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

T RUTH J USTICE A LGOS 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 ∈ 𝑗 𝑇 = σ 𝑕∈𝐻 𝑊 𝑂 , 𝑊 𝑗 𝑕 • Assume additivity unless noted otherwise • An allocation is a partition of the goods, denoted 𝑩 = (𝐵 1 , … , 𝐵 𝑜 ) • Envy-freeness and proportionality are infeasible!

MAXIMIN SHARE GUARANTEE Total: Total: Total: $50 $30 $20 $3 $2 $50 $30 $5 $5 $5

MAXIMIN SHARE GUARANTEE Total: Total: Total: $50 $30 $20 $3 $2 $50 $30 $5 $5 $5 $3 $2 $5 $40 $10 $20 $20 Total: Total: Total: $30 $30 $40

MAXIMIN SHARE GUARANTEE • Maximin share (MMS) guarantee [Budish 2011] of player 𝑗 : 𝑌 1 ,…,𝑌 𝑜 min max 𝑊 𝑗 (𝑌 𝑘 ) 𝑘 • An MMS allocation is such that 𝑊 𝑗 (𝐵 𝑗 ) 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

COUNTEREXAMPLE FOR 𝑜 = 3 17 25 12 1 17 25 12 1 17 25 12 1 17 17 25 25 12 12 1 1 2 22 3 28 2 22 3 28 2 22 3 28 2 22 3 28 2 22 3 28 11 0 21 23 11 0 21 23 11 0 21 11 0 21 23 11 0 21 23 23

COUNTEREXAMPLE FOR 𝑜 = 3 1 1 1 1 17 25 12 1 × 10 6 × 10 3 + + 1 1 1 1 2 22 3 28 1 1 1 1 11 0 21 23 3 -1 -1 -1 3 -1 0 0 3 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 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 𝐵 1 , … , 𝐵 𝑜 is envy free up to one good (EF1) if and only if ∀𝑗, 𝑘 ∈ 𝑂, ∃𝑕 ∈ 𝐵 𝑘 s.t. 𝑤 𝑗 𝐵 𝑗 ≥ 𝑤 𝑗 𝐵 𝑘 \{𝑕} • 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 𝑇 4 𝑇 1 𝑩′ ; clearly EF1 is maintained • #edges in envy graph of 𝑩′ decreased: 𝑇 3 𝑇 2 ◦ Same edges between 𝑂 ∖ 𝐷 ◦ Edges from 𝑂 ∖ 𝐷 to 𝐷 shifted 𝑇 4 𝑇 2 ◦ Edges from 𝐷 to 𝑂 ∖ 𝐷 can only decrease ◦ Edges inside C decreased 𝑇 1 𝑇 3 • Iteratively remove cycles ∎

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 𝐵 𝑘 ≥ 𝑊 𝑘 𝐵 𝑗 = 𝑊 𝑘 𝐶 𝑗 ∖ 𝑕 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 𝐵 𝑗 ≥ 𝑊 𝑗 𝐵 𝑘 − 𝜗 • The foregoing result has interesting implications for cake cutting! Poll 1 ? Complexity of 𝜗 -EF in the RW model? 1 𝑜 • 𝑃 • 𝑃 𝜗 2 𝜗 𝑜 2 1 • 𝑃 • 𝑃 𝜗 2 𝜗

MAXIMUM NASH WELFARE • An allocation 𝑩 is Pareto efficient if there is no allocation 𝑩′ such that ′ ≥ 𝑊 𝑊 𝑗 𝐵 𝑗 𝑗 𝐵 𝑗 for all 𝑗 ∈ 𝑂 , and ′ > 𝑊 𝑊 𝑘 𝐵 𝑘 𝑘 𝐵 𝑘 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 = ෑ 𝑊 𝑗 (𝐵 𝑗 ) 𝑗∈𝑂 • 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 𝑕∈𝐵 𝑘 ,𝑊 𝑗 𝑕 >0 𝑊 𝑘 (𝑕)/𝑊 𝑗 (𝑕) • Move 𝑕 ⋆ from 𝑘 to 𝑗 to obtain 𝑩′ , we will show that NW 𝑩 ′ > NW(𝑩) ′ ) for all 𝑙 ≠ 𝑗, 𝑘 , • It holds that 𝑊 𝑙 𝐵 𝑙 = 𝑊 𝑙 (𝐵 𝑙 ′ = 𝑊 𝑗 𝑕 ⋆ , and 𝑊 𝑗 𝐵 𝑗 𝑗 𝐵 𝑗 + 𝑊 ′ = 𝑊 𝑘 𝑕 ⋆ 𝑊 𝑘 𝐵 𝑘 𝑘 𝐵 𝑘 − 𝑊

PROOF OF THEOREM • NW 𝐵 ′ 𝑊 𝑘 𝑕 ⋆ 𝑊 𝑗 𝑕 ⋆ NW 𝐵 > 1 ⇔ 1 − 1 + > 1 ⇔ 𝑊 𝑘 𝐵 𝑘 𝑊 𝑗 𝐵 𝑗 𝑊 𝑘 𝑕 ⋆ 𝑗 𝑕 ⋆ 𝑊 𝑗 𝐵 𝑗 + 𝑊 < 𝑊 𝑘 𝐵 𝑘 𝑊 𝑗 𝑕 ⋆ • Due to our choice of 𝑕 ⋆ , σ 𝑕∈𝐵 𝑘 𝑊 𝑘 𝑕 ⋆ 𝑘 𝑕 𝑊 𝑗 𝑕 = 𝑊 𝑘 𝐵 𝑘 𝑗 𝑕 ⋆ ≤ σ 𝑕∈𝐵 𝑘 𝑊 𝑊 𝑊 𝑗 𝐵 𝑘 • Due to EF1 violation, we have 𝑗 𝑕 ⋆ < 𝑊 𝑊 𝑗 𝐵 𝑗 + 𝑊 𝑗 𝐵 𝑘 • Multiply the last two inequalities to get the first ∎

TRACTABILITY OF MNW 30 25 20 Time (s) 15 10 5 0 5 10 15 20 25 30 35 40 45 50 Number of players [Caragiannis et al., 2016]

INTERFACE

AN OPEN PROBLEM • An allocation 𝐵 1 , … , 𝐵 𝑜 is envy free up to any good (EFX) if and only if ∀𝑗, 𝑘 ∈ 𝑂, ∀𝑕 ∈ 𝐵 𝑘 , 𝑤 𝑗 𝐵 𝑗 ≥ 𝑤 𝑗 𝐵 𝑘 \{𝑕} • 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