Halloween Treats

[mithrandir]

Possible solution below:
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Ok, clearly if it's Jar Jar's turn, he give it all to himself. So, if it's the goblin's turn, Jar Jar will turn down anything short of everything. Now, if it's the ghoul's turn, the goblin will accept any plan where he gets at least some candy, so the ghoul will give him 1, and take 99 for himself. The ghost has to get three votes for his plan to succeed, the best way to do this is to give 2 to the goblin, who would otherwise get 1, 1 to Jar Jar, who would otherwise get nothing, and 97 to himself. So the fairy princess' best plan is: 97 to self, 0 to ghost, 1 to ghoul (who otherwise gets nothing), 0 to goblin, and 2 to Jar Jar (who gets 1 otherwise), thus getting the majority of votes.

Summary:
Jar Jar: 0,0,0,0,100
Goblin: doesnt matter
Ghoul: 0,0,99,1,0
Ghost: 0,97,0,2,1
Fairy Princess: 97,0,1,0,2

[Ghost Post]

I think that you've got it.

I had originally thought 95, 0, 0 , 3, 2, but 97, 0, 1, 0, 2 is clearly better.

Nick

[Judge Phred]

I got mithrandir's answer first time too.

Then I want back to the question to look for ambiguities, and to work out if any other solution could be argued for. None found.

We seem to have a well written question and a perfect answer. Wow. I bet there won't be ny more posts at all on this topic.

[Judge Phred]

PS - And all before araya could get a word in!

[This message has been edited by Judge Phred (edited 10-29-1999).]

[firemeboy]

Maybe I missed something. Dosen't the puzze say that if there is not a majority, then the proposing person loses their vote, and does not get candy. IF the princess were to propose getting 97, then the others would refuse, the vote would be over. The princess would be out of the loop, and they would propose a new plan starting with the next one. Is this not how it works?

I stand corrected. I have never seen a puzzle like this and the answer was quite interesting. I have gone over it all and see that this is indeed the answer. Cool.

[This message has been edited by firemeboy (edited 10-29-1999).]

[araya]

laff phred, why would I say anything when the first post is the correct answer?

besides, I was in seattle yesterday

[Mulky]

Wait a minute, if the fairy princess decides to give herself the majority of the candy, I don't think the others will agree. I think they will all try and cut as many people out as possible so they can get themselves a bigger share.

Example:

The fairy princess decides to split the candy between herself, the ghost and the ghoul. She gets 34 bars, the other two 33 each. This way she will hopefully get the vote of the majority, because they get more by splitting between three, than five.

If she gave 97 bars to herself, why doesn't the majority say 'no way sister!' and cut her out of the spoils?

If they all play thier cards right, I reckon the goblin will get 99 and jar jar just 1, I disagree jar jar can get the full 100, at most he can get 50 (if the goblin screws up)

[This message has been edited by Mulky (edited 10-31-1999).]

[agraco]

The problem is best solved (IMHO) by going backwards like Mith did. Let me try and explain what he did in step by step fashion.

If you don't understand the solution, take a pencil and paper and follow the steps below. As soon as you get stuck, you can identify it on the board.

1. Lets imagine that JJ is the only vote that counts. He will give himself 100.
(0,0,0,0,100)

2. Now we add the Goblin and JJ. Say the Goblin chooses a 1/99 split in favour of JJ. He still won't get JJ vote because JJ can get 100 by refusing.
(0,0,0,0,100)

3. Ghoul/Goblin/JJ. The Ghoul knows that Goblin is stuck. Ghoul could give any amount to Goblin and still get his vote. One piece of candy would do it and the vote is 2 VS 1.
(0,0,99,1,0) 2 VS 1 vote

4. Ghost/Ghoul/Goblin/JJ. Here is where it gets a bit dicey. The Goblin is gauranteed 1 piece when the Ghoul is in charge. The Ghost will get the Goblin's support by giving TWO pieces. The Ghost still needs another vote. He gives JJ one piece. Why? When the Ghoul is in charge, JJ gets only 0.
(0,97,0,2,1) 3 VS 1 vote

5. FP/Ghost/Ghoul/Goblin/JJ. OK, the Ghost will refuse anything less than 97. There is no point in going after his vote (your solution tried to do this). The Ghoul will get nothing from the Ghost. FP takes advantage of this and gets his vote with one piece. JJ is assured one piece from the Ghost so FP has to pay him off with a premium: two pieces.
(97,0,1,0,2) 3 VS 2 vote


Mith's summary:
Jar Jar: 0,0,0,0,100
Goblin: 0,0,0,0,100
Ghoul: 0,0,99,1,0
Ghost: 0,97,0,2,1
Fairy Princess: 97,0,1,0,2

You can read up on backwards induction in most standard game theory book. This puzzle is a classic case of perfect information and no chance for mistake by any player. cheers!

-Alexandre.

[This message has been edited by agraco (edited 10-31-1999).]

[Ghost Post]

Re: Agraco's posting of 10-31-1999 01:10 PM

I believe the solution to be

Fairy Princess should propose a 98,0,0,1,1 split.

Agraco's logic was fine up until step 4.
Ghost/Ghoul/Goblin/JJ
If the ghost proposes [0,] 98,0,1,1 then there is no reason for the Goblin to vote against it. It will not result in more candy for the Goblin if the Ghost is eliminated. There is no need to offer the inducement of extra candy.

Using the same logic, the Fairy Princess can propose 98, 0, 0, 1, 1 knowing that the Goblin and JJ will vote yes as they can't do better by voting no.


[This message has been edited by Sean (edited 11-01-1999).]

[agraco]

Hi there

You are picking up on a nuance of game theory. Lets go back to step 4.

Ghost/Ghoul/Goblin/JJ where you said: 0, 98,0,1,1

The nuance is this: Goblin is indifferent between this solution and the one in step 3. He has no compelling reason to accept the Ghost's offer. To ensure his vote, Ghost pays a premium to Goblin: 0,97,0,2,1.

The same logic extends to the FP. You don't want people to be indifferent to your offer if they can get the same elsewhere; you want them to regret refusing you to ensure their vote.

-Alexandre.

[Amy]

I think Mulky was confused about the definition of "majority." It means more than 50%--if the votes split evenly, the plan is defeated. There's no Vice President to break the tie.

[vox]

All of the above assumes that side deals are not allowed (not expressly forbidden in problem set-up). For example, in the paradigm:
0,0,0,0,100
0,0,0,0,100
0,0,99,1,0
0,97,0,2,1
97,0,1,0,2
JJ's vote is crucial to the success of both FP's and Ghost's proposals. Suppose JJ makes a deal with Ghoul that he (JJ) will vote against both of the first 2 proposals and for Ghoul's proposal if Ghoul will propose 0,0,2,0,98. This way, Ghoul does better than under FP's or Ghost's plan and will probably accept.

[Andy]

All very well, in principle. What's to keep Ghoul from changing his mind after FP's and Ghost's proposals have been defeated? The puzzle doesn't forbid side deals, but it also doesn't specify that they can be enforced; the participants are greedy and logical, but not necessarily honest - and the actual division of goodies is to be based on the proposal that's made in turn and accepted by a majority, not one that's offered out of turn and cannot be voted on. Maybe I'm just being too cynical.