[Game Theory Notes] Chapter 5 Complete but Imperfect Information Dynamic Game

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Chapter 5 Complete but Imperfect 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 5 Complete but Imperfect Information Dynamic Game

5.1 Dynamic game with imperfect information

5.1.1 Related concepts

Game information limitations
- Incomplete information: The players are asymmetric in the information they gain. Exists in static games and dynamic games
- Imperfect information: The players have asymmetric information about the gameprocess. Only exists in dynamic games
Perfect Information
In a dynamic game, if the player who acts later can observe the progress of the previous game before he takes action, then he has sufficient information about the progress of the game in the previous stage. It is said to have "perfect information".
Dynamic game with complete but imperfect information
A dynamic game in which each player has "complete information" about the benefits, but does not have sufficient information about the game process, is called "complete information". But the dynamic game of imperfect information” is referred to as the “dynamic game of imperfect information”.

5.1.2 Representation of dynamic games with imperfect information

Imperfect Information Representation for Transportation Routing Problems

The two nodes are in the same information set, which means that the businessman cannot know which node he has reached when making a decision, and cannot choose a strategy in a targeted manner, which is represented by a circle.
Imperfect information representation of used car transactions

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Buyer 2 has no way of knowing the path of the first two stages: “Okay – Sell”? "Bad - sell"?

Perform a numerical analysis on this example:

For the seller, the asking price is 2,000. If the car is in good condition, it will be sold normally. If the car is in poor condition, it will cost 1,000 to repair and disguise it before selling it. If you can't sell it, you'll lose 1,000 in vain.
For the buyer, a car in good condition is worth 3,000, and a poor condition is worth 1,000.

In this case, there is a complex interactive decision-making relationship between the game player's strategy, information and judgment, which is the research object of dynamic games with imperfect information.

5.1.3 Subgames of dynamic games with imperfect information

Define a subgame of this game type:

Require:

The original game is not a subgame of its own
Includes all selection nodes and endpoints that follow the initial node, but excludes nodes that do not follow this initial node
Do not split any information set: the subgame must start with a single node information set

Neither Figure 5.4 nor Figure 5.5 is a subgame, because 5.4 divides the information set and 5.5 does not start from a single node.

5.2 Perfect Bayesian Equilibrium

5.2.1 Definition of perfect Bayesian equilibrium

Equilibrium is key to any kind of game analysis, and dynamic games with imperfect information are no exception. For a dynamic game, credibility is always the central issue of equilibrium. The ideal equilibrium must exclude any untrustworthy threats or commitments.

In a dynamic game with complete and perfect information, the equilibrium strategy combination is required to satisfy subgame perfection to ensure that there are no unreliable threats or commitments in the equilibrium strategy.

In a dynamic game with complete but imperfect information, because there are multi-node information sets, some important choices and their subsequent stages do not constitute a sub-game. Therefore, the sub-game perfection requirement cannot completely exclude untrustworthy threats or commitments, and must be introduced. New equilibrium concept - perfect Bayesian equilibrium.

Use this figure to illustrate the requirements for perfect Bayesian equilibrium:

Requirement 1: Probability judgment: At each information set, the selected game player must have a "judgment" on the possibility of the game reaching each node in the information set.

Among them, the "judgment" at the multi-node information set is the probability distribution of the game reaching each node in the information set; the "judgment" at the single-node information set is that the probability of the game reaching the node is 1.

For example, if it is judged that player 1 has a high probability of choosing L -> Gamer 2 will choose U; if it is judged that player 1 has a high probability of choosing M -> Gamer 2 will choose D. The judgment of player 2 is the necessary basis for decision-making.

Requirement 1 is actually the basic premise for solving the dynamic game of imperfect information mentioned above. When it is the turn of the game player to choose in a multi-node information set, he must have a basic judgment on the possibility of reaching each node, otherwise the decision will be made. Without basis, there can be no stability of strategy, let alone equilibrium.
Requirement 2: Sequential rationality: Given the “judgment” of each player, their strategies must be “sequentially rational”

The sequence rationality of requirement 2 is the subgame perfection in the subgame.In the multi-node information set, the beginning does not constitute the subgame. In this part, all game parties are required to abide by the principle of best interests to eliminate unreliable threats or promises in the strategy.

