# Necessary indifference conditions in mixed equilibrium

Suppose we are playing a game where the Action set for Player 1 is $$(a,b)$$, for Player 2 is $$(c,d)$$, and for Player 3 is $$(L,M,R)$$. Assume that for Player 3, the action $$M$$ is weakly dominated by some mix between $$L$$ and $$R$$. I am attempting to find all of the possible mixed equilibria.

Let $$p$$ be the probability that Player 1 chooses $$a$$, let $$q$$ be the probability that Player 2 chooses $$c$$, and let $$r_1, r_2, r_3$$ be the probabilities that Player 3 choose $$L, M, R$$ respectively.

I have two questions:

First, because action $$M$$ is weakly dominated for Player 3, does everyone assume that $$r_2=0$$?

Second, the probabilities $$p$$ and $$q$$ are subject to the constraint that Player 3 be indifferent between his/her own actions. Does Player 3 need to be indifferent between $$L$$, $$M$$, and $$R$$, or just between $$L$$ and $$R$$, since $$M$$ is weakly dominated?

First, because action $$M$$ is weakly dominated for Player 3, does everyone assume that $$r_2=0$$?

Without seeing the game you're considering, it's hard to tell. But the general answer is No. Weakly dominated strategies can be played in a Nash equilibrium (either mixed or pure). In the following game, both $$B$$ and $$C$$ are weakly dominated strategies for both players. But each player randomizing between them in any proportion is still a NE.

$$\begin{array}{|c|c|c|c|} \hline & A & B&C\\\hline A&1,1&0,0&0,0\\\hline B&0,0&0,0&0,0\\\hline C&0,0&0,0&0,0\\\hline \end{array}$$

Does Player 3 need to be indifferent between $$L$$, $$M$$, and $$R$$, or just between $$L$$ and $$R$$, since $$M$$ is weakly dominated?

The indifference condition for MSNE applies to pure strategies in the support of the equilibrium mixed strategy. If $$M$$ is played with positive probability in equilibrium, then you'd need indifference across all three. If $$M$$ is played with zero probability, then you'd need $$u_3(\sigma_1,\sigma_2,L)=u_3(\sigma_1,\sigma_2,R)\ge u_3(\sigma_1,\sigma_2,M)$$

More generally, the necessary and sufficient conditions for a Nash equilibrium (either pure or mixed) are summarized in the following theorem:

A strategy profile $$(\sigma_1,\dots,\sigma_n)$$ is a Nash equilibrium if and only if for all $$i=1,\dots,n$$, $$$$u_i(s_i,\sigma_{-i})= u_i(s_i',\sigma_{-i}), \qquad \forall s_i,s_i'\in\mathrm{supp}(\sigma_i)$$$$ and $$$$u_i(s_i,\sigma_{-i})\ge u_i(s_i',\sigma_{-i}), \qquad \forall s_i\in\mathrm{supp}(\sigma_i)\text{ and }\forall s_i'\notin\mathrm{supp}(\sigma_i).$$$$

Hence, we see that the indifference condition is only required for pure strategies in the support of the equilibrium mixed strategy but not necessarily outside it.