[Game Theory Notes] Chapter 3 Complete and Perfect Information Dynamic Game

This part of the game theory notes are referenced fromEconomic Game Theory (Fourth Edition)/Xie Shiyu and the teacher’s PPT, which are obtained during daily study As well as compiled in the preparation for the final exam, it mainly focuses on sorting out the knowledge points of this chapter and understanding the key knowledge. The details and logic are not yet complete, and may not be suitable for beginners to read (you should understand better after reading the book) O(∩_∩ )Ohaha~). It is now updated on the blog for everyone to browse. I hope it can help everyone who is learning game theory.

Chapter 3 Complete and Perfect Information Dynamic Game

3.1 Representation and characteristics of dynamic games

Definition: A game in which the actions of the game players in selecting strategies are sequential, and the later actors can observe the strategy choices of the first actors and make corresponding strategic choices accordingly.

1. basic concept:

   • Stage: In a dynamic game, a choice of one player or a simultaneous choice of several players is called a "stage" of the game. Dynamic games have at least two stages.

   • Representation method: extended form: nodes represent game parties, line segments starting from nodes represent optional strategies, and arrays at the end of line segments represent benefits. The method of representing dynamic games is called "extended form". Take the following picture as an example:

     

2. Features

   • Feature 1: The strategies and results of dynamic games are different from static games

     • additional concepts

       Action - a strategy choice of the game party in a stage of the dynamic game.
       Strategy - the complete plan for the actions chosen by the players in all stages of a dynamic game.

   • Feature 2: Dynamic games are asymmetric

     • The party that moves first may have the initiative—first-mover advantage.
     • Those who act later can observe the behavior of those who move first and make decisions accordingly - the advantage of being late.

3.2 The credibility of the strategy and the instability of the Nash equilibrium

3.2.1 Credibility issues of camera selection and strategy

The strategies of the players in dynamic games are preset by themselves. Plans for corresponding behavioral choices in various situations at each game stage. These strategies are actually not coercive, and there is a process to implement them. Therefore, as long as there are opportunities that are in line with the immediate interests of the game parties, they can completely implement them during the game. change plan. This situation is called the "camera selection" problem in dynamic games.

Example: The game of opening a gold mine

A mines a gold mine worth 40 million yuan, but lacks 10 million yuan of funds. He wants to persuade B to invest, promising to share the gold in half after mining.

Different legal environments have different policy credibility and therefore different camera options:

in conclusion

• (1) In dynamic gambling, the player will choose according to the situation, that is, make decisions flexibly according to the situation at different stages.
• (2) Dynamic gambling and abandonment. The strategy selection and game results of the gambling and abandonment parties are closely related to the credibility of the strategy.
• (3) The credibility of the strategy is one of the core issues in dynamic game analysis

3.2.2 The instability problem of Nash equilibrium

When legal protection is insufficient (as shown in the figure below):

The previous Nash equilibrium (borrow; beat; score) (this situation is still considered a Nash equilibrium, given the strategy of one of the parties, deviation alone is detrimental to itself) will change to (not borrow). This is because it contains Without credible promises and threats, B's third stage of litigation cannot be truly implemented.

The reason why Nash equilibrium may lack stability in dynamic games is that it cannot rule out untrustworthy behavioral settings that may be included in the player's strategy.

3.2.3 Backward induction method

The analysis method that starts from the behavior of the game players in the last stage of the dynamic game, and gradually works backwards to the analysis of the behavior of the corresponding players in the previous stage, all the way back to the first stage, is called the "backward induction method".

As shown in the figure below, from the final analysis, B chooses not to fight, so he brings this node up.

Through reverse reasoning, the multi-stage dynamic game is simplified into a series of unilateral gambles, and then the unilateral game is analyzed. Afterwards, by summarizing the choices of each player in each stage, the complete strategy of each player in the entire dynamic game can be obtained.

3.3 Subgame and subgame perfect Nash equilibrium

Since Nash equilibrium cannot exclude untrustworthy behavioral choices in dynamic gambling games and is not a truly stable equilibrium concept, a new equilibrium concept needs to be developed to meet the needs of dynamic game analysis.

