Cutting a Pie is Not a Piece of Cake

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Cutting a Pie is Not a Piece of Cake

• Walter Stromquist
• Swarthmore College
• mail_at_walterstromquist.com
• Third World Congress of the Game Theory Society
• Evanston, IL
• July 13, 2008

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Cutting a Pie is Not a Piece of CakeJulius B.
Barbanel, Steven J. Brams, Walter Stromquist

•      Mathematicians enjoy cakes for their own
  sake and as a metaphor for more general fair
  division problems.
• A cake is cut by parallel planes into n pieces,
  one for each of n players whose preferences are
  defined by separate measures. It is known that
  there is always an envy-free division, and that
  such a division is always Pareto optimal. So for
  cakes, equity and efficiency are compatible.
•    A pie is cut along radii into wedges. We
  show that envy-free divisions are not necessarily
  Pareto optimal --- in fact, for some measures,
  there may be no division that is both envy-free
  and Pareto optimal. So for pies, we may have to
  choose between equity and efficiency.

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• This is joint work with
  Julius B. Barbanel (Union
  College)
• Steven J. Brams (New York University)

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• 1. Introduction
• 2. Cakes
• 3. Pies
• 4. Summary

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Some definitions

• Cakes are cut by parallel planes.
• The cake is an interval C 0, m .
• Points in interval possible cuts.
• Subsets of interval possible pieces.
• We want to partition the interval into S1, S2,
  , Sn, where
• Si i-th players piece.
• Players preferences are defined by measures
  v1, v2, , vn
• vi (Sj ) Player is valuation of piece
  Sj.
• These are probability measures.
• We always assume that they are non-atomic (single
  points always have value zero).

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Absolutely continuous

• Sometimes we assume that the measures are
  absolutely continuous with respect to each other.
• In effect, this assumption means that pieces with
  positive length also have positive value to every
  player.

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• 1. Introduction
• 2. Cakes
• 3. Pies
• 4. Summary

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Two playersI cut, you choose
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n playersEverybody gets 1/n

• Referee slides knife from left to right
• Anyone who thinks the left piece has reached 1/n
  says STOP and gets the left piece.
• Proceed by induction. (Banach - Knaster ca.
  1940)

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Envy-free divisions

• A division is envy-free if no player thinks any
  other players piece is better than his own
• vi (Si) ? vi (Sj) for every i and j.
• Can we always find an envy-free division?
• Theorem (1980) For n players, there is always
  an envy-free division in which each player
  receives a single interval.
• Proofs
• (WRS) The division simplex
• (Francis Edward Su) Sperners Lemma

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Two moving knives the squeeze

• A cuts the cake into thirds (by his measure).
• Suppose B and C both choose the center piece.
• A moves both knives in such a way as to keep end
  pieces equal (according to A)
• B or C says STOP when one of the ends becomes
  tied with the middle. (Barbanel and Brams, 2004)

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Undominated allocations

• A division Si S1, S2, , Sn is dominated by
  a division
• Ti T1, T2, , Tn if
• vi(Ti) ? vi(Si) for every i
• with strict inequality in at least one case.
• That is T makes some player better off, and
  doesnt make any player worse off.
• Si is undominated if it isnt dominated by
  any Ti .
• undominated Pareto optimal efficient

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Envy-free implies undominated

• Is there an envy-free allocation that is also
  undominated?
• Theorem (Gale, 1993) Every envy-free division
  of a cake into n intervals for n players is
  undominated (assuming absolute continuity).
• So for cakes EQUITY ? EFFICIENCY.

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Gales proof

• Theorem (Gale, 1993) Every envy-free division
  of a cake into n intervals for n players is
  undominated.
• Proof Let Si be an envy-free division.
• Let Ti be some other division that we think
  might
• dominate Si.
• S2 S3 S1
• T3 T1 T2
• v1(T1) lt v1(S3) ? v1(S1)
• so Ti doesnt dominate Si after all. //

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Cakes without absolute continuity

• First players preference Uniform, EXCEPT on the
  leftmost third of the cake. The first player
  likes only the left half of the leftmost third.
• All other players preferences are uniform.
• The only envy-free divisions involve cutting the
  pie in thirds.
• None of these divisions is undominated.
• Without absolute continuity We may have to
  choose between envy-free and undominated.

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Summary for cakes

• With absolute continuity
• There is always an envy-free division.
• Every envy-free division is also undominated.
• There is always a division that is both
  envy-free and undominated.
• Without absolute continuity
• There is always an envy-free division.
• For some measures, there is NO division that is
  both envy-free
• and undominated. We may have to choose!
• Unless n 2, when there is always an envy-free,
  undominated division, whatever the measures.

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• 1. Introduction
• 2. Cakes
• 3. Pies
• 4. Summary

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Pies

• Pies are cut along radii. It takes n cuts to
  make pieces for n players.
• A cake is an interval.
• A pie is an interval with its endpoints
  identified.

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Pies

• 1. Are there envy-free divisions for pies?
• YES
• 2. Does Gales proof work?
• NO
• Envy-free does NOT imply undominated
• 3. Are there pie divisions that are both
  envy-free and undominated? (Gales
  question, 1993)
• YES for two players
• NO if we dont assume absolute continuity
• NO for the analogous problem with unequal
  claims
• (Brams, Jones, Klamler next talk!)

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Pies

• For n ? 3, there are measures for which there
  does NOT exist an envy-free, undominated
  allocation.
• These measures may be chosen to be absolutely
  continuous.
• So, Gales question is answered in the negative.

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The example

• Partition the pie into 18 tiny sectors.
• Each players preference is uniform, except
• Each player dislikes certain sectors (grayed
  out).
• Each player perceives positive or negative
  bonuses (C)
• or mini-bonuses (?) in certain sectors.
• The measures for three players

C ? C C C ? C ? C C C ?
C
C
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Pies for two players

• Of all envy-free allocations, pick the one most
  preferred by Player 2.
• That allocation is both envy-free and undominated.

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Summary for pies

• With or without absolute continuity
• There is always an envy-free division.
• For some measures, there is NO division that is
  both envy-free
• and undominated. We may have to choose!
• Unless n 2, when there is always an envy-free,
  undominated division, whatever the measures.

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• 1. Introduction
• 2. Cakes
• 3. Pies
• 4. Summary

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SummaryWhen must there be an
envy-free,undominated allocation?
CAKE PIE
2 players YES YES
?3 players YES, assuming absolute continuity (otherwise NO) NO
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Cookies

• This cookie cutter has blades at fixed 120-degree
  angles.
• But the center can go anywhere. Is there always
  an envy-free division of the cookie? Envy-free
  and undominated?

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