Learning incomplete models using dominant markets

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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, $x \in X$ and $r \in R$ , $B_{r} (x) := {y \in X ∣ d (x, y) < r}$ .

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 $X \to R$ with uniform norm.
$T (X) := C (P (X) \times X)$
$B (X)$ is the Borel $σ$ -algebra on $X$ .
$U (X)$ is the $σ$ -algebra of universally measurable sets on $X$ .
Given $μ \in P (X)$ , $supp μ$ denotes the support of $μ$ .

Given $X$ and $Y$ measurable spaces, $K : X mk - \to Y$ is a Markov kernel from $X$ to $Y$ . For any $x \in X$ , we have $K (x) \in P (Y)$ . Given $μ \in P (X)$ , $μ ⋉ K \in P (X \times Y)$ is the semidirect product of $μ$ and $K$ and $K_{*} μ \in P (Y)$ is the pushforward of $μ$ by $K$ .

Given $X$ , $Y$ Polish spaces, $π : X \to Y$ Borel measurable and $μ \in P (X)$ , we denote $μ ∣ π$ the set of Markov kernels $K : Y mk - \to X$ 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 $d : X \times X \to R$ its metric. We denote $Lip (X)$ the Banach space of Lipschitz continuous functions $X \to R$ with the norm

$∥ f ∥_{Lip} := max x | f (x) | + sup x \neq y \frac{| f (x) - f (y) |}{d (x, y)}$

$P (X)$ (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 $d_{KR}$ :

$d_{KR} (μ, ν) := sup ∥ f ∥_{Lip} \leq 1 | E_{μ} [f] - E_{ν} [f] |$

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 $Φ \subseteq P (X)$ convex. We will describe a class of trading strategies that are designed to exploit any $μ \in Φ$ .

#Definition 1

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

$∥ τ (μ) ∥ \leq max (d_{KR} (μ, Φ) - α, 0)$

$E_{ν} [τ (μ)] \geq E_{μ} [τ (μ)] + \frac{1}{2} (d_{KR} (μ, Φ) - α) d_{KR} (μ, Φ)$

#Lemma 1

Consider $Y$ another compact Polish space and $Φ \subseteq Y \times P (X)$ closed. For any $y \in Y$ , define $Φ_{y} \subseteq P (X)$ by

$Φ_{y} := {μ \in P (X) ∣ (y, μ) \in Φ}$

Assume that for any $y \in Y$ , $Φ_{y}$ is convex. Then, for any $α > 0$ , there exists $υ_{α} : Y \to T (X)$ measurable w.r.t. $U (Y)$ and $B (T (X))$ s.t. for all $y \in Y$ , $υ_{α} (y)$ is an $α$ -representative of $Φ_{y}$ .

For each $n \in N$ , let $O_{n}$ be a compact subset of $R^{D (n)}$ ( $D : N \to N$ is arbitrary). Denote

$Y_{n} := \prod m < n O_{m}$

$Y_{n}$ is a compact subset of $R^{\sum_{m < n} D (m)}$ . We will regard it as equipped with the Euclidean metric. Denote

$X = \infty \prod n = 0 O_{n}$

$X$ is a compact Polish space. For each $n \in N$ we denote $π_{n} : X \to Y_{n}$ the projection mapping. Given $n \in N$ and $x \in X$ , we have $x (n) := π_{n} (x) \in Y_{n}$ and $x_{n} \in O_{n}$ .

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

#Definition 2

Consider $Φ \subseteq P (X)$ . For any $n \in N$ and $y \in Y_{n}$ , we define $Φ_{y}^{''} \subseteq P (X)$ by

$Φ_{y}^{''} := {lim r \to 0 (μ ∣ π_{n}^{- 1} (B_{r} (y))) ∣ μ \in Φ}$

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

Denote $Φ_{y}^{'} \subseteq P (X)$ the convex hull of $Φ_{y}^{''}$ and define $Φ_{n}^{'} \subseteq Y_{n} \times P (X)$ by

$Φ_{n}^{'} := {(y, μ) \in Y_{n} \times P (X) ∣ μ \in Φ_{y}^{'}}$

Finally, we define $Φ_{n} \subseteq Y_{n} \times P (X)$ to be the closure of $Φ_{n}^{'}$ . Given $y \in Y_{n}$ , we define $Φ_{y} \subseteq P (X)$ by

