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海盗分金币

Screwy Pirates

专题
Brainteaser / 脑筋急转弯
难度
L2

题目详情

五名海盗抢到 100 枚金币，按规则分配：资历最高者提出方案，所有人投票；若赞成票不少于一半（5 人时至少 3 票）则通过，否则最高者被扔下海，下一位重复。

海盗都完全理性：优先活命，其次金币越多越好；若两种结果自己金币相同，则更偏好船上人数更少。

问：最终金币如何分配？

Five pirates looted a chest full of 100 gold coins. Being a bunch of democratic pirates, they agree on the following method to divide the loot:

The most senior pirate will propose a distribution of the coins. All pirates, including the most senior pirate, will vote. If at least 50% of the pirates (3 out of 5) accept the proposal, the gold is divided as proposed. If not, the most senior pirate will be fed to the sharks, and the process starts again with the next most senior pirate... This is repeated until a plan is approved. You can assume all pirates are perfectly rational: they want to stay alive first and then get as much gold as possible. Finally, being blood-thirsty pirates, they prefer fewer pirates on the boat if they end up equally wealthy in either outcome.

How will the gold coins be divided in the end?

解析

最终分配（按资历从高到低）为 (98,0,1,0,1)(98,0,1,0,1)。

倒推：

  • 2 人时，最高者自己一票即可通过：(100,0)(100,0)。
  • 3 人时，最高者用 1 枚贿赂最年轻者即可：(99,0,1)(99,0,1)。
  • 4 人时，只需贿赂一个在下一轮会拿 0 的人：(99,0,1,0)(99,0,1,0)。
  • 5 人时，需要 3 票（含自己），贿赂两个在下一轮会拿 0 的人，各给 1：(98,0,1,0,1)(98,0,1,0,1)。

Original Explanation

The final distribution is (98, 0, 1, 0, 1) (corresponding to the five pirates in order of seniority). The reasoning goes by backward induction:

  1. If only 1 pirate remains, he takes all 100 coins.
  2. If there are 2 pirates, the senior one needs just his own vote, so he takes all (100, 0).
  3. If there are 3 pirates, the senior pirate can bribe the most junior pirate with 1 coin (99, 0, 1).
  4. If there are 4 pirates, the senior pirate can bribe one of the junior pirates with 1 coin (99, 0, 1, 0).
  5. If there are 5 pirates, the senior pirate needs 2 total votes (including his own). He can bribe the 3rd and 5th pirates with 1 coin each, resulting in (98, 0, 1, 0, 1).