Let them eat cake... fairly.
January 15, 2007 4:09 AM Subscribe
Mathematicians in the 1940s became curious about Fair Division, thus birthing an entire branch of mathematics concerned with cutting cakes. Recently, this man came up with a new method, purported to be the most fair yet. Hard to disbelieve, coming from the topologist who has mastered shoelaces, although arguably he's missing the point.
Why, veedubya, that's what non-standard analysis is for!
posted by eparchos at 4:36 AM on January 15, 2007
posted by eparchos at 4:36 AM on January 15, 2007
Alice cuts [into what she thinks are thirds].
Betty trims one piece [to create a 2-way tie for largest], and sets the trimmings aside.
Let Chuck pick a piece, then Betty, then Alice.
Require Betty to take a trimmed piece if Charlie does not.
Call the person who tooked the trimmed piece T, and the other (of Betty and Chuck) NT.
To deal with the trimmings, let NT cut them [into what she thinks are thirds].
Let players pick pieces in this order: T, Alice, then NT.
What happens if Alice faints from hunger before getting her piece?
posted by three blind mice at 4:55 AM on January 15, 2007
The trick is to just eat the cake while all of the mathematicians are arguing.
</obligatorymathematiciansdontliveintherealworldcomment>
posted by chrismear at 4:55 AM on January 15, 2007
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posted by chrismear at 4:55 AM on January 15, 2007
Wait, if everyone gets at least 50% of the cake, how is there any surplus?
posted by nebulawindphone at 5:31 AM on January 15, 2007
posted by nebulawindphone at 5:31 AM on January 15, 2007
"As they value it". If I think sprinkles are the best and you think icing is the best, if I get 50% of the cake but there's more sprinkles on it, I'll think I got a better deal. If you get 50% of the cake but it has less icing on it than my piece, you'll feel you got less than 50%.
posted by eparchos at 5:33 AM on January 15, 2007
posted by eparchos at 5:33 AM on January 15, 2007
Sure eparchos, but in real life, everyone just likes the icing. :)
posted by Malor at 5:48 AM on January 15, 2007
posted by Malor at 5:48 AM on January 15, 2007
Let me eat cake.
posted by LordSludge at 6:06 AM on January 15, 2007
posted by LordSludge at 6:06 AM on January 15, 2007
I thought personal pan pizza did away with this sort of thing. And isn't the minimum wage supposed to be raised?
posted by Smart Dalek at 6:13 AM on January 15, 2007
posted by Smart Dalek at 6:13 AM on January 15, 2007
But I wanted flapjacks.
On a stick. With a creamy sausage filling.
posted by IronLizard at 6:49 AM on January 15, 2007
On a stick. With a creamy sausage filling.
posted by IronLizard at 6:49 AM on January 15, 2007
For those interested in tying shoelaces, take a look at this.
posted by Enron Hubbard at 7:00 AM on January 15, 2007
posted by Enron Hubbard at 7:00 AM on January 15, 2007
Polster's website had a link to a page with a limerick in the form of an integral. I never saw one before. It's supposedly true, but it scans for crap.
Unfortunately I don't have the HTML skills to fit it into the margin here.
posted by MtDewd at 7:08 AM on January 15, 2007
Unfortunately I don't have the HTML skills to fit it into the margin here.
posted by MtDewd at 7:08 AM on January 15, 2007
An economics textbook that I read in first year discussed the hypothetical situation of having to divide a communally purchased bag of marijuana.
The easiest option: have one person divide it into piles, then let everyone else pick a pile in random order with the person who did the dividing choosing last. This creates the incentive for that person to make the divisions as even as possible, since the best they can do is get a perfectly equal share.
Children use this "cut and choose" procedure, also, but have less Greek symbols to use in analyzing it.
posted by sindark at 7:14 AM on January 15, 2007
The easiest option: have one person divide it into piles, then let everyone else pick a pile in random order with the person who did the dividing choosing last. This creates the incentive for that person to make the divisions as even as possible, since the best they can do is get a perfectly equal share.
Children use this "cut and choose" procedure, also, but have less Greek symbols to use in analyzing it.
posted by sindark at 7:14 AM on January 15, 2007
What is the square root of 69?
Ate something.
posted by LordSludge at 7:27 AM on January 15, 2007 [2 favorites]
Ate something.
posted by LordSludge at 7:27 AM on January 15, 2007 [2 favorites]
Fair division is actually a really interesting subject, but highly impractical.
For instance, to fairly divide a pile of things among n people, you need n-1 rounds of some process. However, at the end of it, everyone has at least 1/n of the value of the pile, as they see it.
The major stumbling block to actually using these, besides the time component, is that people don't generally trust them. It is hard to perceive how 10 people could all receive 80% of the value of something (assuming of course that their interests are varied). People also have the impression that telling the truth in a negotiation makes them a sucker, which is not true in these methods - truth is a minimax strategy (you'll always get at least 1/n of the value of the thing) - but you can do better with other strategies given more information and such.
posted by TypographicalError at 7:47 AM on January 15, 2007
For instance, to fairly divide a pile of things among n people, you need n-1 rounds of some process. However, at the end of it, everyone has at least 1/n of the value of the pile, as they see it.
The major stumbling block to actually using these, besides the time component, is that people don't generally trust them. It is hard to perceive how 10 people could all receive 80% of the value of something (assuming of course that their interests are varied). People also have the impression that telling the truth in a negotiation makes them a sucker, which is not true in these methods - truth is a minimax strategy (you'll always get at least 1/n of the value of the thing) - but you can do better with other strategies given more information and such.
posted by TypographicalError at 7:47 AM on January 15, 2007
TypographicalError
There are some interesting applications of Fair Division, especially recently. In particular, applying these notions of "fairness" and "envy" to dispute resolution (.pdf).
