Here is how I would approach the questions:
(a) To find the pure strategy Nash equilibria, we need to consider each player's best responses and find combinations where the chosen strategies are mutual best responses.
For John, his payoff is always higher choosing x=0 regardless of James' action. So his best response is to always choose x=0.
For James, if x=0, his payoff for Up is 0 and for Down is 1, so Down is optimal. If x>0, his payoff is higher for Up than Down. So his best response is:
- If x=0, choose Down
- If x>0, choose Up
The only combination where the strategies are mutual best responses is when John chooses x=0 and James chooses Down.
Therefore, the only pure strategy Nash equilibrium is (x=0, Down).
Payoff matrix:
| | Up | Down |
|------ |-----------------|----------------|
| x=0 | u1=0, u2=0 | u1=2, u2=1 |
| x>0 | u1=x, u2=x | u1=x, u2=1-x |
(b) Now suppose James randomizes, choosing Up with probability p and Down with probability 1-p, while John still picks a pure strategy x.
James's expected payoff is:
E[u2] = px + (1-p)(1-x)
The optimal mixed strategy for James must make John indifferent between all his pure strategies. If John has a strictly dominant strategy, James's optimal response is to best respond to that dominant strategy.
We know from part (a) that when x=0, John's payoff is highest regardless of James' strategy. So John's dominant strategy is to choose x=0.
Given John chooses x=0, for James to be indifferent, we must have:
E[u2|x] = p*0 + (1-p)*1 = 1-p = 1
-> p = 0
Therefore, the Nash equilibrium is (x=0, p=0) - John chooses 0, James only chooses Down.
Hope it helps