Institute for Mathematical Physics the Axiom of Real Blackwell Determinacy the Axiom of Real Blackwell Determinacy

The theory of infinite games with slightly imperfect information has been focusing on games with finitely and countably many moves. In this paper, we shift the discussion to games with uncountably many possible moves, introducing the axiom of real Blackwell determinacy Bl-AD R (as an analogue of the axiom of real determinacy AD R). We prove that the consistency strength of Bl-AD R is strictly greater than that of AD.


Introduction and background
Infinite perfect information games (called Gale-Stewart games after [4]) play a central rôle in the foundations of mathematics via the investigation of so-called determinacy axioms, among them the Axiom of Determinacy AD and the Axiom of Real Determinacy AD R . In spite of the fact that these two axioms contradict the axiom of choice, their foundational significance can hardly be overestimated.
Blackwell games are the analogue of Gale-Stewart games without perfect information; in these games, we are restricting the lack of perfect information to an infinite sequence of simultaneously made moves (cf. Footnote 5). They were introduced by Blackwell in 1969 [2], and dubbed "games with slightly imperfect information" by him in [3]. These games allowed new proofs of known consequences of determinacy. 1 In [13], Martin proved that in most cases, Blackwell determinacy axioms follow from the corresponding determinacy axioms. Martin conjectured that they are equivalent, and many instances of equivalence have been shown. 2 However, the general question, and in particular the most intriguing instance, viz. whether AD and the axiom of Blackwell determinacy Bl-AD are equivalent, remain open.
In this paper, we turn to the other mentioned determinacy axioms, the stronger AD R and its Blackwell analogues. We shall introduce the Axiom of Real Blackwell Determinacy Bl-AD R and investigate its relationship to AD R . The axiom of real determinacy has been studied by Solovay in his masterful analysis [18]. Its Blackwell analogues were introduced in [9] in two variants, the countable support variant and the Euclidean variant. While we give the definition of both variants below, we shall only be concerned with the countable support variant here, and denote it by Bl-AD R . We follow Solovay's lead and provide the results analogous to [18] for Bl-AD R . Our main result is: Main Theorem 1 Assume Bl-AD R . Then there is a fine normal measure on ℘ ω 1 (R), and hence ℵ 1 is < -supercompact and R # exists, where is the supremum of the ordinals which are surjective images of the reals. In particular, the consistency strength of Bl-AD R is strictly greater than that of AD.
The paper is organized as follows: We first give all necessary definitions to make the proof of Main Theorem 1 self-contained (Sect. 2). In Sect. 3, we reduce the Main Theorem to the existence of a fine normal measure, and then, in Sects. 4 and 5, we give two different proofs for the Main Theorem by proving the existence of a fine normal measure. In the concluding Sect. 6, we discuss some further results without detailed proofs and open questions. 3

Blackwell determinacy
We are using standard notation from set theory and assume familiarity with descriptive set theory throughout the paper. As usual in set theory, we shall be working on Baire space ω ω instead of the ordinary real numbers. Throughout we shall work in the theory ZF + AC ω (R). This small fragment of the axiom of choice is necessary for the definition of axioms of Blackwell determinacy. Using AC ω (R), we can develop the basics of measure theory. If we need more than ZF + AC ω (R) for some definitions and statements, we explicitly mention the additional axioms.
Let X be a set with more than one element and assume AC ω ( ω X ). The case most interesting for us is X = R. Since there is a bijection between ω R and R, the axioms AC ω ( ω R) and AC ω (R) are equivalent. By Prob(X ), we denote the set of all Borel probability measures on X with countable support, i.e., the set of all Borel probability measures p such that there is a countable set C ⊆ X with p(C) = 1. 4 If there is an x ∈ X such that p({x}) = 1, we call p a Dirac measure. From now on, we regard X as a discrete topological space and topologize ω X as the product space. For any finite sequence s of elements in X , let [s] be the basic open set generated by s, Let X Even (X Odd ) be the set of finite sequences in X with even (odd) length. We call a function σ : X Even → Prob(X ) a mixed strategy for player I and a function τ : X Odd → Prob(X ) a mixed strategy for player II. A mixed strategy σ is called pure if for all finite sequences s ∈ dom(σ ), we have that σ (s) is a Dirac measure. Pure strategies correspond to functions σ : X dom(σ ) → X , the standard type of strategies in Gale-Stewart games, and we can define the usual notions of winning strategies. Given mixed strategies σ and τ for players I and II respectively, let ν(σ, τ ) : <ω X → Prob(X ) as follows: For each finite sequence s of elements in X , where lh(s) is the length of s. Since some of the calculations in this paper require a lot of parentheses, let us reduce their number by convention.
x ∈ X and μ ∈ Prob(X ), we write μ(x) for μ({x}). Now, for each finite sequence s of elements in X , define By using AC ω (R× ω X ) (which follows from AC ω ( ω X )), we can uniquely extend μ σ,τ to a Borel probability measure on ω X , i.e., the probability measure whose domain is the set of all Borel sets in ω X . Let us also use μ σ,τ for denoting this Borel probability measure. Let A be a subset of ω X . A mixed strategy σ for player I is optimal in A if for any mixed strategy τ for player II, A is μ σ,τ -measurable and μ σ,τ (A) = 1. Similarly, a mixed strategy τ for player II is optimal in A if for any mixed strategy σ for player I, A is μ σ,τ -measurable and μ σ,τ (A) = 0. We say that A is Blackwell determined if either player I or II has an optimal strategy in A. Finally, Bl-AD X is the statement "for any subset A of ω X , A is Blackwell determined." 5 Remark 2 For any X , AD X implies Bl-AD X . In particular, AD R implies Bl-AD R . If there is an injective map from X to Y , then Bl-AD Y implies Bl-AD X . In particular, Bl-AD R implies Bl-AD := Bl-AD ω .