3.3.1 Subgame

It consists of a subsequent game stage starting from a stage after the first stage of a dynamic game. It has an initial information set and all the information required to play the game. It can form an original game component of the game by itself, which is called a "original dynamic game" subgame".

• First of all, the subgame cannot include the first stage of the original game, which also means that the dynamic game itself is not its own subgame.

• Secondly, the sub-game must have a clear initial information set, which means that the sub-game cannot divide any information set. There may be no sub-game for games with imperfect information sets with multiple nodes (such as the good weather and bad weather game).

  

In the game of opening a gold mine, each level of the dotted box represents a first-level sub-game. Therefore, the third stage of the game of opening a gold mine is the "second-level sub-game" of the original game.

3.3.2 Subgame perfect Nash equilibrium

If a strategy combination of a complete and perfect information dynamic game is satisfied inthe entire dynamic game and all its subgames Nash equilibrium, then the equilibrium is a "subgame perfect Nash equilibrium".

The fundamental difference between subgame perfect Nash equilibrium and Nash equilibrium, which is also the value of this concept, is that it can eliminate unreliable threats or commitments in the equilibrium strategy, so it is truly stable. The reason why subgame perfect Nash equilibrium can exclude untrustworthy behavioral choices is that although the strategy combination including untrustworthy behavioral choices can constitute a Nash equilibrium for the entire gambling rate, untrustworthy behaviors cannot constitute a Nash equilibrium at least in some subgames. Therefore, will be excluded.

In the game of gold mining with insufficient legal protection:

(Borrowing, playing, and dividing) is the Nash equilibrium of the entire game (no one party can change its strategy and benefit. A chooses to divide. If B does not divide and B chooses to play, the income will decrease, so it is still a Nash equilibrium). But it is not a Nash equilibrium in the second-level subgame (because B chooses to play in this small subgame, it is not a Nash equilibrium), so it is not a "subgame perfect Nash equilibrium".

This also verifies that the subgame perfect Nash equilibrium can eliminate untrustworthy threats.

And (no borrowing, no playing, no dividing) is the subgame perfect Nash equilibrium.

At this time, A's choice node in the second stage and B's choice node in the third stage are "not on the choice path", and the choices of these two nodes in the strategies of the two players are "choices not on the equilibrium path".

All in all subgame perfect Nash equilibrium is also a Nash equilibrium. It is a stronger equilibrium concept than Nash equilibrium and is the core concept of dynamic game analysis. Dynamic game analysis must first find their sub-game perfect Nash equilibrium. The basic method to find the sub-game perfect Nash equilibrium of dynamic games is the backward induction method. The backward induction method starts from the last level sub-game of the dynamic game and gradually finds the optimal choice of the game party in all levels of sub-games. The strategies of each player determined by the backward induction method cannot contain untrustworthy behavioral choices, and the equilibrium strategy combination found must be a subgame perfect Nash equilibrium.

3.4 Four classic dynamic games

Subgame perfect Nash equilibrium and backward induction are two major tools for analyzing dynamic games.

3.4.1 Stackelberg model

In the output game between two oligarchs, the stronger party chooses output first, and the weaker party chooses later. The latter knows the former's choice.

The profit functions of the two are:$\begin{array}{c}{in}_1={in}_1(q_1,q_2)=q_{1}P(Q)-c_1{q}_1=q_1[8-(q_1+q_1)]-2q_1=6q_1-q_1{q}_2-q_1^2\end{array}$

${in}_2={in}_2(q_1,q_2)=q_{2}P(Q)-c_2{q}_2=q_2[8-(q_1+q_2)]-2q_2=6q_2-q_1{q}_2-q_2^2$

Backward decision-making, manufacturer 2:${in}_2^{{}^{\prime }}=6-q_1-2q_2=0$,即$q_2=\frac{6-q_1}{2}$

Different from the Cournot model solution method: the former simultaneous equations; the latter substitution method

Manufacturer 1 knows that Manufacturer 2 will choose this way, therefore${in}_1=3q_1-\frac{q_1^2}{2}$, possible$q_1^{\ast }=3$，Inko$q_2^{\ast }=1.5$, profit (4.5,2.25)

Compared with the Cournot model, its output is (2, 2) and the benefit is (4, 4). It can be seen that manufacturer 1 has a first-mover advantage.

