Fundamentals of Game Theory-Jiang Wenhua (Zhejiang University)

Table of contents

1. Overview of Game Theory

2. Basic assumptions

3. The Prisoner's Dilemma

4. Ten thousand yuan trap and smart pig game

V. The game of chicken and the battle of the sexes

6. Mixed strategies and supervised games

7. Ultimatums and bargaining

8. Repeated game and system construction

1. Overview of Game Theory
1. Game: Under the constraints of certain game rules , based on direct interaction , each participant chooses his own strategy (action) based on the information he possesses to maximize the benefits
- 1.1 Direct interaction is the interaction between the participants
- 1.2 Behavior selection: It is related to the situation of the opponent (such as: strong, weak), and the change of the external environment (such as: high or low threat to the participants); the game can be either competition or cooperation
- 1.3 Due to the interaction, each person's benefit depends not only on his own strategy, but also on the strategies of others. The core of the game : Rational empathy based on overall thinking, using other people's benefits (expected results) to speculate on other people's strategies (actions taken to obtain the results), so as to choose the strategy that is most beneficial to oneself
- 1.4 Strategy: A complete action plan ( plan ) prepared by the participants before taking action
  - 1.4.1 Strategies have integrity, diversity, and unobservability
  - 1.4.2 A good emergency plan: classified information, assigned responsibilities, specific measures, and timeliness
  - 1.4.3 Strategy allows us to establish a rapid response mechanism from information to action , so that we can make action choices at the fastest speed
  - 1.4.4 Each of the thirty-six strategies is just a specific action, but how to choose based on information is the strategy
- 1.5 Equilibrium: The optimal strategy combination of all participants. A player cannot change his strategy to gain more (i.e., the player will not make another change)
- 1.6 Rules: Participants (who can participate), actions (what actions can be taken) and results (how to judge) are collectively called the rules of the game
2. note
- 2.1 In the game, you only need to be one step ahead to be ahead of others. sometimes too much
- 2.2 Do not pursue success in a fully competitive market. The lower the barriers to entry, the lower the ultimate returns. No matter what method is used, the chance of getting rich is very small
- 2.3 Choosing the right market (opponent) is more important than choosing the right strategy. Such as: stock speculation VS real estate speculation
- 2.4 Before the game, the game has already started. Before gambling, evaluate the probability of winning
3. Game theory: a theory that studies how people make strategic choices and what the final equilibrium result is
4. Game classification
- 4.1 Cooperative and non-cooperative games
  - 4.1.1 Cooperative game: Participants can reach a binding agreement (cooperation) and choose a strategy that is beneficial to both parties within the scope of the agreement
  - 4.1.2 Non-cooperative game: players cannot reach such an agreement
- 4.2 Static game and dynamic game
  - 4.2.1 Static game: In the game, the players choose at the same time, or choose not at the same time, but simultaneously in logical time
  - 4.2.2 Non-static game: In the game, the actions of the participants have a sequence, and the late mover can observe the action of the first mover
- 4.3 Complete information game and incomplete information game
  - 4.3.1 Complete Information Game: In a game, each player knows exactly about the other players, such as the type of player, the strategy space, and the profit-loss function
  - 4.3.2 Incomplete information game: In the game, there is always some information that not all participants know (information asymmetry)
- 4.4 Zero-sum game and non-zero-sum game
  - 4.4.1 Zero-sum game: the sum of profit and loss after the game is equal to the sum of profit and loss before the game. Such as: Mahjong without pumping
  - 4.4.2 Non-zero-sum game: the sum of profit and loss after the game is not equal to (greater than/less than) the sum of profit and loss before the game (positive sum/negative sum). Such as: there are pumping mahjong (negative sum), stock market (negative sum), lottery (negative sum)

