[Prof. Templeton]

I'm sure this has been posted in some form or another, but in the spirit of Halloween here goes...

A Fairy princess, a Ghost, a Goblin, a Mummy, and a Vampire all sat down to split up their accumulated wealth a Halloween candy. It was agreed that they would proceed alphabetically and make a proposal on how to divide the candy. If the proposal received a majority vote in favor, it was accepted and put into action. If the proposal failed to get a majority of votes the person making the suggestion would lose their voting privilege and the next person alphabetically on the list would get a turn. All persons are completely greedy and completely logical.

How does the Fairy princess’ propose the candy be split?

[Guest]

98, 0, 1, 0, 1

[unreality]

if I was stylish enough to pull off the Fairy Princess look, here's what I would do:

if i propose to keep all of it, I'll be shot down. Clearly I just need to propose enough to get 3 of 5 people to agree with me. That means I only need 2 of my 4 peers to be on board with the plan, getting them equal or more than they could've gotten by voting against.

so it comes down to how much Ghost could get if I was voted down, and his potential candy income comes down to how much Goblin could make, etc. So we have to go all the way down then work back up.

If it ends up as just Mummy and Vampire, Vampire can vote against anything the Mummy says, denying the majority, then the Vampire gets everything. So the Mummy doesn't want the vote to get to him. If the Goblin offers the Mummy just 1/10000 of the candy, the Mummy should accept (since you seem to imply that an irrational lashing back is against their character) the Mummy will accept and Goblin will get 999/1000, essentially all of it, and the Mummy will get an infinitesimal, and the Vampire will get nothing.

Knowing this, if Ghost is in the hot seat, he can't get the Goblin's vote. But Vampire would vote for Ghost if Ghost gives the Vampire even the tiniest amount. But that's only 2/4 so Ghost also needs the Mummy. So Ghost needs to offer an infinitesimal to the Vampire and a miniscule (but larger than infinitesimal) amount to the Mummy. Goblin votes against either way, and it may come down to an auction between Ghost and Goblin trying to get Mummy to vote for or against depending on how much they offer. But they are all perfectly logical, so if Goblin wins the auction, he can offer Mummy a tiny tiny amount (less than promised) and it would still be in the Mummy's best interests to accept, so the Mummy would go with the Ghost no matter what, and so there wouldn't be an auction, the Mummy would just go with the Ghost assuming the Ghost can provide a non-infitesimal amount, but still small, say between 1/10 and 1/100.

So Ghost gets almost all, Mummy gets some, Vampire gets a tiny amount, and Goblin gets none.

So when I, the Fairy Princess, am up first and foremost, I can get the Goblin's vote by offering him an infinitesimal amount. Then I should offer the Vampire a slightly-larger than infintesimal amount, more than the Ghost would just for a vote.

So if I give 98% to myself, 1% to the Vampire and 1% to the Goblin, I can probably get away with it.

The problem here is there are no whole numbers of candy pieces to work with, so it comes down to if a person thinks that a very small fraction would be less or more than another very small fraction.

edit - typo

Edited October 22, 2009 by unreality

[Guest]

I dont know if this is correct but since fairy princess gets the first chance & shud get majority of the votes and everyone is greedy, she can propose that she gets 1/4th of the candy along with 3 others , say the ghost, the goblin and the mummy... and these 3 people will obviously agree since they get 1/4th instead of 1/5th if they split legally... thus fairy princess' proposal can get the majority

[Prof. Templeton]

98, 0, 1, 0, 1

The vampire could do just as well if he votes against this proposal as for it.

[Guest]

The vampire could do just as well if he votes against this proposal as for it.

Didn't get you.

[unreality]

I dont know if this is correct but since fairy princess gets the first chance & shud get majority of the votes and everyone is greedy, she can propose that she gets 1/4th of the candy along with 3 others , say the ghost, the goblin and the mummy... and these 3 people will obviously agree since they get 1/4th instead of 1/5th if they split legally... thus fairy princess' proposal can get the majority

yes but remember that she is both greedy and logical: through logic she knows that she can get even higher than 1/4, and through greed she will go for it.

[Prof. Templeton]

Didn't get you.

If he says no to the F.P.'s plan the ghost will offer him the same thing that she just did, so where's his incentive to agree. I'd give him 2 pieces instead of 1.

[Guest]

If he says no to the F.P.'s plan the ghost will offer him the same thing that she just did, so where's his incentive to agree. I'd give him 2 pieces instead of 1.

The Ghost will offer the mummy 1 candy. Mummy's and his vote will make the Ghost win and Vampire will get nothing.

So FP has to give 1 candy to vampire.

