  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 non-negative 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-implementation of a
desired set of strategy profiles, where k stands for the amount of payment that
need to be actually made in order to implement desirable outcomes. A major
point in k-implementation is that monetary offers need not necessarily
materialize when following desired behaviors. We define and study
k-implementation in the contexts of games with complete and incomplete
information. In the latter case we mainly focus on the VCG games. Our setting
is later extended to deal with mixed strategies using correlation devices.
Together, the paper introduces and studies the implementation of desirable
outcomes by a reliable party who cannot modify game rules (i.e. provide
protocols), complementing previous work in mechanism design, while making it
more applicable to many realistic CS settings.