Requirement 2 is also very necessary. If there is no such requirement in this example, and only Nash equilibrium and subgame perfection are required, player 2 has an equilibrium strategy that can obtain the maximum benefit 3 for itself, but contains an untrustworthy threat, that is, the threat of player 2 is in its turn. When I choose, I will only choose D. If player 2 adopts this strategy, player 1’s best strategy is to directly choose $\mathrm{R}$ Ending the game, the benefits of both parties are $(1, 3)$ 。

The above strategy combination is obviously a Nash equilibrium. Since this game has no subgame, the subgame perfection requirement is automatically satisfied, and it is also a subgame perfect Nash equilibrium.

But player 2 does not choose in player 1 $\mathrm{R}$ , only choose D, and in the game side 1, choose $\mathrm{L}$ is very high, it is obviously an untrustworthy threat, because the expected benefit of player 2 choosing D is much smaller than choosing U, which is not in line with the principle of maximum interest.
Requirement 3: The judgment on the equilibrium path complies with the equilibrium strategy: information on the equilibrium path Set , “judgment” is determined by Bayes’ rule and the determination.
Requirement 4: The judgment on the non-equilibrium path complies with the equilibrium strategy: at the information set that is not on the equilibrium path, the "judgment" must also comply with the equilibrium strategy of all parties
.

There are multi-node information sets in imperfect information games. During the game process, at least some players cannot determine the previous game path in some information sets. The information set "on the equilibrium path" means that if the game proceeds according to the equilibrium strategy, the information set will be reached with a positive probability; the information set "not on the equilibrium path" means that when the game proceeds according to the equilibrium strategy, the probability of reaching is 0.

Pay attention to whether "judgment" and "strategy" are inconsistent

5.2.2 Further understanding of judgment formation

Used car trading model

Analysis:

(1) The probability of the car itself being good or bad: $p (g) 、 p (b), p (g) + p (b) = 1$
(2) After player 1 chooses to sell, the probability of the car being in good or bad condition: $\mid s) , p(b \mid s)$ , $\mid s)+p(b \mid s)=1$ 。

(3) Assumption: It is known that the seller chooses to sell or not sell when the car is in good or bad condition:
When the car is in good condition: $\mid g) , 1-p(s \mid g)$
车况差： $p(\mathbf{s} \mid \mathbf{b}) 、 1 - p(\mathbf{s} \mid \mathbf{b})$

Definitely:
$\begin{aligned} p(g \mid s) & =\frac{p(g)\cdot p(s\mid g)}{p(s)}\\ & =\frac{p(g) \cdot p(s\mid g)}{p(g) \cdot p(s\mid g)+p(b) \cdot p(s\mid b)} \end{ aligned}$
Numerical example:

(1) The probability of good or bad car condition: $p (g) = p (b) = 0.5$
(2) The seller’s strategy choice:
A. If the car is in good condition, the seller will definitely sell it: < /span> $\mathrm{p}(\mathrm{s} \mid \mathrm{g})=1$
B. The poor condition of the car needs to be considered: the buyer’s probability of buying (i.e.The buyer’s strategic choice )
Assumption: The buyer’s probability of buying is 0.5
The seller’s expected profit from selling: $0.5 \times 1+0.5 \times(-1)=0=$ Gains from not selling

(3)Seller’s judgment: mixed strategy $p (s ∣ b) = 0.5$ is consistent with the seller’s equilibrium strategy and also consistent with the buyer’s own equilibrium strategy

(4)买方的判断：
$\mid s) =\frac{0.5 \times 1}{0.5 \times 1+0.5 \times 0.5}=\frac{0.5}{0.75}=\frac{2}{3},p(b \mid s)=1-p(g \mid s)=1-2 / 3=1 / 3$
Under the seller’s strategy [p(s|g)=1, p(s|b)=0.5], the buyer’s information set has a greater probability will be reached, so it is the "information set on the equilibrium path". The buyer’s “judgment” is the judgment that requirement 3 is met.
Three-party three-stage dynamic game with incomplete information

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Backward induction method:
For player 3: the expectation of choosing U is $1 * p + (1 - p) * 2 = 2 - p$ ，choice D's period desired $3 * p + (1 - p) * 1 = 1 + 2 p$ , because $p = 1/3$ is a critical condition

For player 2: Strategy L is strictly the best strategy, so L must be chosen.

