Glamor Economics

Chapter 79

Chapter 11, Section 6 Pirates’ Fund Sharing and Sub-Lease Scheme——Dynamic Game

Pirates' gold distribution is a classic case of dynamic games in economics.Assuming that there are several pirates on the ship, they want to divide the stolen gold coins.Naturally, such issues are resolved by voting.The voting rules are as follows: first the most ferocious pirate proposes a distribution plan, and then everyone votes for one vote. If 50% or more of the pirates agree with the plan, then the plan will be distributed. If less than 50% of the pirates agree , then the pirate who proposed the plan will be thrown into the sea to feed the fish, and then the most vicious pirate among the remaining pirates will propose a plan, and so on.

Let's make some assumptions about the pirates first:
1.Each pirate is ferocious differently, and all pirates know the ferocity of others, that is, each pirate knows his own and others' place in the scheme.In addition, every pirate is a very smart person, and can judge gains and losses very rationally, so as to make choices.Finally, private deals between pirates do not exist, because pirates trust no one but themselves.

2.A gold coin cannot be divided, you can't have half of it and I have half of it.

3.Of course every pirate does not want to be thrown into the sea to feed the fish, this is the most important thing.

4.Of course every pirate wants to get as many gold coins as possible.

5.Every pirate is a utilitarian.If he gets 1 gold coin in one plan, and in the next plan, he has two possibilities, one is to get a lot of gold coins, and the other is to get no gold coins, he will agree to the current plan without fluke .All in all, they believe that a bird in the hand is worse than two birds in the bush.

6.Every pirate loves other pirates being thrown overboard to feed the fish.As long as it doesn't harm his own interests, he will vote for his companions to feed the fish as much as possible.

Now, if there are 5 pirates who want to divide 100 gold coins, what will happen?
Pushing forward from the back, if robbers No. 1 to No. 3 have fed the sharks, and only No. 4 and No. 5 are left, No. 5 must vote against and let No. 4 feed the sharks, so as to swallow all the gold coins.Therefore, No. 4 can only save his life by supporting No. 3.

Knowing this, No. 3 will propose a distribution plan of "100, 0, 0", and will return all the gold coins to No. 4 and No. 5 as his own, because he knows that No. 4 will get nothing but will still vote for it. With his own vote, his plan can be passed.

However, when No. 2 infers No. 3's plan, it will propose a "98, 0, 1, 1" plan, that is, give up No. 3, and give No. 4 and No. 5 each 1 gold coin.Since the plan is more favorable for No. 4 and No. 5 than when No. 3 is assigned, they will support him and don't want him to be out and assigned by No. 3.In this way, No. 2 will take 98 gold coins.

Similarly, the plan of No. 2 will also be understood by No. 1, and No. 1 will propose a plan of (97, 0, 1, 2, 0) or (97, 0, 1, 0, 2), that is, abandon No. 2, And give 3 gold coin to No. 1, and give 4 gold coins to No. 5 (or No. 2) at the same time.Since the plan of No. 1 is better for No. 3 and No. 4 (or No. 5) than when No. 2 is allocated, they will vote for No. 1, plus No. 1's own vote, No. 1 The plan can be passed, and 97 gold coins can easily fall into the bag.This is undoubtedly the plan that No. 1 can get the most benefits from.The answer is: Robber No. 1 distributes 3 gold coin to Robber No. 1, and 4 gold coins to Robber No. 5 or 2, and he gets 97 gold coins for himself.

This seemingly suicidal plan is actually quite precise.The premise is that the five robbers are all scheming and can accurately predict the changes that will occur in each step of the distribution process; Never give up.This is a wonderful game.

There are a few points to master in the game: first, you must grasp the initiative in the competition. Robber #1 looked like the most likely to feed the sharks, but he firmly grasped the first-mover advantage, resulting in not only eliminating the threat of death, but also reaping the most benefits. Robber No. 5 seems to be the safest, without any fear of being thrown into the sea, and may reap the benefits of fishermen, but because he has to act on the face of others, he can only get a small share of the pie, or even empty-handed.Second, the most important thing in competition is the rules.If the rules were changed, the solution to this problem would be a completely different matter.Of course, there are rules, and they must be followed.Third, keep a clear head in the competition.Be good at analyzing gains and losses and learn to be satisfied, otherwise you may get nothing.Of course, this is not about the kind of muddled satisfaction, you must work hard to get what you deserve.

