The question is as follows:
Think of the Golden Ball game. Now player 1 is money-minded and jealous, and player 2 is very good-hearted, so the payoff matrix is follows:
Player 2 SP ST
Player 1 SP ............. 5, 5..........-2, -1
ST 10, -1 0, -1
dots are just to make the matrix neat.
a. Solve for all the mixed Nash equilibrium if any. b. Among the four outcomes here, which outcome(s) are Pareto optimal (in the sense that you cannot find another outcome that makes no player worse off but some player better-off)?
For a, i think the answer is No, but i don't know how to answer it.
I merely think that SP is strictly dominated by ST for player 1, and ST is weakly dominated by SP for player 2. That's all i could immediately figure out. Thus, player 1 should have chosen ST as his pure strategy, and if player 1 choose ST, player 2 would be indifferent on choosing SP and ST since they give her the same payoff -1.
So (10,-1) should be one of the nash equilibrium but then, i find out that (0,-1) should also be the nash equilibrium since both player 1 and 2 could not deviate profitably from this outcome.
After a long explanation, i still have not solved for the mixed nash equilibrium, and i am stucked here. I try to use calculus here to find the mixed strategy for both players, but i could not calculate the probability distribution for both of them since the unknown i set will cancel itself out or having negative probability.
For b, i think the answer for this is not relevant to a, so i try to work it out.
I think that the pareto optimal outcome should be (10,-1) and not (0,-1), although both satisfy what the question is given for pareto optimal, player 1 should be better off having 10 instead of 0 and would like to force player 2 to choose SP. That is why i choose (10,-1).
But i am not sure for that.
Hope anyone could help! Thanks a lot!