static

dynamic

complete information

complete information static game

Nash Equilibrium

complete information dynamic game

Subgame Perfect Nash Equilibrium

incomplete information

static games with incomplete information

Bayesian Nash Equilibrium

dynamic games with incomplete information

Refined Bayesian Nash Equilibrium

2. Basic assumptions
1. Rational assumptions
- 1.1 Cognitive rationality: people are judges of self-interest—with certain preferences
  - 1.1.1 Completeness of preferences: A>B, A~B, A<B
  - 1.1.2 Transitivity of preferences: A>B, B>C, A>C
  - 1.1.3 The Golden Mean: If A~B, C=(A+B)/2, then C>A, C>B
- 1.2 Behavioral Rationality: People are the pursuers of self-interest—the pursuit of maximizing interests
  - 1.2.1 Choose the most important of two benefits and the least of two evils
  - 1.2.2 There needs to be a basis for making decisions (reference)
  - 1.2.3 If you don’t give the other party your two benefits, the other party will probably combine your one benefit with the other’s one benefit, and finally give up on you (do more multiple-choice questions and less judgment questions)
  - 1.2.4 Do not give the other party too many options (2~3 options); the options given to the other party must have obvious differences
2. Common knowledge assumption (common knowledge): the fact that all participants are known in the sense of infinite recursion. That is, everyone knows the event E, everyone knows everyone knows the event E, up to infinite levels
- 2.1 Reaching a consensus is a very difficult thing. It's not enough for everyone to know something, everyone needs to know it for everyone...Infinite recursion
- 2.2 The role of consensus: Through the consensus and the personal information of some people, the personal information of other people can be inferred ("black face" mystery)
- 2.3 In the case of information asymmetry, sometimes the result of the game does not depend on what everyone thinks, but on what everyone thinks ("Emperor's New Clothes")
3. The Prisoner's Dilemma
1. The rules of the game determine the outcome of the game. The result of the game has nothing to do with the truth, but chooses the behavior that is beneficial to oneself based on the rules of the game
2. Definition and cause analysis of the prisoner’s dilemma
- 2.1 Prisoner's dilemma under general conditions
  - 2.1.1 Both sides have possession strategies
  - 2.1.2 There exists a cooperative solution such that the payoffs of both parties are better than their payoffs under the dominant strategy equilibrium. If the two parties choose to cooperate, they can get better results
- 2.2 Reasons
  - 2.2.1 On the surface, it seems that prisoners’ pursuit of their own interests is the cause of the prisoner’s dilemma; in fact, the real reason is that prisoners are pursuing their own interests at the cost of harming the interests of others
  - 2.2.2 Root cause: the difference between private cost and social cost, that is, the negative externality of individual behavior.
- 2.3 If there is a system under which everyone can only achieve the goal of self-interest by benefiting others, it must be a good system. (the logic of the market)
3. How to solve the prisoner's dilemma
- 3.1 Moral education: reduce the proportion of complete egoism and increase the proportion of altruism
- 3.2 System construction (property private ownership): everyone is responsible for their own actions, more work gets more, less work gets less
- 3.3 Learn the new by reviewing the past: let history tell the future (repeated game); learn game theory
4. Ten thousand yuan trap and smart pig game
1. Ten thousand yuan trap
- 1.1 People who fall into the trap usually have two motives:
  - 1.1.1 Economic (rational) motives: eager to win money, want to win back losses, want to avoid more losses, etc.;
  - 1.1.2 Non-economic (emotional) motivations: desire to save face, prove oneself the best player, punish opponent, etc.
- 1.2 Countermeasures
  - 1.2.1 Establish investment limits and pre-agreement, such as how much money or time to invest
  - 1.2.2 Once the limit is established, it is necessary to stick to it (stop loss). Must learn to stop loss in time
  - 1.2.3 Make up your own mind without looking at others
- 1.3 Supplementary Countermeasures
  - 1.3.1 Don't expect to be able to buy cheaply. (Gallows) In a contestable market, it is difficult to make long-term profits
  - 1.3.2 Send the signal of "instant gain" as soon as possible. Break the boat and let others quit
  - 1.3.3 Only fight for steamed buns but not live up to expectations, and lose steamed buns. (Be pragmatic, don't be impulsive) The game between people is won by strength. Strategies can only win a certain war, and the final victory depends on strength
  - 1.3.4 Foresight and foresight. (Think twice)
- 1.4 Seizing the lowest point of cost is the commanding height of victory
2. Smart pig game (more work may not necessarily pay more)
- 1.1 This is a free-rider game. One party pays the corresponding price, and both parties share the obtained benefits
- 1.2 Inspiration
