ = µ 0,..., µ n 1 L + Ω. An event E Ω is weakly assumed under µ if and only if E is Borel and there is 0 j n 1 such that 1 for each i j, µ i E = 1 and for each i j + 1, µ i E = 0; 2 proj S E = i j proj S supp µ i. If µ is a full-support LPS and an event E Ω is weakly assumed under µ, then we say E Ω is weakly assumed under µ. 3 Rationality and Common Weak Assumption of Rationality We consider the universal type spaces, T a and T, constructed in a way similar to Mertens and Zamir 1985, Brandenurger and Dekel 1993, and Heifetz Formally, ased on Heifetz 1993, we have the following result. Lemma 2. There are separale metric spaces T a and T such that T a is homeomorphic to N S T and T is homeomorphic to N S a T a. Let λ a e the homeomorphism etween T a and N S T. Likewise, define λ. Let E S T. Define W A a E {t a T a : E is weakly assumed under λ a t a }. Define W A in similar way. Let R 1 a R a and R 1 R. For each n 1, define R n+1 a and R n+1 inductively y R n+1 a R ] n a [S a W A a Rn and R n+1 R ] n [S W A Ran. By Lemma 3 in the Appendix, each R a n R n is Borel. If s a, s, t a, t n=1 R a n n=1 R n, then we say there is Rationality and Common Weak Assumption of Rationality RCWAR at this state. The main result of this paper is that RCWAR is outcome equivalent to IA in a complete and continuous type structure. Theorem 1. Let S a, S, T a, T, λ a, λ e the complete and continuous type structure defined in this section. For each n, proj s a Ra n proj s R n = S a n S n. 6

8 Moreover, proj S a Ra proj S R = n=1s a n n=1s n. In terms of strategies, IA and RCWAR are identical. RCWAR implies that a player chooses a strategy in IA. On the other hand, for each strategy s in IA, there is RCWAR at a state where s is chosen. The main idea of the proof is to show that there is a fixed s a, t a in each R a n. Since the game is finite, there is a finite m such that S a m = n=1s a n and S m = n=1s n. By Lemma 4 in the Appendix, Ω a \ R m is dense in Ω a. This fact help us to construct a full-support LPS λ a t a = µ 0, µ 1,..., µ m L + m+1 S T such that µ 0 has a finite support in R m and for each i 0, µ i has only atoms in Ω a \ R m. The notion of weak assumption is flexile enough such that for each n m, R m is weakly assumed under λ a t a. Proof of Theorem 1. We prove Theorem 1 in steps. Step 0. proj s a Ra 1 proj R s 1 = S1 a S1. This step is identical to BFK Theorem 9.1 and hence, omitted. Step 1 For each n m, proj s a Ra n proj R s n = Sn a Sn. We prove it y induction. Suppose that for each k n 1, proj R S k = S k and proj S a Ra k = Sk a. Step 1.1. Let s a R n. a We will find a type t a T a such that s a, t a R n a and hence, proj s a Ra n proj R s n Sn a Sn. By Lemma E1 in BFK, for each i {0,..., m 1}, for each s a Sn, a there is ν sa i M S such that supp ν sa i = S i and s a is a est response to ν sa i. Let U n = Ω a \ R 1, U 1 = R n 1 and for each i {2,..., n 1}, let U i = R n i\ R n i+1. By Lemma 3, each U i is Borel. By Lemma 2 separaility of T, each U i has a countale dense suset D i. By assigning positive weight to each point in D i, we may construct an LPS µ 1,..., µ n, where µ i M S T such that marg S µ i = ν sa n i, µ i U i = 1 and supp µ i is the closure of U i. By Lemma 2, there is t a T a such that λ a t a = µ 1,..., µ n. By the construction of µ, µ is a full-support LPS and s a, t a is a rational pair. Moreover, y Lemma 1 and Lemma 3, for each i {1,..., n 1}, R i is weakly assumed under µ. Therefore, s a, t a R a n. Step 1.2. Let s a, t a R a n. We will show that s a S a n and hence, proj s a Ra n proj s R n S a n S n. By hypothesis, and Lemma 1, there is j m such that proj S R n 1 = S n 1 = i j supp marg S µ i. Since s a, t a R a 1, s a is optimal under λ a t a, i.e., π a s a, marg S µ i s m L π a r a, marg S µ i s m for all ra S a. 7

