12.10 Purification: Consider the perturbed Matching Pennies game described in Section 12.5 and show that the Bayesian Nash equilibrium identified is the unique Bayesian Nash equilibrium. (Hint: Some insights can be gained by looking at the solution to the study group example in Section 12.2.2.)
12.5 Mixed Strategies Revisited: Harsanyi’s Interpretation
In particular imagine that the payoffs are given by this ’’perturbed’’ Matching Pennies game:
Player 2
H T
Player 1 H 1 + ε1, -1 + ε2 -1 + ε1, 1
T -1, 1 + ε2 1, -1
and imagine that both ε1 and ε2 are independent and uniformly distributed on the interval [-ε, ε] for some small ε > 0. This means that if εi > 0 is realized then player i has a strict preference for choosing H over T when he believes his opponent is choosing H with probability 1/2, and similarly, if εi < 0 is realized, then player i has a strict preference for choosing T over H when he believes his opponent is choosing H with probability 1/2.
Assume further that the value of εi is known only to player i but that the distribution of the values of εi is common knowledge. This perturbed Matching Pennies game is a Bayesian game of incomplete information with two actions for each player and a continuum of types, similar to the study group example solved in Section 12.2.2. Hence a pure strategy for each player is a mapping si : [-ε, ε] → {H, T} that assigns a choice to every type of player i.
Claim 12.3 In the Bayesian perturbed Matching Pennies game, there is a unique pure-strategy Bayesian Nash equilibrium in which si(εi) = H if and only if εi ≥ 0, and si(εi) = L if and only if εi < 0. This equilibrium converges in outcomes and payoffs to the Matching Pennies game when ε → 0.
It is quite easy to see that the proposed strategies are a Bayesian Nash equilibrium. If they are followed by player i, then because the distribution of εi is uniform over the interval [-ε, ε], it follows that with probability 1/2 player i is playing H, in which case the strategy of player j is a best response. To see that this is the unique Bayesian Nash equilibrium requires more work, but is not too hard.