I am trying to find a zero-sum game with 3 equilibriums. We know from the oddness theory that almost all games are with an odd number of equilibriums. It is easy to find zero-sum games with 1 equilibrium, but I could not find an example for a zero-sum game with 3 equilibrium.
Thanks in advance.
A zero-sum game cannot have exactly three equilibria. The set of equilibria of a zero-sum game corresponds to a pair of convex polyhedra, so there is either one equilibrium (in the generic case that corresponds to the "odd case" for general bimatrix games) or there are an infinite number of equilibria.
You could look for a game that has exactly 3 "extreme equilibria", that is equilibria that are not convex combinations of any others (i.e., all equilibria are convex combinations of the extreme ones). A 1x3 game with identical payoffs is a silly example, so one would want 3 extreme equilibria that are all different for both players. That would not be hard to find.