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I have a game represented by following table:

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It is clear that there is a pure Nash equilibrium at 4,2 (both players do not cooperate, player 1 awarded 4 points and player 2 awarded 2 points).

Now if I used mixed strategy Nash equlibrium algorithm I get a = 1 and b = -1 as the probability of player 1 plays Cooperate and player 2 plays Cooperate respectively. To my understanding, a and b should be equal to 0 so that both players should play the above pure equilibrium strategy.

Why is there such a difference in pure and mixed strategy?

Edit 1: mixed strategy Nash equlibrium algorithm:

  • The expected ultility for player 2 playing cooperate is: a*2 + (1-a)*1 (This means that some of the time player 1 plays Cooperate, so the probability is a).
  • The expected ultility for player 2 playing non-cooperate is: a*2 + (1-a)*2.
  • By equating the two formuala I have (1-a) = 2*(1-a) so a = 1.
  • Simmilarly, the expected ultility for player 1 playing cooperate is: 6*b + 3*(1 - b).
  • The expected ultility for player 1 playing non-cooperate is: 8*b + 4*(1 - b).
  • By equating the two formuala I have 0 = 2*b + (1-b) so b = -1.
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  • $\begingroup$ You should probably go into detail about your "mixed strategy Nash equlibrium algorithm". E.g. what do the starting equations mean, etc. $\endgroup$ – Giskard Jan 3 at 20:35
  • $\begingroup$ @Giskard added to the post. Apologise for not making the question clear. $\endgroup$ – Leo Long Vu Jan 3 at 21:12
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Mixed strategy Nash equilibrium cannot involve strictly dominated strategies. In particular, Cooperate is strictly dominated for player 1 ($6<8$ and $3<4$). Therefore, no $b\in[0,1]$ can make player 1 indifferent between Cooperate and Non-cooperate. You made a mistake by trying to solve for $b$ by equating player 1's expected payoffs from his two pure strategies.

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