9 As in Lemma 7.1 in BFK, there is ν M S such that supp ν = Sn 1 and s a is a est response to ν. Thus, s a is admissile with respect to S a Sn 1. By hypothesis, s a Sn 1. a Therefore, s a Sn a and proj s a Ra n+1 Sn+1. a Step 2. proj S a Ra proj R S = n=1sn a n=1 Sn. Since R n a and R n are shrinking, y Step 1, proj S a Ra proj R S Sm a Sm. It remains to show that for each s a Sm, a there is a type t sa T a such that s a, t sa n=1 Ra n. Step 2.1. Let R 0 a = Ω and R 0 = Ω a. For each s a Sm, a we will find a full-support LPS µ sa 1,..., µ m sa L + Ω a such that for each i {1,..., m}, 1 s a is a est response to each marg S µ sa 2 i j=1 supp µ j is the closure of R m i, and 3 µ i R m = 0. By Lemma E1 in BFK, for each s a Sm, a there is ν sa 0,..., ν sa m M S... M S such that for each i, supp ν sa i = S m i and s a is a est response to ν sa i. Consider a suset of T a : { T a t a T a : λ a t a = µ 0, µ 1,..., µ m L m+1 S T } {, s a Sm, a k 2, s.t. s S,ν } sa 0 s = marg Ω a µ { k 0 s, t } for some t Tk 1. By Lemma 4, T a T a \ T a is dense in T a and Ω S a T a \ R m a is dense in Ω. Likewise, define T a, T a, and Ω a. Let U m = Ω a \ R 1 and for each i {1,..., m 1}, let U i = Ω a R m i \ R m i+1. By Lemma 2, T is separale and each U i has a countale dense suset D i. Similar to Step 1 1.1, y assigning positive weight to each point in D i, for each i {1,..., m}, we may construct µ sa i M S T such that marg S µ i = ν sa i, µ i U i = 1 and supp µ i is the closure of U i. Since m i=1 D i is dense in Ω a, which is dense in Ω a, for each i {1,..., m}, i j=1 supp µ j is the closure of R m i and m j=1 supp µ j = Ω a. Hence, since {D i } m 1 i=1 are disjoint, µ s a 1,..., µ sa m L + Ω a. Likewise, for each s S m, define that i, µ s 1,..., µ s m L + Ω. Step 2.2. For each s a S a m and s S m, we will find t sa T a and t s T such λ t a sa = µ s a sa = µ 0, µ sa 1,..., µ m sa L + Ω a and λ t s = µ s = µ s 0, µ s 1,..., µ s m L + Ω, 8

10 where marg S µ sa 0 = ν sa 0 s.t. µ sa 0 marg S a µ s 0 = ν s 0 s.t. µ s 0 sa µ 0, µ sa 1,..., µ sa s S m { s, µ s} = 1, and s a S a m { s a, µ sa } = 1. Let λ t a sa = µ s a = m. Let margω a k µsa = marg Ω a k µ sa 0,..., marg Ω a k µ m sa. To construct t sa T a, we will specify each of the marginal distriutions marg Ω a k µ sa i. For each i 0 and k 0, let marg Ω a k µ sa i marg Ω a k µ sa i, i.e., µ sa i = µ sa i. Let marg Ω a 1 µ sa 0 ν sa 0. Likewise define each µ s 0. Inactively, for each k 1 define marg Ω a k+1 µ sa 0 M S Tk and marg Ω k+1 µ s 0 M S a Tk a as follows: For each s Sm and s a Sm, a marg Ω a k+1 µ sa 0 marg Ω k+1 µ s 0 s, marg Ω 1 µ s,..., marg Ω k µ s = ν sa 0 s and s a, marg Ω a 1 µsa,..., margω a k µ sa = ν s 0 s a. That is, in each k + 1-th step of constructing µ sa 0, we use the set { s, marg Ω 1 µ s,..., marg Ω k µ s } : s Sm as a finite support of marg Ω a k+1 µ sa 0. By the definition of T and the construction of µ sa, µ sa L + Ω a. By Lemma 2, there is t sa T a such that λ t a sa = µ s a. Likewise } define each t s. Moreover, y the construction of µ sa 0 and Lemma 2, µ sa 0 s S {s, t m s = 1. Likewise for. Thus, each s a i, t sa Ra 1 and each s i, t s R 1. By induction, Lemma 1 and Lemma 3, for each n 2, Ra n R n is weakly assumed under λ t sa λ t a s. Therefore, each s a, t sa s, t s survives in each round of elimination. That is, s a, t sa n=1 Ra n and s i, t s R n=1 n. In defining rationality, as in BFK, we require full support, which says that no event is thought completely impossile y a rational player. If we require full support only on the marginal proailities on strategies, we still have the outcome equivalence etween IA and RCWAR. 1 Appendix Proof of Lemma 1. If part. Let E Ω such that the conditions 1 and 2 are satisfied. We show that Nontriviality and Strict Determination are satisfied. 1 This is the main difference etween this paper and Catonini I thank Amanda Friedenerg for this oservation. The proof in this paper is still valid for this weak version of rationality. 9

