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Game Theory

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Game Theory Books & Ephemera



    The Bargaining Problem," Econometrica, Vol. 18, No. 2, April 1950, pp. 155-162 by Nash, John

    FIRST EDITION IN ORIGINAL WRAPS OF A SEMINAL PAPER IN GAME THEORY - THE MATHEMATICS OF DECISION-MAKING AND STRATEGY -- BY JOHN NASH. "One of the great mathematicians of the 20th century", Nash received the Nobel Prize in Economics and, five days before his death, the Abel prize, the math version of the Nobel Prize (NYT, 5.25.2015). In this paper, "Nash presents a 'bargaining solution' completely unanticipated in the literature. "Breaking with the past, he used a simple yet path-breaking axiomatic approach... aimed to identify the common characteristics of bargaining and its solutions rather than describe the processes involved in bargaining" (Encyclopedia of Economic Sociology). An undergraduate at Carnegie, Nash was accepted for graduate work at Princeton in 1948 based on a "one sentence letter of recommendation from Carnegie that read: 'This man is a genius'" (Kuhn, Nobel Prize Seminar, The Work of John Nash in Game Theory, Dec. 1994, 162). Because Nash's bargaining paper was published during his time at Princeton, it has been assumed that he did the work while there - after all, with Von Neumann and Morgenstern, Princeton was already the capital of game theory's world. This, however, is not true. Early in his undergraduate career at Carnegie, Nash wrote "a paper for an elective course in international economics, possibly the only formal course in economics he has ever had, [and the paper] was done [then] in complete ignorance of the work of von Neumann and Morgenstern. In short, when [Nash] did this work he didn't know that game theory existed" (ibid). With minor revisions, the 1950 'Bargaining Problem' paper is identical to the undergraduate one. Utterly original its thinking, Nash's paper is "a model of theoretical elegance" [that] posits four reasonable axioms (1) Invariant to affine transformations or Invariant to equivalent utility representations (2) Pareto optimality (3) independence of irrelevant alternatives and (4) symmetry. "If these four conditions are satisfied then there is a unique solution, namely, the outcome that maximizes the product of the players' utilities" (ibid, 163). Unaware of game theory, Nash's axiomatic 'bargaining' solutions were a form of game theory even though he didn't know it. His "'Bargaining' was really an as yet unimagined form of game theory... "Unlike the two-person zero-sum game in which the loser loses what the winner wins, a bargaining game offers possible benefits to both sides. In this 'cooperative' game theory, the goal is for all players to do the best they can, but not necessarily at the expense of the others. In a good bargain, both sides gain... [Nash argues] the problem is to find which way maximizes the benefit (or utility) for both sides - given that both players are rational (and know how to quantify their desires), are equally skilled bargainers and are equally knowledgeable about each other's desires. "When bargaining over a possible exchange of resources (Nash's examples were a book, ball, pen, knife, bat, and hat) the two players might assess the values of the objects differently. (To the athletic, a bat might seem more valuable than a book, while the more intellectually oriented bargainer might rank the book more valuable.) Nash showed how to consider such valuations and compute each player's gain in utility for various exchanges, providing a mathematical map for finding the location of the optimal bargain - the one giving the best deal for both" (ibid., 56). At a Nobel Prize seminar on the works of Nash, the Princeton mathematician Harold Kuhn (a close friend of Nash's) argued that even in its brilliance, it is 'clear that this paper was written by a teenager. The evidence is that the objects in the example to be bargained over are a bat, a ball, a toy, and a knife. No amount of urging by his colleagues or Econometrica persuaded John to change this example" (ibid). CONDITION & DETAILS: 4to. 250 x 175mm. Original paper wraps professionally re-backed at the spine. Fine condition.



    Two Person Cooperative Games" WITH "The theory of play and integral equations with skew symmetric kernels," "On games that involve chance and the skill of players," and "On systems of linear skew-symmetric determinant and the genera theory of play" in Econometrica 21, January 1953, pp. 128-141 and pp. 97-118 by Nash, John Forbes with Borel, Emile

    FIRST EDITION IN ORIGINAL PAPER WRAPS of John Nash's fourth and final paper seminal contribution to mathematics and non-cooperative game theory. In his 1950 paper on the subject, "Nash presented a bargaining solution that was completely unanticipated in the literature. Breaking with the past, he used a simple yet path-breaking axiomatic approach and a non-cooperative model of games to derive a bargaining solution between two rational persons. This axiomatic approach aimed to identify the common characteristics of bargaining and its solutions rather than describe the processes involved in bargaining" (Berckert, International Encyclopedia of Economic Sociology, 22). This 1953 paper by Nash set a new entire research agenda that has been referred to as the Nash Program for cooperative games. In the Nash Program, "although players MAY enter into a binding agreement, they need not. If they choose not to, then there is a non-cooperative game in which each player can, adopting the appropriate mixed strategy, be assured of a certain minimum expected payoff; call this outcome the 'disagreement point.' The original cooperative game can thus be conceived as a bargaining problem in which players seek to improve their situation by moving away from the disagreement point to a new, more desirable point conferring greater utility. Exactly which point is selected depends upon the particular arbitration scheme used. An arbitration scheme can be thought of as a function mapping the set of possible outcomes to a single outcome: the solution offered by the arbitrator. A cooperative game, then, can be conceived as an extensive form of a non-cooperative game where the early stages of the game involve the selection of the disagreement point and the arbitration scheme. This approach, of reducing cooperative games to non-cooperative games, is known as the "Nash Program" (The Philosophy of Science: An Encyclopedia, Volume 1, 328). In 1994 Nash shared the Nobel Prize in Economics with Reinhard Selten and John Harsanyi "for their pioneering analysis of equilibria in the theory of non-cooperative games" (Nobel Prize Committee). NOTE: This issue also contains English translations of Emile Borel's three papers on game theory. CONDITION & DETAILS: 4to. 10 X 7 inches (250 x 175mm). Single issue in original wraps in fine condition.

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