Game theory considers the individual's predictive behavior and actual behavior in the game, and studies their optimization strategies. On the surface, different interactions may show similar incentive structures, so they are special cases of the same game. One of the famous and interesting application examples is the prisoner's dilemma.
Competitive or antagonistic behavior becomes game behavior. In this kind of behavior, the parties involved in the struggle or competition have different goals or interests. In order to achieve their respective goals and interests, each party must consider all possible action plans of its opponent and try to choose the most beneficial or reasonable plan for itself. For example, playing chess and cards in daily life. Game theory is a mathematical theory and method to study whether there is the most reasonable behavior scheme for all parties in game behavior and how to find such a reasonable behavior scheme.
Biologists use game theory to understand and predict some results of evolution. For example, the concept of "evolutionary stability strategy" put forward by john maynard smith and George R. Price in the paper published in Nature in 1973 used game theory. You can also refer to evolutionary game theory and behavioral ecology.
Game theory is also applied to other branches of mathematics, such as probability, statistics and linear programming.
The idea of game theory has existed since ancient times, and Sun Tzu's Art of War is not only a military work, but also the earliest monograph on game theory. At first, game theory mainly studied the winning or losing of chess, bridge and gambling. People's grasp of the game situation only stays in experience and has not developed into a theory. It was not until the beginning of the 20th century that it officially developed into a discipline.
The study of game theory began in Zermelo (19 13), Porel (192 1) and von Neumann (1928), and later von Neumann and Oscar. 1944, 1947) systematized and formalized it for the first time (refer to Myerson, 199 1). Subsequently, John Forbes Nash Jr. (1950, 195 1) proved the existence of an equilibrium point by using the fixed point theorem, which laid a solid foundation for the popularization of game theory. In addition, the research of Selton and Hasani also promoted the development of game theory. Today, game theory has developed into a relatively perfect discipline.
"Three Masters" and "Four Gentlemen" in Contemporary Game Theory
The Big Three include John Forbes Nash, John C. Harsanyi and Reinhard Selten. At the same time, these three men won the Swedish Bank Economics Prize (also known as the Nobel Prize in Economics) from 65438 to 0994 for their outstanding contributions to game theory.
The "Four Misters" include Robert J. Auman, ken binmore, David Cripps and ariel rubinstein.
Game elements:
(1) Player: In a game or game, every participant who has the decision-making power becomes a player. A game with only two players is called a "two-player game", and a game with more than two players is called a "multiplayer game".
(2) strategiges: In a game, each player has a practical and complete action plan, that is, the plan is not an action plan at a certain stage, but a plan to guide the whole action, and it is a feasible action plan for the player from beginning to end, which is called the player's strategy in this game. If everyone in a game always has finite strategies, it is called "finite game", otherwise it is called "infinite game".
(3) Revenue: The result at the end of a game is called gain and loss. The gains and losses of each player at the end of a game are not only related to the strategies chosen by the players themselves, but also to a set of policies adopted by the players in the whole situation. Therefore, the "gain and loss" of each participant at the end of a game is a function of a set of policies set by all participants, usually called the payment function.
(4) Order: Every player's decision has priority. If a player wants to make multiple decisions, there will be order problems; Other elements in the same order are different, so the game is different.
(5) The game involves equilibrium: equilibrium is equilibrium, and in economics, equilibrium means that the related quantity is at a stable value. In the relationship between supply and demand, if a commodity market is at a certain price, anyone who wants to buy this commodity at this price can buy it and anyone who wants to sell it can sell it. At this time, we say that the supply and demand of this commodity have reached a balance. The so-called Nash equilibrium is a stable game result.
Nash equilibrium: in a strategy combination, all participants are faced with the situation that his strategy is optimal without others changing his strategy. In other words, if he changes his strategy at this time, his payment will be reduced. At the Nash equilibrium point, every rational participant will not have the impulse to change his strategy alone. The premise of proving the existence of Nash equilibrium point is the concept of "game equilibrium pair" The so-called "balanced couple" means that in a two-person zero-sum game, the authority A adopts its optimal strategy a* and the player B also adopts its optimal strategy b*. If player A still uses b*, but player A uses another strategy A, then player A will not pay more than his original strategy a*. This result is also true for player B.
In this way, "equilibrium pair" is clearly defined as: a pair of strategies a* (belonging to strategy set A) and b* (belonging to strategy set B) are called equilibrium pairs. For any strategy A (belonging to strategy set A) and strategy B (belonging to strategy set B), there is always an even pair (a, b*)≤ even pair (a*, b*)≤.
Non-zero-sum games also have the following definitions: a pair of strategies a* (belonging to strategy set A) and b* (belonging to strategy set B) are called equilibrium pairs of non-zero-sum games. For any strategy A (belonging to strategy set A) and strategy B (belonging to strategy set B), there are always: even pair (a, b*) ≤ even pair (a*, b*) player A; Even pair (a*, b)≤ even pair (a*, b*) of player B in the game.
With the above definition, Nash theorem is immediately obtained:
Any two-person game with finite pure strategy has at least one equilibrium pair. This equilibrium pair is called Nash equilibrium point.
The strict proof of Nash theorem needs fixed point theory, which is the main tool to study economic equilibrium. Generally speaking, finding the existence of equilibrium is equivalent to finding the fixed point of the game.
The concept of Nash equilibrium point provides a very important analysis method, which enables game theory research to find more meaningful results in a game structure.
However, the definition of Nash equilibrium point is limited to any player who doesn't want to change his strategy unilaterally, ignoring the possibility of other players changing their strategy. So many times the conclusion of Nash equilibrium point is unconvincing, and researchers call it "naive and lovely Nash equilibrium point" vividly.
According to certain rules, R Selten eliminated some unreasonable equilibrium points in multiple equilibria, thus forming two refined equilibrium concepts: sub-game complete equilibrium and trembling hand perfect equilibrium.