Purification
A mixed strategy can be read as the limit of pure choices made by slightly different privately known types, so that nobody actually has to randomise.
A mixed strategy can be read as the limit of pure choices made by slightly different privately known types, so that nobody actually has to randomise.
Often misread asDefending mixed equilibria by insisting players really do randomise, which then leaves you explaining why anyone would pick particular odds when at the equilibrium mix they are indifferent between everything.
How to see it
Nobody randomises; the types do it for them
A mixed strategy looks odd because nobody really flips a coin to decide. The purification argument says they do not have to. Give each player a tiny private wobble in their payoffs and each simply picks whichever action is better for the type they turned out to be. From outside, the frequencies look exactly like the mixture.
The checkNobody randomises. Each type plays a strict best response, and the mixture is the modeller's description of the population of types, which is also how the indifference puzzle dissolves.
Try itTake the mixed equilibrium of any two by two game and state its probability. Now describe two tennis players whose fatigue varies slightly day to day, each serving to whichever side is better that day. Check the long run frequency matches the mixture.
Know these first
When it stops being true
- independent-private-values
Easily confused with
Sources
- Harsanyi, J. C. (1973). International Journal of Game Theory 2(1), 1-23.
- Fudenberg, D. & Tirole, J. (1991). MIT Press.
- Osborne, M. & Rubinstein, A. (1994). MIT Press.