[superprismatic]

Assume N pieces of identically desired candy pieces where N

is large compared to 5. Below can be easily modified it

they are not identically desired.

If Mummy gets to propose, Vampire will vote against it no

matter what it is so that Mummy will not get a majority and

then Vampire, being the only one left will, of course,

vote on Vampire's proposal. Vampire will propose that Vampire

get it all. So if (F,Gh,Go,M,V) is a vector representing

who gets how much candy, if Mummy gets to propose it will be

(0,0,0,0,N).

Therefore, Mummy does not want the proposing to get to him.

Goblin knows this. If the proposing gets to Goblin, then,

he should propose that Vampire gets nothing, Mummy gets 1

piece, and Goblin gets the rest. He knows that Mummy will

maximize Mummy's candy by voting for this proposal of

Goblin's. In this case the vector is (0,0,N-1,1,0)

Ghost knows this as does Mummy and Vampire. So, if the

proposing gets to Ghost, He should offer Vampire 1 to

get his vote (otherwise Goblin proposes and Vampire gets

nothing). He should offer 2 to Mummy to insure that

Mummy would rather chose his offer to Goblin's possible

offer of 1. He then would get a majority and the

candy payoff vector in this case is (0,N-3,0,2,1).

Now, everybody know this. Clearly, Fairy princess need

only sweeten the offer to Mummy and Vampire to get a

majority. So, she should sweeten Ghost's possible offer

by offering Vampire 2 and Mummy 3 and get the rest for

herself. The candy payoff vector would then be (N-5,0,0,3,2).

[Prof. Templeton]

The Ghost will offer the mummy 1 candy. Mummy's and his vote will make the Ghost win and Vampire will get nothing.

So FP has to give 1 candy to vampire.

More than 2 votes are required. He'd give 1 to the Mummy and also 1 to the Vampire. The F.P. would have to do better for the Vampire, so he gets 2.

[Guest]

Well if i were the fairy i would arrange an agreement to split it equally between me the ghost and mr dracula, because we are the only ones who can fly, like that we could just fly away with the "gil". Assuming that this are real monsters. If we are taking about cos-play, i would say that the fairy should arrange an agreement with the mummy and the vampyre, because theese two are the ones that could get less candy and are in the worst places. IF this was too bad: sorry first riddle that i'm trying

[Guest]

Superprismatic, your explanation is thorough and easy to follow

If Fairy is perfectly greedy/logical, he/she would see that Goblin will vote for any proposal that will net at least 1 piece. Therefore, Fairy only has to offer Goblin that one piece and then sweeten the deal for Vampire, giving [N-3, 0, 1, 0, 2].

[superprismatic]

Superprismatic, your explanation is thorough and easy to follow

If Fairy is perfectly greedy/logical, he/she would see that Goblin will vote for any proposal that will net at least 1 piece. Therefore, Fairy only has to offer Goblin that one piece and then sweeten the deal for Vampire, giving [N-3, 0, 1, 0, 2].

Thanks, your modification is better than my solution. I just missed it!

[Guest]

More than 2 votes are required. He'd give 1 to the Mummy and also 1 to the Vampire. The F.P. would have to do better for the Vampire, so he gets 2.

Okay. It depends on how you define majority in case of a tie.

Anyways.. The idea is same, we can work our way out as par the definiton.

[Guest]

1/3 for her 1/3 for the Ghost and 1/3 for the Goblin. Majority of votes is granted, since they are greedy and logical enough to prefer 1/3 over 1/5. This will bring the voting outcome to 3 in favor, 2 against.

[tiger_lily111]

The Ghost is incapable of eating candy because it has no substance.

The Mummy can't eat candy because its mouth is bandaged over permanently.

The Vampire won't eat candy because it only drinks blood, and none of the candy contains blood.

The Fairy Princess & the Goblin are the only ones that could eat the candy, but because they're all greedy, everyone wants some of it. Therefore, the Fairy Princess should magic all the candy away from everyone and disappear before they can catch her!

[Guest]

Actually, this could never happen since they are all perfectly logical and therefore all do this analysis and would therefore NEVER agree to the original conditions!!!! Oh and tiger, come on, we are talking about people dressed up, because otherwise how would a ghost even hold a bag to collect the candy and why wouldn't the Goblin eat all its candy as it got it and you know that fairy princess either would never eat candy to maintain her figure ...

[Guest]

Oh and tiger, come on, we are talking about people dressed up, because otherwise how would a ghost even hold a bag to collect the candy

If Patrick Swayze can lift a penny, then Casper can hold a bag

Fairy takes 30% and gives 35% to 2 others, She knows the only way for her to get more then 25% is making 2 others vote with her. She bribes them with a higher ammount then they could ever get with out her