Algebraic structure of games

Game theory and algebra become inextricably intertwined once one recognizes that the notion of a permutation of players gives rise to a representation of the symmetric group in the space of automorphisms of the vector space of games. In this paper the authors utilize this observation to attempt to turn the typical approach to game-theoretic problems on its head by analyzing the space of games from an algebraic viewpoint. In this way we find, quite surprisingly, that it is the notion of inessential, or additive, games, previously thought to be of little or no interest, that is of prime significance in describing the structure of the space of games.