Blackwell determinacy and choice
The third author proved in 2005 that Bl-AD R proves fragments of the axiom of choice that allow us to separate it (in terms of implication, not yet in consistency strength) from Bl-AD. Theorem 3 If X := Y ∪ Z is linearly ordered and AC ω ( ω X ) and Bl-AD X hold, then AC Y (Z ) holds.
Corollary 4 Therefore, Bl-AD R implies AC R (R), and if Bl-AD is consistent, it cannot prove Bl-AD R . Proof The first claim is an immediate consequence of Theorem 3. It is well-known that if R is not wellordered in L(R), then AC R (R) is false in L(R). If V | Bl-AD, then L(R) | Bl-AD and R is not wellordered in L(R) (because every set of reals in L(R) is Lebesgue measurable by the result of Vervoort [19,Theorem 4.3]). Thus we have a model of Bl-AD ∧ ¬Bl-AD R .
Proof Woodin proved that under the assumption of AD and DC, AD R is equivalent to AC R (R) [6,Theorem 32.23]. The claim then follows directly from the assumptions and Corollary 4.

Measures and supercompactness
As usual, Θ := sup{α ; there is a surjection from R onto α}. Let X be a set and κ be an uncountable cardinal. As usual, we denote by ℘ κ (X ) the set of all subsets of X with cardinality less than κ, i.e., subsets A such that there is an α < κ and a surjection from α to A. Let U be a set of subsets of ℘ κ (X ). We say that U is κ-complete if U is closed under intersections of less than κ elements; we say it is non-principal if for any . We say that U is a fine measure if it is a fine κ-complete ultrafilter, and we say that it is a fine normal measure if it is a fine normal κ-complete ultrafilter. It is easy to check that if there is a surjection from X to Y , and there is a fine (normal) measure on ℘ κ (X ), then there is one on ℘ κ (Y ).
In the choice-less context, we can define a cardinal κ to be λ-supercompact if there is a fine normal measure on ℘ κ (λ) and to be λ-strongly compact if there is a fine measure on ℘ κ (λ). In the ZFC-context, this is equivalent to the usual definition (cf. [6,Theorem 22.7]). It is well-known that AD implies that ℵ 1 is ℵ 2 -supercompact [1].

