
How Mathematicians Cut Cake - jonbaer
https://priceonomics.com/how-do-mathematicians-cut-cake/
======
soVeryTired
This article describes how economists cut cake, not mathematicians.
Mathematicians cut cake into a finite number of non-measurable pieces,
rearrange those pieces using only rotation and translation, and end up with
two cakes.

~~~
mhb
No. If economists were cutting the cake, there would be no cake on the table
because someone would already have eaten it.

~~~
ianleeclark
Curious, but why would one of the economists have eaten the cake?

~~~
mhb
[https://financingefficiency.wordpress.com/2011/10/19/the-20-...](https://financingefficiency.wordpress.com/2011/10/19/the-20-bill-
on-the-sidewalk/)

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jzelinskie
I thought this was going to be about slicing a rectangle out the center of the
cake and pushing the two new halves together[0].

[0]: [https://youtu.be/wBU9N35ZHIw?t=131](https://youtu.be/wBU9N35ZHIw?t=131)

~~~
vamin
Me too. I was thinking of this Nature article from 1906:
[http://www.nature.com/nature/journal/v75/n1938/abs/075173c0....](http://www.nature.com/nature/journal/v75/n1938/abs/075173c0.html)

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oli5679
One thing to note is that no allocation mechanism for more than 2 parties
cannot satisfy all 3 of these criteria for general preferences

a) Efficient - guarantees an allocation where making at least one party better
off without making anyone else worse off is possible

b) Balanced budget - guarantees all prices sum to a fixed number eg the total
rent share

c) Incentive compatibility - it always a Nash equilibrium for every party to
report their true preferences for each good.

There is only one allocation mechanism (Vickery-Clarke-Groves) that satisfies
the 1st and 3rd criteria and this will generally not also satisfy the 2nd.

Reference:

[https://en.wikipedia.org/wiki/Vickrey%E2%80%93Clarke%E2%80%9...](https://en.wikipedia.org/wiki/Vickrey%E2%80%93Clarke%E2%80%93Groves_mechanism)

Thus, Su and Spliddit's algorithms by definition must be vulnerable to
strategic reporting.

Example: V=[v1,v2,v3] = preferences for rooms 1,2,3. total rent = 1000

VA = [1000,0,0] VB = [0,1000,0] VC = [0,0,1000]

Truth Telling Spliddit Allocation A->1,B->2,C->3 prices = [333,333,333]

If C reports [500,500,0] and A,B report truth

A->1,B->2,C->3 prices = [500,500,0]

TLDR - fair and efficient allocation mechanisms can never avoid vulnerability
to gaming for some preferences.

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00098345
Another interesting piece: equal pizza slices.

[http://www.popsci.com/cut-better-slice-pizza-with-
math](http://www.popsci.com/cut-better-slice-pizza-with-math)

[http://arxiv.org/pdf/1512.03794v1.pdf](http://arxiv.org/pdf/1512.03794v1.pdf)

~~~
jeffwass
While that solution is cool for preserving shape across all slices, it was
clearly postulated by theoreticians.

An experimentalist would point out that pizza components have a radial
dependence (Ie, crust on the outside), so it's not actually equal slicing in
practice, and the traditional sector slices are really the most optimal ;-)

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thomasthomas
High stakes "I cut you chose" with nba ownership. I love this:

"""The situation could drag on until October of 2017 as Kaplan and Pera have a
deal in which he has the option to make a bid for controlling interest in the
team at a price of their choosing. At that point Pera would have two options:
buy out Kaplan and Straus at that named price, or sell his shares to them
based on the same valuation. """

[http://basketball.realgm.com/wiretap/241181/sale-of-
wolves-h...](http://basketball.realgm.com/wiretap/241181/sale-of-wolves-held-
up-by-grizzlies-ownership-issues)

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swehner
I didn't think “I cut, you choose” is that fair.

Since often it's difficult to cut fairly, so the other person gets to pick a
bigger piece always.

~~~
cyberferret
Interestingly, I've used this technique for a long time with young kids at
parties etc. that I have only just met. It is usually a great indicator of
their personality traits, i.e. who has a compassionate, generous nature, or
who is inherently greedy/selfish.

~~~
yakult
...or who is smart enough to realize there is a social metagame outside the
obvious game of cake optimization.

The REALLY smart ones may have done a cost-benefit analysis and decided that
the payoff from the metagame - an increment in your opinion of them - is not
worth as much utility as the incrementally bigger cake.

