1. The concept of game theory
Game theory, also known as Game Theory, is not only a new branch of modern mathematics, but also an important subject of operations research. Game theory mainly studies the interactions between formulated incentive structures. It is a mathematical theory and method for studying phenomena with the nature of struggle or competition. Game theory considers the predicted and actual behaviors of individuals in the game, and studies their optimization strategies. Game theory has become one of the standard analysis tools of economics.
2. The development process of game theory
Game theory is that two people use each other's strategy to change their confrontation strategy in an equal game to achieve the goal of winning. The idea of game theory has existed since ancient times. Ancient Chinese works such as "The Art of War by Sun Tzu" are not only a military work, but also the earliest work on game theory. Game theory initially mainly studied the problems of winning and losing in chess, bridge, and gambling. People's grasp of the game situation only stayed in experience and did not develop to theorization.
Game theory considers the predicted and actual behaviors of individuals in the game, and studies their optimization strategies.
Modern research on game theory began with Zermelo, Borel and von Neumann.
In 1928, von Neumann proved the basic principles of game theory, thus proclaiming the official birth of game theory. In 1944, the epoch-making masterpiece "Game Theory and Economic Behavior" co-authored by von Neumann and Morgenstein extended the two-person game to the n-person game structure and systematically applied game theory to the economic field, thus laying the foundation for the discipline Basic and theoretical system.
From 1950 to 1951, John Forbes Nash Jr used the fixed point theorem to prove the existence of equilibrium points, laying a solid foundation for the generalization of game theory. Nash's seminal papers "The Equilibrium Point of the N-person Game" (1950), "Non-cooperative Game" (1951), etc., gave the concept of Nash equilibrium and the theorem of equilibrium existence. In addition, the research of Reinhard Zelten and John Hessani also played a role in promoting the development of game theory. Today game theory has developed into a more complete discipline.
Three, the elements of game theory
(1) Player in the game: In a competition or game, each participant with decision-making power becomes a player in the game. 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 in the game has to choose a practical and complete action plan, that is, the plan is not an action plan at a certain stage, but a plan that guides the entire action, one for each player A feasible plan of action from the beginning to the end of the overall plan is called a strategy of the player in this bureau. If everyone has a finite number of strategies in a game, it is called a "limited game", otherwise it is called an "infinite game".
(3) Gains and losses: The result of a game ending is called gains and losses. The gains and losses of each player at the end of a game are not only related to the strategy chosen by the player himself, but also related to a set of strategies adopted by the player in the overall situation. Therefore, at the end of a game, the "wins and losses" of each player in the game is a function of a set of strategies adopted by all players, usually called the payoff function.
(4) For game participants, there is a game result.
(5) The game involves equilibrium: equilibrium means equilibrium. In economics, equilibrium means that the relevant quantity is at a stable value. In the supply and demand relationship, if a commodity market is at a certain price, everyone who wants to buy the commodity can buy it at this price, and everyone who wants to sell it can sell it. At this time, we say that the supply and demand of the commodity has reached balanced. The so-called Nash equilibrium is a stable game result.
Fourth, the purpose of game theory
The solution of game strategy is an important content of game problems. Another important content is the design of game rules: In
other words, assuming that the players in the game are sufficiently rational, how to design a game rule to ensure fairness or achieve the design Best interests of the person. The main difficulty is: the rules are complex and the calculation is large.
Mainly used in:
- Auction bidding: Internet advertising, license plate bidding
- Matching supply and demand: pollution rights, school admissions
- Fair elections: election system, voting system, distribution of seats
5. Stable matchings theory
The theory of stable distribution is a theory founded by Shapley, the 2012 Nobel Prize winner, using the method of cooperative games to study and compare different matching methods. The difficulty of this theory is to ensure that a pair is stable.
The core idea of stable matching is to achieve a stable state. In this state, there are no longer such two market entities at the end of the match. Both of them prefer others to their current counterparts. In reality, the familiar examples of 8-minute blind date, school and student matching, etc. are developed based on the idea of stable market matching theory. In which the bilateral model and delay acceptance algorithm is stable matching two important cornerstone theory.
Bilateral matching model The main function of many markets and social systems is to match the subject with another subject: for example, between students and schools, employees and companies, and between married men and women. This market is divided into match -sided matching market (Single-Sided MarketMatch) and bilateral market matching (Two-Sided MarketMatch) .