3.4.2 Labor-capital game

The union decides the wage W, and the manufacturer decides the number of employees L

Effective function of trade union $in=u(W,L)$ , the manufacturer’s profit function is $π = π (W,L) = R(L) -W× L $

According to the backward reasoning method, the manufacturer makes the decision first${Pi}^{{}^{\prime }}=R(L{)}^{{}^{\prime }}-IN=0$ Get the relationship between the number of people and wages

Then the union chooses W according to the manufacturer's L to maximize utility.

3.4.3 Bargaining game

1. Three-round bargaining game

   A and B share 10,000 yuan. Game rules: (1) A proposes a plan first. If B accepts, the negotiation ends. If B refuses, B will propose a plan. (2) If A accepts, the negotiation ends. If A refuses, A will propose a new plan. B must Accept; (3) For each additional round of negotiation, the cash shared by both parties will be consumed, and the consumption coefficient is δ (eg: 0.98)

   

   According to backward induction: the third round is certain

   Starting from the second round, B’s optimal strategy makes A’s income no less than $d^{2}S$，即$\delta S_2=d^{2}S$,所以$S_2=\delta S$, same time envoy$\delta (10000\text{-}{S}_2)\ge d^2(10000\text{-}S)$,显然成立。

   In the first round, A knows B’s strategy, so he gives B$\delta (10000-S_2)$，即$\delta (10000-\delta S)$,甲剩余$10000(1-\delta )+d^{2}S$

   Further analysis of the sub-game Nash equilibrium: A’s plan B in the third round must accept——》S=10000, so A benefits$S_1=10000(1-d+d^2)$

   final：$[10000(1-d+d^2),10000(\delta -d^2)]$

   ① δ——》 1, A is close to getting all the benefits, B’s benefits are close to 0;
   ② δ——” 0, A is close to getting all the benefits, B’s benefits are close to 0; The benefit is close to 0;
   ③ δ = 0.5, A gets 7500 yuan, B can get the most 2500 yuan, δ=0.5 brings the greatest bargaining power to B;

   The reason why A has the advantage: A. First-mover advantage; B. The privilege of ending the game

2. Infinite round bargaining game

   Both parties are extremely smart and rational, and must have a reverse induction solution, with a benefit of (S, 10000-S). That is, A proposes S in the first round and B accepts it.

   Shaked and Sutton (1984) proved that: the result of the infinite round game starting from the third round is the same as the result of the game starting from the first round - "A proposes S in the third round, B accepts, and gains (S, 1000- S)

   finally：$\mathbf{S}=S_1=10000-10000\delta +d^2\mathbf{S}S=\frac{10000}{1+\delta }$

   A's gain is a decreasing function of δ, and B's gain is an increasing function of δ; δ=0, A has all the gains exclusively; δ=1, both parties share the gains equally.

3.4.4 Principal-agent theory

One issue in the principal-agent relationship is the certainty of the agent's work results, that is, whether the agent's work results are completely determined by his or her work circumstances.

1. Delegation without uncertainty—agency game

   The agent's work product depends on the level of effort, and there is no unintended risk of diminishing the work product. The principal can understand the agent's work based on the work results.

   Three stages of delegation:

   ① When the agent works hard, both parties gain R(E): the principal’s higher income; w(E): the agent’s higher reward; E: the agent’s cost of working hard< a i=1> ② When the agent is lazy, both parties benefit R(S): the principal’s lower benefit; w(S): the agent’s lower reward; S: the agent’s cost of lazy work
   

   

   Use backward induction:

   The third stage: w(E)-E>w(S)-S must be satisfied before the agent will choose to work hard. This constraint is called the incentive compatibility constraint.

   Economic significance: The agent will choose to work hard only if the reward he receives is a compensation based on the reward he gets for being lazy.