$Φ_{y} := {μ \in P (X) ∣ (y, μ) \in Φ_{n}}$

Note that the above definition uses the Euclidean metric on $Y_{n}$ . This is the only place through which the assumption that $O_{n}$ is a compact subset of $R^{D (n)}$ (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 $X$ : ${d_{n} : X \times X \to R}_{n \in N}$ . The reason we need a family rather than a single metric is that convergence in $d_{KR}$ 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 $O_{n}$ are uniformly bounded, we can take

$d_{n} (x^{1}, x^{2}) = max m \in N c_{n m} d (x_{m}^{1}, x_{m}^{2})$

Here, ${c_{n m} > 0}_{n, m \in N}$ are required to satisfy ${lim}_{m \to \infty} c_{n m} = 0$ . To get a non-trivial result, we need the $c_{n m}$ 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 $Φ \subseteq P (X)$ and $α > 0$ . Let $υ$ be a metastrategy s.t. for each $n \in N$ and $y \in Y_{n}$ , $υ_{n} (y)$ is an $α$ -representative of $Φ_{y}$ (relatively to $d_{n}$ ). Then, $υ$ is profitable for any $μ^{*} \in Φ$ .

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

#Corollary 1

Consider $Φ \subseteq P (X)$ , $α > 0$ and $υ$ a metastrategy s.t. for each $n \in N$ and $y \in Y_{n}$ , $υ_{n} (y)$ is an $α$ -representative of $Φ_{y}$ . Let $M$ be a market which dominates $T_{υ}$ . For every $n \in N$ , denote $d_{KR}^{n}$ the Kantorovich-Rubinstein metric associated with $d_{n}$ . Then, for any $μ^{*} \in Φ$ and $μ^{*}$ -almost any $x \in X$ :

$limsup n \to \infty d_{KR}^{n} (M_{n} (x (n)), Φ_{x (n)}) \leq α$

Thus, if we construct a set of traders $T_{υ_{k}}$ as in Corollary 1 where $υ_{k}$ corresponds to $α = \frac{1}{k}$ , then a market dominating all of these traders would have to converge to $Φ_{y_{n}}$ with $μ^{*}$ -probability 1. Putting this together with Lemma 1 (so that $υ_{k}$ as above actually exists) and Theorem B.1 we finally get

#Theorem 1

Consider any $H \subseteq 2^{P (X)}$ countable. Then, there exists a market $M^{H}$ s.t. for any $Φ \in H$ , $μ^{*} \in Φ$ and $μ^{*}$ -almost any $x \in X$ :

$lim n \to \infty d_{KR}^{n} (M_{n}^{H} (x (n)), Φ_{x (n)}) = 0$

This leaves an interesting open question, namely whether the counterpart of Theorem 1 with $d_{KR}$ replaced by total variation distance is true.

##Appendix A

#Proposition A.1

Let $X$ be a compact Polish space, $φ \in Lip (X)^{''}$ and $ϵ > 0$ . Then, there exists $f_{ϵ} \in Lip (X)$ s.t. $∥ f_{ϵ} ∥_{Lip} \leq ∥ φ ∥$ and for all $μ \in P (X)$ , $| E_{μ} [f_{ϵ}] - φ (μ) | < ϵ$ .

#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 $A_{ϵ} \subseteq P (X)$ s.t.

$max μ \in P (X) min ν \in A_{ϵ} d_{KR} (μ, ν) < \frac{ϵ}{4}$

Using the Goldstine theorem, choose $f_{ϵ} \in Lip (X)$ s.t. $∥ f_{ϵ} ∥_{Lip} \leq 1$ and for any $ν \in A_{ϵ}$

$| E_{ν} [f_{ϵ}] - φ (ν) | < \frac{ϵ}{2}$

Consider any $μ \in P (X)$ . Choose $ν \in A_{ϵ}$ s.t. $d_{KR} (μ, ν) < \frac{ϵ}{4}$ . We get

$| E_{μ} [f_{ϵ}] - φ (μ) | \leq | E_{μ} [f_{ϵ}] - E_{ν} [f_{ϵ}] | + | φ (μ) - φ (ν) | + | E_{ν} [f_{ϵ}] - φ (ν) | < \frac{ϵ}{4} + \frac{ϵ}{4} + \frac{ϵ}{2} = ϵ$

#Proposition A.2

Let $X$ be a compact Polish space and $Φ \subseteq P (X)$ convex. Consider any $α > 0$ and $μ_{0} \in P (X)$ s.t. $r := d_{KR} (μ_{0}, Φ) > α$ . Then, there exists $f \in Lip (X)$ s.t.

