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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.

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Notes

  1. 1.

    We do not discuss purification based on perturbing the game as in Harsanyi (1973) and Govindan et al (2003). See Morris (2008) for a comparison of these approaches.

  2. 2.

    See for example Milgrom and Weber (1985) and Balder (1988).

  3. 3.

    A far reaching generalization of Aumann’s framework can be found in Grant et al (2013), where the authors are able to prove the existence of pure strategy equilibria by employing a novel fixed-point theorem from Meneghel and Tourky (2013), in which convexity assumptions are replaced by a decomposability assumption from nonlinear analysis. They obtain pure strategy equilibria in Bayesian games directly without purifying mixed strategy equilibria. It is not clear to us whether a general purification result would hold in their framework, as their assumptions are not directly comparable to ours.

  4. 4.

    Terminology varies. See footnote 4 in Wang and Zhang (2012) for an overview over the various concepts that are equivalent to what we call super-atomless.

  5. 5.

    This result generalizes Theorem 15 in Carmona and Podczeck (2013), which is used there for purifying mixed equilibria in games with a continuum of players.

  6. 6.

    For equilibrium existence results in the framework of states of nature instead of the formulation in terms of types, see Yannelis and Rustichini (1991).

  7. 7.

    We actually never use boundedness, but it ensures that expected utility is well defined. Clearly, weaker assumptions would do.

  8. 8.

    Fu has drawn attention to differences between information that is payoff-relevant and other forms of information, and used this to obtain a purification result in the classical finite-action setting in Fu (2008). He takes the categorization of forms of information to be basic. In our view, payoff-relevance should be derived from the structure of the payoff-functions as we do here.

  9. 9.

    The more involved notion of saturation for adapted stochastic processes introduced in Hoover and Keisler (1984) gives rise to a much more restricted class of probability spaces.

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Correspondence to Michael Greinecker.

Additional information

We are grateful to Rabeè Tourky and Nicholas Yannelis. Discussions with them on existence of pure-strategy equilibria and the relation between randomization and decomposability techniques inspired this research.

Appendix

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:

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

\(\square \)

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Greinecker, M., Podczeck, K. Purification and roulette wheels. Econ Theory 58, 255–272 (2015). https://doi.org/10.1007/s00199-014-0815-1

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Keywords

  • Purification
  • Games with incomplete information
  • Roulette wheels

JEL Classification

  • C72