Necessary indifference conditions in mixed equilibrium

Question

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?

Answer 1

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 $u_3(\sigma_1,\sigma_2,L)=u_3(\sigma_1,\sigma_2,R)\ge u_3(\sigma_1,\sigma_2,M)$

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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 $i=1,\dots,n$, \begin{equation} u_i(s_i,\sigma_{-i})= u_i(s_i',\sigma_{-i}), \qquad \forall s_i,s_i'\in\mathrm{supp}(\sigma_i) \end{equation} and \begin{equation} 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). \end{equation}

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.