INDIVISIBLE GOODS Set of goods Each good is indivisible Players - - PowerPoint PPT Presentation

ALGOS TRUTH JUSTICE

Fair Division V: Indivisible Goods

Teachers: Ariel Procaccia (this time) and Alex Psomas

INDIVISIBLE GOODS

• Set 𝐻 of 𝑛 goods 𝐻
• Each good is indivisible
• Players 𝑂 = 1, … , 𝑜 have valuations 𝑊

𝑗 for

bundles of goods

• Valuations are additive if for all 𝑇 ⊆ 𝐻 and 𝑗 ∈

𝑂, 𝑊

𝑗 𝑇 = σ𝑕∈𝐻 𝑊 𝑗 𝑕

• Assume additivity unless noted otherwise
• An allocation is a partition of the goods,

denoted 𝑩 = (𝐵1, … , 𝐵𝑜)

• Envy-freeness and proportionality are

infeasible!

MAXIMIN SHARE GUARANTEE

$50 $30 $3 $2 $5 $5 $5 Total: $50 Total: $30 Total: $20

MAXIMIN SHARE GUARANTEE

$50 $30 $3 $2 $5 $5 $5 Total: $50 Total: $30 Total: $20 $3 $2 $5 $40 $10 $20 $20 Total: $40 Total: $30 Total: $30

• Maximin share (MMS) guarantee [Budish

2011] of player 𝑗: max

𝑌1,…,𝑌𝑜 min 𝑘

𝑊

𝑗(𝑌 𝑘)

• An MMS allocation is such that 𝑊

𝑗(𝐵𝑗) is at

least 𝑗’s MMS guarantee for all 𝑗 ∈ 𝑂

• For 𝑜 = 2 an MMS allocation always exists
• Theorem [Kurokawa et al. 2018]: ∀𝑜 ≥ 3

there exist additive valuation functions that do not admit an MMS allocation

MAXIMIN SHARE GUARANTEE

COUNTEREXAMPLE FOR 𝑜 = 3

17 25 12 1 2 22 3 28 11 0 21 2 22 3 28 23 17 12 1 2 22 3 28 11 0 21 23 1 25 12 25 17 11 0 21 23 17 25 12 1 2 22 3 28 11 0 21 23 17 25 12 1 2 22 3 28 11 0 21 23

COUNTEREXAMPLE FOR 𝑜 = 3

1 1 1 1 1 1 1 1 1 1 1 1 17 25 12 1 2 22 3 28 11 0 21 23

× 106 × 103

3 -1 -1 -1 0 0 0 0 0 0 0 0 3 -1 0 0

• 1 0 0 0
• 1 0 0 0

3 0 -1 0 0 0 -1 0 0 0 0 -1

Player 1 Player 2 Player 3

+ +

APPROXIMATE ENVY-FREENESS

• Assume general monotonic valuations, i.e.,

for all 𝑇 ⊆ 𝑈 ⊆ 𝐻, 𝑊

𝑗 𝑇 ≤ 𝑊 𝑗(𝑈)

• An allocation 𝐵1, … , 𝐵𝑜 is envy free up to one

good (EF1) if and only if ∀𝑗, 𝑘 ∈ 𝑂, ∃𝑕 ∈ 𝐵𝑘 s.t. 𝑤𝑗 𝐵𝑗 ≥ 𝑤𝑗 𝐵𝑘\{𝑕}

• Theorem [Lipton et al. 2004]: An EF1

allocation exists and can be found in polynomial time

PROOF OF THEOREM

• A partial allocation is an allocation of a

subset of the goods

• Given a partial allocation 𝑩, we have an

edge (𝑗, 𝑘) in its envy graph if 𝑗 envies 𝑘

• Lemma: An EF1 partial allocation 𝑩 can

be transformed in polynomial time into an EF1 partial allocation 𝑪 of the same goods with an acyclic envy graph

PROOF OF LEMMA

• If 𝐻 has a cycle 𝐷, shift

allocations along 𝐷 to obtain 𝑩′; clearly EF1 is maintained

• #edges in envy graph of 𝑩′

decreased:

• Same edges between 𝑂 ∖ 𝐷
• Edges from 𝑂 ∖ 𝐷 to 𝐷 shifted
• Edges from 𝐷 to 𝑂 ∖ 𝐷 can
• nly decrease
• Edges inside C decreased
• Iteratively remove cycles ∎

𝑇1 𝑇2 𝑇3 𝑇4 𝑇2 𝑇3 𝑇1 𝑇4

PROOF OF THEOREM

• Maintain EF1 and acyclic envy graph
• In round 1, allocate good 𝑕1 to arbitrary

agent

• 𝑕1, … , 𝑕𝑙−1 are allocated in acyclic 𝑩
• Derive 𝑪 by allocating 𝑕𝑙 to source 𝑗
• 𝑊

𝑘 𝐶 𝑘 = 𝑊 𝑘 𝐵𝑘 ≥ 𝑊 𝑘 𝐵𝑗 = 𝑊 𝑘 𝐶𝑗 ∖ 𝑕𝑙

• Use lemma to eliminate cycles ∎

ROUND ROBIN

• Let us return to additive valuations
• Now proving the existence of an EF1

allocation is trivial

• A round-robin allocation is EF1:

Phase 1 Phase 2

IMPLICATIONS FOR CAKE CUTTING

• In cake cutting, we can define an allocation

to be 𝜗-envy free if for all 𝑗, 𝑘 ∈ 𝑂, 𝑊

𝑗 𝐵𝑗 ≥ 𝑊 𝑗 𝐵𝑘 − 𝜗

• The foregoing result has interesting

implications for cake cutting!

