Explaining the 2-pirate ending in more detail
Apparently I was insufficiently clear. If it comes down to two pirates, pirate #1 will indeed get all the gold. This can happen in either of two ways.
Originally Posted by icaro
But unfortunately for pirate #2, there is no way to know which of the two ways listed above will be chosen by pirate #1. They are equally attractive to #1 if his only motivations are survival and greed, so from #1's viewpoint he may as well flip a coin and choose one at random. If it comes down to two pirates, #2 is entirely dependent on the whim of #1 to survive, even if he offers all the gold to #1. #2 cannot accept any uncertainty when his life is in the balance. Therefore he must not allow the pirate population to be reduced to two.
- Pirate #2 gives all the gold to pirate #1, and pirate #1 decides to vote in favor, or
- Pirate #2 does whatever he likes, pirate #1 votes against him, pirate #2 is killed, and pirate #1 ends up the only survivor with all the gold.
In some formulations of the puzzle (please read this whole thread for details) the bloodlust motivation has been added to resolve this ambiguity, but it's not really necessary. #2 simply can't allow his life to depend on a coin flip. He would rather #3 get all the gold with a certainty that #2 lives, than #1 get all the gold with a possibility that #2 dies.
I'm not sure what you're getting at here. #3 doesn't have the option to kill #4 if #4 can convince #1 and #2 to vote for #4's proposal, which he can do by offering them each 1 gold. (If they don't vote in favor of #4's proposal they will get nothing because #3 will keep all the gold if it comes down to 3 pirates.)
Originally Posted by icaro
Survival Skews the result
Let's go back and look at pirate #2. If #5 offers him nothing, he might as well vote to kill him because there's a chance #4 will offer him something. Remember, according to the "rules" put out there for survival, a pirate will do anything it takes to live, but once life is assured there's no gaurantee he won't roll the dice trying to get more money. Effectively, #5's only motivation is to make as sure as possible that two pirates vote for his plan. #2 two could easily vote "no" to #5's plan, hoping to get more from pirate #4. To reach a definite answer, a rule must be placed on the voting:
A pirate will vote "yes" to any plan which offers it at least 1 more gold than he would be guaranteed otherwise.
If the pirates are concerned with survival at all, you get all kinds of wacky answers. A better wording of the question is: If at least half of the remaining crew do not vote for a pirates plan, he gets nothing and can not vote on the other plans. This way they are not fearing for their lives, and greed is the only motivation.
The answer with the modifications is, as has already been stated, 97, 0, 1, 0, 2.
How about the modifications? Pirates vote in secret
If a pirate votes "no" to a plan that gets a "yes" overall, he gets nothing
If a pirate votes "yes" to a plan that gets a "no" overall, he's taken out of the rest of the voting.