## ‘A simple three-person poker game,’ pp. 105-116 in: Contributions to the Theory of Games, Volume I. Annals of Mathematics Series, No. 24.

Princeton, NJ: Princeton University Press, 1950.

First edition of Nash and Shapley’s game-theoretic analysis of a simplified three-person poker, **signed by John Nash**. “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*, pp. 25-26). Two-person poker had been analysed from a game theoretic point of view by von Neumann and Morgenstern in their *Theory of Games and Economic Behavior* (1944), and by others, but no one had tried to analyse three-person poker. Nash had originally intended to deal with this case in his doctoral dissertation, but his thesis advisor Albert W. Tucker had recommended that it should not be included. Nash therefore published the work, jointly with his friend and fellow graduate student Lloyd Shapley, in the present paper. 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).

*Provenance*: Signed ‘John F. Nash, Jr.’ on the volume title page; signed ‘W. M. Kincaid’ inside front wrapper. Kincaid (1918-2015) was a professor of mathematics at the University of Michigan, Ann Arbor.

“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).

The present paper works out an interesting example of the theory Nash presented in ‘Non-cooperative games.’ In such a game, when a player executes a particular strategy, there is a well-defined payoff to that player (the amount they win or lose). If there is only one Nash equilibrium, or if the payoffs to each player are the same at all Nash equilibria, the collection of these equilibrium payoffs, one for each player, is called the ‘value’ of the game.

In the present paper, Nash and Shapley consider a simplified version of three-person poker. “The deck contains just two kinds of cards, ‘High’ and ‘Low’, in equal numbers. One card is dealt at random to each of the three players. The deck is so large that the eight possible deals occur with equal probability. Each player antes an amount *a*. The first player has the option of ‘opening’ the bidding with a bet *b*, or of ‘passing’. If he passes, the second player has the same opportunity; then the third. When any player has opened, the other two, in rotation, have the choice of ‘calling’ with a bet *b*, or of ‘folding’ (dropping out), thereby forfeiting the ante money. The payoff rule: If no one opened, (three consecutive passes), the players retrieve their antes. Otherwise the players betting compare cards, and the one with the highest wins the entire accumulation of bets and antes (the ‘pot’). In case of a tie, the winners divide the pot equally … The game turns out to have a well-defined value if the ante does not exceed the amount of the bet, or is more than four times the bet, but no value for at least two transition cases in between” (pp. 105-106).

This volume also contains Brown and von Neumann’s “Solutions of Games by Differential Equations” (pp. 73-79).

Nasar, Introduction to *The Essential John Nash* (Kuhn & Nasar, eds.), 2002.

Large 8vo (253 x 177), pp. xv, 201. Original orange paper wrappers (upper and inner edges of wrappers sunned).

Item #5032

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Price:
$8,500.00
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