% operators that are separated from the operand by a space

% operators that require brackets

% operators that require parentheses

% Paper specific

This post is formal treatment of the idea outlined here.

Given a countable set of incomplete models, we define a forecasting function that converges in the Kantorovich-Rubinstein metric with probability 1 to every one of the models which is satisfied by the true environment. This is analogous to Blackwell-Dubins merging of opinions for complete models, except that Kantorovich-Rubinstein convergence is weaker than convergence in total variation. The forecasting function is a dominant stochastic market for a suitably constructed set of traders.

Appendix A contains the proofs. Appendix B restates the theorems about dominant stochastic markets for ease of reference. Appendix C states a variant of the Michael selection theorem due to Yannelis and Prabhakar which is used in Appendix A.

##Notation

Given X a metric space, x∈X and r∈R, Br(x):={y∈X∣d(x,y)<r}.

Given X a topological space:

P(X) is the space of Borel probability measures on X equipped with the weak* topology.

C(X) is the Banach space of continuous functions X→R with uniform norm.

T(X):=C(P(X)×X)

B(X) is the Borel σ-algebra on X.

U(X) is the σ-algebra of universally measurable sets on X.

Given μ∈P(X), suppμ denotes the support of μ.

Given X and Y measurable spaces, K:Xmk−→Y is a Markov kernel from X to Y. For any x∈X, we have K(x)∈P(Y). Given μ∈P(X), μ⋉K∈P(X×Y) is the semidirect product of μ and K and K∗μ∈P(Y) is the pushforward of μ by K.

Given X, Y Polish spaces, π:X→Y Borel measurable and μ∈P(X), we denote μ∣π the set of Markov kernels K:Ymk−→X s.t. π∗μ⋉K is supported on the graph of π and K∗π∗μ=μ. By the disintegration theorem, μ∣π is always non-empty and any two kernels in μ∣π coincide π∗μ-almost everywhere.

##Results

Consider X any compact Polish metric space and d:X×X→R its metric. We denote Lip(X) the Banach space of Lipschitz continuous functions X→R with the norm

∥f∥Lip:=maxx|f(x)|+supx≠y|f(x)−f(y)|d(x,y)

P(X) (as before, with the weak topology) can be regarded as a compact subset of Lip(X)′ (with the strong topology), yielding a metrization of P(X) which we will call the Kantorovich-Rubinstein metric dKR:

dKR(μ,ν):=sup∥f∥Lip≤1|Eμ[f]−Eν[f]|

In fact, the above differs from the standard definition of the Kantorovich-Rubinstein metric (a.k.a. 1st Wasserstein metric, a.k.a. earth mover's metric), but this abuse of terminology is mild since the two are strongly equivalent.

Now consider Φ⊆P(X) convex. We will describe a class of trading strategies that are designed to exploit any μ∈Φ.

#Definition 1

Let α>0. τ∈T(X) is said to be an α-representative of Φ when for all μ∈P(X), ν∈Φ we have

∥τ(μ)∥≤max(dKR(μ,Φ)−α,0)

Eν[τ(μ)]≥Eμ[τ(μ)]+12(dKR(μ,Φ)−α)dKR(μ,Φ)

#Lemma 1

Consider Y another compact Polish space and Φ⊆Y×P(X) closed. For any y∈Y, define Φy⊆P(X) by

Φy:={μ∈P(X)∣(y,μ)∈Φ}

Assume that for any y∈Y, Φy is convex. Then, for any α>0, there exists υα:Y→T(X) measurable w.r.t. U(Y) and B(T(X)) s.t. for all y∈Y, υα(y) is an α-representative of Φy.

For each n∈N, let On be a compact subset of RD(n) (D:N→N is arbitrary). Denote

Yn:=∏m<nOm

Yn is a compact subset of R∑m<nD(m). We will regard it as equipped with the Euclidean metric. Denote

X=∞∏n=0On

X is a compact Polish space. For each n∈N we denote πn:X→Yn the projection mapping. Given n∈N and x∈X, we have x(n):=πn(x)∈Yn and xn∈On.

The following definition provides a notion of updating an incomplete model by observations:

#Definition 2

Consider Φ⊆P(X). For any n∈N and y∈Yn, we define Φ′′y⊆P(X) by

Φ′′y:={limr→0(μ∣π−1n(Br(y)))∣μ∈Φ}

Note that the limit in the definition above need not exist for every μ∈Φ.