   Second stage: Conditions for agents to participate in entrustment: $w(E)-AND>0,w(S)-S>0$，This nameParticipation promise

   The first stage:

   A. The principal’s choice when the agent works hard: R(E)-w(E) > R(0) - delegation, R(E)-w(E) < R(0) ——No delegation
   B. The principal’s choice when the agent is lazy: R(S)-w(S) > R(0)——Delegation, R(S)-w( S) < R(0)——Not delegated

   Numerical example:

2. “Uncertain but Supervisable” Delegation——Agency Game

   There is no longer a complete alignment between the agent's efforts and results, but they are usually paid based on the performance rather than the results of the work, which means that the risk of output uncertainty is entirely borne by the principal.

   Assumptions:

   Two possible outputs: (20, 10)

   The agent works hard——》The probability of producing 20 is 0.9, and the probability of producing 10 is 0.1
   The agent is lazy—》The probability of producing 20 is 0.1, and the probability of producing 10 is 0.1 Probability 0.9

   Other questions are the same as above.

   Introduce game party 0 weather in representation

It can still be analyzed by backward induction:

Participation incentive constraints: w(E)-E>w(S)-S——》Effort; w(S)-S>w(E)-E——》Lazy
Delegation constraints: w(E)-E>0 and w(S)-S>0——》Accept delegation

The agent works hard and the principal benefits: 0.9× [20-w(E)]+0.1× [10-w(E)]>0 $⟶$ Entrust, otherwise not entrust
The agent is lazy, the principal benefits: 0.1× [20-w(S)]+0.9 × [10-w(S)] $⟶$ commission

At this time, the basis is still the agent's work situation rather than the work results. As long as the agent works hard, he will be paid a high salary.

Specific numerical examples can be used for further analysis.

3. “Uncertain and unmonitorable” delegation—agency game

   The principal cannot fully supervise the agent's work and can only pay wages based on work results rather than work conditions. The uncertainty risk is borne by both parties.

   (The big circle above 0 means that the client does not know which branch the game will take. The client only knows the results of the work, which is incomplete information)

Backward induction:
The third stage: incentive compatibility constraints
$$0.9\times [w(20)-E]+0.1\times [w(10)-E]>0.1\times \left[w\right(20)-S]+0.9\times \left[w\right(10)-S]$$
Second stage: When the agent chooses to work hard in the third stage, push back to the second stage and participate in constraints is
$$0.9\times \left[w\right(20)-E]+0.1\times \left[w\right(10)-E]>0$$
The first stage: Although the principal cannot see the agent’s choices in the third stage, the agent’s decision-making ideas are clear. $E、S、$ $w\left(20\right),\; The\; numerical\; value\; or\; formula\; ofw\left(10\right)$, The principal is fully aware whether the agent will choose to make the effort. Assuming that the principal judges that the agent will choose to work hard, then according to the model setting, the principal's expectation is the delegation condition:
$$0.9\times [20-w(20)]+0.1\times [10-w(10)]>0⟶Delegation$$
The key to incentive mechanism design: determine w(20) and w(10)

4. "Choose Reward and Continuous Effort Level" Delegate

   In the case of uncertainty and unmonitorability, the principal can choose a compensation system (that is, a compensation function) when the agent chooses an effort level in a continuous interval.

   Assumptions:

   ① The agent will have other benefits when he does not accept the entrustment: $U$
   ② The cost of the agent’s effort is a monotonically increasing function of the effort:$C=C(e)$

   ③ Agent’s effort$e$ is distributed in the continuous interval
   ④ Output R is a random function of e:$R=R\left(e\right)$
   ⑤ Commissioner R Payment fee:$In=w(R)=w[R(e)]$

   The gain function is:

   Principal profit function:$R-In=R(e)-w\left[R\right(e)]$
   Agent profit function:$In-C=w\left[R\right(e)]-C\left(e\right)$

   Backward induction:

   Second stage: Participant commitment:$w\left[R\right(e)]-C\left(e\right)\ge IN$

   The first stage: the principal’s profit function:$R\left(e\right)-w\left[R\right(e)]=R\left(e\right)-C\left(e\right)-IN$

On the image, it represents the maximum benefit of the client, that is, the point where the slopes of the two curves are equal.