For player 1: Knowing that 2 chooses L, 3 will choose D later, so it must choose F

During this process, player 3 "judges" p=1, which is consistent with player 2's strategy of choosing L, so requirement 3 is met. Since there is no information set on the non-equilibrium path, requirement 4 is automatically met, so the above solution is Perfect Bayesian Equilibrium

another example

If it is strategy (B, L, U) and "judgment" p=0

First of all, this strategy combination is a Nash equilibrium. In fact, when player 1 chooses $\mathrm{B}$ , the other two players have no choice at all. For player 1, if the strategies of the other two players are $(\mathrm{L}, \mathrm{U})$ , of course, choice B is the most cost-effective. Secondly, when player 3 makes a “judgment” of player 2’s choice $p = When 0$ , (B, L, U) is sequentially rational. Third, because there is no information set that needs to be judged on the equilibrium path, requirement 3 is automatically satisfied. That is to say, the strategy combination (B, L, U) and the "judgment" of player 3 $p = 0$ satisfies the requirement of perfect Bayesian equilibrium $1 - 3 。$

But for requirement 4, player 3’s “judgment” p=0 is inconsistent with player 2’s strategy L—requirement 4 is not met.

So (B, L, U) is not a perfect Bayesian equilibrium

5.3 Single price second-hand car game model

Second-hand car transactions are a typical representative of dynamic game problems with imperfect information. Understanding the game relationships and various equilibria in second-hand car transactions can provide a deeper understanding of such game problems and the construction and analysis of such problems. Provide more ideas.

The latter part of this chapter mainly uses second-hand car transactions as an example for analysis. There are many different situations in second-hand car transactions, such as whether the price is selective, whether bargaining is allowed, whether the buyer can hold the seller accountable and ask for compensation when he finds he has been cheated. The game models in different situations are definitely different. This section discusses the single-price model that has been mentioned many times. The next two sections further discuss the dual-price model with high and low prices, and the model with a money-back guarantee.

5.3.1 Single price used car transaction model

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The letters on the picture represent the benefits under different circumstances: the value of the car in good and bad condition to the buyer: V, W, the seller's selling price: P, and the cost of camouflage: C

Assume ① The price of the car is greater than the cost of disguise: P>C ② The value of the car in good condition is greater than the price: V>P ③ The value of the car in poor condition is less than the price: W<P

Brief analysis
① Both parties’ active choices have risks
② Conservative choices will lose potential benefits
③ Both parties There is no absolute best strategy
④ There are many possible outcomes of the game

5.3.2 Types of equilibrium

When analyzing the second-hand car transaction model, we first introduce the concept of equilibrium type.

Classification based on efficiency differences
- Complete market failure: There are potential trading interests, but all sellers are worried about not being able to sell and choose not to sell. The market is completely inoperable. ——Don’t sell or buy
- The market is completely successful: sellers put good goods on the market and dare not put bad goods. Buyers will buy all the goods on the market to achieve maximum trade benefits. ——Only buy and sell good products
- The market is partially successful: sellers put goods on the market regardless of whether they are good or bad, and buyers buy goods regardless of whether they are good or bad. ——There are good and bad, buy them all
- The market is close to failure: all sellers of good goods put their goods on the market, only some sellers of "bad" goods put their goods on the market, and buyers do not buy them all, but Randomly decide whether to buy with a certain probability, that is, both parties adopt a mixed strategy. ——There are good times and bad times, sell and buy randomly
Combined equalization and separate equalization
- Pooling equilibrium: In market equilibrium, all sellers with perfect information adopt the same strategy regardless of whether the product is good or bad.
  
  For example:
  - The market fails completely—sellers don’t sell any of the different goods
  - The market is partially successful - sellers sell a variety of goods
- Separating equilibrium: In market equilibrium, sellers with different product qualities will adopt completely different strategies.
  
  For example:
  - The market is completely successful - sellers only sell good products, not bad products
① The behavior of the seller in the combined equilibrium cannot reflect any useful information, and the buyer can ignore its behavior when making "judgments".
② In separate equilibrium, the seller’s behavior completely reflects the quality of the goods and can provide a basis for the buyer’s “judgment”.

③ In mixed equilibrium, the seller's behavior can provide certain information to the buyer, but it is not enough for the buyer to make a definite "judgment" and can only obtain a "judgment" of a probability distribution.

