Game theory Concept Game theory, also known as game theory, is a theory and method to study the phenomenon of struggle or competition. It is not only a new branch of modern mathematics, but also an important subject of operational research. The development of game theory The idea of game theory has existed since ancient times. The Art of War in ancient China 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. 1928 von Neumann proved the basic principles of game theory, thus announcing the formal birth of game theory. 1944, the epoch-making masterpiece Game Theory and Economic Behavior written by von Neumann and Morgenstein extended the two-person game to the n-person game structure, and applied the game theory system to the economic field, thus laying the foundation and theoretical system of this discipline. When it comes to game theory, we can't ignore Nash, a genius of game theory, and Nash's groundbreaking papers, Equilibrium Point of N-player Game (1950) and Non-cooperative Game (195 1). The concept of Nash equilibrium and the existence theorem of equilibrium are given. 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. The basic concept of game theory-game elements (1) Players: 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) Strategy: In a game, each player has a feasible 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. A player's feasible action plan from beginning to end 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) Gain and loss: 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) For the participants in the game, there is a game result. (5) The game involves equilibrium: equilibrium is equilibrium. In economics, equilibrium means that the correlation 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, the "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 immediately obtains that 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. Game type (1) cooperative game-study how to distribute the benefits of cooperation when people reach cooperation, that is, income distribution. (2) Non-cooperative game-study how people make decisions to maximize their own interests under the condition of mutual influence of interests, that is, strategic choice. (3) Game between complete information and incomplete information: players have a full understanding of all participants' strategic space and the payment under the strategy combination, which is called complete information; On the contrary, it is called incomplete information. (4) Static game and dynamic game Static game: refers to that the participants take actions at the same time, or although there is a sequence, the latter actor does not know the strategy of the former actor. Dynamic game: refers to the action sequence of both parties, and the latter actor can know the strategy of the former actor. Property distribution and shapley value are considered as a cooperative game: Party A, Party B and Party C vote to decide how to distribute10 million yuan, and they have 50%, 40% and 10% power respectively. According to the rules, a plan can only be passed when more than 50% of the votes are in favor. So how to allocate it is reasonable? According to the distribution of votes, 500,000, B400,000, C65438+100,000 C proposed to A: 700,000, b0, C30,000 B proposed to A: 800,000, B200,000, c0…… ................. Power index: The power of each decision-maker in decision-making is reflected in the number of "key participants" in his winning alliance, "key" Shapley value: the sum of participants' marginal contributions to the alliance divided by various possible alliance combinations under various possible alliance orders. A c a c a b, the key participant of abc acb bac bca cab cba, calculated the Shapley values of A, B and C as 4/6, 1/6 and 1/6 respectively, so A, B and C should get 2/3 of 1 10,000 respectively. Like many other disciplines that use mathematical tools to study social and economic phenomena, the research method of meaning game theory is to abstract the basic elements from complex phenomena, analyze the mathematical model formed by these elements, and then gradually introduce other factors that affect their situation, so as to analyze the results. Based on different levels of abstraction, three game expressions are formed, which can be used to study various problems. Therefore, it is called "Mathematics of Social Science". In theory, game theory is a formal theory to study the interaction between rational actors, but in fact, it is going deep into economics, politics, sociology and so on, and is applied by various social sciences. Game theory refers to the process in which individuals or organizations, under certain environmental conditions and certain rules, choose and implement their chosen behaviors or strategies by relying on the information they have, and obtain corresponding results or benefits from them. Game theory is a very important theoretical concept in economics. What is game theory? As the old saying goes, things are like chess. Everyone in life is like a chess player, and every movement is like putting a coin on an invisible chessboard. Smart and cautious chess players try to figure out and contain each other, and everyone strives to win, playing many wonderful and changeable chess games. Game theory is to study the rational and logical part of chess player's "playing chess" and systematize it into a science. In other words, it is to study how individuals get the most reasonable strategies in complex interactions. In fact, game theory comes from ancient games or chess games. Mathematicians abstract concrete problems and study their laws and changes by establishing a self-complete logical framework and system. This is not an easy task. Take the simplest two-person game as an example. If you think about it, you will know that there is a great mystery. If it is assumed that both sides accurately remember every move of themselves and their opponents, and they are the most "rational" players, then A has to carefully consider B's idea in order to win the game when playing, and B has to consider A's idea when playing, then A has to think that B is considering his idea, and B certainly knows that A has already considered it. How does game theory begin to analyze and solve problems, and how to find the optimal solution of abstract mathematical problems as realistic induction, thus providing the possibility for guiding practice in theory? Modern game theory was founded by Hungarian mathematician von Neumann in the 1920s. His magnum opus Game Theory and Economic Behavior published in 1944 in cooperation with economist Oscar Morgenstein marked the initial formation of modern system game theory. For non-cooperative and purely competitive games, Neumann only solves two-person zero-sum games-just like two people playing chess or table tennis, one person wins one game and the other loses the other, and the net profit is zero. The abstract game problem here is whether and how to find a theoretical "solution" or "balance", that is, the most "reasonable" and optimal specific strategy for both players, given the set of participants (both sides), the set of strategies (all moves) and the set of profits (winners and losers). What is "reasonable"? Applying the "min-max" criterion in traditional determinism, that is, each side of the game assumes that the fundamental purpose of all the advantages and disadvantages of the other side is to make itself suffer the most, and accordingly optimize its own countermeasures, Neumann mathematically proves that every two-person zero-sum game can find a "min-max solution" through certain linear operations. Through a certain linear operation, two competitors randomly use each step in a set of optimal strategies in the form of probability distribution, so as to finally achieve the maximum and equal profit for each other. Of course, the implication is that this optimal strategy does not depend on the opponent's operation in the game. Generally speaking, the basic "rational" thought embodied in this famous minimax theorem is "hope for the best and prepare for the worst".