Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [fair-division]

For various problems related to dividing a resource among several parties.

Fair division, also known as the cake-cutting problem, the sandwich-splitting problem, and the pizza-dividing problem, is the problem of dividing a resource in such a way that all recipients believe that they have received a fair amount.

Various versions of this problems appear as logical puzzles in recreational mathematics.

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Splitting a sandwich and not feeling deceived

This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an hour to an hour, and then just grins at me, and…
VividD
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Optimal strategy for cutting a sausage?

You are a student, assigned to work in the cafeteria today, and it is your duty to divide the available food between all students. The food today is a sausage of 1m length, and you need to cut it into as many pieces as students come for lunch,…
Stenzel
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Is it possible to 'split' coin flipping 3 ways?

When flipping a coin to make important decisions in life you can flip once to choose between 2 possible outcomes. (Heads I eat cake, Tails I eat chocolate!) You can also flip twice to choose between 4 outcomes. (Heads-Heads, Tails-Tails,…
AnnanFay
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What's a fair way to share fees in a group road trip with a personal and a rental car?

I'm planning vacations with a group of friends (12 people), and it involves a ~1200km return trip by car. Only one of us owns a suitable car (4 pax), so we've rented a minivan to transport the other 8, and we're debating on how best to share the…
foucdeg
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Lyapunov Theorem for beginners

I study the subject of fair division (cake-cutting), and many papers contain a reference to a theorem by Lyapunov, which states that the range of any real-valued, non-atomic vector measure is compact and convex. Can you recommend an online resource…
Erel Segal-Halevi
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What's the fairest turn sequence for n players?

For two players, in some sense, the fairest possible turn order is the Thue-Morse-sequence: 01101001100101101001011001101001.... It arises by taking turns taking turns taking..., so: 01|01|01|01|01|01|01|01 -> 01 10|01 10|01 10|01 10 -> 01 10…
kram1032
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Selecting cards to form a fair game

Background In an old card game we draw 2 cards from a pile of 2 red and 2 black cards without replacement. If the two cards have the same color (for instance red and red) you win. However, if the cards have opposite color you lose. This is a rich…
N3buchadnezzar
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Cutting the Cake Problem

The traditional method for two children to divide a piece of cake fairly between them is "you cut, I choose". What process accomplishes this is intuitively clear. Is there a similar process which works for $3$ children? Suppose that three children…
anonymous
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Fair division of an octagon

A land-plot belongs to two partners. Its form is a regular octagon with area 1. They want to divide it such that one gets area $p$ and one gets area $1-p$, where $p \in (0,1)$ is a given constant. One way to do it is just to continuously move a…
Erel Segal-Halevi
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Cutting the cake problem if the value measures are not finitely additive

Background I have (rather recently) dabbled in game theory. I need it to design an algorithm to share chores. Obviously this is a kind of cake-cutting problem. So far, I have fought my way through An Introduction to Game Theory by Martin J. Osborne,…
Count Zero
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The Abel-and-Cain Urn Problem

An urn contains three distinguishable kinds of balls, say $A,B,C$. Abel bets to get, in $t$ trials with replacement, at least one ball of kind $A$ and at least one ball of kind $B$. Cain bets to get, in $t$ trials with replacement, exactly $t$…
user559615
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Weighted division?

Let's say I have 10,000 dollars I want divided among 10 people. With simple division each person gets $1,000. Easy enough. Now suppose each person has a score on a test from 0 to 200. Now I want to divide the money among all the people but…
User
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Cutting a pie with a fork

You baked a pie in the shape of a disc, with some cherries spread unevenly on its top. You want to give each of your two children a piece of cake such that: The pieces are congruent - have the same shape and size; Each piece has the same amount of…
Erel Segal-Halevi
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Why division gives equal parts?

I'm ashamed to ask this question. For a while, I can't pull out of my mind really basic kindergarten questions like: what is actually a number, what is a function and what is the connection between the name (function) and its purpose; what is…
J.Doe
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Equal slicing of my spherical cake

Recently I baked a spherical cake ($3$ cm radius) and invited over a few friends, $6$ of them, for dinner. When done with main course, I thought of serving this spherical cake and to avoid uninvited disagreements over the size of the shares, I took…
user356774
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