5.3.3 ModelPure StrategyPerfect Bayesian Equilibrium

Perfect Bayesian Equilibrium for Market Partial Success

The probability of a missed train $p_b$ It is very small and the cost of camouflage $C$ Relative to car price $P$ is very small, then the following strategy combination and judgment form a perfect Bayesian equilibrium:
- Seller’s strategy: It doesn’t matter whether the car is good or bad $\longrightarrow$ 卖
- Buyer strategy: As long as the seller sells $\longrightarrow$ 买
- One form: $\mathbf{p}(\mathrm{g}\mid \mathbf{ s})=\mathbf{p}_{\mathrm{g}}, \mathbf{p}(\mathbf{b}\mid \mathbf{s})=\mathbf{p}_{\mathbf{b} }$
analyze:

(1) Buyer strategy
Expected profit of buying: $p_g(V-P )+p_b(W-P)$ ,s $\mathbf{V}>\mathbf{P}>\mathbf{W}, \mathbf{p}_{\mathbf{b}}$ 很小 $\longrightarrow$ Expected profit $>\mathbf{0}\longrightarrow$ Purchase selection

(2) Seller’s strategy
The seller knows the buyer’s judgment and knows that he will be able to sell.
- 车好 $\longrightarrow$ 卖的profit $P>\mathbf{0}$ $\longrightarrow$ 卖
- 车差 $\longrightarrow$ Proceeds from sale $\longrightarrow$ 卖
海天买方的美国: $\mathbf{p}(\mathrm{g} \mid \mathbf{s})=\mathbf{p}_{\mathrm{g}}, \mathbf{p}(\mathbf{b} \mid \mathbf{s})=\mathbf{p}_{\mathbf {b}}$ Consistent with seller strategy.

(3) Perfect Bayesian equilibrium judgment

Requirement 1: Existence probability judgment $\left[p(g \mid s)= p_g, p(b \mid s)=p_b\right]$
Requirement 2: Sequential rationality, abide by the principle of best interests and no untrustworthy behavior or threat
Requirement 3: Information on the equilibrium path The set judgment conforms to the equilibrium strategy:
- Seller’s equilibrium strategy (must sell) and buyer’s judgment $\left[\mathbf{p }(\mathrm{g} \mid \mathbf{s})=\mathbf{p}_{\mathrm{g}}, \mathbf{p}(\mathrm{b} \mid \mathrm{s})= \mathbf{p}_{\mathrm{b}}\right]$ match
Requirement 4: The judgment on the unbalanced path must also comply with the balanced strategy. There is no information set on the unbalanced path.

The above four conditions are met, so the above strategy is a perfect Bayesian equilibrium when the market is partially successful.
Perfect Bayesian equilibrium in which the market is completely successful (separating equilibrium)

If: the probability Pb of a bad car is very small, but the disguise cost C >P, then the following strategy combinations and judgments are: Perfect Bayesian Equilibrium
- Seller: The car is in good condition, but not in bad condition.
- Buyer: As long as the seller sells -> buy
- Specifies: $\mid s)=\mathbf{1}, p( b \mid s)=\mathbf{0}$
analyze:

(1) Buyer strategy
Expected profit of buying: $\times(\mathrm{V}-\mathrm{P})+0 \times(\mathrm{W}-\mathrm{P})=\mathrm{V}-\mathrm{P }>0$ $\longrightarrow$ constant purchase

(2) Seller’s strategy
Good car $\longrightarrow$ Proceeds from sale $\mathrm{P}>\mathbf{0}\longrightarrow$ 卖
Car difference $\longrightarrow$ 卖的得益 $\mathbf{P}-\mathbf{C}<\mathbf{0}$ $\longrightarrow$ ふ卖

(3) Buyer’s judgment

$\mid s)=1, \quad p(b \mid s)=0$

(4)Perfect Bayesian equilibrium judgment

Both strategies meet the requirements of sequence rationality. It is not difficult to see that the buyer’s judgment of the information set on the equilibrium path conforms to the equilibrium strategies of both parties and Bayes’ rule, and there is no non-equilibrium path. The set of information that requires judgment. This proves that the above strategy combination and judgment constitute a perfect Bayesian equilibrium.
Perfect Bayesian equilibrium in which the market fails completely (pooled equilibrium)

If the most pessimistic situation occurs, that is, the buyer judges based on past experience that the car must be in poor condition when the seller chooses to sell, then the following combination of strategies and judgments becomes a perfect Bayesian equilibrium:
- Seller: Not selling
- Buyer: Don’t buy
- Buyer’s judgment: $\mid s)=0, p(b \mid s )=1$
analyze:

(1) Buyer strategy

Expected profit from buying: 0× (V-P)+1× (W-P)=W-P <0——》Don’t buy

(2) Seller strategy

The car is good——》Proceeds from the sale=0——》Don’t sell
The car is bad——》The profit from the sale= - C< 0——》Don’t sell< /span>

Meet the requirements of perfect Bayesian equilibrium 1-4

5.3.4 Mixed strategy perfect Bayesian equilibrium of the model

The first three types of markets can all be solved with pure strategies, leaving one type of partial market failure.

