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Game Theory question - URGENT HELP NEEDED!

Hi - if someone could help urgently, it'd be very appreciated!

I have the following question on Game Theory and I'm stuck on part b):

Two friends, John and James, are playing the following game.
John can choose any number x [0, 1] and James can choose between Up and Down. They make their choices simultaneously.
John’s payoffs are u1(x,L) = 2−2x, u1(x,R) = x
James’s payoffs are u2(x,L) = x, u2(x,R) = 1−x

(a) Find all the Nash equilibria in pure strategies in this game. Explain every step of your results and illustrate. [10 marks]
(b) Now consider the case that James randomises between his actions, but John does not which means she just picks a number x [0,1]. Find the Nash equilibrium in this case. Explain and illustrate. [10 marks]

My thinking for part b is that James will randomise his choices with probability p and 1-p, such that John is indifferent between his two payoffs. However when I then maximise john's payoffs, I get that he will choose x=2/3, which then means James will choose left with a probability of 1. However, when James assigns a probability of 1 to a certain strategy, he is no longer randomising. Have I misunderstood or missed something?
(edited 3 months ago)
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

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