Can the same support hosts multiple isolated Nash equilibria?

Question

As from the title, in a strategic game in normal form, can a given support (e.g. actions [1,3,5] for the first player, [2,4] for the second, [1,3] for the third one..) hosts multiple isolated Nash equilibria or there is a guarantee that there is at maximum a single isolated equilibria for that specific support ?

Answer 1

If there exists an isolated NE for a given support, then this must be the unique NE for this support. This is because, in order to make each player indifferent between their actions, one must solve a system of linear equations that can be written in the form $Av=b$, for some matrix $A$ and non-zero vector $b$ for each player. If there is an isolated NE, then $A$ must be full rank, and hence invertible. Thus, these systems of linear equations must have a unique solution for all players. Hence, there can only be a unqiue isolated NE for a given support.