On the team-maxmin equilibria

In this note we extend the result of von Stengel and Koller (Games Econ Behav 21:309–321, 1997) to infinite games. Specifically, we show that every infinite zero-sum game between a team and an adversary admits a pure-strategy team-maxmin equilibrium when (i) the strategy sets are compact, convex subsets of (possibly infinite-dimensional) topological vector spaces, and (ii) the payoff function is bounded, upper-semicontinuous on the team’s strategy-profile set and concave in each team member’s strategy, while being lower-semicontinuous and convex in the adversary’s strategy. Because the vector spaces may have arbitrary dimension, we obtain the following corollary: a mixed-strategy team-maxmin equilibrium exists whenever (i) the strategy sets are compact subsets of metric spaces, and (ii) the payoff function is bounded, measurable, upper-semicontinuous on the team’s strategy-profile set, and lower-semicontinuous on the adversary’s strategy set. The proof employs Sion’s minimax theorem.