$∥ f ∥_{Lip} < r - α$

$inf ν \in Φ E_{ν} [f] > E_{μ_{0}} [f] + \frac{1}{2} (r - α) r$

#Proof of Proposition A.2

Define $Φ^{r} \subseteq Lip (X)^{'}$ by

$Φ^{r} := {μ \in Lip (X)^{'} ∣ inf ν \in Φ ∥ μ - ν ∥ < r}$

By the Hahn-Banach separation theorem, there is $φ \in Lip (X)^{''}$ s.t. for all $ν \in Φ^{r}$ , $φ (ν) > φ (μ_{0})$ . Multiplying by a scalar, we can make sure that $∥ φ ∥ = \frac{3}{4} (r - α)$ . For any $δ > 0$ , we can choose $ζ \in Lip (X)^{'}$ s.t. $∥ ζ ∥ < 1$ and $φ (ζ) > \frac{3}{4} (r - α) - δ$ . For any $ν \in Φ$ , $ν - r ζ \in Φ^{r}$ and therefore

$φ (ν - r ζ) > φ (μ_{0})$

$φ (ν) > φ (μ_{0}) + r φ (ζ) > φ (μ_{0}) + r (\frac{3}{4} (r - α) - δ)$

Taking $δ$ to 0 we get $φ (ν) \geq φ (μ_{0}) + \frac{3}{4} r (r - α)$ . Applying Proposition A.1 for $ϵ = \frac{1}{16} r (r - α)$ , we get $f \in Lip (X)$ s.t.

$∥ f ∥ \leq \frac{3}{4} (r - α) < r - α$

$E_{ν} [f] \geq φ (ν) - \frac{1}{16} r (r - α) \geq φ (μ_{0}) + \frac{11}{16} r (r - α) \geq E_{μ_{0}} [f] + \frac{5}{8} r (r - α) > E_{μ_{0}} [f] + \frac{1}{2} r (r - α)$

#Proposition A.3

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

$U := {μ \in P (X) ∣ d_{KR} (μ, Φ) > α}$

Then, there exists $τ^{'} : U \to Lip (X)$ continuous s.t. for all $μ \in U$

$∥ τ^{'} (μ) ∥_{Lip} < d_{KR} (μ, Φ) - α$

$inf ν \in Φ E_{ν} [τ^{'} (μ)] > E_{μ} [τ^{'} (μ)] + \frac{1}{2} (d_{KR} (μ, Φ) - α) d_{KR} (μ, Φ)$

#Proof of Proposition A.3

Define $F : U \to 2^{Lip (X)}$ by

$F (μ) := {f \in Lip (X) ∣ ∥ f ∥_{Lip} < d_{KR} (μ, Φ) - α, inf ν \in Φ E_{ν} [f] > E_{μ} [f] + (d_{KR} (μ, Φ) - α) d_{KR} (μ, Φ)}$

By Proposition A.2, for any $μ \in U$ , $F (μ) \neq \emptyset$ . Clearly, $F (μ)$ is convex. Fix $f \in Lip (X)$ and consider $F^{- 1} (f) := {μ \in U ∣ f \in F (μ)}$ . Consider any $μ_{0} \in F^{- 1} (f)$ , and take $ϵ > 0$ s.t.

$∥ f ∥_{Lip} < d_{KR} (μ_{0}, Φ) - ϵ - α$

$inf ν \in Φ E_{ν} [f] > E_{μ_{0}} [f] + ϵ + (d_{KR} (μ_{0}, Φ) + ϵ - α) (d_{KR} (μ_{0}, Φ) + ϵ)$

Define $V := {μ \in U ∣ d_{KR} (μ, μ_{0}) < ϵ, E_{μ} [f] < E_{μ_{0}} [f] + ϵ}$ . Obviously $V$ is open and $V \subseteq F^{- 1} (f)$ , hence $F^{- 1} (f)$ is open. Applying Theorem C, we get the desired result.

