tag:blogger.com,1999:blog-5545377529544965534.post4164270115112371282..comments2016-05-03T13:46:55.707-07:00Comments on Math Is Crazy: Pirate Gold ProblemMadhu Sudan Sigdelhttps://plus.google.com/104387178077515574864noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-5545377529544965534.post-60437630785577524132014-04-24T12:59:49.638-07:002014-04-24T12:59:49.638-07:00It might be expected intuitively that Pirate A wil...It might be expected intuitively that Pirate A will have to allocate little if any to himself for fear of being voted off so that there are fewer pirates to share between. However, this is quite far from the theoretical result.<br /><br />This is apparent if we work backwards: if all except D and E have been thrown overboard, D proposes 100 for himself and 0 for E. He has the casting vote, and so this is the allocation.<br /><br />If there are three left (C, D and E) C knows that D will offer E 0 in the next round; therefore, C has to offer E 1 coin in this round to make E vote with him, and get his allocation through. Therefore, when only three are left the allocation is C:99, D:0, E:1.<br /><br />If B, C, D and E remain, B considers what will happen if he is thrown overboard, when he makes his decision. To avoid being thrown overboard, he can simply offer 1 to D. Because he has the casting vote, the support only by D is sufficient. Thus he proposes B:99, C:0, D:1, E:0. One might consider proposing B:99, C:0, D:0, E:1, as E knows he won't get more, if any, if he throws B overboard. But, as each pirate is eager to throw each other overboard, E would prefer to kill B, to get the same amount of gold from C.<br /><br />Assuming A knows all these things, he can count on C and E's support for the following allocation, which is the final solution:<br /><br />A: 98 coins<br />B: 0 coins<br />C: 1 coin<br />D: 0 coins<br />E: 1 coin<br /><br />Also, A:98, B:0, C:0, D:1, E:1 or other variants are not good enough, as D would rather throw A overboard to get the same amount of gold from B.अविरलhttps://www.blogger.com/profile/07734096632696225442noreply@blogger.com