Variable-Sum Games

A game in strategic form is defined by a set of players, a set of strategies for each player, and a payoff function for player defined over all combinations of strategy choices made by the players. In the event that the sum of the payoffs of all of the players is constant, irrespectively of the strategy choices made, the game is constant-sum and, if the constant is zero, the game is zero-sum. All other games, which are the vast majority, are variable-sum.

Two-person zero-sum games have special historical importance, because John von Neumann first proved the existence of an equilibrium in mixed strategies for such games. Von Neumann's result was extended by John F. Nash, who showed that every (variable-sum) game with finite strategy sets has at least one equilibrium in mixed strategies. This is one of the most important theoretical results for noncooperative games.

Variable-sum games are more difficult to analyze than zero-sum ones, and in general it is not possible to find equilibrium strategies for the players: only combinations of strategies correspond to equilibria. The most famous example of this kind is the Battle of the Sexes game (so-called because a couple wish to spend the evening together but one would rather do one activity, say go to the opera, while the other would sooner they both do something else, say go to the ball game). Such a game has two pure strategy equilibria plus one mixed equilibrium.

Different equilibria in variable-sum games, in contrast to constant-sum games, typically give different payoffs to the players. Indeed, an equilibrium may give a payoff to both (or all) players that are inferior to other (equilibrium or nonequilibrium) strategy combinations. The outstanding example of this possibility is the prisoner's dilemma game, which has a unique equilibrium composed of dominant strategies. This problem is found in many strategic interactions, such as the Cournot oligopoly model, the “tragedy of the commons,” and collective action problems.

Repeated interaction may overcome such inefficiency. The so-called folk theorem guarantees that, under appropriate conditions, essentially all of the efficient combinations of payoffs can be achieved as equilibria for the repeated game. The precise result is, however, critically dependent on details like the duration of the game, the fact that this duration is known in advance to the players, and the amount of information available to players on the characteristics of the other players.

The fact that, loosely speaking, all of the efficient results can be achieved as equilibria for repeated games points to another difficulty: the indeterminacy offered by the equilibrium condition. This is a leitmotif for variable sum games that is to be found for many classes of games (among which are signaling games, a special case of games with incomplete information). This difficulty has fostered research agendas, generally known as equilibrium selection or refinement and initiated by Reinhardt Selten, that have yet to resolve this indeterminacy.