Cake-Cutting Indivisible Goods [Some illustrations due to: Ariel - - PowerPoint PPT Presentation
CSC2556 Lecture 6 Fair Division 1: Cake-Cutting Indivisible Goods [Some illustrations due to: Ariel Procaccia] CSC2556 - Nisarg Shah 1 Announcements Reminder Project proposal due by March 3 st by 11:59PM If you want to run your
[Some illustrations due to: Ariel Procaccia]
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➢ Project proposal due by March 3st by 11:59PM ➢ If you want to run your idea by me, this is a good time to
approach me (email me and we’ll setup a time to chat).
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➢ Heterogeneous: it may be valued
differently by different individuals
➢ Divisible: we can share/divide
it between individuals
➢ Almost without loss of generality
➢ A finite union of disjoint intervals
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𝑗 that
is very much like a probability distribution over [0,1]
𝑊
𝑗 𝑌 + 𝑊 𝑗 𝑍 = 𝑊 𝑗 𝑌 ∪ 𝑍
𝑗
0,1 = 1
∃𝑍 ⊆ 𝑌 s.t. 𝑊
𝑗 𝑍 = 𝜇𝑊 𝑗(𝑌)
𝛽 𝜇𝛽 𝛽 β β
𝛽 + 𝛾
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∀𝑗 ∈ 𝑂: 𝑊
𝑗 𝐵𝑗 ≥ 1
𝑜
∀𝑗, 𝑘 ∈ 𝑂: 𝑊
𝑗 𝐵𝑗 ≥ 𝑊 𝑗(𝐵𝑘)
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𝑗 𝐵𝑗 ≥
Τ 1 𝑜
𝑗 𝐵𝑗 ≥ 𝑊 𝑗 𝐵𝑘
proportionality and EF?
1.
Prop ⇒ EF
2.
EF ⇒ Prop
3.
Equivalent
4.
Incomparable
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𝑊
1 𝑌 = 𝑊 1 𝑍 =
Τ 1 2
➢ Why?
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cake-cutting algorithm for 𝑜 players?
length of input encoded as binary.
𝑗, which requires
infinite bits to encode.
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𝑗’s through two
types of queries:
➢ Eval𝑗(𝑦, 𝑧) returns 𝑊
𝑗
𝑦, 𝑧
➢ Cut𝑗(𝑦, 𝛽) returns 𝑧 such that 𝑊
𝑗
𝑦, 𝑧 = 𝛽
𝑦 𝑧
𝛽
eval output cut output
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➢ Eval𝑗 𝑦, 𝑧 = 𝑊
𝑗
𝑦, 𝑧
➢ Cut𝑗 𝑦, 𝛽 = 𝑧 s.t. 𝑊
𝑗
𝑦, 𝑧 = 𝛽
EF allocation when 𝑜 = 2?
➢ Why?
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players
to the right.
1/𝑜 to a player, the player shouts “stop”, gets the piece, and exits.
13
1/3 1/3 ≥ 1/3
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cut query to mark his 1/𝑜 point in the remaining cake.
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Τ 1 3
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Τ 1 3 Τ 1 3
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Τ 1 3 Τ 1 3 ≥ Τ 1 3
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Spanier protocol in the Robertson-Webb model?
1.
Θ 𝑜
2.
Θ 𝑜 log 𝑜
3.
Θ 𝑜2
4.
Θ 𝑜2 log 𝑜
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➢ Assume 𝑜 = 2𝑙 for some 𝑙
𝑊
𝑗
𝑦, 𝑨𝑗 = 1 2 𝑊
𝑗
𝑦, 𝑧
[𝑨∗, 𝑧] with the right 𝑜/2 players.
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➢ Inductive proof. We want to prove that if player 𝑗 is
allocated piece 𝐵𝑗 when [𝑦, 𝑧] is divided between 𝑜 players, 𝑊
𝑗 𝐵𝑗 ≥
Τ 1 𝑜 𝑊
𝑗
𝑦, 𝑧
𝑗
𝑦, 𝑧 = 𝑊
𝑗
0,1 = 1
➢ Base case: 𝑜 = 1 is trivial. ➢ Suppose it holds for 𝑜 = 2𝑙−1. We prove for 𝑜 = 2𝑙. ➢ Take the 2𝑙−1 left players.
𝑗
𝑦, 𝑨∗ ≥ Τ 1 2 𝑊
𝑗
𝑦, 𝑧
𝑗 𝐵𝑗 ≥ 1 2𝑙−1 𝑊 𝑗
𝑦, 𝑨∗ ≥
1 2𝑙 𝑊 𝑗
𝑦, 𝑧
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protocol in the Robertson-Webb model?
