"Game Theory" - where in life is not a game

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This article brings "Game Theory: Everyone Can Become a Decision-Making Expert" written by Jiang Wenhua

什么是博弈 ? In the literal description, the game consists of two words: game and game . A game is a confrontation (competition) between two parties (multi-party), and the confrontation is always carried out under certain rules, and the participants will inevitably consider applying corresponding strategies (tactics), and try to win themselves as much as possible.

The design of game theory is very extensive, and the subsequent development basically follows the following four aspects:

1. Weakening (or generalizing) study of Nash equilibrium
2. Refinement (or screening) study of Nash Equilibrium
3. An in-depth study of the basic assumptions of game theory
4. Applied Research on Game Theory

In fact, game theory has long since developed from the study of economic problems to the study of social problems, and has been widely used in many fields other than economics, such as political science, sociology, and management.

Based on the importance of game theory, Xiaocai made a note summary for the book "Game Theory", believing that game theory will become an important theory for explaining conflicts, establishing cooperation, enhancing trust, and promoting social and economic development.

The article describes a variety of game models, which are close to life and intriguing.

1. Overview

1) Game types and structured thinking

The starting point of science is classification, and only through classification can the difference between things be distinguished.

Clothes will be classified, books will be classified, and even height, income, IQ, etc. can be classified. With classification, things will become orderly.

There are two types of games

1. static game and dynamic game
2. Complete Information Games and Incomplete Information Games

Static and dynamic games

• A static game refers to a game in which the participants choose a single action at the same time or not at the same time. )
• A dynamic game means that in the game, the actions of the participants have a sequence, and the latter actor can observe the action chosen by the first actor.

The above is deduced to real life:

Rock paper scissors and bidding activities are static games

Board games and auctions are dynamic games

A real example:

Each school will ask students to grade the teacher's course after the course is over, and the teacher also needs to grade the student's grade. If the student grades first, it may affect the teacher's grade. If the teacher grades first, it may also It will affect students' course grades. In this way, teachers and students will be mutually restrained, and it is difficult to ensure that the scoring is true and effective. Therefore, a confidential method will be adopted. Students will grade first, but it will be kept secret from the teacher. The teacher can only see the student's grade after a period of time after grading the student's grade. And this is a static game

The difference between the two :

Participants in a static game can only make decisions based on their imagination of the opponent's strategy or action. For example, in the game of "rock-paper-scissors", you can only imagine that the opponent will make a "rock", and then you decide to make a "step".

Participants in a dynamic game make decisions based on the opponent's actual choice.

Complete Information Game and Incomplete Information Game

• Complete information game means that in the game process, each player has an accurate understanding of the other players' types, strategy space, and profit and loss function information.
• Incomplete information games are the opposite of complete information games, where there is always some information that is not known to all players.

The above is deduced to real life:

Education, healthcare and finance are games of incomplete information

By contrast, perfect information games are few and far between and reflect the truth: life is inherently unfair

structured thinking

When something can be classified from two or more different dimensions, if the two dimensions are combined on a plane, a structured combination can be constructed.

Combining the four game types mentioned above, we can get:

2) Expand cognition

Nash Equilibrium

Equilibrium refers to the optimal strategy combination of all game participants.

The definition of Nash equilibrium is: given your strategy, my strategy is the best strategy (or one) for me; given my strategy, your strategy is also the best strategy for you ( Or one), both sides have no incentive to adjust their chosen strategy under the strategy given by the other party.

In short : the two sides have reached the bottom of the heat and cold, and no one is willing to deviate from their given strategic state

strategic inquiry

Three attributes of strategy:

1. Integrity of Policy

The strategy must be complete, which means that the participants fully consider all possible situations, and nothing is left behind (need to be good at using classification, exhaustive method)

2. Variety of strategies

Strategies are always diverse, which means that you actually have a lot of strategies to choose from (A, B plan, keep a hand, be prepared)

3. policy unobservability

Strategy is different from action, action is concrete practice, strategy is the expectation of action. Guessing one's strategy based on one's behavior may be wrong (what you think is always what you think)

Four elements of emergency plan

1. Complete and clear information classification, no omissions
2. The responsibility to the person is clearly stipulated, and the person in charge cannot be without the person in charge
3. Prepare detailed and feasible action plans in advance, so that everyone knows how to deal with it in advance
4. Timeliness, let everyone know the specific time requirements of each action

Generally speaking, strategies rely on four related cognitive elements: diverse knowledge and experience, rich and comprehensive imagination, surprising and even super strong ability to build connections, combing and filtering strategies for operability under realistic conditions ability.

rational assumption

Rationality is a very broad and difficult concept to define

The ideal three criteria:

1. Completeness of Preferences and Situational Rationality

A person's preference for A and B must be based on a specific situation. Take an example: sell a magazine, the magazine has electronic and print versions, so how to price more reasonable?