The client designs the reward function according to "participation constraints" and "incentive compatibility constraints", that is, $w[R(e^*)]-C(e *) ≥ w[R(e)]-C(e) $

5. Example: Shopkeeper and clerk

   R——Profit of the store, e——Effort level of the clerk
   $R=R\left(e\right)=4e+n$，$\eta$——Random disturbance term with mean value 0, the clerk’s cost function: $C=C\left(e\right)={It\; is}^2$, the clerk’s opportunity cost of accepting the job: $IN=1$, clerk salary = fixed salary + profit commission: $S=A+B[R(e)]=A+B[4e+\eta ]$

   For the benefit:

   Shopkeeper profit function:$4e+the-A-B[4e+\eta ]=4(1-B)e+(1-B)\eta -A$
   ShopkeeperEstimated profit: $4(1-B)e-A$
   Store employee profit function:$A+B[4e+\eta ]-{It\; is}^2$
   Store staffEstimated profit:$A+4Be-{It\; is}^2$

   Backward induction: second stage participation constraints:Expected benefit:$A+4Be-{It\; is}^2>=1$,

   $Max(A+4Be-{It\; is}^2)——>{It\; is}^{\ast }=2B$ Meaning: The best effort level of the clerk depends on the profit commission ratio B.

   Shopkeeper’s choice:

   First, the lower limit of the clerk’s participation constraints must be met:$A+B[4e+\eta ]-{It\; is}^2=1$

   Then maximizing shopkeeper's profit: $(4e+\eta )–A-B[4e+\eta ]=4e+the-{It\; is}^2-1$, Expected profit: $4e-{It\; is}^2-1$

   In order to maximize the benefits to the store owner, the clerk’s effort level${It\; is}^{\ast \ast }=2$ ,代入${It\; is}^{\ast }=2B$ 得 $B=1$，之后GET$A=-3$

   Meaning: All profits are given to the store clerk as commission. The store owner does not pay a fixed salary and charges the store clerk a contracting fee or rent of 3 units, that is, contracting system and leasing operation system.

   Subgame perfect Nash equilibrium: (contracting or leasing management system; acceptance; hard work)

3.5 Dynamic games with simultaneous choices

A dynamic game model in which two or more players make simultaneous choices at least in some stage

3.5.1 Standard Model

(1) There are 4 game parties: Gamers 1, 2, 3, and 4;
(2) The first stage: Gamers 1 and 2 are in the strategy set A1 at the same time Choose a1 and a2 from A2;
(3) The second stage: After seeing the choices of players 1 and 2, players 3 and 4 select a3 from the strategy sets A3 and A4 at the same time. and a4;
(4) The benefits of each player depend on the strategies a1, a2, a3, a4 of all players.

3.5.2 Indirect financing and run risk

Assumptions:

(1) The bank issues a loan of 20,000 yuan and attracts deposits at an annual interest rate of 20%.
(2) Two depositors each have 10,000 yuan and deposit it for one year. The bank Loans can be made to enterprises
(3) If the depositor withdraws money in advance and the bank collects the money in advance, the enterprise investment cannot be completed and only 80% of the capital (16,000) can be recovered
(4) If one depositor withdraws money in advance, the bank will repay the entire principal (10,000), and the other depositor can only withdraw the balance (6,000)
(5) If two customers withdraw money in advance at the same time Withdraw money and divide the recovered funds equally (0.8, 0.8)

For savers:

According to the backward induction method, in the second stage, there are two pure strategy Nash equilibria: (advance, advance), (expire, expiration), Pareto optimal equilibrium: (expire, expiry), and risk-advantage equilibrium: (in advance, in advance).

In the first stage, if the game result in the second stage is: (maturity, maturity), the Pareto optimal equilibrium is: (deposit, deposit), indirect financing operates well.

If the game result in the second stage is: (early, ahead of time), the Pareto optimal equilibrium is: (no deposit, no deposit), the risk optimal equilibrium is: (no deposit, no deposit), indirect financing does not work well.

