Can one establish any determinacy in three-player games?
by
Derrick DuBose
University of Nevada, Las Vegas

It is trivial to show determinacy fails for games with at least three players: just consider the game in which player I's first move decides which of the other players wins. Of course, player I should avoid playing such a game. Along similar lines, to establish determinacy, we must consider plays in which the players without winning strategies play in no reasonable manner, and in particular, that players without winning strategies may play moves that result in a position at which another player has a winning strategy, even when such a move need not be played.

Suppose we consider games in which at every position p a move m can be made such that none of the players, other than possibly the player making move m, have a winning strategy at the resulting position p^m. Determinacy holds in such games with a finite number of players in which all the players except one have open payoff, but this is also unsatisfying in that the player with the closed payoff set always wins. We present a determinacy result for multi-player games that is hopefully slightly more satisfying.

Date received: March 20, 2006