# Game Theory!

Game | Theory | 21st August 2021 | Virtual Wire

Game theory studies situations of competition and cooperation between several involved parties by using mathematical methods. This is a broad definition but it is consistent with a large number of applications.

These applications range from strategic questions in warfare to understanding economic competition, from economic or social problems of fair distribution to behaviour of animals in competitive situations, from parlour games to political voting systems—and this list is certainly not exhaustive.

Game theory is an official mathematical discipline (American Mathematical Society Classification code 91A) but it is developed and applied mostly by economists. In economics, articles and books on game theory and applications are found in particular under the Journal of Economic Literature codes C7x.

In terms of applications, game theory is a broad discipline, and it is therefore not surprising that game-theoretic situations can be recognized in the Bible (Brams, 1980) or the Talmud (Aumann and Maschler, 1985). Also, the literature on strategic warfare contains many situations that could have been modelled using game theory: a very early reference, over 2,000 years old, is the work of the Chinese warrior philosopher Sun Tzu (1988).

Early works dealing with economic problems are Cournot (1838) on quantity competition and Bertrand (1883) on price competition. Some of the work of Dodgson—better known as Lewis Carroll, the writer of Alice’s Adventures in Wonderland—is an early application of zero-sum games to the political problem of parliamentary representation, see Dodgson (1884) and Black (1969). One of the first formal works on game theory is Zermelo (1913).

The logician Zermelo proved that in the game of chess either White has a winning strategy (i.e., can always win), or Black has a winning strategy, or each player can always enforce a draw—see Sect. 13.2.5. Up to the present, however, it is still not known which of these three cases the true one is. A milestone in the history of game theory is Von Neumann (1928).

In this article, von Neumann proved the famous minimax theorem for zero-sum games. This work was the basis for the book Theory of Games and Economic Behavior by von Neumann and Morgenstern (1944/1947), by many regarded as the starting point of game theory.

In this book, the authors extended von Neumann’s work on zero-sum games and laid the groundwork for the study of cooperative (coalitional) games. See Dimand and Dimand (1996) for a comprehensive study of the history of game theory up to 1945. The title of the book of von Neumann and Morgenstern reveals the intention of the authors that game theory was to be applied to economics.

Nevertheless, in the 1950s and 1960s, the further development of the Classicalgame theory was mainly the domain of mathematicians. Seminal articles in this period were the papers by Nash1 on Nash equilibrium (Nash, 1951) and on bargaining (Nash, 1950), and Shapley on the Shapley value and the core for games with transferable utility (Shapley, 1953, 1967). See also Bondareva (1962) on the core.

Apart from these articles, the foundations of much that was to follow later were laid in the contributed volumes edited by Kuhn and Tucker (1950, 1953), Dresher et al. (1957), Luce and Tucker (1958), and Dresher et al. (1964). A classical work in game theory is Luce and Raiffa (1957): many examples still used in game theory can be traced back to this source, like the Prisoners’ Dilemma and the Battle of the Sexes.

In the late 1960s and 1970s of the previous century, game theory became accepted as a new formal language for economics in particular. This development was stimulated by the work of Harsanyi (1967/1968) on modelling games with incomplete information and Selten (1965, 1975) on (sub)game perfect Nash equilibrium.

In 1994, Nash, Harsanyi and Selten jointly received the Nobel prize in economics for their work in game theory. Since then, many Nobel prizes in economics have been awarded for achievements in game theory or closely related to game theory: Mirrlees and Vickrey (in 1996), Sen (in 1998), Akerlof, Spence and Stiglitz (in 2001), Aumann and Schelling (in 2005), Hurwicz, Maskin and Myerson (in 2007), and Roth and Shapley (in 2012).

From the 1980s on, large parts of economics have been rewritten and further developed using the ideas, concepts and formal language of game theory. Articles on game theory and applications can be found in many economic journals.

Journals explicitly focusing on game theory include the International Journal of Game Theory, Games and Economic Behavior, and International Game Theory Review. Game theorists are organized within the Game Theory Society, see http://www. gametheorysociety.org/.