Pirate gold distribution is actually a dynamic game.A dynamic game means that the actions of the players have a sequence, and those who act later can know what action the person who acts first will take.

Most of the Chinese students who have just arrived in the United States rent apartments with two or three people, so there is a problem of rent sharing.It is usually discussed with each other, and generally both parties think it is more reasonable.This method is generally feasible, but at most it is "reasonable". Few people think that they have taken advantage of it, and the opposite is not uncommon.

However, someone used the idea of ​​game theory to design such a reasonable rent sharing model.Students A and B decide to rent a one-bedroom, two-bedroom apartment together, and the rent is 550 yuan per month. Room 1 is the master bedroom, spacious and bright, with a separate bathroom in the house. Room No. 2 is relatively small, and it uses the bathroom outside. Of course, it must be used if there are guests. A's economic conditions are slightly better, while B's are poorer.How to apportion the rent of 550 yuan now?
按照模型的第一步，A、B两人各自把自己认为合适的方案写在纸上。A1，A2，B1，B2分别表示两人认为各房间合适的房租。显然，A1+A2=B1+B2=550。第二步，依照两人所写的方案来决定谁住哪个房间。如果A1大于B1（必然B2大于A2），则A住1号B住2号，反之则A住2号B住1号。比如说，A1=310，A2=240；B1=290，B2=260（可以看出，A宁愿多出一点儿住好点儿，而B则相反），所以A住1号，B住2号。第三步，定租。每间房间的租金等于两人所提数字的平均数。A的房租=（310+290）/2=300，B的房租=550-300=（240+260）/2=250。结果：A的房租比自己提的数目小10，B的房租也比自己愿出的少了10，都觉得自己占了便宜。

According to this model of subletting, everyone feels that they have taken advantage of it, and both parties have taken advantage of the same size.The worst case scenario is also "fair and reasonable".If anyone suffers a loss, it must be that he was treacherous and wanted to take advantage of it but failed to take advantage, so it is impossible to say that he suffered a loss.If we use the thinking of game theory to analyze, we can draw the following conclusions.

1.Due to different personal economic conditions and preferences, there will be differences in the subletting plans of the two, and it is difficult to reach an agreement according to ordinary methods.In the model, this difference is the "residual value", which is shared by two people. The greater the difference of opinion, the more dividends, the more satisfied they are.The worst case is that the two agree completely, and no one takes advantage or loses.

2.Telling the truth will never be a disadvantage, the only reason for a disadvantage is to lie.Assuming that A's plan is what he really thinks is reasonable, then no matter what B's plan is, A's rent will definitely be lower than his own plan.The same is true for B.

What kind of situation will A suffer?That is to say, the rent is higher than what you are willing to pay.For example, A guesses that B1 will not be greater than 280, so in order to share more residual value, he writes A1=285, A2=265, then he can only live in room 2, and the rent is 262.5, which is higher than what he actually thought The rent is 22.5 more.But he suffered from being dumb because he wanted to take advantage and didn't tell the truth.

3.Analyzing this model from a game theory perspective may not be the best strategy, especially if you know the opponent's preferences.But if you tell the truth, you will never suffer a loss. If you don’t tell the truth, you will suffer a loss, or you will get more surplus value.

4.This model can also be used to divide houses for more than three people. Each house is occupied by the highest renter, and the average rent is taken.

This case is also a dynamic game.The difficulty of dynamic games is that the optimal decision at the previous moment may not be the optimal decision at the next moment, so there is great difficulty in solving it.There is a sequence of actions in the dynamic game, and different participants act at different time points. The choice of the first mover affects the choice space of the second mover, and the latter mover can observe what choice the first mover made.Therefore, in order to make the optimal choice of action, each player must think about the question: If I choose this way, how will the other party respond?How would I act if I were him?Given his response, what is my best option?In a dynamic game, the actions of each player are obviously based on the actions of the opponent first, just like chess, you take a step, the opponent takes a step, there is a sequence of action strategies, which gives the passive player There is room for initiative from being reactive.

[links to related words]

Dynamic game means that the actions of the participants have a sequence, and the actions of the latter can observe the choices of the earlier actions, and make corresponding choices accordingly.

(End of this chapter)