This paper shows a basic tension in game theory. For many infinite games no ordinary mixed-strategy Nash equilibrium exists. One common fix is to allow finitely additive probabilities — a relaxed kind of probability that need not add up over infinitely many disjoint events in the usual way. But that fix creates a new problem: it can let an equilibrium ignore every single action in an infinite set while placing all probability on that same set. The author proves that insisting on full ‘‘ignore the whole set’’ behavior everywhere is incompatible with a minimal, sensible equilibrium requirement for zero-sum games together with the demand that a solution exist for every game.

A key idea is easy to state. A strictly dominated action is one that some finite lottery (a random choice over a finite list of actions) beats no matter what the opponents do. A finitely additive probability can give each one of infinitely many dominated actions probability zero, yet give probability one to the union of those actions. From the point of view of dominance, every action is ignored individually, but play is entirely concentrated on dominated actions when viewed as a set. That mismatch is the pathology at the heart of the paper.

To make the incompatibility sharp, the author builds a specific two-player zero-sum game, called Γ*. Each player has two “coin” actions, H and T, plus actions labeled by the natural numbers. The number actions are arranged so that n+1 strictly dominates n for each n, so the whole infinite set of number-actions is strictly dominated. After iteratively deleting dominated actions, only H and T remain. Still, any finitely additive evaluation that treats the whole dominated set as negligible leads to a violation of the zero-sum equilibrium coherence condition the paper imposes. This yields a proof that no solution rule can simultaneously (1) always produce some solution, (2) satisfy the minimal zero-sum coherence condition, and (3) treat every entire dominated set as null.