The Computational Complexity of Nash Equilibria

Game theory has grown in importance as academics from all fields have realized the importance of strategy in domains of human behavior ranging from business to the Internet. The computational issues of game theory are important to both a game’s participants and designers. A participant in a game, or their consultant, obviously wants to compute their optimal moves in anticipation of their oppponents’ likely moves. Likewise, someone who wishes to model a situation of conflict with a game needs to calculate its equilibrium in order to test their model’s correctness. In addition, a game designer may wish to use computational intractability to their advantage; where players are unable to design a strategy by computing possible outcomes of the game, they have no incentive not to act according to their own preferences, as might be desired by the game’s designer. For example, although no non-dictatorial voting scheme in an election of more than two candidates is strategy-proof,1 in actuality, the computation needed to determine the ideal strategic vote may not be possible if the election is sufficiently large and the computation sufficiently intractable. A dictatorial election is an election where one person’s vote entirely determines the outcome of the election. The Gibbard-Satterthwaite theorem proves that in an election with more than two candidates, all other voting schemes are vulnerable to strategic voting — that is, voters can potentially gain by voting according to something other than their preverences.