Games in strategic form

A p-person game in strategic form with player (decision maker) set P = {1, 2,…, p} can be described by an ordered 2p-tuple where X _i is the non-empty strategy set of player i ∈ P and K _i : Π _i=1 ^p X _i → ℝ is the payoff function of player i, which assigns to each p-tuple x = (x₁, x₂,…, x _p ) of strategies a real number K _i (x). A play of such a game proceeds as follows. Eventually after pre-play communication, where no binding agreements can be made, each player i chooses independently of the other players one of his possible strategies, say x ⁱ . Then player 1 obtains a payoff K₁(x₁, x₂,…,x _p ),…, and player p a payoff K _p (x₁, x₂,…,x_p). The p-tuple (x₁, x₂,…, x _p ) is called the outcome of the play.