Game Theory Summary

game theory definition

The definition of game theory in the "Game Bible": We call the academic theory that animals use the moving souls of nature to form a three-dimensional equilibrium in the space expected by decision-makers, which is called game theory. [1]
The basic concepts include players, actions, information, strategies, benefits, equilibrium and results, etc. Among them, players, strategies and profits are the most basic elements. The players, actions, and outcomes are collectively called the rules of the game.

development process

Game theory[2] is that two people use each other's strategies to change their own confrontation strategies 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" are not only military works, but also the earliest game theory works. Game theory initially mainly studied the issues of winning and losing in chess, bridge, and gambling. People's grasp of the game situation only remained based on experience and did not develop into theory.
Game theory considers the predicted and actual behavior of individuals in a 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 announcing the official birth of game theory. In 1944, the epoch-making masterpiece "Game Theory and Economic Behavior" co-authored by von Neumann and Morgenstern extended the two-person game to an n-person game structure and systematically applied game theory to the economic field, thus laying the foundation for this discipline. foundation 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 "Equilibrium Points of n-Player Games" (1950), "Non-Cooperative Games" (1951), etc., gave the concept of Nash equilibrium and the existence theorem of equilibrium. In addition, the research of Reinhard Selten and John Harsanyi also played a role in promoting the development of game theory. Today, game theory has developed into a relatively complete discipline.

element

1. Player: In a competition or game, every participant with decision-making power becomes a player. The game phenomenon with only two players is called "two-player game", while the game with more than two players is called "multiplayer game".
　　2. Strategy: In a game, each player has a complete action plan that is actually feasible. That is, the plan is not an action plan for a certain stage, but a plan to guide the entire action. Each player has a feasible plan. An action plan planned from beginning to end is called a strategy of the person in the game. If everyone in a game has a limited number of strategies, it is called a "finite game", otherwise it is called an "infinite game".
　　3. Gains and losses: The results at the end of a game are 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 the set of strategies adopted by the players in the overall game. Therefore, the "gain and loss" of each player at the end of a game is a function of a set of strategies adopted by all players, which is usually called the payoff function.
　　4. For game participants, there is a game result.
　　5. The game involves equilibrium: Equilibrium means balance. In economics, equilibrium means that the relevant quantity is at a stable value. In the relationship between supply and demand, if in the market of a certain commodity at a certain price, everyone who wants to buy the commodity at this price can buy it, and everyone who wants to sell it can sell it, then we say that the supply and demand of the commodity has reached balanced. The so-called Nash equilibrium is a stable game result.
Assumptions of game theory research:

1. The decision-making subject is rational and maximizes his own interests;
2. Perfect rationality is common knowledge;
3. Each participant is assumed to have formed correct beliefs and expectations about the environment and the behavior of other participants. [1

Game type

The classification of games also has different classifications based on different benchmarks.
It is generally believed that games can be mainly divided into cooperative games and non-cooperative games. The difference between cooperative games and non-cooperative games lies in whether there is a binding agreement between the parties interacting with each other. 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 two categories: static games and dynamic games: Static games refer to the game in which the participants make choices at the same time or although they do not choose at the same time, the later actor does not know what the first actor has done. Specific actions; dynamic game means that in the game, the actions of the participants are sequential, and the later actors can observe the actions chosen by the first actors. Popular understanding: "Prisoner's Dilemma" is a static game where decisions are made at the same time; while chess and card games, where decisions or actions are sequential, are dynamic games. According to the participants' understanding of other participants, they are divided into complete information games and incomplete information games
. Complete information game. A complete game means that during the game, each player has accurate information about the characteristics, strategy space and profit function of other players. Incomplete information game means that if the participants do not know the characteristics, strategy space and profit function information of other players accurately enough, or do not have accurate information about the characteristics, strategy space and profit function of all participants, in this kind of game The game played in this situation is an incomplete information game.
The game theory discussed by economists generally refers to non-cooperative games. Since cooperative game theory is more complex than non-cooperative game theory, its theoretical maturity is far less than that of 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, perfect Bayesian equilibrium equilibrium).
There are many classifications of game theory. For example, it can be divided into finite game and infinite game based on the number of times the game is played or the duration; it can also be divided into general type (strategic type) or expansion type based on the form of expression; and it can be divided into different types based on the different logical foundations of the game. It can be divided into traditional games and evolutionary games. [1]

