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Cutting a Pie is Not a Piece of Cake

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mail_at_walterstromquist.com. Third World Congress of the Game Theory Society. Evanston, IL ... The first player likes only the left half of the leftmost third. ... – PowerPoint PPT presentation

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Title: Cutting a Pie is Not a Piece of Cake


1
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

2
Cutting a Pie is Not a Piece of Cake Julius 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.

2

3
  • This is joint work with
    Julius B. Barbanel (Union
    College)
  • Steven J. Brams (New York University)

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

5
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6
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).

6
7
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.

8
  • 1. Introduction
  • 2. Cakes
  • 3. Pies
  • 4. Summary

9
Two players I cut, you choose
9
10
n players Everybody 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)

10
11
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

11
12
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)

12
13
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

13
14
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.

14
15
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. //

15
16
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.

17
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.

18
  • 1. Introduction
  • 2. Cakes
  • 3. Pies
  • 4. Summary

19
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.

19
20
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21
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!)

21
22
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.

23
3
1
2
24
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
24
25
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.

26
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.

27
  • 1. Introduction
  • 2. Cakes
  • 3. Pies
  • 4. Summary

28
Summary When must there be an
envy-free, undominated allocation?
CAKE PIE
2 players YES YES
?3 players YES, assuming absolute continuity (otherwise NO) NO
29
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?

29
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