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# Nonzero-Sum Game

BIBLIOGRAPHY

In game theory, nonzero-sum games (more precisely known as nonconstant-sum games ) include many examples in which the sum of the payoffs to the players varies according to the strategies chosen. These cases are often more relevant to issues in social science and public policy than are zero-sum (and more generally constant-sum ) games. From a mathematical point of view, however, they raise problems of solution that do not arise when the sum of payoffs is a constant.

An example arises in the case of public goods. Suppose that two agents acting independently can each earn payoffs of ten. However, each has the option of contributing three units from his wealth to produce two units of a public good that benefits both players equally. The strategies then are contribute or do not contribute, and the game can be summarized in tabular normal form as shown in Table 1.

The Table is read as follows: Agent 1, by choosing his strategy, chooses the row, and Agent 2 chooses the column. In the chosen cell, the first payoff is to Agent 1 and the second to Agent 2. For example, at the top left, both agents contribute, leaving seven units each of private wealth, but four units of the public good are produced, adding four additional units of benefits for each of the two agents.

A game of this sort admits two broad classes of solution: cooperative and noncooperative. Cooperative solutions allow for the possibility that the agents may form a coalition and enter into a binding agreement to choose a joint strategy. In this case, it seems likely that they would both choose to contribute, because this choice makes both better off, while each would refuse to contribute if the other does not. Noncooperative solutions assume instead that there are no trustworthy commitments to joint action. In the case of the public goods dilemma, each agent can reason that he is better off choosing not to contribute, whatever the other may do, and moreover that the other player can reason in the same way. As a consequence, it seems that each would choose not to contribute. Therefore, do not contribute is the equilibrium of the game. The fact

Table 1. Public Goods Dilemma
Agent 2
contribute do not contribute
Agent 1
contribute 11,117,12
do not contribute 12,710,10

that each agent has an unconditional motive to choose the equilibrium makes this game an instance of a social dilemma. In addition to public goods, noncooperative equilibrium concepts are applied to competition in industry, international rivalries, population dynamics in evolutionary biology, and many other issues.

In cooperative solutions of larger and more complex games, questions may arise with respect to bargaining and the division of the benefits from the common strategy and the stability of coalitions and agreements, leading to a number of alternative concepts of solution. Among non-cooperative solutions, the equilibrium concept is dominant, but ambiguities can arise with respect to information and learning, and some alternative solution concepts have been explored.

A half-century of experimental work in social psychology, political science, and economics has indicated that both cooperative and noncooperative solutions influence human behavior, but also that real human beings often reason less carefully than the hypothetical agents of game theory.

SEE ALSO Equilibrium in Economics; Game Theory; Nash Equilibrium; Nash, John; Public Goods; Von Neumann, John; Zero-sum Game

## BIBLIOGRAPHY

McCain, Roger A. 2004. Game Theory: A Nontechnical Introduction to the Analysis of Strategy. Mason, OH: Thomson South-Western.

Nash, John. 1951. Non-Cooperative Games. Annals of Mathematics 54 (2): 286295.

von Neumann, John, and Oskar Morgenstern. 1944. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.

Roger McCain

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## non-zero-sum game

non-zero-sum game See GAME THEORY.