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Let's say that I have some sort of food item that is shareable. How can it be shared in such a way that all three parties are satisfied with the portion that they get (they feel like their portion is the same size as the others)?

The solution for two people is to let one person divide the food and then have the other person choose which portion they want. How can you do this for three people?

  • Are you defining "fair" as each person has enough control to get approximately the same amount? Please clarify in your post. There's a bit of confusion in places about the problem to be solved. – Robert Cartaino Jan 14 '17 at 7:56
  • @RobertCartaino For this question, fair can mean that everyone gets almost exactly equal portions, or everyone is satisfied with the portion that they get. – 10 Replies Jan 14 '17 at 11:19
  • Search google for "cake cutting" or "fair cake cutting" to see mathematical analysis of this problem. A fair amount of investigation has been invested in this kind of problem – Χpẘ Mar 27 '17 at 22:17
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Have the first person cut off an estimated 1/3, the second person divides the remainder in half, and the third person get first pick, then the second person get second pick.

If the first makes the "1/3" piece too big or too small, then the second and third people would choose the larger slices, so the first person will make a good effort to slice just 1/3.

If the second divides unevenly, then the third would get a larger share, so the second will strive to make the division equitable.

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    This is a pretty good answer, but there is a potential problem because it is not in the second person's (B) best interest to cut the remainder exactly in half if the first person (A) over-estimates the first cut. Example, suppose that A cuts off 40%. Then person B could cut off another 40%, leaving 20%. C will pick one of the 40% pieces, leaving another 40% piece for B and doubly penalizing A for his bad estimate. – James Jan 12 '17 at 14:15
  • @James if the first person cuts an oversize piece, he's ensuring he won't get that one. If he cuts undersize, he's ensuring he will. If the second compensates for what he sees as an inequitable first piece, it all evens out by the time he gets the last remaining piece. This gives the same incentive to cut evenly to both Alice and Bob, while Charlie gets the broadest choice. Collusion, if it occurs, would cheat someone only if it's between Alice and Charlie, and then Alice stands to lose by colluding with Charlie, so there are incentives to all against collusion. – Zeiss Ikon Jan 13 '17 at 12:56
  • @ZeissIkon: I upvoted this answer because I think it's the best one, but I was mostly commenting on "so the second will strive to make the division equitable". I think that assumption is only true if Alice cuts off less than or equal to 1/3. You say "If the second compensates for what he sees as an inequitable first piece, it all evens out by the time he gets the last remaining piece." Are you assuming that Bob has last pick? See the example in my first comment for a situation where Bob does not try to split the remainder in half. – James Jan 13 '17 at 13:24
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    This is not fair if player #2 decides to screw player #1. See my solution as to why — lifehacks.stackexchange.com/a/15024/80 – Robert Cartaino Jan 14 '17 at 7:47
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    If you have friends who collude against you, you need new friends. – k-l Jan 14 '17 at 22:55
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Preface:

DrMoishe Pippik's currently top answer — cut A, B, C; pick C, B, A — does not work because you have to define "fair" as each player either cuts 1/3 to the best of their ability (or maybe accepts less), or the glutton/cheater is going to lose out.

In the top answer, let's say #1 cuts exactly 1/3 (as they should). If #2 cuts a tiny sliver (the bad actor), #3 will select their larger piece, so #2 takes #1's third… and #1 is left with the tiny sliver. FAIL!

So here's a "game" that works

Remember the solution for two people is that one person cuts the portion in half, and the other person gets to pick which one they keep. It's 100% "fair".

You can use that premise to create four slices (half and then half again) but the players only keep three of the slices… and divide up the remaining 1/4th in a similar fashion.

So here's the order:

  • #1 cuts, #2 chooses (the whole becomes 50-50%)
  • #2 cuts their half, #3 choose (that half becomes 25-25%)
  • #3 cuts the other half, #1 chooses (that half becomes 25-25%)
  • Set that last remaining 25% aside and repeat the sequence on the remainder.

After two iterations, the leftover piece will be only 1/16 the original size.
After three iterations, the leftover piece will be only 1/64th.

You either can keep going and live in an infinite-sequence mathematical hell, or just decide the remaining crumbs are too small to be consequential and eyeball it.

Bonus: This is impossible to cheat, collude, or game. You can even accept less than your share (intentionally) if you're not all that hungry.

  • This is not a procedure that terminates, see my answer. – k-l Jan 14 '17 at 22:52
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This get tricky as players could collude. With two there is never an incentive to collude. If First starts with splitting in half then who ever picks first wins big time.

First splits off a portion

Second has the option to take that portion or split the remainder

If Second takes the portion then Third splits and First picks next

If Second splits then pick order is Third, First, Second

  • This seems, on the surface, about equally good as DrMoishe Pippik's answer. I'm not enough of a game theorist to call between them, so I'll +1 this as well. – Zeiss Ikon Jan 13 '17 at 12:58
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    There is no collusion necessary for this method to suck for B. Imagine that A purposefully splits off 10%. Now B either picks the 10%, or splits the 90% and get stuck with the 10% anyway (because he has last pick). B's only revenge against A is to split 10% off the remainder so that C gets 80%, with A and B both getting 10%. – James Jan 13 '17 at 13:32
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Have A split into thirds. Have B and C each rank the pieces by size. There are several possibilities:

  1. B and C agree on everything, A gets what they think is smallest piece, B and C divide-and-choose the rest.
  2. B and C disagree on largest piece, each gets piece they think is largest, A gets the remaining piece.
  3. B and C agree on the largest piece, but disagree on the other two, B and C divide-and-choose the one they think is largest, B and A divide-and-choose the one B thinks is second largest, C and A divide-and-choose the one C thinks is largest.