Denote Φ′y⊆P(X) the convex hull of Φ′′y and define Φ′n⊆Yn×P(X) by

Φ′n:={(y,μ)∈Yn×P(X)∣μ∈Φ′y}

Finally, we define Φn⊆Yn×P(X) to be the closure of Φ′n. Given y∈Yn, we define Φy⊆P(X) by

Φy:={μ∈P(X)∣(y,μ)∈Φn}

Note that the above definition uses the Euclidean metric on Yn. This is the only place through which the assumption that On is a compact subset of RD(n) (rather than an arbitrary compact Polish space) enters. This is used the in the proof of Lemma 2 below, via the Lebesgue differentiation theorem.

Fix a family of metrizations of X: {dn:X×X→R}n∈N. The reason we need a family rather than a single metric is that convergence in dKR is trivial unless we renormalize the metric for each n. On the other hand, completely arbitrary renormalization is allowed. For example, assuming the diameters of On are uniformly bounded, we can take

dn(x1,x2)=maxm∈Ncnmd(x1m,x2m)

Here, {cnm>0}n,m∈N are required to satisfy limm→∞cnm=0. To get a non-trivial result, we need the cnm to fall slower with m as n increases. The stronger we make this trend, the stronger conclusion we get (although it always remains weaker than convergence in total variation).

#Lemma 2

Consider Φ⊆P(X) and α>0. Let υ be a metastrategy s.t. for each n∈N and y∈Yn, υn(y) is an α-representative of Φy (relatively to dn). Then, υ is profitable for any μ∗∈Φ.

Combining Theorem B.2 and Lemma 2, it is easy to get the following:

#Corollary 1

Consider Φ⊆P(X), α>0 and υ a metastrategy s.t. for each n∈N and y∈Yn, υn(y) is an α-representative of Φy. Let M be a market which dominates Tυ. For every n∈N, denote dnKR the Kantorovich-Rubinstein metric associated with dn. Then, for any μ∗∈Φ and μ∗-almost any x∈X:

limsupn→∞dnKR(Mn(x(n)),Φx(n))≤α

Thus, if we construct a set of traders Tυk as in Corollary 1 where υk corresponds to α=1k, then a market dominating all of these traders would have to converge to Φyn with μ∗-probability 1. Putting this together with Lemma 1 (so that υk as above actually exists) and Theorem B.1 we finally get

#Theorem 1

Consider any H⊆2P(X) countable. Then, there exists a market MH s.t. for any Φ∈H, μ∗∈Φ and μ∗-almost any x∈X:

limn→∞dnKR(MHn(x(n)),Φx(n))=0

This leaves an interesting open question, namely whether the counterpart of Theorem 1 with dKR replaced by total variation distance is true.

##Appendix A

#Proposition A.1

Let X be a compact Polish space, φ∈Lip(X)′′ and ϵ>0. Then, there exists fϵ∈Lip(X) s.t. ∥fϵ∥Lip≤∥φ∥ and for all μ∈P(X), |Eμ[fϵ]−φ(μ)|<ϵ.

#Proof of Proposition A.1

Without loss of generality, assume that ∥φ∥=1 (if ∥φ∥=0 the theorem is trivial, otherwise we can rescale everything by a scalar). Using the compactness of P(X), choose a finite set Aϵ⊆P(X) s.t.

maxμ∈P(X)minν∈AϵdKR(μ,ν)<ϵ4

Using the Goldstine theorem, choose fϵ∈Lip(X) s.t. ∥fϵ∥Lip≤1 and for any ν∈Aϵ

|Eν[fϵ]−φ(ν)|<ϵ2

Consider any μ∈P(X). Choose ν∈Aϵ s.t. dKR(μ,ν)<ϵ4. We get

Let X be a compact Polish space and Φ⊆P(X) convex. Consider any α>0 and μ0∈P(X) s.t. r:=dKR(μ0,Φ)>α. Then, there exists f∈Lip(X) s.t.