方案A :

• Digital $59
• Print + Digital Package $125

Based on this package, 68% of the people in the experiment chose the electronic version, while only 32% of the combined version

方案B :

• Digital $59
• Print version $125
• Print + Digital Package $125

The pricing of this kind of plan is inevitably a little confusing. Will anyone buy the print version? The answer is no, but your purpose is to sell the combo version, will the situation be better than plan A , the experiment proves that 84% of people choose the combo version, and 16% choose the electronic version.

The above cases fully demonstrate that people's preferences for things depend not only on themselves, but also on specific situations (the so-called package is the meal that traps you)

All judgments and choices are based on a certain frame of reference, and different frames of reference lead to different judgments and different choices

2. Transitivity of Preference

The transitivity of preferences means that if a person thinks A is better than B and B is better than C, then he must think that A is better than C. The transitivity of preferences ensures that a person can distinguish between the good and the bad from a bunch of things, and pick the one they like best.

If preferences are not transitive, then when faced with more than two options, the choice becomes random and chaotic, and your behavioral choices are unpredictable but subject to artificial interference.

As an example:

If your preference for fruit is as follows: apples to peaches, prefer apples; peaches to lychees, prefer peaches; lychees to apples, but prefer lychees. As long as the other party controls the order in which the fruits are placed, they can control the result of your choice. I show you apples and peaches first, then lychees, and you choose lychees; show you apples and lychees first, then peaches, and you choose peaches; show you peaches and lychees first, and then show you peaches. You look at apples, you choose apples.

3. Moderation of preference

Moderation of preference means that if one thinks A is as good as B, then C, which is between A and B, is a more preferred choice. Moderation of preferences tells us that while everyone's preferences are different, if the two ends are the same, the middle is better.

common knowledge

What is common knowledge?

The Emperor's New Clothes is a household story, but the emperor has no clothes, it's known by everyone Common knowledge , but not Common knowledge , Common knowledge is not only known by everyone (common), but also known by everyone else, And everyone knows everyone else knows everyone knows...

What we usually call consensus is actually common knowledge , and we need to distinguish between the two concepts. (I can't help but think of the famous two generals problem in the Internet. Why does TCP perform three-way handshakes instead of two-way handshakes? It's just to reach a consensus)

The classic model

1) The game of cowards

It is a game where both sides are blocking each other and taking a step back.

Speaking of this game, too many scenes in "The Fast and the Furious" are a reflection of the game of cowards .

In a cowardly game, there are two players A and B, each with two strategies (actions): advance and retreat. The two sides face four outcomes: both advance, both retreat, A advance B retreat, A retreat B advance.

There is a first mover advantage in the game of cowards . Once one party chooses to advance, the other party can only choose to retreat, otherwise it will suffer even greater losses.

In theory, the winning strategy of the game of cowards is: preemptively choose jin, and preemptively tell the other party that he has chosen jin, and at the same time let the other party know that his choice cannot be changed.

In one sentence, three elements are reflected:

• preemptive selection
• Let the other party know that they have chosen to enter
• Let the other party know that there is no way out

Therefore, the first-mover advantage in the cowardly game is both an action-based first-mover advantage and a first-mover advantage based on information transfer.

Winner takes all will lead to excessive competition. In this kind of game, you need to repeatedly evaluate the possibility of winning, and it is generally rational to choose not to participate in the game.

2) Husband and wife game

The game between husband and wife can be said to be constant, and who will go to the new year next year has become the problem of the century.

In the context of this issue, there are mutual interests on both sides. In the game of husband and wife, we can find two pure-strategy Nash equilibria: both go to the man's house for Chinese New Year or both go to the woman's house for Chinese New Year.

From the analysis of real life, there are generally three situations:

1. As in the cowardly game, there is a first-mover advantage, and family status determines the preference for Nash equilibrium
2. Take turns going to each other's house
3. Randomly chosen, this solution is more efficient than a Nash equilibrium, based on mutually agreed rules.

