Purification

Original article (link) posted: 24/07/2005

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