11 Nontriviality. By 1, E. Let s proj S E. We want to show that there are acts x, y A such that x µ E [s] y. By 2, there is i j such that s proj S supp µ i, i.e., µ i [s] > 0. By 1, µ i E = 1 and hence, µ i E [s] = µ i [s] > 0. Therefore, there are acts x, y A such that x µ E [s] y. Strict Determination. Consider acts x, y A satisfying x µ E y, i.e., n 1 n 1 x i ω dµ i ω > L y i ω dµ i ω. ω E By 2, for each i > j, x ω E i ω dµ i ω = y ω E i ω dµ i ω = 0. Hence, we have j j x i ω dµ i ω > L y i ω dµ i ω ω E By 1, for each i j, x ω E i ω dµ i ω = x ω Ω i ω dµ i ω and y ω E i ω dµ i ω = ω Ω y i ω dµ i ω. Thus, x ω Ω i ω dµ i ω j >L y ω Ω i ω dµ i ω j. Therefore, ω Ω x i ω dµ i ω n 1 >L y ω Ω i ω dµ i ω n 1, i.e., x µ y. Only If part. Let E Ω e weakly assumed under µ. We have to show that conditions 1 and 2 are satisfied. Suppose, in negation, that 1 is not satisfied. That is, either a there are i, j such that i < j, µ i E = 0 and µ j E > 0 or there is j such that µ j E 0, 1. In case a, let c = 1 and in case, let c 0, 1 µ j E µ j E ω E ω E.. For each real numer r [0, 1], let r e the constant act assigning r to each state in Ω. By mutual singularity, there are {U i } 0 i n 1 such that for each i, µ i U i = 1 and for each j i, µ i U j = 0. Now, let x c Uj E0 Ω\Uj E and y 0 E 1 Ω\E. However, we have x µ E y and y µ x, contradicting Strict Determination. Given that 1 is satisfied, we show that proj S E = i j proj S supp µ i. Suppose not. Thus, there is s proj S E\ i j proj S supp µ i. Note that for each i j, µ i [s] E µ i [s] = 0. For each i > j, y 1, µ i [s] E µ i E = 0. Therefore, [s] E is null, contradicting Nontriviality. Proof of Lemma 2. We first construct the universal type spaces: T a and T. Let Ω a 1 = S and Ω 1 = S a. We then define the first order elief spaces as T1 a = N S and T1 = N S a. For each metric space A B and each µ = µ 0,..., µ n 1 N A B, define marg A µ = marg A µ 0,..., marg A µ n 1. Inductively, for each k 1 define Ω a k+1 = S T k and T a k+1 = { δ 1,..., δ k, δ k+1 T a k N Ω a k+1 : margω a k δk+1 = δ k}. 10

