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CSC304 Lecture 19
Fair Division 2: Cake-cutting, Indivisible goods
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Recall: Cake-Cutting
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β’ A heterogeneous, divisible good
β’ Represented as [0,1]
β’ Set of players π = {1,β¦ , π}
β’ Each player π has valuation ππ
β’ Allocation π΄ = (π΄1, β¦ , π΄π)
β’ Disjoint partition of the cake
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Recall: Cake-Cutting
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β’ We looked at two measures of fairness:
β’ Proportionality: βπ β π: ππ π΄π β₯ Ξ€1
π
β’ βEvery agent should get her fair share.β
β’ Envy-freeness: βπ, π β π: ππ π΄π β₯ ππ π΄πβ’ βNo agent should prefer someone elseβs allocation.β
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Other Desiderata
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β’ There are two more properties that we often desire from an allocation.
β’ Pareto optimality (PO)β’ Notion of efficiency
β’ Informally, it says that there should be no βobviously betterβ allocation
β’ Strategyproofness (SP)β’ No player should be able to gain by misreporting her
valuation
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Strategyproofness (SP)
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β’ For deterministic mechanismsβ’ βStrategyproofβ: No player should be able to increase her
utility by misreporting her valuation, irrespective of what other players report.
β’ For randomized mechanismsβ’ βStrategyproof-in-expectationβ: No player should be able
to increase her expected utility by misreporting.
β’ For simplicity, weβll call this strategyproofness, and assume we mean βin expectationβ if the mechanism is randomized.
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Strategyproofness (SP)
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β’ Deterministicβ’ Bad news!
β’ Theorem [Menon & Larson β17] : No deterministic SP mechanism is (even approximately) proportional.
β’ Randomizedβ’ Good news!
β’ Theorem [Chen et al. β13, Mossel & Tamuz β10]: There is a randomized SP mechanism that always returns an envy-free allocation.
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Perfect Partition
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β’ Theorem [Lyapunov β40]: β’ There always exists a βperfect partitionβ (π΅1, β¦ , π΅π) of
the cake such that ππ π΅π = Ξ€1
π for every π, π β [π].
β’ Every agent values every bundle equally.
β’ Theorem [Alon β87]: β’ There exists a perfect partition that only cuts the cake at ππππ¦(π) points.
β’ In contrast, Lyapunovβs proof is non-constructive, and might need an unbounded number of cuts.
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Perfect Partition
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β’ Q: Can you use an algorithm for computing a perfect partition as a black-box to design a randomized SP+EF mechanism?
β’ Yes! Compute a perfect partition, and assign the πbundles to the π players uniformly at random.
β’ Why is this EF? o Every agent values every bundle at Ξ€1 π.
β’ Why is this SP-in-expectation?o Because an agent is assigned a random bundle, her expected
utility is Ξ€1 π, irrespective of what she reports.
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Pareto Optimality (PO)
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β’ Definitionβ’ We say that an allocation π΄ = (π΄1, β¦ , π΄π) is PO if there is
no alternative allocation π΅ = (π΅1, β¦ , π΅π) such that
1. Every agent is at least as happy: ππ π΅π β₯ ππ(π΄π), βπ β π
2. Some agent is strictly happier: ππ π΅π > ππ(π΄π), βπ β π
β’ I.e., an allocation is PO if there is no βbetterβ allocation.
β’ Q: Is it PO to give the entire cake to player 1?
β’ A: Not necessarily. But yes if player 1 values βevery part of the cake positivelyβ.
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PO + EF
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β’ Theorem [Weller β85]:β’ There always exists an allocation of the cake that is both
envy-free and Pareto optimal.
β’ One way to achieve PO+EF:β’ Nash-optimal allocation: argmaxπ΄ Οπβπππ π΄πβ’ Obviously, this is PO. The fact that it is EF is non-trivial.
β’ This is named after John Nash.o Nash social welfare = product of utilities
o Different from utilitarian social welfare = sum of utilities
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Nash-Optimal Allocation
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β’ Example:β’ Green player has value 1 distributed over 0, Ξ€2 3β’ Blue player has value 1 distributed over [0,1]
β’ Without loss of generality (why?) suppose: o Green player gets π₯ fraction of [0, Ξ€2 3]
o Blue player gets the remaining 1 β π₯ fraction of [0, Ξ€2 3] AND all of [ Ξ€2
3 , 1].