∥f∥Lip<r−α

infν∈ΦEν[f]>Eμ0[f]+12(r−α)r

#Proof of Proposition A.2

Define Φr⊆Lip(X)′ by

Φr:={μ∈Lip(X)′∣infν∈Φ∥μ−ν∥<r}

By the Hahn-Banach separation theorem, there is φ∈Lip(X)′′ s.t. for all ν∈Φr, φ(ν)>φ(μ0). Multiplying by a scalar, we can make sure that ∥φ∥=34(r−α). For any δ>0, we can choose ζ∈Lip(X)′ s.t. ∥ζ∥<1 and φ(ζ)>34(r−α)−δ. For any ν∈Φ, ν−rζ∈Φr and therefore

φ(ν−rζ)>φ(μ0)

φ(ν)>φ(μ0)+rφ(ζ)>φ(μ0)+r(34(r−α)−δ)

Taking δ to 0 we get φ(ν)≥φ(μ0)+34r(r−α). Applying Proposition A.1 for ϵ=116r(r−α), we get f∈Lip(X) s.t.

By Proposition A.2, for any μ∈U, F(μ)≠∅. Clearly, F(μ) is convex. Fix f∈Lip(X) and consider F−1(f):={μ∈U∣f∈F(μ)}. Consider any μ0∈F−1(f), and take ϵ>0 s.t.

∥f∥Lip<dKR(μ0,Φ)−ϵ−α

infν∈ΦEν[f]>Eμ0[f]+ϵ+(dKR(μ0,Φ)+ϵ−α)(dKR(μ0,Φ)+ϵ)

Define V:={μ∈U∣dKR(μ,μ0)<ϵ,Eμ[f]<Eμ0[f]+ϵ}. Obviously V is open and V⊆F−1(f), hence F−1(f) is open. Applying Theorem C, we get the desired result.

#Proposition A.4

Let X be a compact Polish space, Φ⊆P(X) convex and α>0. Then, there exists τ∈T(X) an α-representative of Φ.

#Proof of Proposition A.4

Let U⊆P(X) be defined by

U:={μ∈P(X)∣dKR(μ,Φ)>α}

Use Proposition A.3 to obtain τ′:U→Lip(X). Define τ:P(X)→Lip(X) by

τ(μ):={τ′(μ) if μ∈U0 if μ∉U
Obviously τ is continuous in U. Consider any μ∉U and {μn∈P(X)}n∈N s.t. μn→μ. We have dKR(μn,Φ)→dKR(μ,Φ)≤α and hence ∥τ(μn)∥Lip≤max(dKR(μn,Φ)−α,0)→0. Therefore, τ is continuous everywhere.

Z′ is closed since it is the projection of Z and P(X) is compact. The Φy are compact, therefore dKR(μ,Φy)=dKR(μ,ζ) for some ζ∈Φy, and hence (y,τ,μ,ν,ξ)∈Z′ if and only if the following two conditions hold:

W is locally closed and in particular Fσ. Define W′⊆Y×T(X) by

W′:={(y,τ)∈Y×T(X)∣∃μ,ν,ξ∈P(X):(y,τ,μ,ν,ξ)∈W}

W′ is the projection of W and P(X)3 is compact, therefore W′ is Fσ. Let A⊆Y×T(X) be the complement of W′. A is Gδ and in particular Borel. As easy to see, (y,τ)∈A if and only if τ is an α-representative of Φy.

For any y∈Y, Ay:={τ∈T(X)∣(y,τ)∈A} is closed since

Ay=⋂μ∈P(X)⋂ν,ξ∈Φy{τ∈T(X)∣(y,τ,μ,ν,ξ)∈Z′}

Moreover, Ay is non-empty by Proposition A.4.

Consider any U⊆T(X) open. Then, A∩(Y×U) is Borel and therefore its image under the projection to Y is analytic and in particular universally measurable. Applying the Kuratorwsi-Ryll-Nardzewski measurable selection theorem, we get the desired result.

#Proposition A.5

Consider X a compact Polish space, D∈N, Y a compact subset of RD, π:X→Y continuous, μ∈P(X) and K∈μ∣π. Then, for π∗μ-almost any y∈Y

limr→0μ∣π−1(Br(y))=K(y)

#Proof of Proposition A.5

Let {fk∈C(X)}k∈N be dense in C(X). For any y∈Y and r>0 we denote χyr:Y→{0,1} the characteristic function of Br(y) and χπyr:X→{0,1} the characteristic function of π−1(Br(y)). For any k∈N and y0∈suppπ∗μ we have μ(π−1(Br(y0)))=π∗μ(Br(y0))>0 and