So deposit insurance system-avoiding inefficient equilibrium

3.5.3 International competition and optimal tariffs

Assumptions:

① 4 game parties: Country 1 and Country 2 decide the import tariff rate;
② The two countries are respectively: Enterprise 1 and Enterprise 2;
③ Consumers in both countries can purchase both domestic products and imported goods, and domestic products and imported goods are completely substitutes;

④ Total quantity of goods on the market in country i: $ Q_i$
⑤ Market clearing price: $P_i=a-Q_i$
⑥ Company i Production: $h_i$Kunai 销、 ${It\; is}_i$供出口——》 $Q_i=h_i+{It\; is}_j$
⑦ Enterprise marginal cost = c, fixed cost = 0——》Enterprise$i$production book=$c(h_i+{It\; is}_i)$

⑧ Nation$The\; tariff\; rate\; of\; j$ is $t_j$——》The total cost of export products of enterprise i$=c{e}_i+t_j{It\; is}_i$, the cost of domestic products is $c{h}_i$

Both countries first set tariff rates at the same time $t_{1}sum{t}_2$——》The two enterprises determine domestic sales and export output at the same time $h_1、e_{1}sum{h}_2、e_2$

企业的得益：
$$\begin{array}{c}{Pi}_i={Pi}_i(t_i,t_j,h_i,h_j,{It\; is}_i,{It\; is}_j)=P_i{h}_i+P_j{It\; is}_i-c(h_i+{It\; is}_i)-t_j{It\; is}_i\\ =\left[\begin{array}{c}a-(h_i+{It\; is}_j)\end{array}\right]h_i+\left[\begin{array}{c}a-(e_i+h_j)\end{array}\right]{It\; is}_i-c(h_i+{It\; is}_i)-t_j{It\; is}_i\end{array}$$
National benefit - total social welfare: consumer surplus + domestic enterprise profits + national tariff revenue
$$\begin{array}{cc}{oh}_i& ={In}_i(t_i,t_j,h_i,h_j,{It\; is}_i,{It\; is}_j)\\ & =\frac{1}{2}(h_i+{It\; is}_j{)}^2+{Pi}_i+t_i{It\; is}_j\end{array}$$
In order to maximize the interests of the enterprise: Max (domestic market profit + foreign market profit)
$$\begin{gathered} \underset{h_i⩾0}{\max }|h_i[a-(h_i+{It\; is}_i^{\ast })-c]| \\ \underset{{It\; is}_i⩾0}{\max }|e_i[a-(e_i+h_j^{\ast })-c]-t_j{It\; is}_i| \end{gathered}$$
Available:

The optimal supply in the domestic market:$h\stackrel{\star }{{}_i}=\frac{1}{2}(a-It\; is\stackrel{\star }{{}_j}-c)$

Optimal supply in foreign markets:$It\; is\stackrel{\ast }{{}_i}=\frac{1}{2}(a-h\stackrel{\ast }{{}_j}-c-t_j)$

Two enterprises (i=1, 2 and j=2, 1), four equations to solve the Nash equilibrium:
$$h_i^{\ast }=\frac{a-c+t_i}{3}{It\; is}_i^{\ast }=\frac{a-c-2t_j}{3}$$
Their internal and external sales only depend on the tariffs imposed by the two countries, $h_i^{\ast }$是$t_i$'s growth function,${It\; is}_i^{\ast }$是$t_j$The decreasing function. It shows that tariffs have the function of protecting domestic enterprises and attacking foreign enterprises.

3.6 Issues and Extended Discussion of Dynamic Game Analysis

3.6.1 Problems with backward induction

1. Only explicitly set games can be analyzed

   • Requires the structure of the game to be very clear
   • Each player understands the game structure
   • Know each other and understand the structure of the game

2. Unable to analyze too complex dynamic games

   • Difficult to use when there are too many paths to choose from

3. Cannot analyze games with equal gains on both paths

4. The rational requirements for gamers are too high

   chestnut:

Subgame perfect Nash equilibrium: player 1 chooses T in the third stage, player 2 chooses N in the second stage, player 1 chooses L in the first stage (cannot be reached in stage 23)

But if 1 chooses R in the first stage, and 2 makes a rational choice in the second stage, he will consider whether 1 will be rational enough to choose T if he chooses N. Game player 2 versus player 1 in the first stage The judgment of making mistakes at each stage is different, and the countermeasures are also different. (Nature of the mistake made by player 1: Accidental mistake? Very low level of rationality? Deliberate mistake?)

