Conditions that enable a player to surely win in sequential quantum games

This paper studies sequential quantum games under the assumption that the moves of the players are drawn from groups and not just plain sets. The extra group structure makes possible to easily derive some very general results characterizing this class of games. The main conclusion of this paper is that the specific rules of a game are absolutely critical. The slightest variation in the rules may have important impact on the outcome of the game. This work demonstrates that it is the combination of two factors that determines who wins: (i) the sets of admissible moves for each player, and (ii) the order of moves, i.e., whether the same player makes the first and the last move. Quantum strategies do not a priori prevail over classical strategies. By carefully designing the rules of the game it is equally feasible either to guarantee the fairness of the game, or to give the advantage to either player.