Potential game

In game theory, a game is said to be a potential game if the incentive of all players to change their strategy can be expressed using a single global function called the potential function. The concept originated in a 1996 paper by Dov Monderer and Lloyd Shapley. In game theory, a game is said to be a potential game if the incentive of all players to change their strategy can be expressed using a single global function called the potential function. The concept originated in a 1996 paper by Dov Monderer and Lloyd Shapley. The properties of several types of potential games have since been studied. Games can be either ordinal or cardinal potential games. In cardinal games, the difference in individual payoffs for each player from individually changing one's strategy, other things equal, has to have the same value as the difference in values for the potential function. In ordinal games, only the signs of the differences have to be the same. The potential function is a useful tool to analyze equilibrium properties of games, since the incentives of all players are mapped into one function, and the set of pure Nash equilibria can be found by locating the local optima of the potential function. Convergence and finite-time convergence of an iterated game towards a Nash equilibrium can also be understood by studying the potential function. We will define some notation required for the definition. Let N {displaystyle N} be the number of players, A {displaystyle A} the set of action profiles over the action sets A i {displaystyle A_{i}} of each player and u {displaystyle u} be the payoff function. A game G = ( N , A = A 1 × … × A N , u : A → R N ) {displaystyle G=(N,A=A_{1} imes ldots imes A_{N},u:A ightarrow mathbb {R} ^{N})} is: where b i ( a − i ) {displaystyle b_{i}(a_{-i})} is the best action for player i {displaystyle i} given a − i {displaystyle a_{-i}} . In a 2-player, 2-strategy game with externalities, individual players' payoffs are given by the function ui(si, sj) = bi si + w si sj, where si is players i's strategy, sj is the opponent's strategy, and w is a positive externality from choosing the same strategy. The strategy choices are +1 and −1, as seen in the payoff matrix in Figure 1. This game has a potential function P(s1, s2) = b1 s1 + b2 s2 + w s1 s2. If player 1 moves from −1 to +1, the payoff difference is Δu1 = u1(+1, s2) – u1(–1, s2) = 2 b1 + 2 w s2.