Follower maximin makes leader’s actions payoff-invariant in 2-action zero-sum games

Prove that in any zero-sum game between a leader and a follower in which the leader has exactly two actions and neither player has a dominated strategy, the follower’s maximin strategy equalizes the leader’s expected payoff across the leader’s two actions, i.e., the leader’s payoff is invariant to its choice of action under the follower’s maximin response.

Background

This conjecture appears in the authors’ exploration of conditions under which full ambiguity may be optimal for the leader in multi-follower settings, particularly in zero-sum structures. Establishing that the follower’s maximin strategy makes the leader indifferent between its actions would support arguments that full ambiguity suffices in certain simplified game classes.

The authors discuss examples where ambiguity can help or hurt, and propose structural lemmas to bound the leader’s value. This conjecture would provide a clean characterization in the 2-action zero-sum case, clarifying when the leader’s choice set can be expanded without affecting the worst-case outcome.