

Game theory studies interactions among strategizing players, where the players can be people, animals, or genes, or more aggregate entities such as businesses, political parties, or nations. The first general method of analysis of games was developed by John von Neumann, who worked on game theory from the mid1920s to the mid1940s. Von Neumann developed his famous maximin method of analysis for twoperson zerosum games, and extended this method to analyze nperson variablesum (non zerosum) games. In 1951, John Nash published a very different theory of nperson variablesum games. Nash supposed that every game yielded a unique 'rational' prediction, and, on this basis, asserted that players would be led to play in accordance with what he called an equilibrium point. Over time, Nash equilibrium became the dominant form of analysis in game theory. This has been especially true in applications of game theory to economics, which has been a major user of gametheoretic methods.
A new book, published by World Scientific, describes a third method of analysis of games  one that began in the 1980s and has now reached a point of maturity. This third approach starts with the idea that a game is played under conditions of uncertainty, where each player is uncertain about what strategies the other players adopt. Players think about what strategies other players choose, but they may be correct or incorrect in what they think. The viewpoint is quite different from the Nash theory, in which players deduce their way to a unique correct answer in a game.
The book, titled The Language of Game Theory: Putting Epistemics into the Mathematics of Games, written by Adam Brandenburger (New York University, USA), contains an introduction to the epistemic (meaning "pertaining to knowledge and belief") approach to game theory. This is followed by eight papers written by Brandenburger and his coauthors over the past 25 years that develop the approach. A key feature of the approach is the construction of chains of "I think you think..." reasoning in games. Players engage in firstorder reasoning (they think about what strategies other players choose), secondorder reasoning (they think about what other players think), and so on to higher orders. The book describes the mathematical construction of such chains of reasoning and the implications of this view of games for how games are played. As befits a new theory, epistemic game theory contains the classical Nash theory as a special case, and the book explains this relationship.
Game theory built in epistemic fashion connects with several other fields. There are exciting developments in experimental game theory that test how many levels of reasoning actual players employ. Neuroscience is able to point to areas of the brain employed in such reasoning. Connections among epistemic game theory, experimental game theory, cognitive science, and neuroscience can be expected to grow rapidly. On the mathematical level, there are close connections among epistemic game theory, computer science, and logic.
More information on the book and the purchase link, can be found at: http://www.worldscientific.com/worldscibooks/10.1142/8844. The 300page book retails for US$98 / £65. Its ebook edition retails for US$127 / £85.
The book is Volume 5 of the World Scientific Series in Economic Theory, edited by Nobel Laureate Eric S. Maskin (Harvard University, USA). More information on the series and also on other forthcoming titles, can be found at http://www.worldscientific.com/series/wsset.
About the Author
Adam Brandenburger received B.A., M.Phil., and Ph.D. degrees from the University of Cambridge. Following his Ph.D., he was on the faculty at Harvard for fifteen years before moving to New York University in 2002. He was one of the early workers in developing what has come to be called "epistemic game theory"  an approach that studies the effect of how players in a game reason about one another (including their reasoning about other players' reasoning) on how the game is played. He also developed (with his colleague Harborne Stuart) the concept of a "biform game"  a model of strategy as moves that affect the amount of value created, and its division, among the players in a game. Recently, he has been working on the implications for game theory of giving players in a game access to quantum rather than classical information resources. More details on his research can be found at http://adambrandenburger.com.
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