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by Robert J. Aumann
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Mathematics
  • Author:
    Robert J. Aumann
  • ISBN:
    0262011549
  • ISBN13:
    978-0262011549
  • Genre:
  • Publisher:
    The MIT Press (February 18, 2000)
  • Pages:
    720 pages
  • Subcategory:
    Mathematics
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  • Rating:
    4.3
  • Votes:
    413
  • Formats:
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Robert J. Aumann is Professor of Mathematics and a member of the Center for Rationality and Interactive Decision Theory, The Hebrew University of Jerusalem.

Robert J. Series: MIT Press (Book 1).

Collected Papers, Vol. 1 book. Robert Aumann's groundbreaking career in game theory has spanned.

Robert Aumann's groundbreaking career in game theory has spanned over 35 years. Threaded through all of Aumann's work (symbolized in his thesis on knots) is the study of relationships between different ideas, between different phenomena, and between ideas and phenomena. When you look closely at one scientific idea," writes Aumann, "you find it hitched to all others. It is these hitches that I have tried to study. Aumann, 1995. Aumann & S. Hart (e., 1994.

Robert Aumann's groundbreaking career in game theory has spanned over 35 years

Robert Aumann's groundbreaking career in game theory has spanned over 35 years. Robert Aumann's groundbreaking career in game theory has spanned over 35 years.

MIT Press began publishing journals in 1970 with the first volumes of Linguistic Inquiry and the Journal of Interdisciplinary History. Collected Papers, Volume 1. By Robert J. Aumann. Today we publish over 30 titles in the arts and humanities, social sciences, and science and technology.

Threaded through all of Aumann's work (symbolized in his thesis on knots) is the study of relationships between different ideas, between different phenomena, and between ideas and phenomena. The papers are organized in several categories: general, knot theory, decision theory (utility and subjective probability), strategic games, coalitional games, and mathematical methods.

Explore Further: Topics Discussed in This Paper. Publications citing this paper.

Robert Aumann's groundbreaking career in game theory has spanned over 35 years. These two volumes provide convenient access to all of his major researchfrom his doctoral dissertation in 1956 to papers as recent as January 1995. Threaded through all of Aumann's work (symbolized in his thesis on knots) is the study of relationships between different ideas, between different phenomena, and between ideas and phenomena. "When you look closely at one scientific idea," writes Aumann, "you find it hitched to all others. It is these hitches that I have tried to study."The papers are organized in several categories: general, knot theory, decision theory (utility and subjective probability), strategic games, coalitional games, and mathematical methods. Aumann has written an introduction to each of these groups that briefly describes the content and background of each paper, including the motivation and the research process, and relates it to other work in the collection and to work by others. There is also a citation index that allows readers to trace the considerable body of literature which cites Aumann's own work.

THOMAS
Game theory has been of major importance to some economists, and completely repudiated by others. This book is the first volume of a collection of papers from a person of the former category, and gives an interesting and helpful overview of his work. Game theory could properly be characterized as mathematical economics, since there is usually no attempt to empirically verify its conclusions, but its use does give a perspective on what might be possible when groups of individuals come together and seek to maximize their gain.

The author is clearly aware of the skepticism by some regarding the applicability of game theory to economics. In the first article of the book, entitled `What is Game Theory Trying to Accomplish," the author outlines his apology for game theory, with his main premise being that game theory offers insights, or `solution concepts' into social processes and that its use should be judged by how well it performs in applications. He is definitely (and correctly) against philosophical speculation regarding game theory, realizing that such activities complicate matters rather than offer clarification.

Four of these `solution concepts' are discussed in this article, giving the reader not only a look at the author's viewpoints, but also a historical/conceptual overview of game theory. One of these solution concepts, the famous `Nash equilibrium' is without doubt the most extensively used in applications, such as in evolutionary biology and bandwidth allocations in telecommunications networks, to name a few. Another solution concept, called the `core' is also discussed in some detail. It represents when outcomes can `dominate' other outcomes, i.e. when there is a coalition that can achieve an outcome by its own efforts, with each member preferring that outcome to another. The core of the game is the set of all undominated outcomes, and encapsulates the notion of unrestricted competition that leads to stability. Studies of the core led to the `von Neumann-Morgenstern stable set, also discussed in this article, and which is a collection of outcomes that are not dominated by any element of this collection. The stable sets represent social organization and are globally stable, unlike the core outcomes or Nash equilibria. The author views the N-M stable sets as being "moderately successful" due in part to the difficulties in calculating them. The last solution concept discussed is the `Shapley value', which is viewed by the author as a measure of the power of a particular game or coalition, and one that is the most successful of all concepts in cooperative games. Extensive references are given for readers interested in applications of these solution concepts.

One of the papers in the book, entitled "Irrationality in Game Theory," the author takes a somewhat radical position if compared with the usual assumptions that are made in game theory. The author considers the content of the article is being part of the `current frontier' in game theory research, and it calls into question the rationality assumption. This implies that the players do not necessarily engage in utility maximization. When approaching this paper one is naturally curious as to what can actually be accomplished in game theory when the rationality hypothesis is dropped. The author wants to study what would happen if one considered environments where players may be both rational and irrational. His study involves the use of an `information system', which consists of a strategic form n-person game, a (finite) collection of `information states' assigned to each player, an assignment of a pure strategy to each information state, and for each information state of a particular player an assignment of a probability distribution on the information states of the other players. The information system represents the uncertainties of the players. One speaks of the `surmise' of a player as his (probabilistic) estimate of the information states of the other players, and her `belief' as a surmise on the pure strategies that the other players choose. A player is called `rational' at a given information state if the strategy she chooses maximizes her payoff according to her belief at that state.

In the article, the author studies information systems where some players are irrational at some states. The payoff in this study is to resolve certain paradoxes in backward-induction, such as the repeated prisoner's dilemma. A player who is not rational would be expected to introduce behavior in the game that would not exist if the players were all rational. The usual results in game theory would not be expected to occur. Such an expectation is discussed in this article in the context of `crazy perturbations.' The author briefly reviews the literature, which in general involves finding the Nash equilibria for a repeated game where one of the players has a non-zero probability of playing irrationally. This "crazy" behavior on the part of an irrational player has the effect of changing the behavior of the rational players. It is found that the rational players wind up mimicking the irrational player to a degree that is optimal for them. The rational players act irrational in order to force other players to act in certain ways. These developments are very interesting and further work on irrational players will no doubt have many more surprises. One would like to know for example what the upper bound would be on the number of irrational players before game theory analysis would become useless. In addition, it would be interesting to find out to what degree a rational player has to act irrational in order to maximize her utility.
Malhala
Very good quality book.