Complexity of 𝜗-EF in the RW model?

• 𝑃

1 𝜗

• 𝑃

𝑜 𝜗2

• 𝑃

1 𝜗2

• 𝑃

𝑜2 𝜗

Poll 1

?

MAXIMUM NASH WELFARE

• An allocation 𝑩 is Pareto efficient if

there is no allocation 𝑩′ such that 𝑊

𝑗 𝐵𝑗 ′ ≥ 𝑊 𝑗 𝐵𝑗 for all 𝑗 ∈ 𝑂, and

𝑊

𝑘 𝐵𝑘 ′ > 𝑊 𝑘 𝐵𝑘 for some 𝑘 ∈ 𝑂

• Round Robin is not efficient
• Is there a rule that guarantees both EF1

and efficiency?

MAXIMUM NASH WELFARE

• The Nash welfare of an allocation 𝑩 is the

product of values NW 𝑩 = ෑ

𝑗∈𝑂

𝑊

𝑗(𝐵𝑗)

• The maximum Nash welfare (MNW) solution

chooses an allocation that maximizes the Nash welfare

• For ease of exposition we ignore the case of

NW 𝑩 = 0 for all 𝑩

• Theorem [Caragiannis et al. 2016]: Assuming

additive valuations, the MNW solution is EF1 and efficient

PROOF OF THEOREM

• Efficiency is obvious, so we focus on EF1
• Assume for contradiction that 𝑗 envies 𝑘 by

more than one good

• Let 𝑕⋆ ∈ argmin𝑕∈𝐵𝑘,𝑊𝑗 𝑕 >0 𝑊

𝑘(𝑕)/𝑊 𝑗(𝑕)

• Move 𝑕⋆ from 𝑘 to 𝑗 to obtain 𝑩′, we will

show that NW 𝑩′ > NW(𝑩)

• It holds that 𝑊

𝑙 𝐵𝑙 = 𝑊 𝑙(𝐵𝑙 ′ ) for all 𝑙 ≠ 𝑗, 𝑘,

𝑊

𝑗 𝐵𝑗 ′ = 𝑊 𝑗 𝐵𝑗 + 𝑊 𝑗 𝑕⋆ , and

𝑊

𝑘 𝐵𝑘 ′ = 𝑊 𝑘 𝐵𝑘 − 𝑊 𝑘 𝑕⋆

PROOF OF THEOREM

• NW 𝐵′

NW 𝐵 > 1 ⇔ 1 −

𝑊𝑘 𝑕⋆ 𝑊𝑘 𝐵𝑘

1 +

𝑊𝑗 𝑕⋆ 𝑊𝑗 𝐵𝑗

> 1 ⇔

𝑊𝑘 𝑕⋆ 𝑊𝑗 𝑕⋆

𝑊

𝑗 𝐵𝑗 + 𝑊 𝑗 𝑕⋆

< 𝑊

𝑘 𝐵𝑘

• Due to our choice of 𝑕⋆,

𝑊

𝑘 𝑕⋆

𝑊

𝑗 𝑕⋆ ≤

σ𝑕∈𝐵𝑘 𝑊

𝑘 𝑕

σ𝑕∈𝐵𝑘 𝑊

𝑗 𝑕 = 𝑊 𝑘 𝐵𝑘

𝑊

𝑗 𝐵𝑘

• Due to EF1 violation, we have

𝑊

𝑗 𝐵𝑗 + 𝑊 𝑗 𝑕⋆ < 𝑊 𝑗 𝐵𝑘

• Multiply the last two inequalities to get the first ∎

TRACTABILITY OF MNW

5 10 15 20 25 30 5 10 15 20 25 30 35 40 45 50

Time (s) Number of players

[Caragiannis et al., 2016]

INTERFACE

AN OPEN PROBLEM

• An allocation 𝐵1, … , 𝐵𝑜 is envy free up to

any good (EFX) if and only if ∀𝑗, 𝑘 ∈ 𝑂, ∀𝑕 ∈ 𝐵𝑘, 𝑤𝑗 𝐵𝑗 ≥ 𝑤𝑗 𝐵𝑘\{𝑕}

• Strictly stronger than EF1, strictly weaker

than EF

• An EFX allocation exists for two players

with monotonic valuations

• Existence is an open problem for 𝑜 ≥ 3

players with additive valuations