Nonzero-sum game

A nonzero-sum game is a situation in which one participant’s gain does not have to be another’s loss: players may cooperate to increase what they receive together. In The Selfish Gene, the idea distinguishes situations that reward mutual benefit from contests in which every advantage for one side comes at the other’s expense. Its importance is both analytical and practical: the structure of a game, and the information available to its players, can make cooperation advantageous even in a world shaped by selfish interests.

Definition and contrast

Game theorists distinguish zero-sum games from nonzero-sum games by whether the players’ interests are necessarily opposed. Chess is zero-sum: one player’s victory entails the other’s defeat. The Prisoner’s Dilemma, by contrast, is nonzero-sum because a banker pays out money and both players can do better by cooperating than by trying only to outscore each other. They can, in the book’s image, link arms and laugh all the way to the bank.

The game’s basic arrangement makes the distinction concrete. A banker adjudicates and pays; each of two players holds cards marked cooperate and defect, chooses one without seeing the other’s choice, and lays it face down. Because the choices are simultaneous, neither player can adapt to the other’s move before committing. The banker then reveals both cards, and the players’ winnings depend on the pair of choices rather than on either choice alone. The possibility of a shared reward means that the other player need not be an enemy: both can benefit from the same outcome.

Cooperation in human affairs

The concept changes how apparently adversarial situations can be understood. Divorce, for example, may begin with the end of a cooperative marriage, but the former partners can still have reasons to cooperate. Child welfare is one; preserving family finances from the cost of two lawyers is another. Treating the separation as a zero-sum contest may obscure those shared interests.

The legal arrangements described in the book can nevertheless force the couple into opposing camps. In England and, until recently, all fifty states of the United States, lawyers’ professional rules required a lawyer to accept only one member of a couple as a client. The other was turned away or had to seek separate advice. Lawyers could then speak as though their own side were set against the other, and the court case might be listed as “Smith versus Smith,” whether or not the couple wished to remain amicable. Under those conditions the clients risked a zero-sum struggle while the lawyers could benefit from the fees it generated. The example shows that conflict need not follow inevitably from the parties’ interests: institutions and professional rules can shape the incentives that make a situation adversarial.

The book extends the question beyond divorce to wage bargaining and pay differentials. People negotiating raises may be motivated by envy, seeking to do better than others, or may be able to cooperate in pursuing greater real income. The important distinction is between a situation’s actual structure and how its participants perceive it. A contest treated as zero-sum may not be one; the book raises the possibility without claiming to resolve every case.

A football match changes its logic

The 1977 Bristol City–Coventry City match provides a dramatic instance of a game changing character while it is under way. On the final day of the season, May 18, two of the three First Division relegations had already been settled, but the last remained in contention among Sunderland, Bristol, and Coventry. Bristol and Coventry were playing each other, while Sunderland played a separate match. If Sunderland lost, a draw between Bristol and Coventry would keep both clubs in the First Division; if Sunderland won, one of the two clubs would be relegated. Although the decisive matches were theoretically simultaneous, the Bristol–Coventry game was running five minutes late, so the Sunderland result arrived before it ended.

For most of the match, Bristol and Coventry played fiercely, and the score had reached 2–2 by the eightieth minute. Two minutes before the end, news came that Sunderland had lost. The Coventry manager had the result displayed on the stadium’s electronic message board. Once the players knew that a draw would save both clubs, attacking became a danger: pushing players forward to score could leave the defence exposed and risk relegation. Both sides instead concentrated on holding the draw.

The change was visible to the crowd as well as the players. Supporters who had been fierce rivals joined in a shared celebration after Don Gillies scored an equaliser for Bristol, while referee Ron Challis watched the players pass the ball with little challenge. A piece of outside information had transformed the match’s incentives. The teams no longer needed to defeat each other; Sunderland’s result created the equivalent of an external banker, making a draw beneficial to both. What began as a zero-sum contest became, in practical terms, a nonzero-sum game.

Iteration and the possibility of cooperation

The football example also sharpens the distinction between spectator sport and life beyond the pitch. Matches are usually zero-sum because crowds find it more exciting to watch opponents strive against one another than to watch them cooperate. But real human, animal, and plant life is not arranged for spectators. Many real situations can instead resemble nonzero-sum games, with nature playing the role of banker and allowing individuals to benefit from one another’s success.

Cooperation, however, depends on interaction continuing. In an iterated Prisoner’s Dilemma, players must know—or act as if they know—that the current encounter will not be their last. If both players know a game has exactly 100 rounds, the final round becomes a one-off dilemma, making defection the rational choice under the stated assumptions. Reasoning backward makes the ninety-ninth round equivalent to a final round, then the ninety-eighth, until defection is predicted throughout. For that reason, analysis of repeated games assumes that the ending is unpredictable, or known only to the banker.

Even when the exact end is unknown, players may estimate how near it is. A banker looking at a watch could suggest that the game is about to end; a player who notices may be tempted to defect, and may suspect the other has noticed too. The expected duration of interaction therefore affects strategy. The concept’s wider significance lies in this combination: shared gains can make cooperation possible, but the structure of the interaction, the rules that govern it, the information participants receive, and their expectations about its continuation all help determine whether that possibility is realized.