Nash Equilibrium

Nash Equilibrium: In a strategy combination, all participants face a situation where, when others do not change their strategies, his strategy at this time is the best. That is, if he changes his strategy at this point his payout will decrease. At the Nash equilibrium point, no rational player has the urge to change his strategy alone. The premise of proving the existence of Nash equilibrium point is the introduction of the concept of "game equilibrium couple"

The so-called "equilibrium couple" means that in a two-person zero-sum game, when player A adopts his optimal strategy a*, player B also adopts his optimal strategy b*. If player A still adopts b*, and player B also adopts his optimal strategy b*, Player A adopts another strategy a, so the payout of player A will not exceed the payout of player A using the original strategy a . This result is also true for player B.
In this way, the "equilibrium pair" is clearly defined as: a pair of strategy a (belonging to strategy set A) and strategy b* (belonging to strategy set B) is called an equilibrium pair. For any strategy a (belonging to strategy set A) and strategy b (belonging to strategy set B), there are always: pairs (a, b*) ≤ pairs (a*, b*) ≥ pairs (a*, b).
There is also the following definition for non-zero-sum games: a pair of strategies a* (belonging to strategy set A) and strategy b* (belonging to strategy set B) is called the equilibrium pair of a non-zero-sum game. For any strategy a (belonging to strategy set A) ) and strategy b (belonging to strategy set B), there are always: pairs of players A in the game (a, b*) ≤ pairs (a*, b*); pairs of players B in the game (a*) , b) ≤ even pair (a*, b*).
With the above definition, we immediately get Nash's theorem:
any two-player game with finite pure strategies has at least one equilibrium pair. This equilibrium is called the Nash equilibrium point.
The strict proof of Nash's theorem requires the use of fixed point theory, which is the main tool for economic equilibrium research. In layman's terms, finding the existence of an equilibrium point is equivalent to finding the fixed point of the game.
　　The concept of Nash equilibrium point provides a very important analytical method, allowing 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 does not want to unilaterally change strategies, and ignores the possibility of other players changing strategies. Therefore, in many cases, the conclusion of Nash equilibrium point is unconvincing. Researchers It is vividly called the "innocent and cute Nash equilibrium point".
R. Selten eliminated some equilibrium points that were unreasonable according to certain rules from multiple equilibria, thereby forming two refined concepts of equilibrium: subgame complete equilibrium and trembling hand perfect equilibrium. [1]

Case 1 Prisoner's Dilemma

In game theory, a famous example of a dominant strategic equilibrium is the "prisoner's dilemma" game model given by Tucker. This model tells us a story about a policeman and a thief in a special way. Suppose two thieves A and B jointly commit a crime and break into a private house and are caught by the police. The police put the two people in two different rooms for interrogation. For each suspect, the police policy is: if both suspects confess the crime and hand over the stolen goods, then the evidence is conclusive. Both were found guilty and each was sentenced to 8 years in prison; if only one suspect confesses and the other denies instead, he will be sentenced to an additional 2 years for obstructing official duties (because there is evidence that he is guilty). The person who confessed was meritoriously sentenced to an eight-year sentence reduction and was immediately released. If both of them deny the crime, the police will not be able to convict them of theft due to insufficient evidence, but they can be sentenced to one year in prison each for entering a private house. The table below gives the payoff matrix for this game.
Prisoner's dilemma game [Prisoner's dilemma]

For A, although he does not know what B will choose, he knows that no matter what B chooses, his choice of "confession" is always optimal. Obviously, based on symmetry, B will also choose to "confess", and the result is that both of them are sentenced to 8 years in prison. However, if they all choose to "deny", each will only be sentenced to one year in prison. Among the four action choice combinations in Table 2.2, (deny, deny) is Pareto optimal, because any other action choice combination that deviates from this action choice combination will make at least one person worse off. However, "confession" is the dominant strategy of any criminal suspect, and (confession, confession) is a dominant strategic equilibrium, that is, a Nash equilibrium. It is not difficult to see that there is a conflict between Nash equilibrium and Pareto here.
From a purely mathematical point of view, this theory is reasonable, that is, all choices are frank. But it is obviously inappropriate in the sociological field where multi-dimensional information works together. Just as in ancient China, bribery among officials was called "bad rules" instead of trying every means to investigate them. This is because the social system restrains people's behavior and forces people to change their decision-making. For example, from a psychological point of view, the cost of choosing to confess will be greater. If one party confesses and the other party is guilty, then the subsequent revenge behavior and the fact that he will not be easily "betrayed" among the surrounding insiders will cause him losses. More. The increase rate between 8 and 10 years will be diluted, and human dignity will make people feel revengeful and slightly break the "business rules". We are in the era of big data. If we want to deal with something closer to the facts, we must master as much relevant information as possible and reasonably weight the analysis. Human activities have complex dynamics, so the Prisoner's Dilemma can only be used as a reference for simplified models and specific decisions. It needs to be analyzed in detail. [5]