3.6.2 Trembling hand equalization and inductive induction

Two methods to understand the nature of game players’ mistakes: trembling hand equilibrium and logical induction.

1. The idea of ​​trembling hand equilibrium: Treat the errors of each stage of the game as unrelated small probability events.

   (1) A trembling hand equilibrium game

   

   Nash equilibrium at this time: (D, L), (U, R)

   For (D, L): Player 1: Plans to choose D——》 Considering that Player 2 may deviate from L and choose R——》 Player 1’s benefits are reduced For 2——》The best choice for player 1 is U. Player 2: Choose R after considering Player 1’s idea——》 (D, L) is unstable.

   For (U, R): Gamer 1: To choose U——》 Regardless of whether Gamer 2 will deviate from R——》Gamer 1 does not need to deviate from U ; Gamer 2: Plan to choose R——》The probability of Gamer 1 deviating from U is very small (no more than 2/3)——》Gamer 2 will not change its strategy.

   (U,R) is stable to small-probability accidental deviations - "trembling hand equilibrium", while (D,L) is not "trembling hand equilibrium"

   (2) A game of equilibrium between two trembling hands

   Slightly modify the benefits of player 1:

   

   For (D, L): Player 1: To choose D——》 As long as it is judged that the probability of player 2 deviating from L is very small (no more than 20%)— —》 Gambler 1 will insist on choosing D; Gambler 2: Choose L after considering Gambler 1’s idea——》 (D, L) is a trembling hand equilibrium;

   (U, R) is also a trembling hand equilibrium.

   (3) The conditions for a strategy combination to become a trembling hand equilibrium:

   • Must be a Nash equilibrium
   • Cannot contain any "weak strategies"
     • Weak and inferior strategy - a strategy that will not cause losses to the player even if the player deviates from the strategy. (Strategy D of player 1 in (1))
     • The Nash equilibrium containing the "weakly inferior strategy" cannot withstand any irrational disturbance and lacks stability under limited rationality conditions.

   (4) Trembling hand equilibrium in dynamic games

   

   Two subgame perfect Nash equilibrium paths:

   • Player 1 chooses L to end the game (the initiative in the first step lies with player 1)
   • R—N—T—V

   The second is not a trembling hand equilibrium path. If player 1 chooses R, player 2 doubts its rationality and may choose M. Player 1 considers Player 2’s thoughts and will not choose R.

   (5) Trembling hand equilibrium in the modified dynamic game

   

   R-N-T-V is the only path of sub-bo abandoning the perfect Nash equilibrium, and it is also the trembling hand equilibrium.

   (3,3) will attract each player to persist until the end. As long as the probability of each player deviating from the equilibrium path is relatively small, the player will persist.

   (6) Analyze the original problem based on trembling hand equilibrium

   If player 1 mistakenly chooses R in the first stage, according to the idea of ​​trembling hand equilibrium: the errors of the player in each stage are regarded as unrelated small probability events. Bo Qifang 2 will still choose N based on the balanced thinking of shaking hands.

2. Sequential induction - solving intentional mistakes

The analytical method of inferring their thinking based on the behavior of the players in the previous stages, including deviations from a specific equilibrium path, and providing a basis for the game in subsequent stages is called "sequential induction." (The deviation from the path may be intentional)

(1) Van Damme Game (1989)

If the game proceeds to the second stage, how should we explain it?

• Explanation 1: Gambling side 1 made an accidental mistake in the first phase of selection
• Explanation 2: Gambler 1 deliberately chose the wrong R in the first stage

Explanation 2 is more convincing than explanation 1: when player 1 chooses R, Rw is a strictly bad move (compared to Rs and D). He will not choose it. His best choice in the second stage is w. If Gambling Party 1 believes in Gambling Party 2's analytical ability, he can expect that after he chooses R, Gambling Party 2 will choose W, achieving greater benefits (s, w).