Agree on the distribution mechanism of cooperation benefits in advance : In all cooperative relationships, the distribution of cooperation benefits will be involved. Different distribution models have their own rationality. The key is to voluntarily agree before cooperation, which can greatly reduce the cost of cooperation.

The husband and wife game is a win-win cooperation game, but there are differences in the distribution of the benefits of cooperation. Agreeing on the distribution of cooperation benefits in advance is an effective means to avoid conflicts.

3) Prisoner's Dilemma

The prisoner's dilemma, also known as the "prisoner's dilemma", is one of the most classic theoretical models in game theory.

Prisoners' dilemma is not uncommon in modern TV dramas. The general scenario is: two criminals are interrogated separately. If both of them remain silent, they can only be convicted on the basis of the facts of the crime, and the crime that may be sentenced is relatively light. But if one person can confess first, then this person can be released on the spot, and the other person suffers a heavier punishment. If both of them confess, not only will they not be released, but the punishment will be increased.

At this time, the Prisoner's Dilemma appears. The best result for both of them is that both sides choose to resist, but for individuals, it is to confess first, which is the dominant strategy.

Dominant strategy : The result of adopting this strategy is better than other strategies, and the equilibrium formed by the combination of dominant strategies of all players in the game is the dominant strategy equilibrium

The above situation is that when the two sides cannot communicate, after the conditions are relaxed, two people are allowed to bring them together for 10 minutes, and then decide whether to confess.

Then the direction of the game at this time will become whether the two sides abide by the agreement. Of course, the best choice is still not to abide by the agreement. The final result also depends on whether one party can have the ability to psychologically not betray the other party alone.

The prisoner's dilemma is ubiquitous in reality. The reason for the prisoner's dilemma is generally understood as the pursuit of self-interest by the participants. Therefore, where there is a conflict of interest, it is often a prisoner's dilemma. So how to get out of the prisoner's dilemma?

1. Try to change the rules of the game. Strive to find everyone's dominant strategy. If you can't make it beneficial to others, you must also achieve mutual benefits and others.
2. Seek moral strength. Human society is no different from the animal world if there is no moral code governing human behavior
3. With the help of historical memory. Turn the memory of the past into the imagination of the future

The ultimate reason for the prisoner's dilemma is that everyone increases his own interests at the expense of others

Inspiration from the Prisoner's Dilemma: The equilibrium result of the game has nothing to do with the truth, but only depends on the rules of the game

4) The million-dollar trap

This game brings up a very interesting case:

There is a "dollar auction game". In the game, Subic uses 1 dollar bills as auction items and auctions them in public. The auction rules are: the person with the highest bid wins the lot, and the person with the second highest bid also pays the price. payments.

What does that mean? Now we will auction 10,000 yuan to everyone, please bid each other, with 100 yuan as the price increase unit, until no one increases the price. The highest bidder gets 10,000 yuan for the price he bid, and the second highest bidder also has to pay me the amount of money he bid.

If none of the people present participate in the auction, then you can get 10,000 yuan for 100 yuan, and make a net profit of 9,900 yuan.

Of course, the above is the best ideal result. If there are two people who are constantly increasing the price, when you get to 1000 and the other gives 1100, are you willing to give up? If you give up at this time, you will lose 1000 yuan. That is the only way It is to continue to increase the price. The heating and cooling continues. When you have added to 9,900 yuan, the other party has already bid 10,000 yuan. At this time, both parties are unwilling to give up, so both parties continue to increase the price, even until bankruptcy.

This is the famous million-dollar trap , and these traps usually have three characteristics:

1. an obvious bait
2. The road to the bait is one-way, in and out
3. The more the participants try to break free, the deeper they sink, the more they become unable to extricate themselves, and finally pay a heavy price for this

So how to avoid the million-dollar trap ?

1. Establish the limits of your investment and pre-agreed
2. Once the limit is established, it must be adhered to to the end, and must not be changed easily
3. Make up your own mind, try not to be influenced by others, and avoid herd mentality
4. Keep reminding yourself the cost of continuing to invest

Interesting bidding method: Don't think about the pie in the sky, just open the price of 9,900 yuan, only 100 yuan of profit, under normal circumstances, no one will continue to increase the price, unless it makes trouble, but if someone really increases the price to 10,000 yuan, if you If you decide to continue to increase the price, then take the action of breaking the boat and increase the price directly from 9,900 to 19,900 yuan, so that the other party realizes that if the price is increased to 20,000 yuan again, it will also lose 10,000 yuan. You will also be faced with continuing to increase prices and increase losses.