12 Likewise for. Let T a and T e the projective limits of the spaces {Tk a} k=1 and { Tk}, k=1 respectively. Note that T a and T are constructed to meet coherency requirement: Different order of eliefs shouldn t contradict one another. Let { δ k} {T a k=1 k } k=1 such that δ1 = δ 1 0,..., δn 1 1 N S. For each i {0,..., n 1}, let δ i { } δ k { {δ i and k=1 a k } {T a k=1 k } k=1 : δ1 M S }. Then for each i {0,..., n 1}, δ i a. Suppose that Y, d is a separale metric space. We may have a complete metric space Y, d such that Y is a completion of Y and d preserves identical metric on Y. For the formal proof, see Kolmogorov and Fomin 1970, Theorem 4, page 62. Hence, Y, d is Polish. We refer to the metric space Y without specifying the metric d. Since Ω a 0 is a Polish, the completion T1 a is also Polish. Hence, T1 a, as a suspace of the separale metric space T1 a, is also separale. By induction, each Tk a is also a separale metric space. So is T a. Hence, each proaility measure on Ω a k is tight Parthasarathy 1967, Chapter II, Theorem 3.2 and regular Parthasarathy 1967, Chapter II, Theorem 3.1 in the sense of Heifetz Thus, y Theorem 8 in Heifetz 1993, for each i {0,..., n 1}, there exists a unique µ i M S T such that for each k 1, marg Ω a k µ i = δ k i. Therefore, µ µ1,..., µ n 1 N S T. On the other hand, for each µ = µ 0,..., µ n 1 N S T, each µ i M S T. Since every metric space is normal, y Theorem 5* of Heifetz 1993, the topology used in Definition 2 of Heifetz 1993 is weak topology Billingsley 1968, Appendix III, which can e metrizale y Prohorov metric. Hence, y Theorem 9 in Heifetz 1993, a is homeomorphic to M S T. Likewise, define, which is homeomorphic to M S a T a. By the homeomorphism etween a and M { S T, each margω a µ k i} k=1 a. Therefore, under the topology generated y Prohorov metric in this paper, T a is homeomorphic to N S T. Lemma 3. For each natural numer n, R n a is Borel in S a T a. Proof. Let N + Ω e the set of full-support sequences of proaility measures, N n Ω { µ N Ω : µ = } µ 0,..., µ n 1, Ln Ω L Ω N n Ω and L + n Ω L + Ω N n Ω. By separaility and BFK s proof for Lemma C.2 and Corollary C.1, for each natural numer n, the sets N n Ω, N n + Ω, L n Ω and L + n Ω are Borel and the sets N + Ω, L Ω and L + Ω are also Borel. Let j n e a natural numer and E Ω e a Borel set. By the proof of BFK for Lemma C.3, the set of µ = µ 0,..., µ n 1 in L + n Ω such that condition 1 in Lemma 1 holds is Borel. Note that, given condition 1, condition 2 in Lemma 1 11

13 is equivalent to the following statement: For each s a proj S ae, there is i j such that marg µ i {s a } T a > 0. Thus, y fact ii in BFK s Appendix C, the set of µ = µ 0,..., µ n 1 in L + n Ω such that E is weakly assumed under µ is Borel. Similar to the proof of BFK for Lemma C.4, R n a is Borel in S a T a. Lemma 4. Let µ 1,..., µ m L m S T and ν sa 0 M S. Consider a suset of T a : { T a t a T a : λ a t a = µ 0, µ 1,..., µ m L S T } {, s a Sm, a k 2, s.t. s S,ν } sa 0 s = marg Ω a µ { k 0 s, t } for some t Tk 1. Then T a T a \ T a is dense in T a. Also, Ω \ R a m is dense in Ω. Thus, Ω S a T a \ R a m is dense in Ω. Proof. To prove this, y Lemma 2, it remains to show that for each t a T a, we may find a sequence {t n } T a such that lim n t n = t a. Suppose that λ a t a = µ 0, µ 1,..., µ m. Let X S T e the set of atoms of µ 0,..., µ m. Since T is uncountale and X is countale, S T \X. By Lemma 2, there is t T a such that λ a t = µ 0, µ 1,..., µ m and µ 0 µ 0 is a degenerate p.m. with mass at a point in S T \X. Then the sequence of {t n } n=1 can e found y letting λ a t n = 1 1 µ n 0 + 1n µ 0, µ 1..., µ n 1. Also, Ω \ R m a is dense in Ω. To prove this, y Lemma 2, it remains to show that for each s a, t a R m, a we may find a sequence {t n } T a such that lim n t n = t a and each s a, t n Ω \ R m. a Suppose that λ a t a = µ 0, µ 1,..., µ k L + k Ωa. Let X S T e the set of atoms of µ 0,..., µ k. Since Ω a \ R 1 is uncountale and X is countale, Ω a \ R 1 \X. By Lemma 2, there is t T a such that λ a t = µ 0, µ 1,..., µ k and µ 0 is a degenerate p.m. with mass at a point in can e found y letting λ a t n = Ω a \ R µ n 0 + 1n µ 0, µ 1..., µ k. \X. Then the sequence of {t n } n=1 Each s a, t n Ω \ R m a ecause, y Lemma 1, R 1 is not weakly assumed under λ a t n. Therefore, Ω S a T a \ R m a is dense in Ω. 12

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