An alternative definition of the axiom of real Blackwell determinacy
In [9], the second author started the investigation of the axiom of real Blackwell determinacy, and gave two alternative definitions of Bl-AD R . The second definition was the source of the definitions of long Blackwell games in [12, § 9.2]. For the sake of completeness, we give this definition here.
Instead of considering R as discretely topologized, we use the usual topology on R. In this setting, we do not require strategies to have countable support. A function assigning an arbitrary Borel probability measure on R to each finite sequence s of reals is called an E-mixed strategy. If σ and τ are E-mixed strategies for players I and II respectively, we define Clearly, the sequence of measures μ k σ,τ ; k ∈ ω coheres, i.e., and thus generate a Borel measure μ E σ,τ on ω R by the Kolmogorov consistency theorem.
For a subset A ⊆ ω R, we define the other notions as before: An E-mixed strategy σ for player I is E-optimal in A if for any E-mixed strategy τ for player II, A is μ E σ,τ -measurable and μ E σ,τ (A) = 1. Similarly, an E-mixed strategy τ for player II is E-optimal in A if for any E-mixed strategy σ for player I, A is μ E σ,τ -measurable and μ E σ,τ (A) = 0. We say that A is Euclidean Blackwell determined if either player I or II has an E-optimal strategy in A. Finally, EBl-AD R is the statement "for any subset A of ω R, A is Euclidean Blackwell determined".
It is easy to see that EBl-AD R implies Bl-AD. The proofs of Sect. 2.2 all go through under the assumption of EBl-AD R , so it is strictly stronger than Bl-AD. We do not know what the exact relationship between EBl-AD R and Bl-AD R is.

Solovay's analysis: reducing the main theorem to the existence of a fine normal measure
In [18], Solovay provided the foundations of the theory of AD R while at the same time gaining some understanding of the relationship between determinacy axioms and fragments of the axiom of choice. The set R # encodes a truth definition of L(R) in the same sense that 0 # encodes a truth definition of L. It yields a proper class of indiscernibles for L(R) such that they and the reals have L(R) as their Skolem hull in L(R) while it can be recovered from such a class of indiscernibles (for details, cf. [18, § 4]). In particular, if L(R) | T , then the existence of R # implies Cons(T ), and thus by Gödel's incompleteness theorem, T "R # exists". Fix A ⊆ ℘ ω 1 (R) and consider the following game G A : Players alternately play reals producing an infinite sequence − → x = (x i ; i ∈ ω). Then player II wins the game G A if {x n ; n ∈ ω} ∈ A, otherwise player I wins.
The following two theorems are the core of Solovay's analysis: Theorem 6 (Solovay) The axiom AD R implies that there is a fine normal measure on ℘ ω 1 (R).
Proof Solovay defined a family U ⊆ ℘ (℘ ω 1 (R)) as follows: A ∈ U if and only if player II has a winning strategy in G A .
Assuming AD R , Solovay then proved that this is a fine normal measure (cf. [ From Theorems 6 and 7, Solovay deduced that AD R is strictly stronger (in terms of consistency strength) than AD (as L(R) | AD). In Sects. 4 and 5, we shall give two proofs of the existence of a fine normal measure on ℘ ω 1 (R) under the assumption of Bl-AD R . If κ < Θ, then there is a surjection from R onto κ witnessing this. This surjection allows us to pull back the fine normal measure on ℘ ω 1 (R) to ℘ ω 1 (κ), and so ω 1 is κ-supercompact for every κ < Θ.

Theorem 8 (Martin, Neeman, Vervoort) If V=L(R) and Bl-AD holds, then AD holds.
Assume that Bl-AD R holds. By Remark 2, we also have Bl-AD which pulls back to L(R), so L(R) | Bl-AD, and hence L(R) | AD by Theorem 8. But by Theorem 7, we get R # , and thus Cons(AD).

Existence of a fine normal measure: the first proof
In this section, we shall give the first proof of the existence of a fine normal measure on ℘ ω 1 (R). We assume Bl-AD R and prove the (pure strategy) determinacy of games whose payoff sets are range invariant. A Note that the games G A defined in Sect. 3 are range invariant.

Lemma 9
Assume Bl-AD R . Then every range invariant subset of ω R is determined.
Proof (of Lemma 9) Let A be an range invariant subset of ω R. We show that if there is an optimal strategy for player I in A, then so is a (pure) winning strategy for player I in A. 6 The case for player II is similar and we shall skip it.
Let us first introduce some notation. Given a function f : For a pure strategy σ : R Even → R for player I, a ∈ ℘ ω 1 (R) is closed under σ if σ "a Even ⊆ a. For a function F : <ω R → ℘ ω 1 (R), a ∈ ℘ ω 1 (R) is closed under F if for any s ∈ <ω a, F(s) ⊆ a.