The most effective strategy to avoid the 10,000 yuan trap is to stop losses in time, and don’t get caught up in it. Second, don’t believe that the pie will fall from the sky. Li Xiaocai will not cause the opponent’s strong jealousy and covetousness. If there is a lot of blood, it will attract wolves.

5) Smart Pig Game

What is the smart pig game? The classic model is as follows:

Suppose there are two pigs in the pigpen, one large and one small. There is a trough on one side of the pigpen, and a button to control the supply of pig food is installed on the other side. When you press it, 10 units of pig food will go into the trough, but the cost of pressing the button is 2 units of pig food.

Case 1: If two pigs press the button together and run to eat together, the big pig will eat 7 units and the little pig will eat 3 units. Minus the cost, Big Pig: 5, Little Pig: 1

Scenario 2: If the big pig goes to press the button and the little pig is waiting by the trough to eat, then the big pig will eat 6 units and the little pig will eat 4 units. Minus the cost, Big Pig: 4, Little Pig: 4

Scenario 3: If the little pig presses the button and the big pig is waiting by the trough to eat, then the big pig will eat 9 units and the little pig will eat 1 unit. Minus the cost, Big Pig: 9, Little Pig: -1

The three situations are obvious. For the little pig, waiting is the dominant strategy, then in the case that the little pig must wait, if the pig wants to eat pig food, the big pig can only press it, then the Nash equilibrium in this case is ( Big pig press, little pig, etc.)

But this kind of equilibrium is not a dominant strategy for the big pig. After all, more work is not more rewarding, and the final profit is (big pig: 4, little pig: 4). In this game model, for the little pig, the button will not be pressed in any case. Whether the big pig presses the button will only affect the profit and loss of the little pig, and will not affect the strategy or behavior choice of the little pig.

Therefore, this model can also be called a free- rider game , where one party pays the corresponding price and both parties share the gains. From this model we can get the following inspirations:

1. Whether individual rationality and collective rationality conflict with each other or are consistent depends on the institutional arrangement itself
2. Unequal income distribution will contribute to the conflict between individual rationality and collective rationality. That is, letting some people get rich first helps to help those who are not rich.
3. The rich are willing to do good deeds without the praise of the poor. If you want the big pig to press the button, the little pig can choose to eat less, so that the big pig will not feel too unbalanced
4. With great power comes great responsibility

There are two very important concepts in guessing games: pure strategy and mixed strategy

Pure strategy: players choose only one specific action for each given information situation

Mixed strategy : players randomly choose different actions with a certain probability distribution given information

For example, "If people do not offend me, I will not offend others; if others offend me, I will offend others" is a pure strategy. "If someone doesn't commit me, I have an 80% chance of not committing; if someone commits me, I have a 90% chance of committing" is a mixed strategy

6) Deer hunting game

Another interesting game model

The two of them went out hunting for deer and rabbits.

Scenario 1 : If two people go to hunt rabbits alone, they can hit 2 rabbits in one day

Situation 2 : If you want to fight deer, you need two people to go together, and one is indispensable. (The value of a deer is equivalent to 8 rabbits, and each person is equivalent to 4 rabbits) Of course, if only one person goes to hunt the deer and the other does not, then the person who goes to hunt the deer will gain 0

The analysis found that there are two Nash equilibria under this model, that is, either go to the deer together or go to the rabbit together.

But to fight deer, you need to take the risk of one party not going to fight deer. In this case, it is necessary to communicate in advance

If the communication cost is less than the final profit cost, then it is necessary to communicate in advance and cooperate with each other to maximize the benefits. The lower the cost of communication and coordination, the easier it is to build partnerships between people

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Life is full of games. Although game theory was born very late, its role is ubiquitous. Learning a little bit of game theory can allow us to look at problems from a forward perspective, that is, to make rational decisions about what is about to happen. Deduction, so as to find the greatest possibility, and then let yourself make a relatively good decision.

Don't talk empty-handed, don't be lazy, let's be a programmer who brags about the architecture with Xiaocai. Please pay attention to be a companion, so that Xiaocai is no longer alone. See you below!

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