#Proposition A.4

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

#Proof of Proposition A.4

Let $U \subseteq P (X)$ be defined by

$U := {μ \in P (X) ∣ d_{KR} (μ, Φ) > α}$

Use Proposition A.3 to obtain $τ^{'} : U \to Lip (X)$ . Define $τ : P (X) \to Lip (X)$ by

$τ (μ) := {\begin{matrix} τ^{'} (μ) if μ \in U 0 if μ \notin U \end{matrix}$ Obviously $τ$ is continuous in $U$ . Consider any $μ \notin U$ and ${μ_{n} \in P (X)}_{n \in N}$ s.t. $μ_{n} \to μ$ . We have $d_{KR} (μ_{n}, Φ) \to d_{KR} (μ, Φ) \leq α$ and hence $∥ τ (μ_{n}) ∥_{Lip} \leq max (d_{KR} (μ_{n}, Φ) - α, 0) \to 0$ . Therefore, $τ$ is continuous everywhere.

#Proof of Lemma 1

Define $Z_{1, 2, 3} \subseteq Y \times T (X) \times P (X)^{4}$ by

$Z_{1} := {(y, τ, μ, ν, ξ, ζ) \in Y \times T (X) \times P (X)^{4} ∣ (y, ζ) \in Φ}$

$Z_{2} := {(y, τ, μ, ν, ξ, ζ) \in Y \times T (X) \times P (X)^{4} ∣ ∥ τ (μ) ∥ \leq max (d_{KR} (μ, ξ) - α, 0)}$

$Z_{3} := {(y, τ, μ, ν, ξ, ζ) \in Y \times T (X) \times P (X)^{4} ∣ E_{ν} [τ (μ)] \geq E_{μ} [τ (μ)] + \frac{1}{2} max (d_{KR} (μ, ζ) - α, 0) d_{KR} (μ, ζ)}$

Define $Z := Z_{1} \cap Z_{2} \cap Z_{3}$ . It is easy to see that $Z_{1, 2, 3}$ are closed and therefore $Z$ also. Define $Z^{'} \subseteq Y \times T (X) \times P (X)^{3}$ by

$Z^{'} := {(y, τ, μ, ν, ξ) \in Y \times T (X) \times P (X)^{3} ∣ \exists ζ \in P (X) : (y, τ, μ, ν, ξ, ζ) \in Z}$

$Z^{'}$ is closed since it is the projection of $Z$ and $P (X)$ is compact. The $Φ_{y}$ are compact, therefore $d_{KR} (μ, Φ_{y}) = d_{KR} (μ, ζ)$ for some $ζ \in Φ_{y}$ , and hence $(y, τ, μ, ν, ξ) \in Z^{'}$ if and only if the following two conditions hold:

$∥ τ (μ) ∥ \leq max (d_{KR} (μ, ξ) - α, 0)$

$E_{ν} [τ (μ)] \geq E_{μ} [τ (μ)] + max (d_{KR} (μ, Φ_{y}) - α, 0) d_{KR} (μ, Φ_{y})$

Define $W \subseteq Y \times T (X) \times P (X)^{3}$ by

$W := {(y, τ, μ, ν, ξ) \in Y \times T (X) \times P (X)^{3} ∣ ν, ξ \in Φ_{y} \land (y, τ, μ, ν, ξ) \notin Z^{'}}$

$W$ is locally closed and in particular $F_{σ}$ . Define $W^{'} \subseteq Y \times T (X)$ by

$W^{'} := {(y, τ) \in Y \times T (X) ∣ \exists μ, ν, ξ \in P (X) : (y, τ, μ, ν, ξ) \in W}$

$W^{'}$ is the projection of $W$ and $P (X)^{3}$ is compact, therefore $W^{'}$ is $F_{σ}$ . Let $A \subseteq Y \times T (X)$ be the complement of $W^{'}$ . $A$ is $G_{δ}$ and in particular Borel. As easy to see, $(y, τ) \in A$ if and only if $τ$ is an $α$ -representative of $Φ_{y}$ .