Claim 10
The following are equivalent: 1. There is a winning strategy for player I in A, 2. there is a function F : <ω R → ℘ ω 1 (R) such that if a is closed under F, then any enumeration of a belongs to A, and 3. there is a function f : <ω R → R such that if a is closed under f , then any enumeration of a belongs to A.
Proof (of Claim 10) The direction from (1) to (2) is easy to see: Given such a σ , let f : <ω R → R be any extension of σ and for s ∈ <ω R, let F(s) = { f (s)}. This F works for our purpose. We show the direction from (2) to (3): Given such an F, it suffices to show that there is an f such that if a is closed under f , then a is also closed under F. Fix a bijection π : R → ω R. Let g : <ω R → R be such that ran π(g(s)) = F(s) for each s (this is possible because every relation on the reals can be uniformized by a function by Corollary 4). Let h : <ω R → R be such that h(s) = π s(0) (lh(s) − 1) when s = ∅, where lh(s) is the length of s, if s = ∅ let h(s) be an arbitrary real.
It is easy to see that if a is closed under g and h, then so is under F: Fix an s ∈ <ω a. We have to show that F(s) ⊆ a. Consider g(s). By the closure under g, g(s) is in a. By choice of g, we know that ran(π(g(s))) = F(s), so it is enough to show that x is in a for any x in ran(π(g(s)). Suppose x is the nth bit of π(g(s)). Consider the finite sequence t = g(s), ..., g(s) of length n + 1. Then h(t) = π(t (0))(lh(t)) = π(g(s))(n) = x. But g(s) is in a and a was closed under h, so x is in a. Now it is easy to construct an f such that if a is closed under f , then so is under g and h.
We finally show the direction from (3) to (1). Given such an f , we can arrange a pure strategy σ for player I such that if x is a run of the game following σ , then the range of x is closed under f : Given a finite sequence of reals (a 0 , . . . , a 2n−1 ), consider the set of all finite sequences s from elements of {a 0 , · · · a 2n−1 } and all the values f (s) from this set. What we should arrange is to choose σ (a 0 , . . . , a 2n−1 ) in such a way that the range of any run of the game via σ will cover all such values f (s) when (a 0 , . . . , a 2n−1 ) is a finite initial segment of the run for any n in ω moves. But this is possible by a standard book-keeping argument. By the property of f , this implies that x is in A and hence σ is winning for player I.
(Claim 10) Therefore, to obtain a winning strategy for player I in A, it suffices to show (2) in Claim 10, i.e., there is a function F : <ω R → ℘ ω 1 (R) such that if a is closed under F, then any enumeration of a belongs to A. Let σ be an optimal strategy for player I in A. Let F be as follows: Then F is as desired: If a is closed under F, then enumerate a to be a n ; n ∈ ω and let player I follow σ and let player II play the Dirac measure for a n at her nth move. Then the probability of the set {x ∈ ω R ; ran(x) = a} is 1 and since σ is optimal for player I in A, there is an x such that the range of x is a and x is in A. But by the range invariance of A, any enumeration of a belongs to A.
(Lemma 9) We shall now show that the family U defined in Solovay's proof of Theorem 6 is a fine normal measure. Inspection of Solovay's proof shows that the games used to show that U is a fine measure are all (equivalent to the games) of the type G A which are range invariant and thus by Lemma 9, determined. Therefore, U is a fine measure.
We shall now show that U is normal. For this, let {A x ; x ∈ R} be a family of sets in U . We show that x∈R A x is in U . Consider the following game G: Player I moves x, then player II passes, and after that, they play the game G A x . This game is Blackwell determined by Bl-AD R . Player I cannot have an optimal strategy: Since A x ∈ U , player II has a winning strategy τ x in G A x ; thus, any mixed strategy that gives positive measure to some x will not play optimally against τ x . By Blackwell determinacy, player II has an optimal strategy, call it τ . Let F : <ω R → ℘ ω 1 (R) be as follows: The analogue of Claim 10 shows that in order to show that player II has a winning strategy in G x∈R A x , we only need to show that if a is closed under F, then a ∈ x∈R A x .
Suppose a = {x i ; i ∈ ω} is closed under F. Fix an arbitrary x ∈ a. We shall show that a ∈ A x , by considering a run of the game G. In this run, let player I start with x and then play the Dirac measure concentrating on x n at the nth move, and let player II follow τ . By construction, player I will with probability 1 only play elements of a, and thus, by definition of F and the fact that a is closed under F, player II will with probability 1 only play elements of a either. Consequently, with probability 1, the set produced by players I and II is just a. But τ was optimal for player II, so a ∈ A x . This proves that U is a fine normal measure, and thus, we have finished the first proof of our Main Theorem.

