Regularity of the minmax value and equilibria in multiplayer Blackwell games

Abstract A real-valued function φ that is defined over all Borel sets of a topological space is regular if for every Borel set W , φ ( W ) is the supremum of φ ( C ), over all closed sets C that are contained in W , and the infimum of φ ( O ), over all open sets O that contain W . We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player’s minmax value is regular. We then use the regularity of the minmax value to establish the existence of ε -equilibria in two distinct classes of Blackwell games. One is the class of n -player Blackwell games where each player has a finite action space and an analytic winning set, and the sum of the minmax values over the players exceeds n − 1. The other class is that of Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values. For the latter class, we obtain a characterization of the set of equilibrium payoffs.