The last topic of chapter 6 of

**FT**is a

**purification**.

6.7 shows the theorem by

**Harsanyi (1973)**, which states;

*Mixed-strategy equilibrium of complete-information games can usually be interpreted as the limits of pure-strategy equilibria of slightly perturbed games of incomplete information.*

We should note that

**full measure condition**w.r.t. a set of payoff is sufficient. Moreover, given this condition, the following statement holds;

*A single sequence of perturbed games can be used to "purify" all the mixed equilibria of the limit game.*

Ex 6.7 gives the intuition that how purification argument works. Looking at this example, you can easily understand the idea.

In 6.8, the other kind of purification result due to

**Milgrom and Weber (1986)**is shown.

*Under certain regularity conditions, pure-strategy equilibria do exist in games with an atomless distribution over types. The idea is that the effects of mixing can be duplicated by having each type play a pure strategy.*

In short, quite wide range of incomplete information games has pure-strategy equilibria. It is interesting, because pure-strategy equilibria need not exist in games of complete-information.

The proof is given by using a "distributional strategy" introduced by Milgrom and Weber, which is motivated by

**Aumann (1964)**.

**References**

**Aumann (1964)**"Mixed vs. behavior strategies in infinite extensive games"

*Annals of Mathematics Studies, 52*

**Harsanyi (1973)**"Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points"

*IJGT, 2*

**Milgrom and Weber (1986)**"Distributional strategies for games with incomplete information"

*Mathematics of Operation Research, 10*

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