#### Transcript A Brief History of Game Theory

A Brief History of Game Theory From various sources The first known solution to such a twoperson zero-sum game is contained in a letter dated 13 November 1713 from James Waldegrave to Pierre-Remond de Montmort concerning a two player version of the card game le Her. De Montmort wrote to the mathematician Nicolas Bernoulli (perhaps best known, at least among economists, for his formulation of the St. Petersburg Paradox). The great French mathematician and probabilist, Emile Borel published four notes on strategic games between 1921 and 1927. Borel gave the first formal definition of what we shall call a mixed strategy and demonstrated the existence of the minimax solution to two player zero-sum games with either three or five possible strategies. He initially conjectured that games with more strategies would not have such a solution but, being unable to find a counter example, he later considered that to be an open question. That question was answered by the Hungarian (and later American) mathematician John von Neumann in 1928. Wikipedia describes von Neumann as having “made major contributions to a vast range of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science, numerical analysis, hydrodynamics (of explosions), and statistics, as well as many other mathematical fields.” In 1913 the German mathematician Ernst Zermelo, most famous for his axiomatisation of set theory, gave a statement concerning a result about what we would now call extensive form games. In a game like chess either white can guarantee that he wins or black can guarantee that he wins or both players can guarantee at least a draw. There were a number of substantial advances in the book by von Neumann and Morgenstern: the axiomatic development of the theory of expected utility; the formal definition of normal form games and extensive form games; the elaboration of the minmax theorem for twoperson zero sum games; and the definition of what are now called cooperative or coalitional games. In the early 1950s John Nash [1950,1951] proposed a definition of equilibrium, that we now call the Nash equilibrium, that has become the central solution concept for noncooperative game theory. Nash as an older man, after he won the Nobel Prize in Economics. That’s not Nash. That’s Russel Crowe in one of his best performances in a film about Nash’s life. 1994 Nobel Prize in Economics was award to John Nash, John C. Harsanyi and Reinhard Selten "for their pioneering analysis of equilibria in the theory of non-cooperative games". In two papers, Extensive Games (1950) and Extensive Games and the Problem of Information (1953), H. W. Kuhn included the formulation of extensive form games which is currently used, and also some basic theorems pertaining to this class of games. Lloyd Shapley in his paper A Value for NPerson Games characterised, by a set of axioms, a solution concept that associates with each coalitional game,v, a unique outcome, v. This solution in now known as the Shapley Value. In the same year, 1953, Shapley published a paper on “Stochastic Games,” introducing a new and important model of games. In 2012, Shapley won the Nobel Prize in Economics with Al Roth "for the theory of stable allocations and the practice of market design". Robert J. Aumann's greatest contribution was in the realm of repeated games, which are situations in which players encounter the same situation over and over again. Aumann was the first to define the concept of correlated equilibrium in game theory, which is a type of equilibrium in non-cooperative games that is more flexible than the classical Nash equilibrium. Furthermore, Aumann has introduced the first purely formal account of the notion of common knowledge in game theory. He collaborated with Lloyd Shapley on the Aumann-Shapley value. He is also known for his agreement theorem, in which he argues that under his given conditions, two Bayesian rationalists with common prior beliefs cannot agree to disagree. In 2005 Aumann won the Nobel Prize in Economics with Thomas C. Schelling "for having enhanced our understanding of conflict and cooperation through game-theory analysis".