Mastering multi-player games

We consider multi-player games, and the guarantees that a master player that plays on behalf of a set of players can offer them, without making any assumptions on the rationality of the other players. Our model consists of an (n + l)-player game, with m strategies per player, in which a master player M forms a coalition with nontransferable utilities among n players, and the remaining player is called the independent player. Existentially, it is shown that every game admits a product-minimax-safe strategy for M - a strategy that guarantees for every player in M's coalition an expected value of at least her product minimax value (which is at least as high as her minimax value and is often higher). Algorithmically, for any given vector of values for the players, one can decide in polytime whether it can be ensured by M, and if so, compute a mixed strategy that guarantees it. In symmetric games, a product minimax strategy for M can be computed efficiently, even without being given the safety vector. We also consider the performance guarantees that M can offer his players in repeated settings. Our main result here is the extension of the oblivious setting of Feldman, Kalai and Tennenholtz [ICS 2010], showing that in every symmetric game, a master player who never observes a single payoff can guarantee for each of its players a similar performance to that of the independent player, even if the latter gets to choose the payoff matrix after the fact.