If only pirate 5 were left he would get everything.
If only pirate 4 and 5 were left then pirate 4 has no hope, regardless of what he suggests pirate 5 will vote to throw him overboard, and overboard he will go. Thus it doesn't make any difference what pirate 4 suggests.
If only pirates 3,4, and 5 were left then pirate 3 could count on pirate 4's support as long as pirate 4 got at least 1 coin. This is because if pirate 3 goes overboard pirate 4 knows he will get nothing. So pirate 3 would suggest (0,1,999) to which he and pirate 4 would vote 'yes.' Note that according to the problem pirate 4 would vote 'no' to (0,0,1000), even though a 'yes' vote would save his own life his pleasure of throwing pirate 3 over would be worth his own life.
If only pirates 2,3,4, and 5 were left then pirate 2 could count on pirate 3 voting 'no' unless pirate 3 got more than 999 coins, which would leave everybody else with 0, which obviously wouldn't pass. So pirate 2 should not count on pirate 3's support so he shouldn't waste any coins on him. If pirate 2 goes overboard he knows the money will be distributed (0,1,999). So if he offers 2 coins to pirate 4 and 1 coin to pirate 5 then he will get their support since they will come out better voting 'yes.' Thus pirate 2's suggestion would be (1,2,0,997) to which pirates 2, 4 and 5 would vote 'yes.'
Thus if all five pirates were left pirate 1 could count on pirate 2 voting 'no' since pirate 2 would get 997 if pirate 1 went overboard. So pirate 1 should not waste any coins on pirate 2 since he will vote 'no' anyway and concentrate on winning 2 votes from the other 3. If pirate 1 goes overboard the money will be distributed (1,2,0,997). So it would cost 2 coins to get pirate 5's support, 3 coins to get priate 4's support, and 1 coin to get pirate 3's support. Since he only needs to buy two votes the cheapest two are pirates 5 and 3. Thus pirate 1 would suggest (2,0,1,0,997) to which pirate 1, 3, and 5 would vote 'yes'.
Thanks to The Ultimate Puzzle Site which has also posted this problem, click here to see their solution if this one was too confusing.
Michael Shackleford, A.S.A.