β’ Greenβs utility = π₯, blueβs utility = 1 β x β 23+
1
3=
3β2π₯
3
β’ Maximize: π₯ β 3β2π₯
3β π₯ = Ξ€3 4 ( Ξ€3 4 fraction of Ξ€2 3 is Ξ€1 2).
0 1ΰ΅2 3
Allocation 0 1
ΰ΅1 2
Each playerβs utility = Ξ€3 4
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Problem with Nash Solution
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β’ Difficult to compute in generalβ’ I believe it should require an unbounded number of
queries in the Robertson-Webb model. But I canβt find such a result in the literature.
β’ Theorem [Aziz & Ye β14]:β’ For piecewise constant valuations, the Nash-optimal
solution can be computed in polynomial time.
0 1
The density function of a piecewise constant valuation looks like this
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Indivisible Goods
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β’ Goods cannot be shared / divided among playersβ’ E.g., house, painting, car, jewelry, β¦
β’ Problem: Envy-free allocations may not exist!
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Indivisible Goods: Setting
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8 7 20 5
9 11 12 8
9 10 18 3
We assume additive values. So, e.g., π , = 8 + 7 = 15
Given such a matrix of numbers, assign each good to a player.
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Indivisible Goods
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β’ Envy-freeness up to one good (EF1):
βπ, π β π, βπ β π΄π βΆ ππ π΄π β₯ ππ π΄π\{π}
β’ Technically, we need either this or π΄π = β .
β’ βIf π envies π, there must be some good in πβs bundle such that removing it would make π envy-free of π.β
β’ Does there always exist an EF1 allocation?
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EF1
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β’ Yes! We can use Round Robin.
β’ Agents take turns in cyclic order: 1,2,β¦ , π, 1,2,β¦ , π, β¦
β’ In her turn, an agent picks the good she likes the most among the goods still not picked by anyone.
β’ Observation: This always yields an EF1 allocation.β’ Informal proof on the board.
β’ Sadly, on some instances, this returns an allocation that is not Pareto optimal.
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EF1+PO?
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β’ Nash welfare to rescue!
β’ Theorem [Caragiannis et al. β16]:β’ The allocation argmaxπ΄ Οπβπππ π΄π is EF1 + PO.
β’ Note: This maximization is over only βintegralβ allocations that assign each good to some player in whole.
β’ Note: Subtle tie-breaking if all allocations have zero Nash welfare.o Step 1: Choose a subset of players π β π with largest |π| such that
it is possible to give a positive utility to every player in πsimultaneously.
o Step 2: Choose argmaxπ΄ Οπβπππ π΄π
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8 7 20 5
9 11 12 8
9 10 18 3
Integral Nash Allocation?
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8 7 20 5
9 11 12 8
9 10 18 3
20 * (11+8) * 9 = 3420 is the maximum possible product
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Computation
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β’ For indivisible goods, Nash-optimal solution is strongly NP-hard to computeβ’ That is, remains NP-hard even if all values in the matrix
are bounded
β’ Open Question: If our goal is EF1+PO, is there a different polynomial time algorithm? β’ Not sure. But a recent paper gives a pseudo-polynomial
time algorithm for EF1+POo Time is polynomial in π, π, and max
π,πππ π .
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Stronger Fairness
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β’ Open Question: Does there always exist an EFxallocation?
β’ EF1: βπ, π β π, βπ β π΄π βΆ ππ π΄π β₯ ππ π΄π\{π}β’ Note: Or π΄π = β also allowed.
β’ Intuitively, π doesnβt envy π if she gets to remove her most valued item from πβs bundle.
β’ EFx: βπ, π β π, βπ β π΄π βΆ ππ π΄π β₯ ππ π΄π\{π}β’ Note: βπ β π΄π such that ππ π > 0.
β’ Intuitively, π doesnβt envy π even if she removes her least positively valued item from πβs bundle.
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Stronger Fairness
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β’ To clarify the difference between EF1 and EFx:β’ Suppose there are two players and three goods with
values as follows.
β’ If you give {A} β P1 and {B,C} β P2, itβs EF1 but not EFx.o EF1 because if P1 removes C from P2βs bundle, all is fine.
o Not EFx because removing B doesnβt eliminate envy.
β’ Instead, {A,B} β P1 and {C} β P2 would be EFx.
A B C
P1 5 1 10
P2 0 1 10