Pirate's Gold (Nash equilibrium)

Problem:

There is a "pirate divide" model in economics, it is said that 5 pirates robbed 100 gold coins, they are in the order of the lottery to give the plan: first by the 1th distribution plan, and then 5 votes, more than half of the agreed program was passed, otherwise he will be thrown into the sea to feed the sharks, and so on. Pirates are happy to see other pirates thrown into the sea to feed the sharks on the premise of maximizing their profits, assuming that every pirate is smart and sensible, then what is the first pirate's plan to do to maximize his own profits?

Answer:

Backward push, from the forward, if 1 to 3rd robbers were fed sharks, only 4th and 5th, the number 5th must vote against the 4th to feed the sharks, to take all the gold coins. So, number 4th only supports number 3rd to survive.
3rd know this, will put forward the "100,0,0" distribution scheme, 4th, 5th Skinflint and all the gold coin as already has, because he knew that 4th no gain but will vote in favour of the vote, plus his own vote, his plan can be passed.
However, the 2nd inferred from the 3rd plan, will be proposed "98,0,1,1" plan, that is, to give up 3rd, and give 4th and 5th each one gold coin. Since the scheme is more advantageous for numbers 4th and 5th than in distribution 3rd, they will support him rather than expect him to be out and be allocated by number 3rd. In this way, number 2nd will take 98 gold coins.
Similarly, the 2nd program will also be understood by the 1th, 1th and will propose (97,0,1,2,0) or (97,0,1,0,2) scheme, that is, give up 2nd, and give 3rd a gold coin, while giving 4th (or 5th) 2 gold coins. As a result of this programme for number 1th, for 3rd and 4th (or 5th), compared to the 2nd distribution, they will vote 1th for the vote, plus 1th of their own tickets, 1th of the program can be passed, 97 gold coins can easily fall into the bag. This is no doubt that number 1th will be able to obtain the maximum benefit of the program. The answer is: 1th Bandits to 3rd 1 gold coins, divided to 4th or 5th robbers 2, their own alone 97.

Allocation scheme can be written (97,0,1,2,0) or (97,0,1,0,2)

What happens if the terms "more than half of the consent scheme is passed" to "more than half or just half of the consent scheme is passed"?

Again, if 1 to 3rd robbers were fed to the Sharks, the number 4th proposed (100,0) could gain maximum benefit.

Push backwards, number 3rd will put forward (99,0,1) let 5th support him.

Push backwards, number 2nd will put forward (99,0,1,0) let 4th support him.

Forward backwards, number 1th will propose (98,0,1,0,1) let 3rd and 5th support him.

Allocation scheme is (98,0,1,0,1)

In fact these 2 kinds of situation are similar, the difference is not big.

But intuitively, we feel the difference between the scene and the life, the result is very surprised, this is why.

First of all, if you disagree, throw someone out to feed the sharks, which is certainly not the same as daily life, but that's not the point.

In fact, even if this condition is removed, the result will not be much different, as the following details are:

If you get rid of the original title and you don't agree, throw someone out and feed the sharks.

If 1 to 3rd robbers were fed the Sharks, number 4th could propose any plan, but only (0,100) would pass

Push backwards, number 3rd will be raised (99,1,0)

Push backwards, number 2nd will be raised (99,0,0,1)

Push backwards, number 1th will be raised (98,0,1,1,0)

If the term "more than half of the consent scheme is passed" to "more than half or just half of the consent scheme is passed", if you don't agree, you throw someone out and feed the sharks. This condition is the same as not getting rid of this condition, the final result is 1th will be proposed (98,0,1,0,1)

Get rid of those who disagree. Feed the sharks, assuming that every pirate is smart and rational, although not exactly the same as the reality, but for this simple logical reasoning, there are many people can think of transparent, so that this is not the key difference.

The real reason for the surprising result, the key difference between the subject and the reality, is that the 5 people in this topic are independent, that is to say, the game is a kind of equilibrium.

Although not explicitly, but in this topic, this is a rule.

Otherwise, if there is no such rule, 2, 3, 4, 5th can fully discuss the countermeasures, each person to 25, natural than agreed 1 of the scheme is much better.

But in this case, the problem becomes much more complicated, because different people can discuss the matter in private, and end up endless.