Implementation theory is an area of game theory

Game theory

Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

 closely related to mechanism design

Mechanism design

Mechanism design is a field in game theory studying solution concepts for a class of private information games...

 where an attempt is made to add into a game a mechanism such that the equilibrium of the game conforms to some concept of social optimality (for instance Pareto optimality).

In a game where multiple agents are to report their preferences (or their type), it may be in the best interest of some agents to lie about their preferences. This may improve their payoff

Outcome (Game theory)

In game theory, an outcome is a set of moves or strategies taken by the players, or their payoffs resulting from the actions or strategies taken by all players. The two are complementary in that, given knowledge of the set of strategies of all players, the final state of the game is known, as are...

, but it may not be seen as a fair outcome to other agents. In order to implement a more "fair" outcome, in a repeated game, the other players may choose to punish any "cheaters".

The conditions of a repeated game where an arbitrary outcome may be enforced are set out in theorems often known as folk theorems

Folk theorem (game theory)

In game theory, folk theorems are a class of theorems which imply that in repeated games, any outcome is a feasible solution concept, if under that outcome the players' minimax conditions are satisfied. The minimax condition states that a player will minimize the maximum possible loss which he...

. If a game is not repeated, it may only be possible to implement outcomes which are Nash equilibria

Nash equilibrium

In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...

 or satisfy some other equilibrium

Equilibrium selection

Equilibrium selection is a concept from game theory which seeks to address reasons for players of a game to select a certain equilibrium over another...

concept.