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, operations 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).
Game theory is basically the study of mathematical models and equations of conflict and cooperation between intelligent and coherent rational decision-makers. Game theory is mainly used in psychology political science, economics, as well as computer, science logic and biology. In the current status, it addressed zero-sum games, in which one player's gains result in losses for the other players. Today, game theory applies to a wide range of behavioral and componential relations, and is now a widely used term for the science of logical decision making in humans, computer and animals. Game theory is the study of cooperation and human conflict within a competitive scenario. In some aspects, game theory is the science of strategy and how to specialize in those strategies, or at least the optimal decision-making of independent and competing actors in a challenging setting. The key pioneers of game theory were mathematicians John Nash and John von Neumann, as well as economist Oskar Morgenstern.
The modern take on the game theory began with the idea concerning the existence of mixed-strategy equilibria in two-person zero-sum games and its proof submitted by John von Neumann. Von Neumann's original proof used the simplified Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a basic method in game theory and also in mathematical economics. After completing his paper, he followed with the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. Game theory creates formal structure and a language of analysis for making logical and reasonable decisions in competitive environments. The term “game” can be misinterpreted. Even though game theory applies to recreational games, the concept of “game” simply means any interactive situation in which independent participants share more-or-less formal rules and regulations. The game theory was developed majorly in the 1950s by many prominent scholars. Game theory was later explicitly applied to biology in the 1970s, but certain same developments go back to as far as 1920s. Game theory is recognized as an important tool in many fields of works and services. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2014, a total of eleven game-theorists have now won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to the field of biology. Game theory has a wide range of applications, including evolutionary biology, psychology, war, economics, politics and business. Despite its many advances, game theory is still a benevolent and developing science.
A game theory brought a revolution and a new age in old school economics as well as modern economics by addressing crucial problems in prior mathematical economic models. For instance, neoclassical economics struggled to understand the entrepreneurial anticipation and couldn't handle imperfect competition. Game theory turned attention away from steady-state equilibrium and toward the market process. In game theory, every decision-maker must anticipate the reaction of those affected by the decision. In business, this means economic agents must anticipate the reactions of rivals, employees, customers and investors.
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Nash Equilibrium is basically a solution concept in the game theory. This solution concept is a non-cooperative game which involves two or more individuals or players. In this, each player is assumed to be aware of the equilibrium strategies of the other individuals or players, and no one has anything particular to gain by changing the desired strategy. The optimal outcome of the game is considered where no player has an incentive or an extra advantage to deviate from his or her chosen strategy after considering the choice of their opponent. On the whole, a player can receive no extra benefit from changing actions, taking in consideration that other players remain constant in their strategies. Sometimes, a game may have none Nash Equilibria at all or multiple. If every player has chosen a strategy and in contrary, no player can take advantage or gain benefit by changing strategies in the game run, while the other individuals keep their unchanged and unaltered, then the current set of strategies and choices related to them constitutes a Nash equilibrium.
The corresponding pay offs also constitutes a Nash Equilibrium. Nash Equilibrium is considered one of the foundational concepts of the game theory. The main expectation from a Nash Equilibrium is to test a game using some experimental economics methods as well as techniques. Read more about Nash Equilibrium
Prisoner’s dilemma is also a part of the game theory and Nash Equilibrium. Prisoner’s dilemma is an economic term which is primarily used in the game theory which shows two completely rational elements might not cooperate, even if it is the best course of action for them. It is basically a paradox in the decision analysis. This includes when two individuals or elements are acting for their personal gain and self-interest, for pursuing a specific course of action, which can or cannot result in the ideal outcome. Due to the purely logical thought process, both the elements find their self-interest in a worse state than if they had worked with each other in the discussion process. Prisoner’s dilemma is a good part of the Nash Equilibrium and the example of the two prisoners is the best to understand it. Read more about Prisoner’s dilemma
Games, as studied by economists and real-world game players, are generally finished in finitely many moves. Highly qualified mathematicians are not so constrained within any boundaries and set theorists in particular study games which last for infinitely many moves and set pieces, with the winner or another payoff, not known until after all those set pieces are completed.
The focus of attention is usually on the best winning strategy one player or participant has, but rather not on the best way to play such a game. (It can be proven, using the axiom of choice, that there are games – even with perfect information and where the only outcomes are "win" or "lose" – for which neither player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.
This game theory is the study of strategic decision making and set pieces in very large populations of small interacting agents or elements. This class of problems was considered in the economics literature by Robert W. Rosenthal and Boyan Jovanovic, in the engineering literature by Peter E. Caines and also by popular mathematician Pierre-Louis Lions and Jean-Michel Lasry.
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