Consider any Φ∈H and a positive integer k. By Lemma 1, for any n∈N, there exists υΦkn:Yn→T(X) measurable s.t. for all y∈Yn, υΦkn(y) is a 1k-representative of Φ. We fix such a υΦkn(y) for each Φ and k. It is easy to see that the dnKR-diameter of P(X) is at most 2, therefore ∥υΦkn(y)∥≤2 and {υΦkn}n∈N is a trading metastrategy. By Theorem B.1, there is a stochastic market MH that dominates TυΦk for all Φ and k. By Corollary 1, for any Φ∈H, μ∗∈Φ, positive integer k and μ∗-almost any x∈X

limsupn→∞dnKR(MHn(x(n)),Φx(n))≤1k

It follows that

limn→∞dnKR(MHn(x(n)),Φx(n))=0

##Appendix B

The following are the theorems about dominant stochastic markets proven before.

#Theorem B.1

Given any countable set of traders R, there is a market M s.t. M dominates all T∈R.

#Theorem B.2

Consider μ∗∈P(X), {Kn∈μ∗∣πn}n∈N, υ a trading metastrategy profitable for μ∗ and M a market. Assume M dominates Tυ. Then, for μ∗-almost any x∈X:

% operators that are separated from the operand by a space

% operators that require brackets

% operators that require parentheses

% Paper specific

This post is formal treatment of the idea outlined here.Given a countable set of incomplete models, we define a forecasting function that converges in the Kantorovich-Rubinstein metric with probability 1 to every one of the models which is satisfied by the true environment. This is analogous to Blackwell-Dubins merging of opinions for complete models, except that Kantorovich-Rubinstein convergence is weaker than convergence in total variation. The forecasting function is a dominant stochastic market for a suitably constructed set of traders.

Appendix A contains the proofs. Appendix B restates the theorems about dominant stochastic markets for ease of reference. Appendix C states a variant of the Michael selection theorem due to Yannelis and Prabhakar which is used in Appendix A.

##Notation

Given X a metric space, x∈X and r∈R, Br(x):={y∈X∣d(x,y)<r}.

Given X a topological space:

P(X) is the space of Borel probability measures on X equipped with the weak* topology.

C(X) is the Banach space of continuous functions X→R with uniform norm.

T(X):=C(P(X)×X)

B(X) is the Borel σ-algebra on X.

U(X) is the σ-algebra of universally measurable sets on X.

Given μ∈P(X), suppμ denotes the support of μ.

Given X and Y measurable spaces, K:Xmk−→ Y is a Markov kernel from X to Y. For any x∈X, we have K(x)∈P(Y). Given μ∈P(X), μ⋉K∈P(X×Y) is the semidirect product of μ and K and K∗μ∈P(Y) is the pushforward of μ by K.

Given X, Y Polish spaces, π:X→Y Borel measurable and μ∈P(X), we denote μ∣π the set of Markov kernels K:Ymk−→X s.t. π∗μ⋉K is supported on the graph of π and K∗π∗μ=μ. By the disintegration theorem, μ∣π is always non-empty and any two kernels in μ∣π coincide π∗μ-almost everywhere.

##Results

Consider X any compact Polish metric space and d:X×X→R its metric. We denote Lip(X) the Banach space of Lipschitz continuous functions X→R with the norm

∥f∥Lip:=maxx|f(x)|+supx≠y|f(x)−f(y)|d(x,y)

P(X) (as before, with the weak topology) can be regarded as a compact subset of Lip(X)′ (with the strong topology), yielding a metrization of P(X) which we will call the Kantorovich-Rubinstein metric dKR:

dKR(μ,ν):=sup∥f∥Lip≤1|Eμ[f]−Eν[f]|

In fact, the above differs from the standard definition of the Kantorovich-Rubinstein metric (a.k.a. 1st Wasserstein metric, a.k.a. earth mover's metric), but this abuse of terminology is mild since the two are strongly equivalent.

Now consider Φ⊆P(X) convex. We will describe a class of trading strategies that are designed to exploit any μ∈Φ.

#Definition 1

Let α>0. τ∈T(X) is said to be an

α-representative of Φwhen for all μ∈P(X), ν∈Φ we have∥τ(μ)∥≤max(dKR(μ,Φ)−α,0)

Eν[τ(μ)]≥Eμ[τ(μ)]+12(dKR(μ,Φ)−α)dKR(μ,Φ)

#Lemma 1

Consider Y another compact Polish space and Φ⊆Y×P(X) closed. For any y∈Y, define Φy⊆P(X) by

Φy:={μ∈P(X)∣(y,μ)∈Φ}

Assume that for any y∈Y, Φy is convex. Then, for any α>0, there exists υα:Y→T(X) measurable w.r.t. U(Y) and B(T(X)) s.t. for all y∈Y, υα(y) is an α-representative of Φy.

