The modern game-theoretic concept of Nash Equilibrium is instead defined in terms of mixed strategies, where players choose a probability distribution over possible actions. The concept of the mixed strategy Nash Equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior. However, their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash Equilibrium will exist for any zero-sum game with a finite set of actions. The contribution of John Forbes Nash in his 1951 article Non-Cooperative Games was to define a mixed strategy Nash Equilibrium for any game with a finite set of actions and prove that at least one (mixed strategy) Nash Equilibrium must exist in such a game.
Einj said:Thank you. I'll definitely look at it, but isn't Nash's theory been developed after this book?
Nash's game theory is a mathematical framework used to analyze and understand strategic decision-making in situations where the outcome of one's own choices is affected by the choices of others. It was developed by mathematician John Forbes Nash Jr. in the 1950s and has applications in various fields such as economics, political science, and evolutionary biology.
The key concepts in Nash's game theory include the notion of players, strategies, and payoffs. Players are individuals or entities making decisions in the game, strategies are the possible choices each player can make, and payoffs are the outcomes resulting from the combination of strategies chosen by all players.
Nash's equilibrium is a solution concept in game theory that predicts the outcome of a game based on the assumption that each player is rational and seeks to maximize their own payoff. It is calculated by finding the set of strategies where no player can improve their payoff by unilaterally changing their strategy.
Nash's game theory has been used to explain and predict various phenomena in the real world, such as the behavior of firms in a market, the decision-making of countries in international relations, and the evolution of animal behavior. It has also been applied in fields such as computer science, sociology, and psychology.
While Nash's game theory has been widely applied and studied, it does have its limitations. Some critiques point out that it assumes players are rational and have perfect information, which may not always be the case in real-world situations. Additionally, it may not be applicable in games with a large number of players or in situations where players have varying levels of power or influence.