This paper discusses an interested party who wishes to influence the behavior of agents in a game (multi-agent interaction), which is not under his control. The interested party cannot design a new game, cannot enforce agents’ behavior, cannot enforce payments by the agents, and cannot prohibit strategies available to the agents. However, he can influence the outcome of the game by committing to non-negative monetary transfers for the different strategy profiles that may be selected by the agents. The interested party assumes that agents are rational in the commonly agreed sense that they do not use dominated strategies. Hence, a certain subset of outcomes is implemented in a given game if by adding nonnegative payments, rational players will necessarily produce an outcome in this subset. Obviously, by making sufficiently big payments one can implement any desirable outcome. The question is what is the cost of implementation? In this paper we introduce the notion of k-implementa...