It’s Friday evening and Jon and Ygritte have some free time. Both are invited to

Question

It’s Friday evening and Jon and Ygritte have some free time. Both are invited to join Ygritte’s friends, who are going out this evening. Jon wants to spend his evening with Ygritte. He doesn’t really like Ygritte’s friends, and prefers to go to the beach and watch the sunset. However, if Ygritte goes out with her friends, then he would prefer to join them. Ygritte is angry at Jon and does not want to spend her evening with him. She would prefer to either go out with her friends without Jon, or watch the sunset without him. Their payoffs are shown in the following matrix:

Ygritte

Jon

Go out

Watch sunset

Go out

1,-1

-1,1

Watch sunset

0,3

3,-3

Question: The game doesn't have a Nash equilibrium in pure strategies. Compute a mixed-strategy Nash equilibrium.

Ygritte

Jon

Go out

Watch sunset

Go out

1,-1

-1,1

Watch sunset

0,3

3,-3

Explanation / Answer

Answer

If Jon Choose Go out Ygritte will choose Watch Sunset. If Ygritte choose Watch Sunset then Jon Will also choose Watch Sunset. If Jon choose watch sunset then Ygritte will choose Go Out. Hence There is no Pure Strategy Nash Equilibrium.

Suppose Jon choose GO OUT with probability p . Then we hill choose Watch Sunset with probability 1-p. Also, Suppose Ygritte choose GO OUT with probability q . Then we hill choose Watch Sunset with probability 1-q.

Suppose Ygritte choose go out. Then Expected Utility(Payoff) of Jon = 1*p + 0*(1-p) = p

Suppose Ygritte choose Watch Sunset. Then Expected Utility(Payoff) of Jon = -1*p + 3*(1-p) = 3-4p

Hence we have to find p such that both the expected utilities are equal.

Hence p = 3-4p => p = 3/5

Now,

Suppose Jon choose go out. Then Expected Utility(Payoff) of Ygritte = (-1)*q + (1)*(1-q) = 1 - 2q

Suppose Jon choose Watch Sunset. Then Expected Utility(Payoff) of Ygritte = 3*q + (-3)*(1-q) = 6q-3

Hence we have to find q such that both the expected utilities are equal.

Hence 1 -2q = 6q - 3 => q = 1/2.

Hence Mixed Strategy Nash Equilibrium is Strategy (3/5 , 2/5) for Jon and (1/2 , 1/2) for Ygritte  

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