  - 1.2.1 The conflict/consistency between individual rationality and collective rationality depends on institutional arrangements (rules of the game)
  - 1.2.2 Solving the conflict between individual rationality and collective rationality is not by negating individual rationality, but by modifying the system (rules of the game), so as to realize collective rationality on the basis of satisfying individual rationality
  - 1.2.3 It can also be found from the smart pig game that when the net income is > 0, it is a collective rational choice for either party to pay the price. The uneven distribution of income will help reduce the conflict between individual rationality and collective rationality
V. The game of chicken and the battle of the sexes
1. Game of chickens (competitive relationship)
- 1.1 Cockfighting game: the result of equilibrium is that one party advances and the other retreats
- 1.2 Hawk-Dove Game
  - 1.2.1 When A>C, the equilibrium result when all adopt the eagle strategy
  - 1.2.2 When A<C, the probability of both parties adopting the eagle strategy p=A/C is the result of equilibrium
  - 1.2.3 Inference: When A increases, the two sides are more likely to adopt the eagle strategy; when C increases, the two sides are more likely to adopt the dove strategy
- 1.3 The winner takes all industry is prone to excessive competition
- 1.4 Competitive sports are often a 10,000 yuan trap, unless it can bring you happiness in itself
2. Public Goods
- 2.1 Characteristics of public goods: non-excludable and non-rivalrous
- 2.2 Who will provide: Prisoner’s Dilemma (need to be provided by the government), smart pig game, cockfighting game
3. Battle of the Sexes (Partnerships)
- 3.1 The result of the balance: advance together or retreat together
4. Extended Analysis
- 4.1 Farmers’ product selection (scale and cooperation): Within a certain planting scale, cooperation is better; when the scale exceeds the capacity of the region, it is better for farmers to plant different types of crops
- 4.2 Vacation system (reunion and travel): separate vacations for the purpose of tourism; centralized vacations for the purpose of reunion
- 4.3 Effective Demand (Long Tail Theory)
5. The Existence and Multiplicity of Equilibrium
- 5.1 Existence theorem: Every finite game has at least one Nash equilibrium (pure strategy or mixed strategy)
- 5.2 Multiplicity ("cluster point" equilibrium)
  - 5.2.1 Culture, history (reception)
  - 5.2.2 Morality (going out): Priority is given to the elderly, weak, sick and disabled
  - 5.2.3 Law (Right line, zebra crossing): balance between vehicles and people (vehicles let others)
6. Mixed strategies and supervised games
1. Mixed strategy
- 1.1 Pure strategy: Players choose only one specific action for each given information situation (a special case of mixed strategy)
- 1.2 Mixed strategy: Participants randomly choose different actions with a certain probability distribution given information
- 1.3 Rock-paper-scissors winning strategy (human instinctive response, win, stay, lose)
  - 1.3.1 If you are the loser, next turn use the shot that beats your opponent
  - 1.3.2 If you are the winner, do not use the original shot in the next round
- 1.4 After playing the game for a long time, it is difficult to make money. reached a state of equilibrium
- 1.5 A system (technology) that can reduce coordination costs is a better system (technology). Such as the transition from traffic police to traffic lights
2. Supervision game
- 2.1 Mixed Strategy Nash Equilibrium
  - 2.1.1 Differentiation method
  - 2.1.2 Using the concept of balance
  - 2.1.3 When the equilibrium is reached: the probability of the employer checking H/(W+F), the probability of the employee being lazy C/(W+F)
- 2.2 Inference
  - 2.2.1 The greater the labor cost H, the easier it is for the employer to check
  - 2.2.2 The greater the supervision cost C, the easier it is for employees to be lazy
  - 2.2.3 The greater the employee's salary W, the less lazy the employee is, and the less the employer checks
  - 2.2.4 The larger the initial deposit F, the less lazy the employee is and the less the employer checks
- 2.3 Benefits
  - 2.3.1 The employer's expected benefit: VW-VC/(W+F). When the employer reduces the monitoring cost C, more benefits can be obtained. (V: employee contribution)
  - 2.3.2 When the wage paid by the employer is W=(CV)^(1/2)-F, the employer will benefit the most. At this time, the employer's expected income is: T=V-2(CV)^(1/2)+F
  - 2.3.3 Expected benefits of employees: T=WH
  - 2.3.4 The total expected income of both parties: Ttotal=VH-CV/(W+F). When W=(CV)^(1/2)-F, the total expected income of both parties is Ttotal=VH-(CV)^(1/2)
- 2.4 Inference
  - 2.4.1 The employer's expected income will increase with the increase of F
  - 2.4.2 The employee's expected income will decrease with the increase of F
  - 2.4.3 The total expected return has nothing to do with the wage W paid and the size of the mortgage F, but only with the employee contribution V, employee labor cost H and supervision cost C
7. Ultimatums and bargaining
1. Ultimatum
- 1.1 The Ultimatum Game