" Unilateral market matching " means that there is only one set in the market, and the individuals in the set match each other according to their preferences. However, the "roommate" phenomenon in unilateral market matching can lead to instability in matching. When suppose there are four "roommates" {1, 2, 3, 4}, among which 1 prefers 2, 2 prefers 3, 3 prefers 1, and they rank 4 as the least preferred. In this case, any pairwise grouping cannot be stable, because the person who is with 4 points will end the current match and match the already matched person again, and this new match will succeed, making the market unable to To achieve stability (Gale&Shapley, 1962).
The " bilateral matching model " was first proposed by Gale and Shapley (1962) from the study of student application to school model and marriage stability. The so-called "two-sided market" refers to the existence of a market in which there are two types of individual sets. Individuals in the first set can only match individuals in the second set. They proved that in such a two-sided market, as long as the individual's preferences are complete and transmittable, and the market is free enough to allow individuals to perform any potential matching, there will always be a stable match in the market. Similarly, taking 4 roommates as an example, suppose any 2 people sleep on the upper bunk and 2 people sleep on the lower bunk. Now it is required that only people sleeping in different bunks match each other. At this time, a bilateral market matching model is formed. At the same time, Gale and Sha-pley pointed out that the following two conditions are met when market matching is stable: (1) There are no two individuals from different categories in the market that can match each other in preference, but there is no match; (2) Already Individuals who are successfully paired will not try to end the current pairing, and will try to match individuals from another type that have been successfully matched.
Bilateral matching model has the characteristic of stable matching, which makes it receive extensive attention in theory and practice. One of the important applications is labor market matching. Shapley and Shubik (1972) used mathematical models to abstract a two-sided market full of indivisible commodities. Every participant in the market is both a demander and a supplier of commodities. They found that the nature of matching stability in this more generalized market remains robust.
Roth first studied the application of bilateral matching models in solving practical problems. He realized that Shapley's theories and calculations on stable market matching could make the operation of the market clearer. In the 1950s, the organization of the primary labor market for physicians in the United States was able to ensure the success of most individual matching, but this matching lacked stability. The follow-up experimental research of Roth (1984) applied Shapley's matching design to the primary labor market of physicians. His research results show that this matching method can reduce the instability of matching and other existing disorder in the original organization. problem.
GS algorithm (Gale-Shapley)
There are different algorithms in rule design, such as the GS algorithm:
In life, people usually encounter decision-making problems related to resource matching (such as job hunting, application for admission, etc.). These situations that require two-way selection are called It is a bilateral matching problem. In the bilateral matching problem, the two parties need to meet each other's needs to reach a match.
In 1962, American mathematician David Gale and game theorist Shapley proposed a solution algorithm for the bilateral stable matching problem and applied it to the solution of the stable marriage problem.
The stable marriage problem (stable marriage problem) refers to the search for a stable match in two groups of members given the preference of the members. Since this kind of matching is not simply obtained by the higher land price, the matching solution should consider the wishes of both parties.
The stable solution to the problem of stable marriage means that two people who have not reached a match are more inclined to choose each other than their current match.
Top-Trading Cycle algorithm
Among the matching problems, there is also a kind of matching problem of the indivisible subject matter of exchange, which is called unilateral matching problem, such as bartering of objects in ancient times, or the allocation of beds in dormitories.
In 1974, Shapley and Sifu proposed a stable matching algorithm for the unilateral matching problem: the maximum trading circle algorithm (TTC). The algorithm process is as follows:
first, each trader connects an edge pointing to his favorite subject, and From each subject matter to its occupant or trader with the highest priority.
At this time, a directed graph is formed, and there is a trading circle. For traders in the trading circle, each person points to the subject represented by the node. At the same time, the trader gives up the original possession of the subject, the occupant and the match. The successful target leaves the matching market and
then repeats the trading circle matching between the remaining traders and the target, until the trading circle cannot be formed, and the algorithm stops.
Roommate matching problem
6. Types of game theory
The classification of the game also has different classifications according to different benchmarks.
It is generally believed that games can be divided into cooperative games and non-cooperative games. The difference between a cooperative game and a non-cooperative game is whether there is a binding agreement between the interacting parties. If there is, it is a cooperative game; if not, it is a non-cooperative game.
