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GAME THEORY
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INTRODUCTION TO GAME THEORY
Definition of Game Theory
The study of mathematical models of conflict and cooperation between intelligent rational decision-makers.
Game theory is a method of studying strategic decision making. More formally, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." An alternative term suggested "as a more descriptive name for the discipline" is interactive decision theory. Game theory is mainly used in economics, political science, and psychology, as well as logic and biology. The subject first addressed zero-sum games, such that one person's gains exactly equal net losses of the other participant(s). Today, however, game theory applies to a wide range of class relations, and has developed into an umbrella term for the logical side of science, to include both human and non-humans, like computers. Classic uses include a sense of balance in numerous games, where each person has found or developed a tactic that cannot successfully better his results, given the other approach.
Modern game theory began with the idea regarding the existence of mixed-strategy equilibrium 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 Behaviour, with Oskar Morgenstern, which considered cooperative games of several players. 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 Memorial Prize in Economic Sciences, and John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.
History
The Danish mathematician Zeuthen proved that a mathematical model has a winning strategy by using Brouwer's fixed point theorem. In his 1938 bookApplications aux Jeux de Hasard and earlier notes, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix was symmetric. Borel conjectured that non-existence of a mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false.
Game theory did not really exist as a unique field until John von Neumann published a paper in 1928. His paper was followed by his 1944 book Theory of Games and Economic Behavior, with Oskar Morgenstern, which considered cooperative games of several players. Von Neumann's work in game theory culminated in the 1944 book Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. During this time period, work on game theory was primarily focused on cooperative game theory, which analyses optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.
In 1950, the first discussion of the prisoner's dilemma appeared, and an experiment was undertaken on this game at the RAND Corporation. Around this same time, John Nash developed a criterion for mutual consistency of players' strategies, known as Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. This equilibrium is sufficiently general to allow for the analysis of non-cooperative games in addition to cooperative ones.
In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge were introduced and analysed.
In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing an equilibrium coarsening, correlated equilibrium, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.
In 2007, Leonid Hurwicz, together with Eric Maskin and Roger Myerson, was awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory." Myerson's contributions include the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict (Myerson 1997). Hurwicz introduced and formalized the concept of incentive compatibility.
BASICS OF GAME THEORY
Game theory is the process of modelling the strategic interaction between two or more players in a situation containing set rules and outcomes. While used in a number of disciplines, game theory is most notably used as a tool within the study of economics. The economic application of game theory can be a valuable tool to aide in the fundamental analysis of industries, sectors and any strategic interaction between two or more firms. Here, we'll take an introductory look at game theory and the terms involved, and introduce you to a simple method of solving games, called backwards induction.
Game
A conflict in interest among n individuals or groups (players). There exists a set of rules that define the terms of exchange of information and pieces, the conditions under which the game begins, and the possible legal exchanges in particular conditions. The entirety of the game is defined by all the moves to that point, leading to an outcome.
Move
The way in which the game progresses between states through exchange of pieces and information. Moves are defined by the rules of the game and can be made in either alternating fashion ,occur simultaneously for all players. Moves may be choice or by chance. For example , choosing a card from a deck or rolling a die is a chance move with known probabilities. On the other hand ,asking for cards in blackjack is a choice move.