Existence of a fine normal measure: the second proof
In our second proof, we shall be formulating the Blackwell analogue of Solovay's fine normal measure from Theorem 6 and prove that this is a fine normal measure.
We define a family U ⊆ ℘ (℘ ω 1 (R)) by using the games G A defined in the proof of Theorem 6, and defining U by A ∈ U if and only if player II has an optimal strategy in G A .
The object U is the obvious Blackwell analogue of Solovay's normal measure. In the following, we shall show that under the assumption of Bl-AD R , it is a fine normal measure, thus finishing the proof.
A few properties of U are obvious: For instance, we see readily that ∅ / ∈ U and that ℘ ω 1 (R) ∈ U , as well as the fact that U is closed under taking supersets. In order to see that U is a fine family, fix a real x, and let player II play {x} with probability 1 in her first move; this is an optimal strategy for G {a ; x∈a} . Note that under the assumption of Bl-AD R , all of the games G A are Blackwell determined.
In order to prove the other required properties of a normal measure, we need to develop the appropriate transfer technique (as discussed and applied in [10]) for the present context. Let π ⊆ ω be an infinite and co-infinite set. We think of π as the set of rounds in which player I moves. We identify π with the increasing enumeration of its members, i.e., π ={π i ; i ∈ ω}. Similarly, we writeπ for the increasing enumeration of ω\π , i.e., ω\π ={π i ; i ∈ ω}. For notational ease, we call π a I-coding if no two consecutive numbers are in π and a II-coding if no two consecutive numbers are in ω\π .
Fix A ⊆ ℘ ω 1 (R) and define two variants of G A with alternative orders of play as determined by π . If π is a I-coding, the game G π,I A is played as follows: If π is a II-coding, then we play the game G π,II A as follows: In both cases, player II wins the game if {x n ; n ∈ ω} ∈ A. Obviously, we have where Even is the set of even numbers.

Lemma 11
Let A be a subset of ℘ ω 1 (R) and π be a I-coding. Then there is a translation σ → σ π of mixed strategies for player I such that if σ is an optimal strategy for player I in G A , then σ π is an optimal strategy for player I in G π,I A . Similarly, if π is a II-coding, there is a translation τ → τ π of mixed strategies for player II such that if τ is an optimal strategy for player II in G A , then τ π is an optimal strategy for player II in G π,II A .
Proof We prove only the claim for the games G π,I A , the other proof being similar. Fix a I-coding π . Let π * be the permutation of ω defined by: Let h : ω R → ω R be the range-preserving homeomorphism given by setting If s is a position in G π,I A of length π n , let g(s) be the corresponding position of length 2n in G A , i.e., for i < 2n let g(s)(i) = s π * (i) . (In g(s), player II has made only n of the moves corresponding to the π n − n moves that II has made in s.) If σ is a mixed strategy for I in G A , let σ π be the mixed strategy for I in G π,I A given by σ π (s) = σ g(s) .
We shall show that σ π is optimal for player I in G A assuming σ is optimal for player I in G π,I A . Let τ be any mixed strategy for player II in G A . We show that the set { − → x ; ran( − → x ) ∈ A} has μ σ π ,τ -measure 1.
If s is a position in G A of length 2n + 1, let f (s) be the corresponding position of lengthπ n in G π,I A , letf (τ ) be the mixed strategy for II in G A given bŷ By induction on length, one can easily show that, for all s ∈ <ω R, Since h is range-preserving, Since μ is optimal for player I in G π,I A , the left-hand side of the above formula is equal to 1. Therefore, the set { − → x ; ran( − → x ) ∈ A} has μ σ π ,τ -measure 1, as desired.
(Lemma 11) Based on Lemma 11, we can now finish the proof of the Main Theorem.
Proof If player II does not have an optimal strategy in G A , then player I does. Let π := Odd, the set of odd numbers. Then in G π,I A , the rôles of players I and II are switched. By Lemma 11, there is an optimal strategy σ π for player I in the game G π,I A , but this is optimal for player II in G ℘ ω 1 (R)\A .

Claim 14 For a family {A x ; x ∈ R} ⊆ U , the diagonal intersection x∈R A x is in U .
Proof (of Claim 14) As in [18], this is the most intricate part of the proof.