1.
Θ 𝑜
2.
Θ 𝑜 log 𝑜
3.
Θ 𝑜2
4.
Θ 𝑜2 log 𝑜
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proportional protocol needs Ω(𝑜 log 𝑜) operations in the Robertson-Webb model.
provably optimal!
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algorithms for finding envy-free allocations?”
➢ [Brams and Taylor, 1995] give an unbounded EF protocol. ➢ [Procaccia 2009] shows Ω 𝑜2 lower bound for EF. ➢ Last year, the long-standing major open question of
“bounded EF protocol” was resolved!
➢ [Aziz and Mackenzie, 2016]: 𝑃(𝑜𝑜𝑜𝑜𝑜𝑜
) protocol!
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desire from an allocation.
➢ Notion of efficiency ➢ Informally, it says that there should be no “obviously
better” allocation
➢ No player should be able to gain by misreporting her
valuation
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➢ “Strategyproof”: No player should be able to increase her
utility by misreporting her valuation, irrespective of what
➢ “Strategyproof-in-expectation”: No player should be able
to increase her expected utility by misreporting.
➢ For simplicity, we’ll call this strategyproofness, and
assume we mean “in expectation” if the mechanism is randomized.
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➢ Bad news! ➢ Theorem [Menon & Larson ‘17] : No deterministic SP
mechanism is (even approximately) proportional.
➢ Good news! ➢ Theorem [Chen et al. ‘13, Mossel & Tamuz ‘10]: There is a
randomized SP mechanism that always returns an envy- free allocation.
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➢ There always exists a “perfect partition” (𝐶1, … , 𝐶𝑜) of
the cake such that 𝑊
𝑗 𝐶 𝑘 = Τ 1 𝑜 for every 𝑗, 𝑘 ∈ [𝑜].
➢ Every agent values every bundle equally.
➢ There exists a perfect partition that only cuts the cake at
𝑞𝑝𝑚𝑧(𝑜) points.
➢ In contrast, Lyapunov’s proof is non-constructive, and
might need an unbounded number of cuts.
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perfect partition as a black-box to design a randomized SP-in-expectation+EF mechanism?
➢ Yes! Compute a perfect partition, and assign the 𝑜
bundles to the 𝑜 players uniformly at random.
➢ Why is this EF?
1 𝑜.
➢ Why is this SP-in-expectation?
utility is Τ
1 𝑜, irrespective of what she reports.
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➢ We say that an allocation 𝐵 = (𝐵1, … , 𝐵𝑜) is PO if there is
no alternative allocation 𝐶 = (𝐶1, … , 𝐶𝑜) such that
𝑗 𝐶𝑗 ≥ 𝑊 𝑗(𝐵𝑗), ∀𝑗 ∈ 𝑂
𝑗 𝐶𝑗 > 𝑊 𝑗(𝐵𝑗), ∃𝑗 ∈ 𝑂
➢ I.e., an allocation is PO if there is no “better” allocation.
part of the cake positively”.
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➢ There always exists an allocation of the cake that is both
envy-free and Pareto optimal.
➢ Nash-optimal allocation: argmax𝐵 ς𝑗∈𝑂 𝑊
𝑗 𝐵𝑗
➢ Obviously, this is PO. The fact that it is EF is non-trivial. ➢ This is named after John Nash.
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➢ Green player has value 1 distributed over 0, Τ
2 3
➢ Blue player has value 1 distributed over [0,1] ➢ Without loss of generality (why?) suppose:
2 3]
2 3] AND all of [ Τ 2 3 , 1].
➢ Green’s utility = 𝑦, blue’s utility = 1 − x ⋅ 2 3 + 1 3 = 3−2𝑦 3 ➢ Maximize: 𝑦 ⋅ 3−2𝑦 3
⇒ 𝑦 = Τ
3 4 ( Τ 3 4 fraction of Τ 2 3 is Τ 1 2).
1
ൗ 2 3
Allocation 1
ൗ 1 2
Green has utility 3
4
Blue has utility
1 2
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➢ I believe it should require an unbounded number of
queries in the Robertson-Webb model. But I can’t find such a result in the literature.
➢ For piecewise constant valuations, the Nash-optimal
solution can be computed in polynomial time.
1
The density function of a piecewise constant valuation looks like this
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➢ Each good can be divided fractionally between the agents
➢ Homogeneous = agent doesn’t care which “part”
➢ Line up the goods on [0,1] → piecewise uniform
valuations
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Maximize σ𝑗 log 𝑉𝑗 𝑉𝑗 = Σ 𝑦𝑗, ∗ 𝑤𝑗, ∀𝑗 Σ𝑗 𝑦𝑗, = 1 ∀ 𝑦𝑗, ∈ [0,1] ∀𝑗,
➢ Polynomial time solvable
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➢ E.g., house, painting, car, jewelry, …
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We assume additive values. So, e.g., 𝑊 , = 8 + 7 = 15 Given such a matrix of numbers, assign each good to a player.