  - 1.1.1 Two people share a fixed amount of money, one proposes a plan, and the other votes
  - 1.1.2 If the voters agree, then the proposal will be divided
  - 1.1.3 If they disagree, the two will have nothing
- 1.2 Note
  - 1.2.1 It is difficult for a person without faith to understand the thinking and behavior of a person with faith
  - 1.2.2 The most fundamental conflict among people is the conflict of values
  - 1.2.3 The more mature the organization (society), the more the ultimatum game is used in management
  - 1.2.4 An ultimatum can save a lot of negotiation costs on the one hand, and on the other hand, it can give people a sense of fairness
  - 1.2.5 The price is determined by: greed and fear
- 1.3 Basis of power (attribution of distribution rights)
  - 1.3.1 Random: play luck
  - 1.3.2 Intelligence test: spelling ability
  - 1.3.3 The job promotion system directly affects the management efficiency of the organization. Establish a promotion system based on performance and ability as much as possible, and adhere to the principle of open competition
  - 1.3.4 Who Bears the Consequences of Decisions and Who is Responsible for Decisions
2. The dictator game
- 2.1 Game design
  - 2.1.1 Two people form a group of 100 yuan, the proposer proposes a distribution plan, distributes X yuan to the responder, and keeps (100-X) yuan for himself
  - 1.1.2 Regardless of whether the responder agrees or not, the proposer will get his own (100-X) yuan
- 2.2 There are very few people who take all the money. Most people will choose to distribute some money to the responders (because they think from the perspective of the responders and hope to get the money), but this amount is less than the ultimatum the amount of money in
- 2.3 Three principles
  - 2.3.1 People do not consider economic interests alone when deciding their actions. Although this may be the main consideration, they also consider some moral and social norms , such as the principle of fairness, "do unto others as you would have them do unto you"
  - 2.3.2 If a society can give people more rights to refuse in terms of institutional arrangements , then this society will have more fairness and even more efficiency improvements
  - 2.3.3 A society has given institutions (officials) more powers to refuse (such as the administrative approval system) in terms of institutional arrangements, so the institutions with approval powers must be able to benefit from it (such as social maintenance fees)
3. Bargain
- 3.1 Also known as bargaining or negotiation, it mainly refers to the distribution of interest disputes resolved by participants through negotiation
  - 3.1.1 When it is called bargaining , it mainly emphasizes its action or process
  - 3.1.2 When referred to as a negotiation , emphasize its status or outcome
- 3.2 Equilibrium solution
  - 3.2.1 Let X represent the share obtained by participant 1, (1-X) is the share obtained by participant 2, Xi and (1-Xi) are the respective shares obtained by the two participants in period i
  - 3.2.2 Assume that the discount factors of the two participants (convert the future money into the current money, multiplication. Due to the time cost, 0<discount factor<1) are and $\delta_{1}$ respectively $\delta_{2}$ . Generally speaking, the more "patient" the person, the larger the discount factor
  - 3.2.3 Thus, if the game ends at time t: the discounted value paid by player 1 : $W_{1}=\delta_{1}^{t-1}X_{t}$ ; the discounted value paid by player 2 : $W_{2}=\delta_{2}^{t-1}(1-X_{t})$
  - 3.2.4 After the two parties have gone through an infinite game, the Nash equilibrium solution obtained is: the share obtained by participant 1 $X^{*}=(1-\delta_{2})/(1-\delta_{1}\delta_{2})$
  - 3.2.5 $\delta _{1}=\delta _{2}=\delta$ At that time , $X^{*}=1/(1+\delta )>1/2$ . Explain that player 1 holds more shares, that is, the first-mover advantage
  - 3.2.6 When player 1 is more patient, $\delta _{1}=1, \delta _{2}<1$ immediately , $X^{*}=1$ , player 1 takes all shares
  - 3.2.7 When player 2 is more patient, $\delta _{2}=1, \delta _{1}<1$ immediately , $X^{*}=0$ , player 2 takes all shares
  - 3.2.8 In bargaining, an equilibrium solution can always be found
4 discount factor
- 4.1 What factors determine a person's comparison of immediate interests and long-term interests (discount factor)?
  - 4.1.1 Lifespan: the longer the lifespan, the longer-term benefits
  - 4.1.2 Wealth: The less wealth you have, the more you pay attention to the immediate interests (far away can’t quench your near thirst)
  - 4.1.3 Certainty of future income: There is uncertainty in future income, and the greater the uncertainty, the more immediate interests are valued. This emphasizes the importance of a society ruled by law and encourages people to invest in the long run
  - 4.1.4 Educational level: The higher the educational level, the more important the long-term interests are
- 4.2 Organizations have longer life expectancy than individuals, thus increasing people's discount factor
8. Repeated game and system construction
1. Repeated game