For each n∈N, let On be a compact subset of RD(n) (D:N→N is arbitrary). Denote

Yn:=∏m<nOm

Yn is a compact subset of R∑m<nD(m). We will regard it as equipped with the Euclidean metric. Denote

X=∞∏n=0On

X is a compact Polish space. For each n∈N we denote πn:X→Yn the projection mapping. Given n∈N and x∈X, we have x(n):=πn(x)∈Yn and xn∈On.

The following definition provides a notion of updating an incomplete model by observations:

#Definition 2

Consider Φ⊆P(X). For any n∈N and y∈Yn, we define Φ′′y⊆P(X) by

Φ′′y:={limr→0(μ∣π−1n(Br(y)))∣μ∈Φ}

Note that the limit in the definition above need not exist for every μ∈Φ.

Denote Φ′y⊆P(X) the convex hull of Φ′′y and define Φ′n⊆Yn×P(X) by

Φ′n:={(y,μ)∈Yn×P(X)∣μ∈Φ′y}

Finally, we define Φn⊆Yn×P(X) to be the closure of Φ′n. Given y∈Yn, we define Φy⊆P(X) by

Φy:={μ∈P(X)∣(y,μ)∈Φn}

Note that the above definition uses the Euclidean metric on Yn. This is the only place through which the assumption that On is a compact subset of RD(n) (rather than an arbitrary compact Polish space) enters. This is used the in the proof of Lemma 2 below, via the Lebesgue differentiation theorem.

Fix a family of metrizations of X: {dn:X×X→R}n∈N. The reason we need a family rather than a single metric is that convergence in dKR is trivial unless we renormalize the metric for each n. On the other hand, completely arbitrary renormalization is allowed. For example, assuming the diameters of On are uniformly bounded, we can take

dn(x1,x2)=maxm∈Ncnmd(x1m,x2m)

Here, {cnm>0}n,m∈N are required to satisfy limm→∞cnm=0. To get a non-trivial result, we need the cnm to fall slower with m as n increases. The stronger we make this trend, the stronger conclusion we get (although it always remains weaker than convergence in total variation).

#Lemma 2

Consider Φ⊆P(X) and α>0. Let υ be a metastrategy s.t. for each n∈N and y∈Yn, υn(y) is an α-representative of Φy (relatively to dn). Then, υ is profitable for any μ∗∈Φ.

Combining Theorem B.2 and Lemma 2, it is easy to get the following:

#Corollary 1

Consider Φ⊆P(X), α>0 and υ a metastrategy s.t. for each n∈N and y∈Yn, υn(y) is an α-representative of Φy. Let M be a market which dominates Tυ. For every n∈N, denote dnKR the Kantorovich-Rubinstein metric associated with dn. Then, for any μ∗∈Φ and μ∗-almost any x∈X:

limsupn→∞dnKR(Mn(x(n)),Φx(n))≤α

Thus, if we construct a set of traders Tυk as in Corollary 1 where υk corresponds to α=1k, then a market dominating all of these traders would have to converge to Φyn with μ∗-probability 1. Putting this together with Lemma 1 (so that υk as above actually exists) and Theorem B.1 we finally get

#Theorem 1

Consider any H⊆2P(X) countable. Then, there exists a market MH s.t. for any Φ∈H, μ∗∈Φ and μ∗-almost any x∈X:

limn→∞dnKR(MHn(x(n)),Φx(n))=0

This leaves an interesting open question, namely whether the counterpart of Theorem 1 with dKR replaced by total variation distance is true.

##Appendix A

#Proposition A.1

Let X be a compact Polish space, φ∈Lip(X)′′ and ϵ>0. Then, there exists fϵ∈Lip(X) s.t. ∥fϵ∥Lip≤∥φ∥ and for all μ∈P(X), |Eμ[fϵ]−φ(μ)|<ϵ.

