Bayesian Nash Equilibrium Examples

282.1 Opponent of unknown strength First, write down the game. An equilibrium will be a triple of an action for player 1, and action for player 2 in the case that player 2 is strong, and an action for player 2 in the case that player 2 is weak. Player 1 has 2 strategies: {F}, {Y}. Player 2 has 4 strategies, {FY}, {FF}, {YF}, and {YY}. Consider best responses to these strategies. Payoffs to player 1 from player 2’s possible strategies: FY FF YF YY F α(-1) + (1- α)1 α(1) + (1- α)1 Y

Opponent of unknown strength 2 Payoffs to player 2 from player 1’s possible strategies: To find best responses, we need to compare these in magnitude. This depends on the size of α. FY FF YF YY F α(1) + (1- α)0 α(1) + (1- α)(-1) α(0) + (1- α)(-1) Y 1 α(0) + (1- α)1

Opponent of unknown strength 3 Suppose α > ½. Then, the payoffs to player 1 are: Payoffs to player 2 are: With best responses in bold. So the unique Bayesian NE is [Y,FF] FY FF YF YY F (1 - 2α) 1 Y FY FF YF YY F α 2α – 1 - (1 – α) Y 1 (1- α)1

Opponent of unknown strength 4 Now suppose α < ½. Then, the payoffs to player 1 are: Payoffs to player 2 are: With best responses in bold. So now the unique Bayesian Nash equilibrum is [F,FY]. FY FF YF YY F (1 - 2α) 1 Y FY FF YF YY F α 2α – 1 - (1 – α) Y 1 (1- α)1

First price auction with private information valuations Consider the first price auction with 2 players where each player’s valuation is private information known only to them, and valuations are uniformly distributed between [0,1]. Assume that the equilibrium bid function is of the form bi = avi. Now let’s find a. Consider i’s profit maximisation problem: maxbi Probwin*Payoff if win = maxbi P[bi > bj](vi – bi) = maxbi P[bi > avj](vi – bi) = maxbi P[bi/a > vj](vi – bi)

FP auction 2 = maxbi (bi/a)(vi – bi) = maxbi vibi/a – bi2/a Take a FOC wrt bi , set = 0. vi/a – 2bi/a = 0 Solve to find bi = vi/2. Which is of the linear form we assumed. Solve this for n players in the problem set: almost the same.