In case one, A can't complain, because A believes the pieces are equal, B and C can't complain either, because they each believe they got at least half of what they believe is at least two thirds. In case two, A can't complain because A believes the pieces are equal, B and C can't complain because they each got the piece they believe is the largest. In case three, A can't complain, because A got a fair share of two pieces, and A believes them to all be equal, B can't complain, because B got a fair share of what B believes to be at least two thirds, and C gets the same deal as B, so C can't complain.

  • Your is better than mine. More clear, short and grounded :). – Sasha Jan 14 '17 at 23:28
  • @Sasha The underlining method is very similar (same idea basically), +1 on your solution. – k-l Jan 14 '17 at 23:30
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At the risk of getting downvoted by the pedants, I'm going to explain a game called Who Gets This. Our family first learned about this technique while reading about Capt Bligh's survival voyage after he and his crew were put at sea on an overcrowded lifeboat in the wake of the Bounty mutiny. It's how they divided bird meat in the lifeboat during their 4,000-mile trek, and we adopted the same technique in our family with four children.

Here's how it works: one person divides the food into N roughly even portions. Then, one-by-one, someone takes one of the items, and says, "Who gets this?"

Then another member of the party – one who is not allowed to see the portions being chosen – calls out a name (either their own name, or the the name of someone else in the group). This happens N times (in your case, N = 3).

For example, if Leah, Mia, and Liam all had to share three pieces of a brownie, it might go something like this. First Liam cuts the brownie into three pieces. Then Mia turns her back. Liam picks one of the three pieces, and says, "Who gets this?"

Mia: Liam does.

Liam (picking up a second piece): Who gets this?

Mia: I do.

Because there is only one person left, Leah gets the remaining piece.

Advantages of this methodology:

  1. It can be used with any number of people.
  2. The process simple to conduct and easy to understand.
  3. It can be used even when there is no cutting involved (for example, when the teller at the bank hands you four different-colored lollipops to distribute to your four kids).
  4. Because the person cutting or dividing has no idea when his or her name will be called, there is incentive to cut the pieces equally.
  5. It's fun. After the very first time we tried this, it immediately became a well-liked and oft-used family tradition.

I mentioned the downvote at the beginning because this method doesn't necessarily guarantee that everyone "will feel like their portion is the same size as the others." Still, there is incentive to divide roughly equally, and at least the apportionment process is fair. Moreover, this technique kept over a dozen men alive under the most perilous conditions, so how bad can it be?

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    +1: I like this method even though it doesn't exactly answer the question as you've mentioned in your last paragraph. I do like it enough to give it a try with my four kids. Thank you! – James Jan 27 '17 at 16:43
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  1. Let one of them to try cutting ⅓ out of pie (let's call that part α and remainder βγ).
  2. Two others vote by saying either "α is really smaller than or equal to ⅓", or "α is really larger than or equal to ⅓" (if a voter thinks that α=⅓, he's free to choose any variant).
  3. Then:
    • If both said α≤⅓: The "cutter" takes α, two others share βγ.
    • If one said α≤⅓ and other said α≥⅓: The latter takes α, the former shares βγ with the "cutter".
    • If both said α≥⅓:
      1. The "cutter" tries to cut ⅓ (of original size) again, but now out of βγ (let's call that part β and remainder γ).
      2. Two others vote for either "β≤⅓", or "⅓≤β≤α", or "α≤β".
      3. Then:
        • If both said β≤⅓: The "cutter" takes β, two others share αγ.
        • If one said β≤⅓ and other said α≤β: The latter takes β, the former takes α, the "cutter" takes γ.
        • Otherwise: The "cutter" takes γ, others share αβ.

This is actually more puzzle solution, than for real life.
Unlike Robert Cartaino's answer, this procedure is not recursive: maximum 3 cuts are done.
Whey I say that some two share something, I mean the standard cut-then-choose between them.

And, as Anthony Joseph said, there are some researches on fair cake-cutting problem in game theory. Neither the my procedure above, nor the standard cut-then-choose (for two participants) are ideal. Maybe better to ask at Mathematics with Game Theory tag (or is it too simple for them)? Also there exists Surplus procedure (Wikipedia article, direct link).

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Being the youngest in a family with four kids, we had a lot of sharing and often enough just between three if one of us was not there.

We always had one cutting and then the others chose.
The chosing would be in random order if the cutting had been done fairly in our opinion. If we were not satisfied we would insist on a draw (straws or hidden number) to get the order of chosing or we would insist on further cutting to get the parts more equal.

We would never go to the effort of two people cutting, if you did cut that bad you would be boo'ed, rediculed and told to do a better job next time. In the worst cases you would be told to make a few more cuts to devide better.

If the people sharing are all adults and the sharing will be just once, you can do the #1 cuts first, #2 cuts second and #3 choses first.
But when teaching kids living together, better teach them from the start to cut as good as they can and pick wisely.

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If you are having a dinner party, setting up the food yourself on the plates with a good presentation and then serve it can be another solution. A nice presentation will make the guests happy. If you have leftovers, make sure the guests feel welcome to take seconds if they want. This avoids the problem with the third being polite and not taking an equal size as the others so as to leave nothing left.

  • This is not a situation at a dinner party. This situation only applies when you are going out with your friends and sharing food. – 10 Replies Jan 13 '17 at 18:46
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This answer is based on the answer from @Paparazzi, but should work well in all situations if there is no collusion between persons A and B.

A cuts off a piece.

B either takes the piece, or forces A to take it.

The two remaining people who don't yet have a piece do the standard two-person splitting as described in the original question.

A will try to cut exactly 1/3 of the original. If he cuts less, he is stuck with it. If he cuts more, B gets it.

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