#Proof of Proposition A.1

Without loss of generality, assume that ∥φ∥=1 (if ∥φ∥=0 the theorem is trivial, otherwise we can rescale everything by a scalar). Using the compactness of P(X), choose a finite set Aϵ⊆P(X) s.t.

maxμ∈P(X)minν∈AϵdKR(μ,ν)<ϵ4

Using the Goldstine theorem, choose fϵ∈Lip(X) s.t. ∥fϵ∥Lip≤1 and for any ν∈Aϵ

|Eν[fϵ]−φ(ν)|<ϵ2

Consider any μ∈P(X). Choose ν∈Aϵ s.t. dKR(μ,ν)<ϵ4. We get

|Eμ[fϵ]−φ(μ)|≤|Eμ[fϵ]−Eν[fϵ]|+|φ(μ)−φ(ν)|+|Eν[fϵ]−φ(ν)|<ϵ4+ϵ4+ϵ2=ϵ

#Proposition A.2

Let X be a compact Polish space and Φ⊆P(X) convex. Consider any α>0 and μ0∈P(X) s.t. r:=dKR(μ0,Φ)>α. Then, there exists f∈Lip(X) s.t.

∥f∥Lip<r−α

infν∈ΦEν[f]>Eμ0[f]+12(r−α)r

#Proof of Proposition A.2

Define Φr⊆Lip(X)′ by

Φr:={μ∈Lip(X)′∣infν∈Φ∥μ−ν∥<r}

By the Hahn-Banach separation theorem, there is φ∈Lip(X)′′ s.t. for all ν∈Φr, φ(ν)>φ(μ0). Multiplying by a scalar, we can make sure that ∥φ∥=34(r−α). For any δ>0, we can choose ζ∈Lip(X)′ s.t. ∥ζ∥<1 and φ(ζ)>34(r−α)−δ. For any ν∈Φ, ν−rζ∈Φr and therefore

φ(ν−rζ)>φ(μ0)

φ(ν)>φ(μ0)+rφ(ζ)>φ(μ0)+r(34(r−α)−δ)

Taking δ to 0 we get φ(ν)≥φ(μ0)+34r(r−α). Applying Proposition A.1 for ϵ=116r(r−α), we get f∈Lip(X) s.t.

∥f∥≤34(r−α)<r−α

Eν[f]≥φ(ν)−116r(r−α)≥φ(μ0)+1116r(r−α)≥Eμ0[f]+58r(r−α)>Eμ0[f]+12r(r−α)

#Proposition A.3

Let X be a compact Polish space, Φ⊆P(X) convex and α>0. Let U⊆P(X) (an open set) be defined by

U:={μ∈P(X)∣dKR(μ,Φ)>α}

Then, there exists τ′:U→Lip(X) continuous s.t. for all μ∈U

∥τ′(μ)∥Lip<dKR(μ,Φ)−α

infν∈ΦEν[τ′(μ)]>Eμ[τ′(μ)]+12(dKR(μ,Φ)−α)dKR(μ,Φ)

#Proof of Proposition A.3

Define F:U→2Lip(X) by

F(μ):={f∈Lip(X)∣∥f∥Lip<dKR(μ,Φ)−α,infν∈ΦEν[f]>Eμ[f]+(dKR(μ,Φ)−α)dKR(μ,Φ)}

By Proposition A.2, for any μ∈U, F(μ)≠∅. Clearly, F(μ) is convex. Fix f∈Lip(X) and consider F−1(f):={μ∈U∣f∈F(μ)}. Consider any μ0∈F−1(f), and take ϵ>0 s.t.

∥f∥Lip<dKR(μ0,Φ)−ϵ−α

infν∈ΦEν[f]>Eμ0[f]+ϵ+(dKR(μ0,Φ)+ϵ−α)(dKR(μ0,Φ)+ϵ)

Define V:={μ∈U∣dKR(μ,μ0)<ϵ,Eμ[f]<Eμ0[f]+ϵ}. Obviously V is open and V⊆F−1(f), hence F−1(f) is open. Applying Theorem C, we get the desired result.

#Proposition A.4

Let X be a compact Polish space, Φ⊆P(X) convex and α>0. Then, there exists τ∈T(X) an α-representative of Φ.

#Proof of Proposition A.4

Let U⊆P(X) be defined by

U:={μ∈P(X)∣dKR(μ,Φ)>α}

Use Proposition A.3 to obtain τ′:U→Lip(X). Define τ:P(X)→Lip(X) by

τ(μ):={τ′(μ) if μ∈U0 if μ∉U Obviously τ is continuous in U. Consider any μ∉U and {μn∈P(X)}n∈N s.t. μn→μ. We have dKR(μn,Φ)→dKR(μ,Φ)≤α and hence ∥τ(μn)∥Lip≤max(dKR(μn,Φ)−α,0)→0. Therefore, τ is continuous everywhere.