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∀𝑗, 𝑘 ∈ 𝑂, ∃ ∈ 𝐵𝑘 ∶ 𝑊
𝑗 𝐵𝑗 ≥ 𝑊 𝑗 𝐵𝑘\{}
➢ Technically, we need either this or 𝐵𝑘 = ∅. ➢ “If 𝑗 envies 𝑘, there must be some good in 𝑘’s bundle such
that removing it would make 𝑗 envy-free of 𝑘.”
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➢ Agents take turns in cyclic order: 1,2, … , 𝑜, 1,2, … , 𝑜, … ➢ In her turn, an agent picks the good she likes the most
among the goods still not picked by anyone.
➢ Informal proof on the board.
that is not Pareto optimal.
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➢ The allocation argmax𝐵 ς𝑗∈𝑂 𝑊
𝑗 𝐵𝑗 is EF1 + PO.
➢ Note: This maximization is over only “integral” allocations
that assign each good to some player in whole.
➢ Note: Subtle tie-breaking if all allocations have zero Nash
welfare.
it is possible to give a positive utility to every player in 𝑇 simultaneously.
𝑗 𝐵𝑗
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strongly NP-hard to compute
➢ That is, remains NP-hard even if all values in the matrix
are bounded
different polynomial time algorithm?
➢ Not sure. But a recent paper gives a pseudo-polynomial
time algorithm for EF1+PO
𝑗, 𝑊 𝑗
.
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➢ Generalization of “cut and choose” for 𝑜 players ➢ MMS value of player 𝑗 =
➢ Let 𝒬
𝑜 𝑁 denote the family of partitions of the set of
goods 𝑁 into 𝑜 bundles. 𝑁𝑁𝑇𝑗 = max
𝐶1,…,𝐶𝑜 ∈𝒬𝑜 𝑁
min
𝑙∈ 1,…,𝑜 𝑊 𝑗(𝐶𝑙) .
➢ An allocation is 𝛽-MMS if every player 𝑗 receives value at
least 𝛽 ∗ 𝑁𝑁𝑇𝑗.
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➢ [Procaccia, Wang ’14]:
There is an example in which no MMS allocation exists.
➢ [Procaccia, Wang ’14]:
A Τ
2 3 - MMS allocation always exists.
➢ [Ghodsi et al. ‘17]:
A Τ
3 4 - MMS allocation always exists.
➢ [Caragiannis et al. ’16]:
The Nash-optimal solution is
2 1+ 4𝑜−3 −MMS, and this is
the best possible guarantee.
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allocation?
𝑗 𝐵𝑗 ≥ 𝑊 𝑗 𝐵𝑘\{}
➢ Intuitively, 𝑗 doesn’t envy 𝑘 if she gets to remove her most
valued item from 𝑘’s bundle.
𝑗 𝐵𝑗 ≥ 𝑊 𝑗 𝐵𝑘\{}
➢ Note: Need to quantify over such that 𝑊
𝑗
> 0.
➢ Intuitively, 𝑗 doesn’t envy 𝑘 even if she removes her least
positively valued item from 𝑘’s bundle.
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➢ Suppose there are two players and three goods with
values as follows.
➢ If you give {A} → P1 and {B,C} → P2, it’s EF1 but not EFx.
➢ Instead, {A,B} → P1 and {C} → P2 would be EFx.
A B C P1 5 1 10 P2 1 10
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➢ Let 𝑑𝑗,𝑐 be the cost of player 𝑗 for bad 𝑐.
➢ EF: ∀𝑗, 𝑘 𝐷𝑗 𝐵𝑗 ≤ 𝐷𝑗 𝐵𝑘 ➢ PO: There should be no alternative allocation in which no
player has more cost, and some player has less cost.
➢ EF + PO allocation always exists, like for divisible goods.
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➢ EF1: ∀𝑗, 𝑘 ∃𝑐 ∈ 𝐵𝑗 𝑑𝑗 𝐵𝑗\ 𝑐
≤ 𝑑𝑗 𝐵𝑘
➢ EFx: ∀𝑗, 𝑘 ∀𝑐 ∈ 𝐵𝑗 𝑑𝑗 𝐵𝑗\ 𝑐
≤ 𝑑𝑗 𝐵𝑘
➢ Open Question 1:
➢ Open Question 2:
➢ More open questions related to relaxations of
proportionality