## ‘Two-person cooperative games,’ pp. 128-140 in: Econometrica, Vol. 21, No. 1, January 1953.

Baltimore: Waverley Press, 1953.

First edition, journal issue in original printed wrappers, **signed by John Nash**, of this important paper which set a new entire research agenda that has been referred to as the "Nash program" for cooperative games. "In Nash's (1953) analysis of bargaining, the innovation was to use the notion of equilibrium to characterize players' rational strategies in an explicit, noncooperative model of the bargaining process. In his model, bargainers make simultaneous, once-and-for-all demands, expressed as utilities. If their demands are feasible, taken together, bargaining ends with a binding agreement that yields them the utilities they demanded; otherwise the process ends in disagreement... Nash's (1953) noncooperative analysis of bargaining is important because it explains, within the theory, why bargaining is a problem, and thus provides a framework in which the influence of the environment on bargaining outcomes can be evaluated. The Nash program—the analysis of noncooperative models of cooperation—has since greatly improved our understanding of efficient and inefficient outcomes in bargaining and many other aspects of economic life." (Crawford.)

“During a three-year period, from 1948 to 1951, Nash wrote a doctoral dissertation and four research papers that revolutionized game theory, the branch of mathematics dealing with the study of competition and cooperation. In the 1920s Hungarian mathematician John von Neumann had analysed two-person, zero-sum games in which two competing participants made choices that resulted in a payoff for one player and a penalty of equal magnitude for the other … Nash broadened the scope of game theory to include situations with more than two participants and the analysis of general strategies for games in which players can either cooperate or compete with one another. He introduced concepts, tools, and techniques that became fundamental components in the full development of game theory and that enabled game theory to be broadly applied to evolutionary biology, economic theory, and political strategies” (*Mathematics Frontiers* (2006), pp. 25-26). In this paper, Nash treats cooperative games by showing that they can be reduced in many cases to non-cooperative games, by including the players’ strategies of cooperation into the structure of a suitable non-cooperative game. Nash was awarded the 1994 Nobel Memorial Prize in Economic Sciences “for his pioneering analysis of equilibria in the theory of non-cooperative games” (the prize was shared with John Harsanyi and Reinhard Selten). No copies in auction records.

*Provenance*: Signed ‘John F. Nash, Jr. | 29 November 2011’ on the first page of the article and also on the first page of the next article in the issue (not by Nash).

“Nash made his breakthrough at the start of his second year at Princeton … After wrangling for months with Tucker, his thesis adviser, Nash provided an elegantly concise doctoral dissertation … In his thesis, ‘Non-Cooperative Games,’ Nash drew the all-important distinction between non-cooperative and cooperative games, namely between games where players act on their own ‘without collaboration or communication with any of the others,’ and ones where players have opportunities to share information, make deals, and join coalitions. Nash’s theory of games—especially his notion of equilibrium for such games (now known as Nash equilibrium)—significantly extended the boundaries of economics as a discipline.

“All social, political, and economic theory is about interaction among individuals, each of whom pursues his own objectives (whether altruistic or selfish). Before Nash, economics had only one way of formally describing how economic agents interact, namely, the impersonal market. Classical economists like Adam Smith assumed that each participant regarded the market price beyond his control and simply decided how much to buy or sell. By some means—i.e., Smith’s famous Invisible Hand—a price emerged that brought overall supply and demand into balance.

“Even in economics, the market paradigm sheds little light on less impersonal forms of interaction between individuals with greater ability to influence outcomes. For example, even in markets with vast numbers of buyers and sellers, individuals have information that others do not, and decide how much to reveal or conceal and how to interpret information revealed by others. And in sociology, anthropology, and political science, the market as explanatory mechanism was even more inadequate. A new paradigm was needed to analyze a wide array of strategic interactions and to predict their results.

“Nash’s solution concept for games with many players provided that alternative. Economists usually assume that each individual will act to maximize his or her own objective. The concept of the Nash equilibrium, as Roger Myerson has pointed out, is essentially the most general formulation of that assumption. Nash formally defined equilibrium of a non-cooperative game to be ‘a configuration of strategies, such that no player acting on his own can change his strategy to achieve a better outcome for himself.’ The outcome of such a game must be a Nash equilibrium if it is to conform to the assumption of rational individual behavior. That is, if the predicted behavior doesn’t satisfy the condition for Nash equilibrium, then there must be at least one individual who could achieve a better outcome if she were simply made aware of her own best interests.

“In one sense, Nash made game theory relevant to economics by freeing it from the constraints of von Neumann and Morgenstern’s two-person, zero-sum theory. By the time he was writing his thesis, even the strategists at RAND had come to doubt that nuclear warfare, much less post-war reconstruction, could usefully be modeled as a game in which the enemy’s loss was a pure gain for the other side. Nash had the critical insight that most social interactions involve neither pure competition nor pure cooperation but rather a mix of both” (Nasar, pp. xvii-xix).

Nash had made a start on the cooperative theory with his paper on the Bargaining Problem, to some extent conceived while he was still an undergraduate (‘The bargaining problem,’ *Econometrica* 18 (1950), 155–162), but presented the first detailed account in the present paper.

“An earlier paper by the author [i.e., ‘Non-cooperative games’] treated a class of games which are in one sense the diametrical opposites of the cooperative games. A game is non-cooperative if it is impossible for the players to communicate or collaborate in any way. The non-cooperative theory applies without change to any number of players, but the cooperative case, which is analysed in this paper, has only been worked out for two players.

“We give two independent derivation for our solution of the two-person cooperative game. In the first, the cooperative game is reduced to a non-cooperative game. To do this, one makes the players’ steps of negotiation in the cooperative game become moves in the non-cooperative model. Of course, one cannot represent all possible bargaining devices as moves in the non-cooperative game. The negotiation process must be formalized and restricted, but in such a way that each participant is still able to utilize all the essential strengths of his position.

“The second approach is by the axiomatic method. One states as axioms several properties that it would seem natural for the solution to have and then one discovers that the axioms actually determine the solution uniquely. The two approaches to the problem, via the negotiation model or via the axioms, are complementary; each helps to justify and clarify the other” (pp. 100-101).

“[Nash] had originally intended to present the ideas from this paper in a section of his dissertation, but his adviser, Tucker, had recommended that he remove the topic from an early draft of the work. In this paper he more fully developed his ideas on Nash bargaining solutions for games with fixed threats that he had discussed in his ‘Bargaining’ paper and presented Nash bargaining solutions for games with variable threats. Nash showed that with rational players, a variable threats game – a game in which a player can select one of a choice of penalties when the opponent deviates from the agreed-upon strategy – reduces to a fixed threat game, with each player employing an optimal threat strategy. In contrast to traditional economic theory, Nash’s paper showed that the rational division of an economic surplus leads to a unique outcome rather than being dependent on the players’ negotiation skills” (*Mathematics Frontiers* (1998), pp. 28-29).

Nasar, Introduction to *The Essential John Nash* (Kuhn & Nasar, eds.), 2002; Crawford, John *Nash and the Analysis of Strategic Behavior, *2000.

Large 8vo (253 x 175 mm), pp. [ii], 231, [9]. Original printed wrappers.

Item #5030

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Price:
$13,500.00
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