Problem 23. Dividing the Gold Coins
Halter Hight and four of his friend wish to divide 100 gold coins among themselves (5 people).
Each time they do the following:
The oldest person decides how many coins should be given to whom.
Everyone (including the oldest person) votes to accept the division or reject it.
If at least half of the votes are positive, the division will be accordingly, otherwise, they will kill the oldest person and repeat the process.
Assuming that everyone decides optimally, how many coins does each person receive?
Everyone’s priorities are as follows:
To survive (highest priority)
To receive more money (second priority)
If the above 2 are equal, make a decision that kills the most people.
Link to the problem on Twitter: https://twitter.com/Riazi_Cafe/status/1689531391149842432
Let the people involved in this process be numbered from 1 to 5 from the youngest to the oldest. The correct answer is the following: The fifth person (the oldest) takes 98 coins for himself, and gives one coin to each of the first and third persons.
The solution is based on a bottom up analysis:
If there are 2 people left, one vote is enough for the second person to survive. This would be his own vote. So the oldest person can take all 100 coins for himself.
If there are 3 people left, the oldest person needs 2 votes to survive. The first person also knows that in the case of 2 people, nothing will be given to him; Thus he would be happy with receiving only one coin. Therefore, the third person takes 99 coins for himself and gives one coin to the first person.
If there are 4 people left, with the same reasoning, the second person will be happy with receiving one coin, and thus the fourth person will get 99 coins for himself. He takes, one for the second person, and the first and third person will receive nothing.
For 5 people, the first and the third persons know that if the fifth person dies, they will receive nothing. Thus, they are happy with receiving a single coin. Therefore, the fifth person takes 98 coins for himself, and gives one to each of the first and third persons.