Case 2: Smart Pig Game

1. "Pigs' payoffs" in economics This example is:
Suppose there is a big pig and a small pig in the pigsty. There is a pig trough at one end of the pigsty (both pigs are at the trough end), and a button to control the supply of pig food is installed at the other end. When the button is pressed, 10 units of pig food will enter the trough, but on the way to the trough, There will be physical energy consumption of two units of pig food on the road. If the big pig arrives at the trough first, the profit ratio of the small pig from eating food is 6:4; if they act at the same time (press the button), the profit ratio is 7:3; Go to the slot first, and the profit ratio is 9:1. So, on the premise that both pigs are wise, the final result is that the little pig chooses to wait.
The "smart pig game" was proposed by Nash in 1950. In fact, the reason why Little Pig chooses to wait and let Big Pig press the control button, and chooses to "take a boat" (or hitchhike) himself is very simple: on the premise that Big Pig chooses to act, if Little Pig chooses to wait, Little Pig will It can get 4 units of net income, and if the little pig acts, it can only get 1 unit of net income left by the big pig, so waiting is better than acting; under the premise that the big pig chooses to wait, if the little pig acts If so, the piggy's income will not be equal to the cost, and the net income will be -1 unit. If the piggy also chooses to wait, then the piggy's income will be zero and the cost will be zero. In short, waiting is still better than action.
The reward matrix in game theory can be used to more clearly depict the piggy's choice:
it can be seen from the matrix that when the big pig chooses to act, if the piglet acts, its profit is 1, and if the piglet waits, the profit It is 4, so the little pig chooses to wait; when the big pig chooses to wait, if the little pig acts, its profit is -1, and if the little pig waits, the profit is 0, so the little pig also chooses to wait. Taken together, no matter whether Big Pig chooses to act or wait, Little Pig's choice will be to wait, that is, waiting is Little Pig's dominant strategy.
In small business management, learning how to "free ride" is the most basic quality of a shrewd professional manager. At some point, it is a wise choice to wait and let other large companies develop the market first. This is when you can do something by not doing anything!
Smart managers are good at taking advantage of various favorable conditions to serve themselves. "Free riding" is actually another option for professional managers to face every expense. Paying attention to and studying it can save enterprises a lot of unnecessary expenses, thereby putting the management and development of enterprises on a better path. new level. This phenomenon is very common in economic life, but it is rarely familiar to managers of small businesses.
In the smart pig game, although the piggy's "picking up ready-made" behavior is morally disgusting, isn't the main purpose of the game strategy to use strategies to maximize one's own interests? [5]

Case 3 Beauty’s Coin

A strange beauty comes over to strike up a conversation with you and asks to play a game with you. The beauty suggested: "Let us each show one side of the coin, either heads or tails. If we are both heads, then I will give you 3 yuan, if we are both tails, I will give you 1 yuan, and you will give me the rest. Just 2 yuan." Sounds like a good suggestion. If I were a male, I would play anyway, but the economic considerations are another matter. Is this game really fair enough?
Suppose our probability of getting heads is x, and the probability of tails is 1-x. In order to maximize the benefits, our income should be equal when the opponent shows heads or tails. Otherwise, the opponent can always change the probability of heads and tails to reduce our total income. The equation listed here is 3x+(-2) (1-x)=(-2) x+1 (1-x)
In layman's terms, this equation means that the benefits you get when your opponent keeps showing heads are the same and the largest as the benefits your opponent gets when he keeps showing tails. Solving the equation yields x=3/8, which means that on average, 3 heads and 5 tails every eight times is our optimal strategy. Substitute x=3/8 into the income expression 3 x+(-2) (1-x) to get the expected income each time, and the calculation result is -1/8 yuan.
Similarly, assuming that the probability of a beautiful woman getting heads is y, and the probability of tails being 1-y, the equation -3y+2(1-y)=2y+(-1) (1-y) is solved to find that y is also equal to 3/8
, The expected profit of the beauty each time is 2(1-y)-3y=1/8 yuan. This tells us that when both parties adopt the optimal strategy, the beauty wins on average 1/8 yuan each time. In fact, as long as the beauty adopts the plan (3/8, 5/8), no matter what plan you adopt, it will not change the situation. If all heads come up, the expected profit each time is (3+3+3-2-2-2-2-2)/8=-1/8 yuan
If all come out tails, the expected profit each time is also (-2-2-2+1+1+1+1+1)/8=-1/8 yuan. Any strategy is nothing more than a linear combination of the above two strategies, so the expectation is still -1/8 yuan. But when you also adopt the best strategy, you can at least ensure that you lose the least. Otherwise, you will definitely be targeted by the strategies adopted by beautiful women and lose more. It seems that this game model is of little use, but in fact it may involve the most important model in financial market pricing: the pricing weight model.
In general, the essence of "game theory" is to express the competitive contradictions in daily life in the form of games, and to use mathematical and logical methods to analyze the operating rules of things. Since there are game participants, there must also be game rule makers. An in-depth understanding of the nature of competitive behavior helps us analyze and grasp the relationship between things in competition, and makes it easier for us to formulate and adjust rules so that they can ultimately operate in accordance with our expected purposes. [5]