#Proof of Lemma 1

Define Z1,2,3⊆Y×T(X)×P(X)4 by

Z1:={(y,τ,μ,ν,ξ,ζ)∈Y×T(X)×P(X)4∣(y,ζ)∈Φ}

Z2:={(y,τ,μ,ν,ξ,ζ)∈Y×T(X)×P(X)4∣∥τ(μ)∥≤max(dKR(μ,ξ)−α,0)}

Z3:={(y,τ,μ,ν,ξ,ζ)∈Y×T(X)×P(X)4∣Eν[τ(μ)]≥Eμ[τ(μ)]+12max(dKR(μ,ζ)−α,0)dKR(μ,ζ)}

Define Z:=Z1∩Z2∩Z3. It is easy to see that Z1,2,3 are closed and therefore Z also. Define Z′⊆Y×T(X)×P(X)3 by

Z′:={(y,τ,μ,ν,ξ)∈Y×T(X)×P(X)3∣∃ζ∈P(X):(y,τ,μ,ν,ξ,ζ)∈Z}

Z′ is closed since it is the projection of Z and P(X) is compact. The Φy are compact, therefore dKR(μ,Φy)=dKR(μ,ζ) for some ζ∈Φy, and hence (y,τ,μ,ν,ξ)∈Z′ if and only if the following two conditions hold:

∥τ(μ)∥≤max(dKR(μ,ξ)−α,0)

Eν[τ(μ)]≥Eμ[τ(μ)]+max(dKR(μ,Φy)−α,0)dKR(μ,Φy)

Define W⊆Y×T(X)×P(X)3 by

W:={(y,τ,μ,ν,ξ)∈Y×T(X)×P(X)3∣ν,ξ∈Φy∧(y,τ,μ,ν,ξ)∉Z′}

W is locally closed and in particular Fσ. Define W′⊆Y×T(X) by

W′:={(y,τ)∈Y×T(X)∣∃μ,ν,ξ∈P(X):(y,τ,μ,ν,ξ)∈W}

W′ is the projection of W and P(X)3 is compact, therefore W′ is Fσ. Let A⊆Y×T(X) be the complement of W′. A is Gδ and in particular Borel. As easy to see, (y,τ)∈A if and only if τ is an α-representative of Φy.

For any y∈Y, Ay:={τ∈T(X)∣(y,τ)∈A} is closed since

Ay=⋂μ∈P(X)⋂ν,ξ∈Φy{τ∈T(X)∣(y,τ,μ,ν,ξ)∈Z′}

Moreover, Ay is non-empty by Proposition A.4.

Consider any U⊆T(X) open. Then, A∩(Y×U) is Borel and therefore its image under the projection to Y is analytic and in particular universally measurable. Applying the Kuratorwsi-Ryll-Nardzewski measurable selection theorem, we get the desired result.

#Proposition A.5

Consider X a compact Polish space, D∈N, Y a compact subset of RD, π:X→Y continuous, μ∈P(X) and K∈μ∣π. Then, for π∗μ-almost any y∈Y

limr→0μ∣π−1(Br(y))=K(y)

#Proof of Proposition A.5

Let {fk∈C(X)}k∈N be dense in C(X). For any y∈Y and r>0 we denote χyr:Y→{0,1} the characteristic function of Br(y) and χπyr:X→{0,1} the characteristic function of π−1(Br(y)). For any k∈N and y0∈suppπ∗μ we have μ(π−1(Br(y0)))=π∗μ(Br(y0))>0 and

Eμ[fk∣π−1(Br(y0))]=Eμ[χπy0rfk]μ(π−1(Br(y0)))=Ey∼π∗μ[EK(y)[χπy0rfk]]π∗μ(Br(y0))=Ey∼π∗μ[χy0rEK(y)[fk]]π∗μ(Br(y0))

In particular, the above holds for π∗μ-almost any y0∈Y. Applying the Lebesgue differentiation theorem, we conclude that for π∗μ-almost any y0∈Y

limr→0Eμ[fk∣π−1(Br(y0))]=EK(y0)[fk]

