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Among game theoretic concepts, mixed strategy is arguably the most difficult to digest. We don’t see people tossing coins all the time, and it’s difficult to justify rational decision as based on Lady Fortuna’s unpredictable caprices. The case of Nash Equilibrium is especially disturbing — if you are indifferent between a couple of strategies then why bother randomizing between them according to the mixture prescribed by the equilibrium. Just pick one of these strategies arbitrary and get it over with.

I know of two types of answers that game theory gives for this conundrum. One, which may be called `interpretation of mixed strategies’ is arguing that the mixed strategies in Nash equilibrium do not reflect an actual randomization performed by the players: Epistemic game theory interprets mixed strategies as opponent’s beliefs about a player’s (non-randomized) strategy; Harsanyi’s Purification approach embeds a normal form game in a larger game with incomplete information and pure equilibrium. The other type of answers is identifying classes of games which admit pure equilibrium, such as games with strategic complementarity and potential games.

In my new paper with Yaron(pdf) we suggest another class of games which admit pure ${\epsilon}$-equilibrium, which means that no player gains more than ${\epsilon}$ from deviating. These are games in which a player’s payoff does not change much if one of her opponents changes his strategy:

Math and open problems below the fold…