Basic characteristics of market near failure:

(1) $P > C$ Only the seller of the time-lag car will sell the car;
(2) The buyer’s expected benefit from buying the car $< 0$ 即 $p (g) (V - P) + p (b) (W - P) < 0$

In this case, if the strategies of both parties are limited to pure strategies, the buyer has no choice but not to buy, the seller has no choice but not to sell, and the market fails completely. Only a mixed strategy can avoid such a result.
The mixed strategy here is that the seller of a poor car randomly chooses to sell or not to sell with a certain probability. For example, the seller of a car chooses to sell, and the buyer also randomly chooses to buy or not to buy with a certain probability. If this is an equilibrium, then it is a market approaching a failure type of equilibrium.

Number example child：

Assume $① V=3000 ② W=0③ P=2000④ C=1000⑤ p_g=p_b=0.5 $

At this time, it has been verified that the two conditions for the market to be close to failure are met.

Strategy and Judgment
A. Seller: If the car is in good condition, sell; if the car is in poor condition, there is a 0.5 probability of selling or not selling at random
B. Buyer: 0.5 probability of randomly buying or not buying
C. Buyer’s judgment: $\mid s)=2 / 3, p(b \mid s)=1 / 3$

Perfect Bayesian Equilibrium Judgment

Requirement 1: Existence probability judgment: $\mid s)= 2 / 3, p(b \mid s)=1 / 3$

Requirement 3: Does the buyer’s judgment comply with the seller’s strategy and Bayes’ rule?
Known: $\mathrm{p}_{\mathrm{g}}=\mathrm{p}_{\mathrm{b}}=0.5, \mathrm{p} (\mathrm{s} \mid \mathrm{g})=1, \mathrm{p}(\mathrm{s} \mid \mathrm{b})=0.5$
The probability that the car is in good condition when the seller chooses to sell
$\mid s) =\frac{p_g \cdot p(s \ mid g)}{p_g \cdot p(s \mid g)+p_b \cdot p(s \mid b)} =\frac{0.5 \times 1}{0.5 \times 1+0.5 \times 0.5}=\frac {0.5}{0.75}=\frac{2}{3}$
Consistent with the buyer’s judgment

Requirement 2: Sequential rationality

For the buyer: The buyer’s expected profit from buying: $ p(g \mid s)(V-P)+p(b \mid s)(W-P) = \frac{2}{3} \times 1000+ \frac{1}{3} \times(-2000)=0 $The same benefit as not buying$ \longrightarrow$ The buyer’s mixed strategy is sequentially rational
For sellers:
- When the car is in good condition: Under the buyer’s strategy, the seller’s expected profit is: 0.5× 2000+0.5× 0=1000>0 $\longrightarrow$ 卖
- When the car is in poor condition: Expected profit from selling after disguise: $0.5 \times(2000-1000) +0.5 \times(-1000)=0$ The benefit is the same as not selling, so random selection is consistent with sequence rationality

To sum up, the strategy combination of both parties and the buyer's judgment constitute a perfect Bayesian equilibrium.

5.3.5 Summary of market types

5.4 Double-priced second-hand car trading game

There are two types of car condition: good and bad. In both cases, it may be sold at a high or low price.

Basic calculation：

Only when the car is in poor condition and you want to sell it at a higher price, there is a camouflage fee C
$\mathbf{V}>\mathbf{W}, \mathbf{P}_{\mathrm{h}}>\mathbf{P}_1$
$V-P_1>V-P_h>W-P_1>0>W-P_h$

analyze

The buyer cannot judge the condition of the car based on the price $\longrightarrow$ It must be judged based on the seller’s strategy, experience and Bayes’ rule
If $\mathrm{C} \longrightarrow \mathbf{0} $ then all sellers will bid higher. So in order to let the price reflect the condition of the car $\longrightarrow$ $\mathbf{C} \neq \mathbf{0}$

Only discuss the strategy combination and judgment of both parties if the market is completely successful:

Seller: If the car is in good condition, ask for a high price; if the car is in poor condition, ask for a low price;
Buyer: buys a car;
Buyer’s judgment: p(g|h)=1, $\mid h)=0, p(g \mid l)=0, p(b \mid l)=1$

