Mixed strategy Nash equilibrium - what dose a negative probability mean? Consider the game table below for a zero-sum game with two players, a row player (who wants to maximize the payoff), and a column player (who wants to minimize the payoff). I am looking for a mixed strategy Nash Equilibrium. If row player selects row i with probability pi, finding a mixed strategy that makes column player indifference across his actions results in negative p2. Since the row 2 is not dominated by other rows, how should I interpret the negative probability? The same thing happens to the column player.
\begin{bmatrix}1.5197&1.6569&1.8113\\1.1784&0.4629&2.0625\\1.9041&1.1683&1.0211\end{bmatrix}
 A: Indifference only holds among the actions for which the strategy has non-integer probability. If the probability is $0$ or $1$, the player need not be indifferent between this action and other actions: She may want to reduce (or increase) the weight of the action, but she can’t, since it’s already $0$ (or $1$).
Thus, by assuming indifference for all actions, you assumed that all probabilities are non-integers. That this yields a negative probability proves by contradiction that there is no Nash equilibrium that satisfies this assumption. You can now try subsets of the actions and set the probabilities for the remaining actions to $0$. Since a mixed Nash equilibrium is guaranteed to exist, you should find some subset of the actions such that assuming indifference among them and $0$ probability for the remaining actions yields non-negative probabilities. Then to show that this is actually an equilibrium, you should check that the actions with zero probability are not better responses to the other player’s strategy than the ones with non-zero probability.