- 1.1 Definition: A game with the same structure is repeated many times, and each game is called a "stage game"
- 1.2 Repeated game is an important part of dynamic game , it can be a game with complete information or a game with incomplete information
- 1.3 Features
  - 1.3.1 The game in the previous stage does not change the game structure in the next stage
  - 1.3.2 All participants can observe the past history of the game
  - 1.3.3 The total profit of the participant is the sum of the discounted value of the profit and loss of all stages
- 1.4 Strategies (taking the prisoner's dilemma as an example)
  - 1.4.1 Lei Feng's strategy: in repeated games, always choose to cooperate. Otherwise, the opponent is likely to choose to defect, so as to obtain the maximum benefit
  - 1.4.2 Cao Cao's strategy: in repeated games, always choose to defect. Then, the opponent must choose to defect
  - 1.4.3 Relentless strategy: also known as trigger strategy. Choose to cooperate at the beginning, if the other party chooses to cooperate, then continue to cooperate for the second time; if the other party chooses to betray at a certain stage, then choose to betray from then on
  - 1.4.4 Soft-hearted strategy: choose to cooperate at the beginning, if the opponent only betrays once, then choose to cooperate (give the opponent a chance to correct); if the opponent betrays twice in a row, then choose to betray from then on
  - 1.4.5 Tit for tat (Tit for tat): Except for the first time, every time is consistent with the opponent's last choice
  - 1.4.6 Human-Ghost Strategy: (Talking human words to people, talking nonsense to ghosts) Look at all the past behaviors of the other party. If there is more cooperation, choose cooperation; if there are more betrayals, choose betrayal
  - 1.4.7 Bullying the weak and fearing the strong: also called the tester strategy. In the first round of defection, if the opponent chooses to cooperate, continue to betray; if the opponent chooses to betray, then choose to cooperate later (used to deal with the soft-hearted strategy)
  - 1.4.8 Tranquilizer strategy: first establish a trust relationship, and then find ways to take advantage of it
  - 1.4.9 Psychopath: Also known as the drunkard strategy, randomly choose cooperation or defection. The characteristic of the person who chooses this strategy is that he does not know what is good or bad, so the opponent must choose Cao Cao's strategy and always betray
2. Tit for Tat
- 2.1 Four characteristics of Tit for Tat
  - 2.1.1 It is kind (choose cooperation for the first time) (no enemies): prevent getting into trouble of non-cooperation
  - 2.1.2 It is provocative (anti-betrayal): to ensure the prudence of the other party's betrayal behavior
  - 2.1.3 It is forgiving (gives a chance): helps to restart cooperation after the other party defected
  - 2.1.4 It is clear (readable): Simple and clear rules are easy to understand, leading to long-term cooperation. People prefer to cooperate with people who are easy to understand, so don't give others the impression that you are inscrutable
- 2.2 Four points of advice
  - 2.2.1 Don't be jealous: No matter what strategy is adopted, the final score cannot be higher than Tit for Tat
  - 2.2.2 Don't be the first to betray
  - 2.2.3 Report both cooperation and defection
  - 2.2.4 Don't use cleverness
- 2.3 Two revelations
  - 2.3.1 The basis of cooperation is not trust, but the continuity of the relationship
  - 2.3.2 Organizations tend to have a longer life expectancy than individuals, which increases the persistence of relationships
3. Brutal strategy
- 3.1 The total income of the other party who chooses not to cooperate is R=5
- 3.2 The total income R=3+3+3+...+3+... if the other party chooses continuous cooperation; consider the discount factor $R=3+3\delta +3\delta ^{2}+...+3\delta ^{n}\approx 3/(1-\delta )$
- Another meaning of 3.3 $\delta$ is the possibility of the next cooperation
- 3.4 How to deal with noise? (accidental betrayal)
  - 3.4.1 Tolerance. If the other party always chooses to cooperate, but suddenly betrays, then tolerate the other party and choose to continue to cooperate
  - 3.4.2 Repentance. Due to negligent choice of defection, the other party chooses defection. At this time, one should continue to choose cooperation instead of defection.
  - 3.4.3 When the noise frequency is low (0.1~1), repentance is better than tolerance
  - 3.4.4 When the noise frequency is high (1~10), tolerance is better than repentance
  - 3.4.5 Noise leads to forgiveness, but too much forgiveness can be exploited. So we should be merciful
4. System building
- 4.1 To change a phenomenon, it is necessary to change people's behavior; and to change people's behavior, it is necessary to continuously improve the system
- 4.2 Don't blame others for their selfishness. If a self-interested behavior leads to bad consequences, it is the system that needs to be changed
- 4.3 A good manager is not to eliminate people's self-interest, but to make full use of people's self-interest. Self-interest can make people kill and help others; it can make people lie and tell the truth