1. Bayesian Nash Equilibrium
Examples

2. 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

3. 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

4. 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

5. 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

6. 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)

7. 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.