Positive Quiddity: Game Theory

This article is about the mathematical study of optimizing agents. For the discipline of studying games, see Game studies.

In mathematics, game theory models strategic situations, or games, in which an individual's success in making choices depends on the choices of others (Myerson, 1991). It is used in the social sciences (most notably in economics, management, operaions research, political science, and social psychology) as well as in other formal sciences (logic, computer science, and statistics) and biology (particularly evolutionary biology and ecology). While initially developed to analyze competitions in which one individual does better at another's expense (zero sum games), it has been expanded to treat a wide class of interactions, which are classified according to several criteria. Today, "game theory is a sort of umbrella or 'unified field' theory for the rational side of social science, where 'social' is interpreted broadly, to include human as well as non-human players (computers, animals, plants)." (Aumann 1987).

Traditional applications of game theory define and study equilibria in these games. In an equilibrium, each player of the game has adopted a strategy that cannot improve his outcome, given the others' strategy. Many equilibrium concepts have been developed (most famously the Nash equilibrium) to describe aspects of strategic equilibria. These equilibrium concepts are motivated differently depending on the area of application, although they often overlap or coincide. This methodology has received criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models in the social sciences.

Mathematical game theory had beginnings with some publications by Emile Borel, which led to his 1938 book Applications aux Jeux de Hasard. However, Borel's results were limited, and his conjecture about the non-existence of a mixed-strategy equilibria in two-person zero-sum games was wrong. The modern epoch of game theory began with the statement of the theorem on the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann*. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior, with Oskar Morgenstern, which considered cooperative games of several player. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.

This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. Eight game-theorists have won the Nobel Prize in Economic Sciences, and John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.

     --  http://en.wikipedia.org/wiki/Game_theory . This link goes on to discuss detail of game theory in sections on history, representation of games (including extensive form, normal form, characteristic function form, and partial function form). general and applied uses are also discussed (including description and modelling, prescriptive or normative analysis, economics and business, political science, biology, computer science and logic, and philosophy. The link further describes types of games, including cooperative or non-cooperative, symmetric or asymmetric, zero-sum and non-zero-sum, simultaneous and sequential, perfect information and imperfect information, combinatorial games, infinitely long games, discrete and continuous games, many-player and population games, stochastic outcomes, and metagames.

The blog author is particularly interested in the intimate relevance that quiddity has with games of perfect information and imperfect information. Here is part of what the Wikipedia article says about these two kinds of defining information:

“An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players. Thus, only sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others. Most games studied in game theory are imperfect-information games, although there are some interesting examples of perfect-information games, including the ultimatum game and centipede game. Recreational games of perfect information games include chess, go, and mancala. Many card games are games of imperfect information, for instance poker or contract bridge.

“Perfect information is often confused with complete information, which is a similar concept. Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions...”

The blog author regards chess, poker and contract bridge as peculiarly analogous to real life decision making.

*John von Neumann was accidentally told about the existence of computers at the Princeton, New Jersey train depot during World War II. He used computers and their speed to calculate the “explosive lens” for the atomic bomb. Von Neumann also, in 1947, invented the “core memory” architecture for computers, a design still in nearly universal use today. Some scientists insist on calling computers “von Neumann machines.”