1 Consider the following game where the row player is Player

1. Consider the following game, where the row player is Player 1 and the column player is Player 2. (a) Let p be the probability (belief) that Player 2 plays x. What should Player 1 do if she believes that p > 2/3 ? If p

Solution

Let the probability that palyer 2 plays x is p.

Then prob. that player 2 plays y is (1-p).

Then, if player 1 plays a, her expected payoff = 12p + 0(1-p) = 12p

if player 1 plays b, her expected payoff = 11p + 1(1-p) = 10p+1

if player 1 plays c, her expected payoff = 10p + 4(1-p) = 6p+4

if player 1 plays d, her expected payoff = 9p + 6(1-p) = 3p+6

Now, if p=2/3, her expected payoff is 8 for a,c and d.

now, if in case of a, expected payoff of player 2 = 6(1-p)

in case of c, expected payoff of player 2 = 2 + 2(1-p)

in case of d, expected payoff of player 2 = 3p.

So, if she thinks p>2/3, she should play a as in that case expected payoff of player 2 is less than 2.

if she thinks p<2/3, she should play d as in that case expected payoff of player 2 is less than 2.

if p=2/3, she can play any of a,c,d as in all cases the expected payoff of player 2 is 2.

b. player 1 should never choose strategy b, no matter what her beliefs.

c. let player 1 plays pure strategy and player 2 plays mixed strategy.

if player 1 plays a, player 2 plays x with probability p and y with prob (1-p).

then for equilibrium case, p = 1/3.

then, expected pay off for both = 4.

if player 1 plays b, player 2 plays x with prob p and y with prob (1-p),

then for equilibrium case, p=2/7.

then, expected payoff for both = 27/7.

other two cases will be no equilibrium case.

so, the equilibrium case is a for player 1 and for player 2, x with prob 1/3 and y with prob 2/3.

d. this part is out of my knowledge.