Perfect Bayesian Equilibrium Proof

(1) Buyer’s strategic choice

In the case of a slightly different, slightly smoother function: $\mathbf{P}(\mathrm{g}\mid \mathbf{h})\left(\mathbf{V}-\mathbf{P}_{\mathbf{h}}\right)+\mathbf{p }(\mathbf{b} \mid \mathbf{h})\left(\mathbf{W}-\mathbf{P}_{\mathrm{h}}\right)=\mathbf{V}-\mathbf{ P}_{\mathbf{h}}>0$
Note that one, the following equation: $\mathbf{P}(\mathbf{g} \mid \mathbf{l})\left(\mathbf{V}-\mathbf{P}_{\mathbf{P}}\right)+\mathbf{p }(\mathbf{b} \mid \mathbf{l})\left(\mathbf{W}-\mathbf{P}_{\mathbf{1}}\right)=\mathbf{V}-\mathbf{ P}_1>\mathbf{0}$
$\therefore$ buy

(2) Seller’s strategic choice

The car is in good condition, and $\mathbf{P}_{\mathrm{h}}>\mathbf{P}_1 \longrightarrow$ required
interpret, remove $\mathbf{P}_1>\mathbf{0}>\mathbf{P}_{\mathrm{n}}-\mathbf{C} \longrightarrow$ requiredlow
unique sequential rational strategy

(3) Buyer’s judgment
The buyer’s judgment is consistent with the seller’s strategy.

Conclusion: The above strategy combinations and judgments pass the test of perfect Bayesian equilibrium
At the same time, this situation is also the only perfect Bayesian equilibrium of this model - the market is completely successful and separated. Balance

For other cases, if C approaches 0, then $\mathbf{p}_ {\mathrm{g}}\left(\mathbf{V}-\mathbf{P}_{\mathrm{h}}\right)+\mathbf{p}_{\mathrm{b}}\left(\ mathbf{W}-\mathbf{P}_{\mathrm{h}}\right)<0$

That is, if the buyer's expected profit from buying is less than 0, the buyer's inevitable choice is not to buy, and the seller will of course not be able to sell. In this way, the market is completely paralyzed, and all sellers have to withdraw from the market, and no one believes in buying high-quality goods. This mechanism, in which inferior products drive away high-quality products and bring down the entire market under conditions of imperfect information, was first proposed by George Akerlof when discussing lemon market transactions, and is called the "lemon mechanism". principle".

Summary

basic concept
- Definition: Each player has "complete information" aboutbenefit situation, but does not know about the game Dynamic game of process without sufficient information
- The subgame is roughly consistent with the previous requirement
Perfect Bayesian Equilibrium
- Require:
  - Probability judgment: At each information set, the selected game player must have a "judgment" on the possibility of the game reaching each node in the information set. Without judgment, the basis for making decisions will be lost.
  - Sequential Rationality: Eliminating Uncredible Threats or Promises from Strategies
  - The judgment on the equilibrium path conforms to the equilibrium strategy
  - The judgment on the non-equilibrium path conforms to the equilibrium strategy
- example:
  - Second-hand car transaction example: The buyer’s judgment is derived through Bayesian and is in line with the equilibrium strategy
  - Three-party three-stage game: Gamer 3’s “judgment” p>1/3 conforms to Gamer 2’s strategy.
Single price used car game model
- market type
  - Total failure - no selling, no buying
  - Total success - only good products
  - Partial success – buy them all, good or bad
  - Near failure - sell all the good ones, sell the bad ones, sell randomly and buy
- Merge equalization, separate equalization
- The strategies and judgments under the four examples respectively correspond to the four market types.
  - Partial success: probability of missing car $p_b$ It is very small and the cost of camouflage $C$ Relative to car price $P$ 很小
  - Completely successful: the probability Pb of a bad car appearing is very small, butthe disguise cost C >P
  - Complete failure: Based on past experience, the buyer judged that the vehicle must be in poor condition when the seller chose to sell it.
  - Near failure: an example:
    
    A. Seller: If the car is in good condition, sell; if the car is in poor condition, there is a 0.5 probability of randomly selling or not selling
    B. Buyer: 0.5 probability of randomly buying or not buying
    
    In this way, the buyer's expected profit from buying is 0, which is the same as not selling.
Double-priced second-hand car trading: only one example of complete success in the market