Two-Person Zero-Sum Game is Non-Cooperative

Theorem
A two-person zero-sum game necessarily has to be non-cooperative.

Proof
A cooperative game is one where players form coalitions against the other players.

If the players in a two-person zero-sum game were to form a coalition, there would be no other players to form it against.

Further, as the total payoff is zero, there would be no benefit in collaborating on using one strategy over another, as when they pool their payoffs they are back where they started.