How to divide a cake fairly among a group of people

Well first of all, you institute the dictatorship of the proletariat…

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So the change you’re proposing is to go in reverse order after the last person cuts? That still doesn’t seem to work, as in:

1 cuts 20% piece
2 cuts 26.6% piece
3 takes 26.6% piece
4 cuts 26.6% piece (and a second 26.6% by default)
3 is out
2 takes 26.6% piece
1 takes 26.6% piece
4 takes 20% piece

And you can’t give 4 two actions in a row, or they’ll just cut and take a huge piece.

N-1 one people each make a cut. The nth person then selects a piece. Everyone elses name goes in a hat when their name is drawn they pick their piece.

You are right of course. I think the mistake then in my algorithm was passing the cutter role forward always. If instead of 4 cutting after 3 takes, cutter role stays with 2, then it still works out, because that way the person at the end of the line is always a chooser, never a cutter. So in your example, it would be 2 cutting (his second) (and third, I guess) 26.6% piece, 4 picks one, 2 gets the other, and 1 ends up with his own piece.

HOWEVER having given this now entirely too much thought, I think there are some issues not yet considered here. What if someone makes a cut which they think is fair, but at least two others think is on the large side? one of those two or more will now believe there is not enough to go around so how can they guarantee that they don’t wind up with the short end of the stick? After all the original question centers on belief.

1 cuts 25% according to himself
2 and 4 both think that it’s more like 26%
2 takes the piece
1 cuts another 25% that everyone agrees is 25%
3 takes that piece

at this point, 1 is happy to split the last into 2 equal slices and consider himself fairly treated, but 4, who until this point has not even gotten a chance to pick, considers himself unfairly treated because he has to choose between two 24.5% pieces! It is OK of course that 2 feels he got a good deal because that fits the requirement but what about 4?

The cutter moves the knife, but doesn’t cut until someone else shouts “stop”—then the shouter gets the resulting piece and becomes the cutter.

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Easy. I call in Rob @beschizza and get him to do it.

Or I call @maggiek and give her a box of rubber bands.

Or I call @john_brownlee and tell him to bring his calibrated plate.

Or Mark @frauenfelder and his Fair Division Calculator.

Or Dan Ruderman and the Spliddit app.

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In this solution, 1 and 2 can collude to get more cake as long as 1 trusts 2:

First, 1 cuts a much larger than fair slice (let’s say 45% of the cake even though there are 10 people), then 2 accepts it. After everyone else (including 1) gets a small piece, 2 shares some of the very large piece with 1 so the two of them come out ahead of everyone else.

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Similar to this is the white elephant method: each player in order choose to CUT a slice for themselves, or TAKE someone else’s slice. If your slice is taken, you are inserted to the front of the queue. A slice that is taken cannot be re-taken until a new slice is cut. Go through the queue until each player has a slice.

No player has an incentive to cut a larger than fair slice, as it will be stolen from them. You can still collude with the final player, however (who is guaranteed to get the largest larger than fair slice)

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this wouldn’t guarantee that each person is satisfied with their piece of cake though. Even if everyone is attempting to create a fair piece, some people may consider certain pieces larger than others. e.g. if 3 out of 4 people agree that 2 pieces are a little larger than fair and 2 are a little smaller than fair (totally possible, if the 1st person cuts the first a little large but he thinks it’s fair, 2 will take it. then he cuts the 2nd the same way, and 3 takes it. Now 1 cuts the remainder exactly in half and is happy with his piece. 4 doesn’t want the last piece because the first 2 are larger, so he takes from either 2 or 3. then whichever was taken from, takes from the other. then lastly 2 (or 3) is stuck with the final piece of cake, which by his reckoning is smaller than fair.

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The thing is, in order to be able to guarantee that a given person is getting what they consider a ‘fair’ piece, they have to either a) have cut it themselves or b) have chosen it from a set of pieces that contains at least 1 piece that they consider to be fair. I am not sure there is a way to have people make cuts based on their idea of what a fair piece is which might not jive with other peoples and still satisfy those requirements. So the answer has to be more along the auction style that was suggested earlier where people basically bid on how small a piece they are willing to take.

[quote=“rethfernhim, post:10, topic:77643, full:true”]
For three people, the first person cuts a piece. The second person gets to choose between taking that piece, or dividing the remaining cake into two pieces. If person 2 takes the first slice, we’re back to the n=2 problem. If person 2 slices, then person 3 gets to choose their slice, then person 1 chooses a slice, and person 2 is left with the last slice.

This should work for all n…unless I’m missing something. Thoughts?
[/quote]Person one has bad eyesight, so cuts the cake into more like 40/60, but thinks it’s 33/67. Person 2 takes the 40% slice. Person 1 cuts again, correctly this time into 30/30. Person 3 demands person 2’s slice since it’s obviously larger. Fisticuffs ensue.

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Crumble-down Cake-onomics?

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Related to this is “equitable sequencing”. Let’s say we’re divvying up baseball cards. Alternating choices favors the first person, but you can mix things up using the Thue-Morse sequence…

  1. Person 1 cuts n slices.
  2. Then all others pick one slice, in any order.
  3. Then person 1 gets last slice.

If all parts were not equal, person 1 would get the smallest => person 1 will cut equal slices.

Or what am I missing?

Cut the cake as equitably as your eyeball allows. If anyone complains, they get no cake at all. Problem solved.

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There is only one way to determine the correct answer, and that is by lots and lots of testing.

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The richest guy in the room takes 90% of the cake, then crumbles the rest onto the table for the others to work hard to try and get…

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If only 90%…

The Charles Koch (neoliberalism poster boy) approach:
“I want my fair share – and that’s all of it.”

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Person 1 forms alliance with Person 2, hands them whole cake, flips of Persons 3 - n
Welcome to Congress!

Simply invite people with sufficient combinations (at least one per person) of:
• dietary restrictions: medical, religious, whatever
• caloric intake caps
• personal preferences

With an appropriately constructed cake (including, for example, GMO flour, gluten, milk, bacon)

And you get to eat all the cake.