I need to solve following logical task:
10 people sit on the Board of Directors of a company that recieved 1 share as a profit. There is a linear hierarchy among directors: from the most senior (chief) to the most junior. The profits are shared as follows: the chief proposes who to give a share to, then everyone, including the chief, votes "for" or "against". If at least half of the directors vote "for", the decision of the chief is accepted, but if not, the chief is excluded from the Board of Directors, the next in the hierarchy becomes the chief and the process starts from the beginning according to the same principle. Everyone wants to stay on the Board of Directors, and also wants to get a share. All other things being equal, a director will vote for the option that leaves fewer colleagues on the Board. How many directors will remain on the Board after profit sharing?
My answer is 4, because by $\geqslant5$ all, except chief and new share holder would vote "against". And by 4 two of them would vote "for" to get the share and not to be excluded.
But I can't prove it more clearly. Any ideas?
Assume n directors, n >= 1.
If n = 1 or n = 2, the chief wins the vote and gets the share.
If n = 3 the chief can’t propose himself because he will just be voted out. If he proposes the lowest ranking board member, he’ll get two votes. He’ll keep his job, and the lowest ranking director gets the share. If he proposes the middle board member, he’ll vote for this, the middle member will vote against, and since the lowest board member wants to reduce the numbers, he’ll vote against. The chief loses the job and #2 will first become chief and then get the share. The best the chief can do is vote for the lowest board member and keep his job, while the lowest board member gets the share.
If n = 4, the board consists of chief, x, y, and z. If the chief gets voted out, x becomes chief and z gets the share; y will get nothing. So if the chief proposes y, chief and y will vote for it, chief stays chief and y gets the share. If he proposes z then z knows he gets the share anyway, so z votes against. y votes against because he can’t get anything, and x votes against, so x becomes chief and z gets the share. If he proposes x, then y and z will vote against. x has the choice of voting “for” and getting the share, or voting against and becoming chief, he’ll choose what he values more. So the safe bet for the chief is proposing y and remain chief, while y gets the share.
If n >= 5 then a majority will vote the chief out. This repeats, so the end result is that the board will be reduced to the four lowest ranking board members chief, x, y and z. Chief will remain chief, and y gets the share.
So you were almost correct, except it will be the second lowest board member who gets the share.