I'm trying to figure out a Nash Equilibrium for a 3x3 zero-sum game, and it's not following normal patterns (or I'm making a huge oversight, in which case I'll feel stupid!). Can anyone help me?
The payoff matrix for P1 is (additive inverses for P2):
0.0 0.0 1.0
1.5 3.0 -0.5
-1.5 -2.0 1.5
As far as I can tell, nothing is dominated for either player. Doing the usual calculations where you find probabilities each player makes each play such that the other player is then indifferent to his plays yields negative probabilities though...not sure what's wrong with what I'm doing.
Thanks in advance!
Edit: Some more thinking has led me to believe that I don't think I'm doing this wrong, and that there's a reason I wasn't explicitly taught to do this. It seems to be equivalent to a LP problem in the 3x3 case (and in the general nxn case) where no strategy is strictly dominated, and where there's no pure strategy equilibrium. My confusion arose from the fact that I know a Nash equilibrium is guaranteed to exist -- I guess I was taking that to mean that I should be able to calculate one easily. :)
There exists a mixed-strategy Nash equilibrium, but some strategies have $0$ weight in that equilibrium.