Game Theory

3. 9 pirates are trying to share 100 recently plundered gold coins. Pirate sharing rules are complicated, and involve seniority and voting. The pirates are ordered by seniority so that the captain (player 1) goes first, followed by the first-mate (player 2), through to the lowly cabin-boy (player 9). The rules of the vote are that the most senior pirate currently aboard ship proposes a split of the coins among the other pirates (coins can not be cut in half or divided further). Everyone votes, and so long as a majority (a fraction equal to or greater than one half) vote "Aye!" then that share is enacted and the gold is split. If more than a half vote "nay" then the pirate who proposed the share is thrown overboard, and the next-most-senior pirate proposes a split among those remaining. Pirates are assumed to be a mercenary lot. Their worst outcome is being thrown overboard, but long as they're alive they enjoy two things: i) Gold coins and ii) secing their friends thrown overboard. However, they value gold disproportionately and would not trade a single coin for the enjoyment of seeing all their friends thrown overboard. (a) What is the sub-game perfect outcome of this game? (b) What is the sub-game perfect outcome for a ship with 100 pirates aboard? (c) Suppose instead that a strict majority were required to pass a split (so if a half or more vote 'nay' then the proposer is thrown overboard). What is the sub-game perfect outcome for 5 pirates now