Game Harmony: A Short Note

Strategic uncertainty in game theory may have two di¤erent general sources, either alone or in combination: uncertainty because of the existence of a coordination problem, and uncertainty because of a con‡ict between one own and the other n players’ interests. Game harmony is conceived as a generic game property that describes how harmonious (non-con‡ictual) or disharmonious (con‡ictual) the interests of the n players are, as embodied in the game payo¤s. Pure coordination games are examples of games with maximal game harmony; zero sum games are examples of games with very low game harmony. This note brie‡y describes attempts to measure game harmony simply as a real-valued number. These attempts do not focus just on equilibrium payo¤s. This is not just because this would make the measurement of game harmony dependent on the reasonableness of the equilibrium concept for any given game, but also and more fundamentally because a restriction on equilibrium payo¤s would hide any disharmony in the interests among players that does not derive from the equilibrium payo¤s. For example, in the case of the standard Prisoner’s Dilemma, the equilibrium payo¤ gives the same amount to each player, and so no disharmony of interests emerges while it should. A simple way to think about game harmony measurement would be to ask ourselves to what extent the payo¤s of each player co-vary. In the case of a pure coordination game, they co-vary one-to-one and positively, whereas in the case of a zero-sum game they co-vary one-toone but negatively. Therefore, parametric or nonparametric correlation measures, such as Pearson or Spearman respectively, may be used as game harmony measures. Unfortunately, these measures can only be used with n = 2. Some form of extension is required if there are more than two players. Let k be the number of binary combinations among players: