Infinite two players zero sum game and Nash equilibrium

Question

Consider a two player zero sum game (finite strategy space) with mixed strategies. Let the game value is $v$ which is obtained at a Nash equilibrium say $(p,q)$ where $p$ and $p$ are probability law associated with the strategies of Player 1 and 2 respectively. Suppose there exist another pair of probability laws $(p_1,q_1)$ for which the game value is same as Nash Value $v$. Can we say that $(p_1,q_1)$ is also a Nash equilibrium. Is this result also true for infinite game? That is if another set of strategies has same value as Nash equilibrium then that will also be a Nash equilibrium?

Answer 1

This can fail even in very simple finite games. Here, BR is a Nash equilibrium in pure strategies with a value of $0$. TL has the same value but is not a Nash equilibrium.

\begin{array} {|r|r|}\hline & L & R \\ \hline T & 0,0 & -1,1 \\ \hline B & 1,-1 & 0,0 \\ \hline \end{array}