I try those methods taught by the textbook, but I still can't figure out this game. It has 4 types of 2 players. However, I find most of the videos or textbooks only use their PBE methods to solve the game about 3 types of 2 players, one player has 2 types and the other has only 1 type. It only requires one of the players to guess what type is the others by using Bayes'rule：
Assume that the expected utility of different choices is equal, and calculate the conditional probability of a certain type after observing a certain behavior. Finally, find all equations and work out the mixed strategies of each type for various situations.
When this method is applied to this game, there will be strange cases where the conditional probability is equal to two absolutely unequal formulas.
Here's the text version of the game:
The computer randomly pairs two experimenters into a group and assigns one of them to be A and the other to be B. The computer assigns a given probability(0.2--long-term, 0.8--short-term) of whether the two parties are long-term or short-term. This probability is publicly published public information. Each person was told his or her own type, but not the exact type of the other. (Only know the probability of the other person's type). Each party will get an initial resource of $\$10$.
A decides whether to transfer the initial fund ($\$10$) it has to B. If A chooses not to transfer, then the game is over, and each of the two sides gets their initial funds, that is, their income is ($\$10$, $\$10$). If A chooses to hand over to B, the total value of B's assets will increase to $\$25$ (equivalent to B's product after production), and then B's decision will be made.
B decides whether to return the product (worth $\$25$) to A. If B doesn't, then the game is over, and B gets $25$ dollars, and A gets $0$ dollars. If B chooses to hand over to A, the total value of A's assets will increase to $\$30$ (equivalent to the value of the product sold), and then A's decision will be made.
A decides whether to share the total value of $\$30$ with B. If PERSON A decides not to share, then the game is over, and person A gets $30$ dollars, and person B gets $0$ dollars.
If A decides to share, the cooperation is successful, and the income of A and B is $\$15$ respectively. At the same time, according to the type of player, decide the additional income. No matter A or B, the long-term type will get an additional income of $\$45$ (that is, if the cooperation is successful, the total income of the long-term type will be $\$60$ ). However, the short-term type does not get additional benefits, that is, if the cooperation is successful, the total income of the short-term type is $\$15$.
For both A and B, the probability of being a long-term type is 20%.
Here's a picture version of the game:
What is the Bayesian perfect equilibrium for this game？
What is the method for us to deal with 4 types of 2 players in PBE?
At stage $3$, if $B$ is short-term, the $\$25$ is more than the $\$15$ she could get if she shares, so she doesn’t share.
At stage $2$, $A$ assigns probability $0.8$ to $B$ being short-term and knows that in that case $B$ wouldn’t share and $A$ would get nothing. If $A$ is short-term, he gains at most $\$30$ if $B$ shares, which happens with probability at most $0.2$. Thus he expects at most $0.2\cdot\$30=\$6$ if he shares, so he doesn’t share.
Thus if $A$ does share at stage $2$, $B$ knows for certain that $A$ is long-term. $B$ will then share if and only if she is long-term herself. Thus, if $A$ is long-term, he expects $0.8\cdot\$0+0.2\cdot\$60=\$12$ if he shares, which is more than the $\$10$ he gets if he doesn’t share; so he shares.
Thus the payoffs according to the types are:
\begin{array}{c|cc} A\backslash B&S&L\\\hline S&(10,10)&(10,10)\\ L&(0,25)&(60,60) \end{array}