The meaning of game theory

The research methods of game theory are the same as many other disciplines that use mathematical tools to study social and economic phenomena. They abstract basic elements from complex phenomena, analyze the mathematical models composed of these elements, and then gradually introduce them to affect the situation. other factors to analyze the results.
　　Based on different abstraction levels, three game expression methods are formed, standard type, extended type and characteristic function type. Using these three expression forms, various problems can be studied. Therefore, it is called "mathematics of social sciences." In theory, game theory is a formal theory that studies the interaction of rational actors. In fact, it is penetrating into economics, political science, sociology, etc., and is used by various applied to the social sciences.
　　Game theory means that an individual or organization, facing certain environmental conditions and under certain rules, relies on the information they have to choose and implement their respective behaviors or strategies, and obtain corresponding results from each. The process of results or benefits, game theory is a very important theoretical concept in economics.
　　What is game theory? As the old saying goes, everything in the world is like chess. Everyone in life is like a chess player, and every action they take is like laying a piece on an invisible chessboard. Smart and cautious chess players figure each other out and check each other. Everyone competes to win, and many wonderful and varied moves are made. chess game. Game theory is the study of the rational and logical part of chess players' "playing" moves, and systematizes it into a science. In other words, it is a study of how individuals arrive at the most reasonable strategies amid intricate interactions. In fact, game theory is derived from ancient games or games such as chess, poker, etc. Mathematicians abstract specific problems and study their laws and changes by establishing self-complete logical frameworks and systems. This is not an easy thing. Take the simplest two-player game as an example. If you think about it for a moment, you will know that there is a lot of mystery in it: assuming that both parties accurately remember every move made by themselves and their opponents and are the most "rational" For a chess player, when A makes a move, in order to win, he has to carefully consider B's thoughts, and B also has to consider A's thoughts when he makes a move, so A also has to think that B is thinking about his thoughts, and of course B also knows that A I thought of what he was thinking about A...
　　Faced with such confusion, how can game theory analyze and solve problems, and how can it find optimal solutions to abstract mathematical problems that are a summary of reality, thereby providing the possibility to guide practice in theory? Modern game theory was founded by the great Hungarian mathematician von Neumann in the 1920s. In 1944, he co-published the masterpiece "Game Theory and Economic Behavior" with economist Oscar Morgenstern, which marked the development of modern systems. The initial formation of game theory. For non-cooperative, purely competitive games, Neumann can only solve two-person zero-sum games – just like two people playing chess or table tennis. If one person wins one move, the other person must lose another move, and the net gain is The profit is zero. The abstracted game problem here is, given the set of players (two parties), the set of strategies (all moves), and the set of profits (winners and losers), can and how to find a theoretical "solution" Or "balance", that is, the most "reasonable" and optimal specific strategy for both parties involved? What is "reasonable"? Applying the "minimum max" criterion in traditional determinism, that is, each party in the game assumes that the fundamental purpose of all the opponent's strategies is to maximize their own losses, and optimize their own strategies accordingly. Neumann mathematically proved , through certain linear operations, a "minimum-maximum solution" can be found for every two-person zero-sum game. Through certain linear operations, both competing parties randomly use each step in an optimal strategy in the form of a probability distribution, and can ultimately achieve maximum and equal profits for each other. Of course, the implicit meaning is that this optimal strategy does not depend on the opponent's operations in the game. In layman's terms, the basic "rational" thought embodied in this famous minimum-maximum theorem is "hope for the best and prepare for the worst."