Subclaim 15
There is a choice function x → τ x picking an optimal strategy for G A x .
Proof (of Subclaim 15) The argument is exactly the same as the one in the proof of normality of U in the last section.
The properties of ρ make sure that this strategy is well-defined; we shall now prove that τ is an optimal strategy for player II in G x∈R A x . Pick any mixed strategy σ for player I in G x∈R A x , and define strategies σ n for G π n ,II . Let m = ρ(k, ), then σ n (x 0 , . . . , x 2m−1 ) := σ (x 0 , . . . , x 2m−1 ), and σ n (x 0 , . . . , Note that for each n ∈ ω and s ∈ n+1 R, μ σ,τ and μ σ n ,(τ s(n) ) πn agree below [s], i.e., for any set A ⊆ [s], μ σ,τ (A) = μ σ n ,(τ s(n) ) πn (A). The payoff set (for player II) for G x∈R A x is A : it suffices to check that the sets A n := { − → x ; ran( − → x ) ∈ A x n } has μ σ,τ -measure 1. for all n ∈ ω. But μ σ n ,(τ s(n) ) πn ([s] ∩ A n ) = μ σ n ,(τ s(n) ) πn ([s]). Then, using the fact that the measures have countable support, we get ( Claim 14) Note that together with the trivial properties of U mentioned at the beginning of this section, Claims 12, 13, and 14 are all we need to show: The non-principality of U follows from the fineness of U , and the σ -completeness follows from the fact that every set of reals is Lebesgue measurable (an ultrafilter failing σ -completeness defines a non-principal ultrafilter on ω and hence a non-Lebesgue measurable set). Thus, we have finished the second proof of the Main Theorem. Main Theorem 1 shows that the consistency strength of Bl-AD R is strictly greater than that of Bl-AD. In unpublished work, Woodin and the first author made steps towards computing the exact consistency strength of Bl-AD R [5]. They conjecture that Bl-AD R and AD R are equiconsistent [5, Conjecture 6.2] and show that this conjecture follows from plausible inner model theoretic statements.
The following argument gives an indication of what lower bounds we currently know how to prove: We give a sketch of a proof of AD L(R,R # ) from Bl-AD R . This is a stronger assumption than AD L(R) +"R # exists". 7 By Theorems 8 and 1, we have AD L(R) and that R # exists. By a result of Solovay (cf. [17, p. 117]), this implies that every set of reals in L(R) is Suslin in L(R, R # ). If we let W n,m be the set of n formulas true in L(R) with m indiscernibles, and F n,m the set of n formulas false in L(R) with m indiscernibles, then each of these sets is in L(R) and thus Suslin, and furthermore, R # = n,m∈ω W n,m and R\R # = n,m∈ω F n,m . Assuming AC R (R) in V, we have DC in L(R, R # ) because every set in L(R, R # ) can be coded by an ordinal, a real and R # . We use DC to pick witnesses for the Suslinness of W n,m and F n,m and can thus show that R # is both Suslin and co-Suslin.
Let ( ‡) denote the statement "all sets that are both Suslin and co-Suslin are determined"; in [7, Theorem 1.2], Kechris and Woodin show that ( ‡) implies that L(R) | AD. For any set X that is both Suslin and co-Suslin, this easily generalizes to L(R, X ) | AD. But we assumed Bl-AD R , and thus by [15,Lemma 4.1], we get that ( ‡) holds. Collecting all of the components, we obtain L(R, R # ) | AD.
The same argument can now be applied to L(R, R # ) to give AD L(R,R ## ) , or more inductively.

Getting a fine measure from Bl-AD
It is well-known that Solovay's proof of the existence of a fine normal measure on ℘ ω 1 (R) under AD R can be modified to work under AD by a simple coding argument (cf. [8, § 4.5]) if you give up normality, thus giving a proof of the following theorem:
Our proofs in Sects. 4 and 5 do not seem to allow us to prove the Blackwell analogue of Theorem 16 (essentially, since the modified games are not range invariant anymore, as they rely on the coding used).
6.3 Bl-AD R and the cofinality of Θ Solovay proved that the consistency strength of AD R + cf(Θ) > ω is strictly bigger than that of AD R . This proof uses model constructions based on the Wadge hierarchy (cf. [18,Theorems 2.5 and 5.7]). In order to prove the appropriate analogues in the Blackwell context, we should need to make use of the Blackwell analogue of the Wadge hierarchy, the Blackwell Wadge hierarchy (cf. [12, § 7.3]). Unfortunately, we do not know how to prove that this hierarchy is wellfounded (not even under the assumption of Bl-AD R ).