For any $y \in Y$ , $A_{y} := {τ \in T (X) ∣ (y, τ) \in A}$ is closed since

$A_{y} = ⋂ μ \in P (X) ⋂ ν, ξ \in Φ_{y} {τ \in T (X) ∣ (y, τ, μ, ν, ξ) \in Z^{'}}$

Moreover, $A_{y}$ is non-empty by Proposition A.4.

Consider any $U \subseteq T (X)$ open. Then, $A \cap (Y \times U)$ 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, $D \in N$ , $Y$ a compact subset of $R^{D}$ , $π : X \to Y$ continuous, $μ \in P (X)$ and $K \in μ ∣ π$ . Then, for $π_{*} μ$ -almost any $y \in Y$

$lim r \to 0 μ ∣ π^{- 1} (B_{r} (y)) = K (y)$

#Proof of Proposition A.5

Let ${f_{k} \in C (X)}_{k \in N}$ be dense in $C (X)$ . For any $y \in Y$ and $r > 0$ we denote $χ_{y r} : Y \to {0, 1}$ the characteristic function of $B_{r} (y)$ and $χ_{y r}^{π} : X \to {0, 1}$ the characteristic function of $π^{- 1} (B_{r} (y))$ . For any $k \in N$ and $y_{0} \in supp π_{*} μ$ we have $μ (π^{- 1} (B_{r} (y_{0}))) = π_{*} μ (B_{r} (y_{0})) > 0$ and

$E_{μ} [f_{k} ∣ π^{- 1} (B_{r} (y_{0}))] = \frac{E_{μ} [χ_{y_{0} r}^{π} f_{k}]}{μ (π^{- 1} (B_{r} (y_{0})))} = \frac{E_{y \sim π_{*} μ} [E_{K (y)} [χ_{y_{0} r}^{π} f_{k}]]}{π_{*} μ (B_{r} (y_{0}))} = \frac{E_{y \sim π_{*} μ} [χ_{y_{0} r} E_{K (y)} [f_{k}]]}{π_{*} μ (B_{r} (y_{0}))}$

In particular, the above holds for $π_{*} μ$ -almost any $y_{0} \in Y$ . Applying the Lebesgue differentiation theorem, we conclude that for $π_{*} μ$ -almost any $y_{0} \in Y$

$lim r \to 0 E_{μ} [f_{k} ∣ π^{- 1} (B_{r} (y_{0}))] = E_{K (y_{0})} [f_{k}]$

Now consider any $f \in C (X)$ . For any $ϵ > 0$ , there is $k \in N$ s.t. $∥ f - f_{k} ∥ < ϵ$ and therefore

$limsup r \to 0 E_{μ} [f ∣ π^{- 1} (B_{r} (y_{0}))] \leq limsup r \to 0 E_{μ} [f_{k} ∣ π^{- 1} (B_{r} (y_{0}))] + ϵ = E_{K (y_{0})} [f_{k}] + ϵ \leq E_{K (y_{0})} [f] + 2 ϵ$

Similarly

$liminf r \to 0 E_{μ} [f ∣ π^{- 1} (B_{r} (y_{0}))] \geq E_{K (y_{0})} [f] - 2 ϵ$

Taking $ϵ$ to 0, we conclude that

$lim r \to 0 E_{μ} [f ∣ π^{- 1} (B_{r} (y_{0}))] = E_{K (y_{0})} [f]$

$lim r \to 0 μ ∣ π^{- 1} (B_{r} (y_{0})) = K (y_{0})$

#Proof of Lemma 2

Fix any $μ^{*} \in Φ$ and ${K_{n} \in μ^{*} ∣ π_{n}}_{n \in N}$ . By Proposition A.5, for $π_{n *} μ$ -almost any $y \in Y$

$lim r \to 0 μ^{*} ∣ B_{r} (y) = K_{n} (y)$

Therefore, $K_{n} (y) \in Φ_{y}^{''} \subseteq Φ_{y}$ . Since $υ_{n} (y)$ is an $α$ -representative of $Φ_{y}$ , for any $μ \in P (X)$

$E_{K_{n} (y)} [υ (y, μ)] - E_{μ} [υ (y, μ)] \geq \frac{1}{2} max (d_{KR}^{n} (μ, Φ_{y}) - α, 0) d_{KR}^{n} (μ, Φ_{y}) \geq \frac{1}{2} ∥ υ (y, μ) ∥ d_{KR}^{n} (μ, Φ_{y})$