~~~
optimiz3
Alternately the greedy ones have concluded your opinion of them is worth less
than the extra cake :).

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agentgt
Fair voting is almost exactly analogous (I think it is even the same math).

I had a fantastic math teacher in high school who showed how it is fairly
difficult to make any voting scheme fair.

~~~
buzzdenver
s/fairly difficult/impossible/

[https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theore...](https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem)

~~~
agentgt
I was going to say impossible as the teacher even said that but I think for
small numbers it is possible IIRC.

There was some joke I think by Kenneth that the only fair voted government is
a dictatorship.

~~~
thaumasiotes
Joke? Non-dictatorship is one of the mutually impossible criteria of Arrow's
impossibility theorem. "The only fair voted government is a dictatorship" is
an accurate statement of the theorem, as far as it goes.

~~~
soVeryTired
Yes, but there's some equivocation there. The meaning of 'dictator' isn't
quite what you might think it is.

~~~
thaumasiotes
The meaning of "dictator", as used in Arrow's theorem, is that the outcome of
an election is exactly what the dictator says it should be. More technically,
the election outcome considers only the dictator's vote, and is independent of
the votes of everyone who is not the dictator.

I don't see how that differs at all from the colloquial understanding. It's a
perfect reflection of the principle of One Man, One Vote, as described for
Ankh-Morpork ("The Patrician was the Man; he had the Vote").

~~~
soVeryTired
Bu then why does the theorem only hold with more than two candidates?

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hendler
Surprised there's no mention of Nash. It's mentioned often in their book and a
popular culture reference many would recognize.

[https://books.google.com/books?id=cLUA-
sRhJ5QC&q=nash#v=snip...](https://books.google.com/books?id=cLUA-
sRhJ5QC&q=nash#v=snippet&q=nash&f=false)

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wtbob
> In an overview of fair division applications published in Nautilus, science
> writer Erica Klarreich offers one such example. In the event of the
> contentious breakup of a marriage or relationship, the algorithm, which Ring
> and Brams call “Fair Buy-Sell” requires each partner to simultaneously
> propose a price.

> “If John proposes $110,000 and Jane proposes $100,000 then John, the higher
> bidder, will buy out Jane for $105,000,” explains Klarreich. “Each
> participant ends up with something—either money or the business—at a price
> that is better than his or her offer.”

I'm curious how that would work in a marriage, given that the couple's assets
are generally commingled (how can I spend our assets to pay you for our
assets?).

I think that'd work better when the husband & wife each have their own
personal assets, and only the common property is held in common.

~~~
PeterisP
If there are many assets, then it's just to decide on value when splitting
them; if a large item (e.g. house) is the majority of everything, then one
gets the asset at the decided value; and mortgages it to pay half of that
value (minus the value of other smaller assets) in cash.

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kejaed
Along these same lines, Five Thirty Eight had a neat Riddler last week about
how to choose the heights of a 3 layer cake to fit under a glass cone:

[http://fivethirtyeight.com/features/can-you-bake-the-
optimal...](http://fivethirtyeight.com/features/can-you-bake-the-optimal-
cake/)

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losteverything
Excellent article

Fair does not mean equal and equal does not mean fair.

I wonder how mathematicians eat corn on the cob? Across left to right? In
circles? I was told by future in-laws that I was "confident" Because I ate
left to right like a typewriter. They eat in circles. Ever since then I eat
randomly. Bite here.. Row right to ledt there. .....

~~~
tamana
Plover has an article about his unnscientific survey of about 6
mathematicians, eating corn by the style of tbeir subfield of math.

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jobigoud
Splitting a sandwich and not feeling deceived

[https://math.stackexchange.com/questions/637728/splitting-a-...](https://math.stackexchange.com/questions/637728/splitting-
a-sandwich-and-not-feeling-deceived)

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bitwize
"Suppose I cut a piece. It's... crumbly, but good..."

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gbog
I always cut more pieces than eaters and unevenly, because some people want
just one small piece, some want a big piece, and some will be happy to have
another piece later.

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mattybrennan
How do you cut a cake in a way that gives me a piece that is only large enough
to satisfy the host's expectation that I will like the cake they made?

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Ffaine
How Mathematicians eat a cake? Do they calculate the optimal angle of fork,
optimal mouth opening? Im more interested in to eat than to cut.

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fatdog
Some design patterns for founder equity allocation based on these principles
would be useful.

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quickben
See, that headline would have been awesome if there was 'pie' somewhere in it
:)

