Games with ordered outcomes

We present a brief review of the most important concepts and results concerning the games in which the goal structure is formalized by binary relations called preference relations. The main part of the work is devoted to games with ordered outcomes, i.e., game-theoretic models where preference relations of players are given by partial orders on the set of outcomes. We discuss both antagonistic games and $n$-person games with ordered outcomes. Optimal solutions in games with ordered outcomes are strategies of players, situations, or outcomes of the game. In the paper, we consider noncooperative and certain cooperative solutions. The special attention is paid to an extension of the order on the set of probabilistic measures since this question is substantial for constructing the mixed extension of the game with ordered outcomes. The review covers works published since 1953 until now.