At this time (Rs, w) is a stable sub-Bo abandon perfect Nash equilibrium and a trembling hand equilibrium.

3.6.3 Centipede Game

Multi-stage dynamic game: Players 1 and 2 take turns to choose, with a total of 198 stages.

If according to backward induction method:

(1) The last stage is the best choice$⟶$ d, gain (98, 101)
(2) In the penultimate stage, give up side 1 and choose D, gain (99, 99)
(3) The third to last stage: Player 2 chooses d
(4) ……., Player 1 is in the third In the first stage, choose D to end the game and gain (1,1)

The analysis results of the backward induction method are very inconsistent with people's intuition, but the results of the specific human simulation experiment show that in the experiment, people will not follow the backward induction method, but tend to go backward, which is similar to people's intuition.

The source of the contradiction between theory and practice:

① In practice, players will expect greater potential benefits - risky speculation

② The spirit of cooperation will be tested in practice

③ Trial cooperation and interaction in practice

Analysis: The closer the game is to the end - "The potential benefits of cooperation between the two parties will be smaller -" The logic of backward induction will work at some point, but it is difficult to predict when the logic of backward induction will work. kick in.

Corollary from this analysis: reducing the number of stages to 3-5, cooperation is much less likely to begin with. Because the potential benefits of choosing to cooperate are much reduced, but the initial risks are indeed the same.

Summary

This section is used to quickly sort out and memorize what you have learned.

1. Introduction to basic concepts (easy to understand)

• Action - a strategy choice of the player in a stage of a dynamic game
• Strategy - the complete plan for the actions chosen by the players in all stages of a dynamic game.

2. The credibility of strategies and the instability of Nash equilibrium

   • Camera selection: refers to the fact that the game players may change their plans based on immediate interests (when certain strategies are not credible) during the game, such as the game of opening a gold mine.
   • Instability of Nash Equilibrium: Nash Equilibrium Will Change Due to Untrustworthy Strategies
   • Backward induction method: through backward deduction, the multi-stage dynamic game is simplified into a series of unilateral gambles

3. Subgames and Subgame Perfect Nash Equilibrium

   • The concept of subgame: excluding the first stage, there must be a clear initial information set
   • Subgame perfect Nash equilibrium: It is a Nash equilibrium in every subgame, overcoming the problem of unreliable strategies.

4. Four classic models

   • Stark Game: Similar to Cournot, except that there is a priority of choice

   • Labor-capital game: The union decides the wage W after the union, and the manufacturer first decides the number of employees L

   • bargaining game

     • Three rounds of bargaining: backward induction
     • Infinite rounds: The result of the infinite round game starting from the third round is the same as the result of the game starting from the first round

   • principal-agent theory

     • Delegation without uncertainty
     • Delegation with uncertainty but can be monitored (the store's profits are not necessarily high if the clerk works hard, but wages are still based on the clerk's efforts)
     • There is uncertainty and cannot be supervised (wages are paid based on the store’s revenue)
     • "Choose Reward and Continuous Effort Level" Delegate

     The above four types are mainly based on three aspects: incentive compatibility constraints (the clerk's benefits are higher if he works hard than if he doesn't work hard), participation constraints (the clerk will not have negative returns if he accepts the job), and delegation constraints (the store owner's net profit after entrustment cannot be negative). analyze.

5. Dynamic games with simultaneous choices

   • Financing and run models
   • Tariff model: first combine the two, and then substitute them into the solution

6. Dynamic game expansion analysis method

   • Problems with backward induction: very demanding
   • Two ways to analyze the nature of mistakes:
     • Shaking hand equilibrium: Existence conditions: firstly, it is a Nash equilibrium, and secondly, it cannot include any weak and inferior strategies (the player will not cause losses if he deviates). Stability for accidental deviations with a small probability is called a shaking hand equilibrium.
     • Sequential induction: intentional deviation from the equilibrium path
     • Centipede Game: The backward induction method does not work. Both parties try to cooperate with each other for greater potential benefits.

All in all, using subgame perfect Nash equilibrium and backward induction method for analysis, plus several methods of expansion