Most math is highly impractical, and the funny thing about fair division in cake-cutting algorithms is that it was originally conceived as a sort of joke, but it seems to actually be much more practical than most areas of math. Certainly it's doing well for an only 50-year-old field.
posted by eparchos at 8:38 AM on January 15, 2007
There are some interesting applications of Fair Division, especially recently. In particular, applying these notions of "fairness" and "envy" to dispute resolution (.pdf).
Most math is highly impractical, and the funny thing about fair division in cake-cutting algorithms is that it was originally conceived as a sort of joke, but it seems to actually be much more practical than most areas of math. Certainly it's doing well for an only 50-year-old field.
posted by eparchos at 8:38 AM on January 15, 2007
hah, my mom used the fair division algorithm for me and my sister when we were little. I remember once when we were really little, my sister divided the candy bar like 80% 20%, then got all disappointed when I took the 80% piece.
posted by delmoi at 8:39 AM on January 15, 2007 [1 favorite]
posted by delmoi at 8:39 AM on January 15, 2007 [1 favorite]
Our childhood non-mathematical technique for dividing treats of any kind between two people, was to have one person divide it in half and the other person choose their piece first. With a particularly desirable treat, I doubt you would find more precise cutting from an eye surgeon...
posted by fairmettle at 8:51 AM on January 15, 2007
posted by fairmettle at 8:51 AM on January 15, 2007
Death, please.
posted by jeversol at 9:53 AM on January 15, 2007 [1 favorite]
posted by jeversol at 9:53 AM on January 15, 2007 [1 favorite]
The problem is gaming. If someone casually lets it be known that they desire the icing portion with the cherry on it, then someone else gets the notion that the cherry notion is the best part and makes it harder to get. Or someone else might exploit this free information to bluff the person for sacrificing more for the cherry. So it involves second-guessing and the most skilled bargainer with the most information complicates the process. I think that dividing the cake (or estate as it is practically presumed to be), should be done by artificial points awarded equally, and then a blind bidding process is employed. But this gets into another theoretical problem area, because some suggest that the most fair bid to award is to the highest bid winner, but who pays the second-highest bid price.
posted by Brian B. at 9:59 AM on January 15, 2007 [1 favorite]
posted by Brian B. at 9:59 AM on January 15, 2007 [1 favorite]
I think folks should be grateful to have cake at all, and that they live in a time when how much fucking cake you get is an acceptable use of scientists' problem-solving skills.
posted by eustacescrubb at 10:06 AM on January 15, 2007
posted by eustacescrubb at 10:06 AM on January 15, 2007
> Our childhood non-mathematical technique for dividing treats of any kind between two people, was
> to have one person divide it in half and the other person choose their piece first. With a particularly
> desirable treat, I doubt you would find more precise cutting from an eye surgeon...
As soon as they make me ruler of the universe, I'm going to solve Israel/Palestine exactly that way. Anyone who won't play along, that's what the thunderbolts are for.
posted by jfuller at 10:19 AM on January 15, 2007
> to have one person divide it in half and the other person choose their piece first. With a particularly
> desirable treat, I doubt you would find more precise cutting from an eye surgeon...
As soon as they make me ruler of the universe, I'm going to solve Israel/Palestine exactly that way. Anyone who won't play along, that's what the thunderbolts are for.
posted by jfuller at 10:19 AM on January 15, 2007
You laugh now, but you'll be sorry when this comes up in your Google interview.
posted by StrangerInAStrainedLand at 10:43 AM on January 15, 2007
posted by StrangerInAStrainedLand at 10:43 AM on January 15, 2007
Related problem discussed on AskMe a while back.
posted by DevilsAdvocate at 11:59 AM on January 15, 2007
posted by DevilsAdvocate at 11:59 AM on January 15, 2007
I've been waiting for an excuse to use the Knaster inheritance procedure, since it looks interesting. Maybe it would make "secret Santa" more fun.
posted by ctmf at 6:45 PM on January 15, 2007
posted by ctmf at 6:45 PM on January 15, 2007
Indeed tadellin, everything fun is a waste of time. I actually found the rented house problem more interesting. I wondered how to use a ranking system that didn't game, ending up with a system that ranked, and then awarded the lowest of the ranker's high, price adjusted as a percentage of the total.
Fair distro of rented house by ordered room rankings
A B C D E F G H (rooms)
8 7 6 5 4 3 2 1
7 8 5 4 3 6 1 2
7 6 8 1 4 3 2 5
8 6 7 5 4 3 2 1
6 7 8 5 2 3 1 4
6 7 8 4 5 3 2 1
7 6 8 5 4 3 2 1
5 6 8 7 3 1 4 2
highest in column, least in ranked row
Total=47, 8 pays .17 of total, etc.
posted by Brian B. at 10:21 AM on January 16, 2007
Fair distro of rented house by ordered room rankings
A B C D E F G H (rooms)
8 7 6 5 4 3 2 1
7 8 5 4 3 6 1 2
7 6 8 1 4 3 2 5
8 6 7 5 4 3 2 1
6 7 8 5 2 3 1 4
6 7 8 4 5 3 2 1
7 6 8 5 4 3 2 1
5 6 8 7 3 1 4 2
highest in column, least in ranked row
Total=47, 8 pays .17 of total, etc.
posted by Brian B. at 10:21 AM on January 16, 2007
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posted by veedubya at 4:11 AM on January 15, 2007 [1 favorite]