Purification and roulette wheels

Abstract

We use concepts introduced by Aumann more than 30 years ago to throw new light on purification in games with extremely dispersed private information. We show that one can embed payoff-irrelevant randomization devices in the private information of players and use these randomization devices to implement mixed strategies as deterministic functions of the private information. This approach gives rise to very short and intuitive proofs for a number of purification results that previously required sophisticated methods from functional analysis or nonstandard analysis. We use our methods to prove the first general purification theorem for games with private information in which a player’s payoffs can depend in arbitrary ways on events in the private information of other players and in which we allow for shared information in a general way.

Appendix

Proof of Lemma 8

As in the proof of Theorem 1, choose a $\Sigma _{2}$-measurable map $w:\varOmega \rightarrow [0,1]$ so that $\mu _{w}=\lambda$, and by Lemma 3, choose a ${\fancyscript{I}}\otimes {\fancyscript{B}}$-measurable $h:\varOmega \times [0,1]\rightarrow X$ so that $f_{x}(\omega )(B_{x})=\lambda (h(\omega ,\cdot )^{-1}(B_{x}))$ for each $B_{x}\in {\fancyscript{B}}(X)$ and each $\omega \in \varOmega$. Let $g_{x}=h\circ (\iota _{\varOmega },w)$ and note that $g_{x}$ is ${\fancyscript{I}}$-measurable, as $\Sigma _{2}\subseteq {\fancyscript{I}}$.

Now pick any $\Sigma _{1}$-measurable function $f_{y}:\varOmega \rightarrow {\fancyscript{M}}(Y)$. Note that by Lemma 2, $\mu _{(\iota _{\varOmega },f_{y},w)}\!\upharpoonright \!\Sigma _{1}\otimes {\fancyscript{B}}(Y)\otimes {\fancyscript{B}}=\bigl (\mu _{(\iota _{\varOmega },f_{y})}\!\upharpoonright \!\Sigma _{1}\otimes {\fancyscript{B}}(Y)\bigr )\otimes \lambda$.

Now given any rectangle $A\times B_{y}\times B_{x}\in \Sigma _{1}\otimes {\fancyscript{B}}(Y)\otimes {\fancyscript{B}}(X)$, we can calculate as follows, where the sixth equality follows by Fubini’s theorem, the seventh by the generalized version of Fubini’s theorem, and where $\tilde{h}:\varOmega \times Y\times [0,1]\rightarrow X$ is given by setting $\tilde{h}(\omega ,y,r)=h(\omega ,r)$, and $\delta$ is used to denote a Dirac measure:

$$\begin{aligned}&\mu _{(\iota _{\varOmega },f_{y},g_{x})}(A\times B_{y}\times B_{x})\\&\quad =\int \limits _{A}f_{y}(\omega )(B_{y})\delta _{g_{x}(\omega )}(B_{x})\mathrm{d} \mu (\omega )\\&\quad =\int \limits _{A}f_{y}(\omega )(B_{y})\delta _{h(\omega ,w(\omega ))}(B_{x})\mathrm{d} \mu (\omega )\\&\quad =\int \limits _{A}f_{y}(\omega )(B_{y})\delta _{(\iota _{\varOmega }(\omega ),w(\omega ))}(h^{-1}(B_{x}))\mathrm{d} \mu (\omega )\\&\quad =\int \limits _{A}f_{y}(\omega )(B_{y})\delta _{\iota _{\varOmega }(\omega ))}\otimes \delta _{w(\omega )}(h^{-1}(B_{x}))\mathrm{d} \mu (\omega )\\&\quad =\mu _{(\iota _{\varOmega },f_{y},w)}\!\upharpoonright \!\Sigma _{1}\otimes {\fancyscript{B}}(Y)\otimes {\fancyscript{B}}\bigl (\tilde{h}^{-1}(B_{x})\cap (A\times B_{y}\times [0,1])\bigr )\\&\quad =\bigl (\mu _{(\iota _{\varOmega },f_{y})}\!\upharpoonright \!\Sigma _{1}\otimes {\fancyscript{B}}(Y)\bigr )\otimes \lambda \bigl (\tilde{h}^{-1}(B_{x})\cap (A\times B_{y}\times [0,1])\bigr )\\&\quad =\int \limits _{A\times B_{y}}\lambda (h(\omega ,\cdot )^{-1}(B_{x}))\mathrm{d}\mu _{(\iota _{\varOmega },f_{y})}\\&\quad =\int \limits _{A}f_{y}(\omega )(B_{y})\lambda (h(\omega ,\cdot )^{-1}(B_{x}))\mathrm{d}\mu (\omega )\\&\quad =\int \limits _{A}f_{y}(\omega )(B_{y})f_{x}(\omega )(B_{x})\mathrm{d}\mu (\omega )\\&\quad =\mu _{(\iota _{\varOmega },f_{y},f_{x})}(A\times B_{y}\times B_{x}). \end{aligned}$$

$\square$