When $d_{KR}^{n} (μ, Φ_{y}) \leq α$ , $υ (y, μ) = 0$ , hence for any $μ \in P (X)$ , $∥ υ (y, μ) ∥ d_{KR}^{n} (μ, Φ_{y}) \geq ∥ υ (y, μ) ∥ α$ . We get

$E_{K_{n} (y)} [υ (y, μ)] - E_{μ} [υ (y, μ)] \geq \frac{α}{2} ∥ υ (y, μ) ∥$

#Proof of Corollary 1

By Lemma 2, $ν$ is profitable for $μ^{*}$ . By Theorem B.2, for $μ^{*}$ -almost any $x \in X$

$lim n \to \infty (E_{K_{n} (x (n))} [υ (x (n), M_{n} (x (n)))] - E_{M_{n} (x (n))} [υ (x (n), M_{n} (x (n)))]) = 0$

By Proposition A.5, $K_{n} (x (n)) \in Φ_{x (n)}$ and therefore

$E_{K_{n} (x (n))} [υ (x (n), M_{n} (x (n)))] - E_{M_{n} (x (n))} [υ (x (n), M_{n} (x (n)))] \geq \frac{1}{2} (d_{KR}^{n} (M_{n} (x (n)), Φ_{x (n)}) - α) d_{KR}^{n} (M_{n} (x (n)), Φ_{x (n)})$

We get

$limsup n \to \infty (d_{KR}^{n} (M_{n} (x (n)), Φ_{x (n)}) - α) d_{KR}^{n} (M_{n} (x (n)), Φ_{x (n)}) \leq 0$

$limsup n \to \infty d_{KR}^{n} (M_{n} (x (n)), Φ_{x (n)}) \leq α$

#Proof of Theorem 1

Consider any $Φ \in H$ and a positive integer $k$ . By Lemma 1, for any $n \in N$ , there exists $υ_{k n}^{Φ} : Y_{n} \to T (X)$ measurable s.t. for all $y \in Y_{n}$ , $υ_{k n}^{Φ} (y)$ is a $\frac{1}{k}$ -representative of $Φ$ . We fix such a $υ_{k n}^{Φ} (y)$ for each $Φ$ and $k$ . It is easy to see that the $d_{KR}^{n}$ -diameter of $P (X)$ is at most 2, therefore $∥ υ_{k n}^{Φ} (y) ∥ \leq 2$ and ${υ_{k n}^{Φ}}_{n \in N}$ is a trading metastrategy. By Theorem B.1, there is a stochastic market $M^{H}$ that dominates $T_{υ_{k}^{Φ}}$ for all $Φ$ and $k$ . By Corollary 1, for any $Φ \in H$ , $μ^{*} \in Φ$ , positive integer $k$ and $μ^{*}$ -almost any $x \in X$

$limsup n \to \infty d_{KR}^{n} (M_{n}^{H} (x (n)), Φ_{x (n)}) \leq \frac{1}{k}$

It follows that

$lim n \to \infty d_{KR}^{n} (M_{n}^{H} (x (n)), Φ_{x (n)}) = 0$

##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 $T \in R$ .

#Theorem B.2

Consider $μ^{*} \in P (X)$ , ${K_{n} \in μ^{*} ∣ π_{n}}_{n \in N}$ , $υ$ a trading metastrategy profitable for $μ^{*}$ and $M$ a market. Assume $M$ dominates $T_{υ}$ . Then, for $μ^{*}$ -almost any $x \in X$ :

$lim n \to \infty (E_{K_{n} (x (n))} [υ (x (n), M_{n} (x (n)))] - E_{M_{n} (x (n))} [υ (x (n), M_{n} (x (n)))]) = 0$

##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 $F : X \to 2^{Y}$ is s.t.

(i) For each $x \in X$ , $F (x) \neq \emptyset$ .

(ii) For each $x \in X$ , $F (x)$ is convex.

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

Then, there exists $f : X \to Y$ continuous s.t. for all $x \in X$ , $f (x) \in F (x)$ .

LESSWRONG
LW

LESSWRONG
LW

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Learning incomplete models using dominant markets

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