From the time sequence of behavior, game theory is further divided into static games and dynamic games: static games are in which players choose at the same time or not at the same time, but later actors do not know what the first actor has taken Specific actions; dynamic game means that in the game, the actions of the participants have a sequence, and the later actors can observe the actions chosen by the first actor. Popular understanding: "Prisoner's Dilemma" refers to simultaneous decision-making, which belongs to a static game; while chess and card games and other decision-making or actions are sequential, which belong to a dynamic game.
According to the participants' understanding of other participants, it can be divided into complete information game and incomplete information game. A complete game means that in the game process, each participant has accurate information about the characteristics, strategy space and profit function of other participants. Incomplete information game refers to if the participants do not know the characteristics, strategy space and profit function information of other participants accurately enough, or do not have accurate information about the characteristics, strategy space and profit function of all participants, in this case The game played under the circumstances is an incomplete information game.
The game theory that economists talk about generally refers to non-cooperative games. Since cooperative game theory is more complicated than non-cooperative game theory, its theoretical maturity is far less than non-cooperative game theory. Non-cooperative games are further divided into: static games with complete information, dynamic games with complete information, static games with incomplete information, and dynamic games with incomplete information. The equilibrium concepts corresponding to the above four games are: Nash equilibrium, subgame perfect Nash equilibrium, Bayesian Nash equilibrium, and perfect Bayesian Nash equilibrium. Bayesian Nash equilibrium).
There are many classifications of game theory. For example, the number or duration of the game can be divided into limited game and infinite game; in the form of expression, it can also be divided into general type (strategic type) or expansion type; the logic basis of the game is different and It can be divided into traditional game and evolutionary game.
Here are some game models that we often mention, which can be used as introductory interest mentors——
Wisdom pig game-take a good ride and take advantage of your strength
Boxed pigs game, a famous example of Nash equilibrium
The Gunners Game——Comparative relationship and strategy determine strength
Prisoner's Dilemma-Individual Reason and Collective Irrationality
Prisoner's Dilemma, is a representative example of non-zero sum games in game theory, reflecting that the best choice of individuals is not the best choice of groups
Cockfighting game-the brave who meets on a narrow road may not win
Chicken Game, also known as Grass Chicken Game, Coward Game, Coward Game
Cake Sharing Game-Bargaining Strategy
Tooth for tooth-there is a kind of wisdom called forgiveness
Tit for tat, is a very effective strategy used in game theory's Reiterated Prisoner's Dilemma
Eagle and Pigeon Game——A New Solution to the Law of Path Dependence
Hawk Dove game-path dependence
in evolution The two pure strategy equilibriums of this model are similar to the cowardly game, while the mixed strategy equilibrium derives the concept of evolutionary stable strategy. In addition, there is a version of Bayesian game with incomplete information.
Centipede game-reasoning from back to front
Centipede game
Deer hunting game-cooperation is the last word
Stag Hunt Game, also known as Stag Hunt Model, Pareto efficiency of hunters
Bar game-the wisdom of seeking common ground while reserving differences
Bar Problem
Catfish effect-there is competition to develop
Catfish Effect
Repeated games-conflict and cooperation can only be shared
Repeated Games means that the game of the same structure is repeated many times, and each game is called "stage games". Repeated game is an important content in dynamic game. It can be a repeated game with complete information or a repeated game with incomplete information.
Repeated game refers to a special type of extended form game (extensive form game). This type of game includes a base game-called a stage game; in the entire repeated game, the stage game will be repeated a certain number of times. The stage game is generally a familiar game (such as the prisoner's dilemma). Similarly, non-repetitive games can also be called single stage games or single shot games.
Concord fallacy-the wrong of wanting to stop adds to the wrong
Coordination Problem, that is, something that has been invested in a certain cost and progressed to a certain level and then found that it is not suitable to continue, but suffers from various reasons and makes mistakes and cannot stop.
Information screening-wine is not afraid of deep alleys
The hostage dilemma-the prisoner's dilemma worse
Dirty face games-all caused by common knowledge
Cost game-a strategy to get rid of the fetters of sunk costs
The law of watches-different standards make different conclusions
Watch Law
Balanced strategy-no one is to blame
strategy equilibrium
Moiety selected from paper
Author: deep learning and advanced intelligent decision
link: https: //juejin.im/post/5e33cf9f5188252c5232b039
Source: Nuggets
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