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!
Best Answer
Your explanation about strict dominance of ST over SP for player 1 is correct. Therefore SP cannot be a best response for this player and hence, it cannot be part of any Nash equilibria (either in pure of mixed strategies). This leaves us with player 1 choosing ST with probability one in any Nash equilibria.
Turn now to player 2. Since player 2 is indifferent between playing ST or SP when she expects player 1 to play ST, her best response is to play ST with any probability between 0 and 1. Suppose she chooses to play ST with probability 1/2. Then, player 1's best response is to play ST and, given her indifference, player 2's best response is to play ST with probability a half. Therefore, playing ST for sure for player 1 and playing ST with prob a half for player 2 is a Nash equilibrium for your game. Since the above is true for any prob between 0 and 1, we conclude that playing ST for sure for player 1 and playing ST with any prob between 0 and 1 describes the whole set of mixed strategy Nash equilibria of the game.
For part b), the unique outcome that cannot be considered Pareto optimal is (SP,ST) because moving to (ST,SP) improves player 1 without hurting player 2. Out of the 3 remaining, any of them satisfies the definition of Pareto optimality given in the question. However, observe that (SP,SP) has the additional characteristics of making the sum of payoff the highest.