Now consider any f∈C(X). For any ϵ>0, there is k∈N s.t. ∥f−fk∥<ϵ and therefore

limsupr→0Eμ[f∣π−1(Br(y0))]≤limsupr→0Eμ[fk∣π−1(Br(y0))]+ϵ=EK(y0)[fk]+ϵ≤EK(y0)[f]+2ϵ

Similarly

liminfr→0Eμ[f∣π−1(Br(y0))]≥EK(y0)[f]−2ϵ

Taking ϵ to 0, we conclude that

limr→0Eμ[f∣π−1(Br(y0))]=EK(y0)[f]

limr→0μ∣π−1(Br(y0))=K(y0)

#Proof of Lemma 2

Fix any μ∗∈Φ and {Kn∈μ∗∣πn}n∈N. By Proposition A.5, for πn∗μ-almost any y∈Y

limr→0μ∗∣Br(y)=Kn(y)

Therefore, Kn(y)∈Φ′′y⊆Φy. Since υn(y) is an α-representative of Φy, for any μ∈P(X)

EKn(y)[υ(y,μ)]−Eμ[υ(y,μ)]≥12max(dnKR(μ,Φy)−α,0)dnKR(μ,Φy)≥12∥υ(y,μ)∥dnKR(μ,Φy)

When dnKR(μ,Φy)≤α, υ(y,μ)=0, hence for any μ∈P(X), ∥υ(y,μ)∥dnKR(μ,Φy)≥∥υ(y,μ)∥α. We get

EKn(y)[υ(y,μ)]−Eμ[υ(y,μ)]≥α2∥υ(y,μ)∥

#Proof of Corollary 1

By Lemma 2, ν is profitable for μ∗. By Theorem B.2, for μ∗-almost any x∈X

limn→∞(EKn(x(n))[υ(x(n),Mn(x(n)))]−EMn(x(n))[υ(x(n),Mn(x(n)))])=0

By Proposition A.5, Kn(x(n))∈Φx(n) and therefore

EKn(x(n))[υ(x(n),Mn(x(n)))]−EMn(x(n))[υ(x(n),Mn(x(n)))]≥12(dnKR(Mn(x(n)),Φx(n))−α)dnKR(Mn(x(n)),Φx(n))

We get

limsupn→∞(dnKR(Mn(x(n)),Φx(n))−α)dnKR(Mn(x(n)),Φx(n))≤0

limsupn→∞dnKR(Mn(x(n)),Φx(n))≤α

#Proof of Theorem 1

Consider any Φ∈H and a positive integer k. By Lemma 1, for any n∈N, there exists υΦkn:Yn→T(X) measurable s.t. for all y∈Yn, υΦkn(y) is a 1k-representative of Φ. We fix such a υΦkn(y) for each Φ and k. It is easy to see that the dnKR-diameter of P(X) is at most 2, therefore ∥υΦkn(y)∥≤2 and {υΦkn}n∈N is a trading metastrategy. By Theorem B.1, there is a stochastic market MH that dominates TυΦk for all Φ and k. By Corollary 1, for any Φ∈H, μ∗∈Φ, positive integer k and μ∗-almost any x∈X

limsupn→∞dnKR(MHn(x(n)),Φx(n))≤1k

It follows that

limn→∞dnKR(MHn(x(n)),Φx(n))=0

##Appendix B

The following are the theorems about dominant stochastic markets proven before.

#Theorem B.1

Given any countable set of traders R, there is a market M s.t. M dominates all T∈R.

#Theorem B.2

Consider μ∗∈P(X), {Kn∈μ∗∣πn}n∈N, υ a trading metastrategy profitable for μ∗ and M a market. Assume M dominates Tυ. Then, for μ∗-almost any x∈X:

limn→∞(EKn(x(n))[υ(x(n),Mn(x(n)))]−EMn(x(n))[υ(x(n),Mn(x(n)))])=0

##Appendix C

The following variant of the Michael selection theorem appears in Yannelis and Prabhakar (1983) as Theorem 3.1.

#Theorem C

Consider X a paracompact Hausdorff topological space and Y a topological vector space. Suppose F:X→2Y is s.t.

(i) For each x∈X, F(x)≠∅.

(ii) For each x∈X, F(x) is convex.

(iii) For each y∈Y, {x∈X∣y∈F(x)} is open.

Then, there exists f:X→Y continuous s.t. for all x∈X, f(x)∈F(x).