Competitive Fair Division of Goods and Bads Herv Moulin University - - PowerPoint PPT Presentation

Competitive Fair Division of Goods and Bads

Hervé Moulin University of Glasgow and HSE St Petersburg June, 2016

joint work with Anna Bogomolnaia, U of Glasgow and HSE St Petersburg Fedor Sandomirskyi, HSE St Petersburg Elena Yanovskaia, HSE St Petersburg

the fair division problem interpreting common property under heterogenous responsible preferences

• equal shares is surely fair
• efficiency exploits differences in preferences through unequal shares
• goal : define a concept of fairness compatible with efficiency, hence fair

unequal shares

procedural fairness change common property into equal private property rights here, give1/n-th of the resources to everyone let participants themselves capture the surplus opportunities by direct decen- tralized trades fair game, but → cost of enabling direct transactions could be large → uncertain outcomes: multi-valued predictions → rewards morally irrelevant strategic skills

end-state fairness use a mechanical rule (benevolent dictator) to select a fair outcome → minimal transaction costs → eliminates uncertainty, role of strategic skills → allows a single-valued predictable recommendation difficulty: find compelling normative justifications for the chosen rule

two types of rules welfarist, resourcist

• Egalitarian rule: equalize some (paternalistic) measure of individual wel-

fares

• Competitive rule: privatize resources virtually, then select the allocation

that the trading game “should” produce

here: fair allocation of divisible “items”, all goods or all bads when side-payments are ruled out (“moneyless”) under linear preferences ⇐ ⇒ additive utilities → all bads: production inputs; tasks bewteen substitutable workers: household chores, job shifts, customers’ orders → all goods: family heirlooms, assets between divorcing partners, computing resources in peer-to-peer platforms

"compulsory" substitutability linear preferences are easy to elicit (realistic complexity) participants spread 100 points over the goods/objects proof of the pudding is in the eating: Adjusted Winner, Spliddit

Model N i the agents, A a the items ui ∈ RA

+ the linear utilities or disutilities

• rdinal content only: ui λui if λ > 0

feasible allocations: z = (zi)i∈N , zi ∈ [0, 1]A ,

N zi = eA

(convention: one unit of each good) final utilities/disutilities: Ui = ui · zi

division rule: defined in utility terms, does not distinguish two Pareto indifferent allocations single- or multi-valued: picks for each problem a single utility vector U = (Ui, i ∈ N)

• r

a set of feasible utility vectors {U} = ⇒ requires downstream negotiation

the oldest and most compelling test of fairness Fair Share Guarantee (FSG) ui · zi ≥ ui · (1 neA), resp. ui · zi ≤ ui · (1 neA) → private rights interpretation: everyone can veto any allocation and enforce the default equal split → uniform preference externalities: differences in preferences are to everyone’s advantage

the Egalitarian rule equalizes ex post welfare, measured as relative utilities the allocation z is Efficient and for all i, j ∈ N ui · zi ui · eA = uj · zj uj · eA if some uia = 0 use the leximin (goods) or leximax (bads) refinment → the simplest definition → existence and single-valuedness guaranteed, goods or bads → meets FSG

the Competitive rule

• 1. each agent chooses from the same budget set: equal opportunities ex ante

allocation z; price p ∈ RA

+ such that A pa = n and

goods: zi ∈ arg max

yi∈RA

+

{ui · yi|p · yi ≤ 1} for all i bads: zi ∈ arg min

yi∈RA

+

{ui · yi|p · yi ≥ 1} for all i

• 2. =

⇒ No Envy: ui · zi ≥ ui · zj for all i, j = ⇒ FSG

• 3. Core Stability: stand alone trades by coalitions not profitable (core from

equal split)

the Competitive rule → existence is guaranteed → characteristic first order KKT conditions: goods: {zia > 0 = ⇒ uia Ui = pa} and uib Ui ≤ pb, for all i, a, b bads: {zia > 0 = ⇒ uia Ui = pa} and uib Ui ≥ pb, for all i, a, b recall Ui = ui · zi

two alternative formulations:

• (Kelly) for any feasible allocation z with U

i = ui·z i: N U

i

Ui ≤ n (goods)

;

N U

i

Ui ≥ n (bads)

• z is a critical point of the Nash product of utilities ΠNui · zi in the set of

feasible allocations

example: a b c u1 2 1 4 u2 1 1 5 Competitive good a b c u1 1 1 1/8 u2 7/8 Egalitarian good a b c u1 1 1 2/9 u2 7/9 Competitive bad a b c u1 7/10 u2 1 1 3/10 Egalitarian bad a b c u1 1 1 7/9 u2 2/9

U1 U2 7 3 2 4 5 5 6 7 2 1 U

1 U

U

1 U

Figure 1:

take-home point #1: first key difference between goods and bads

dividing goods

• (Eisenberg Gale) the Competitive rule maximizes the Nash Product ΠNui·

zi in the feasible set = ⇒ single-valued

dividing bads

• the Competitive rule is multivalued

an example with five CEEI allocations bad a b c u1 3 2 8 u2 6 3 2

z1 = bad a b c price 12/11 6/11 4/11 z1 11/12 z2 1/12 1 1 z5 = bad a b c price 6/13 4/13 16/13 z1 1 1 3/16 z2 13/16 z3 = bad a b c price 18/19 12/19 8/19 z1 1 1/12 z2 11/12 1 z2 = bad a b c price 1 3/5 2/5 z1 1 z2 1 1 z4 = bad a b c price 3/5 2/5 1 z1 1 1 z2 1

U1 U2 8 10 13 5 3 11 9 6 2 5 U 4 U 5

1 U4 1 U5

U 1 U 2 U 3

1 U1 1 U2 1 U3

Figure 2:

what is the largest possible number of different Competitive allocations ?

• if n = 2 it is 2p − 1
• if p = 2 it is 2n − 1
• for general n, p it is no less than 2min{n,p} − 1

bad a1 a2 · · · an−1 u1 1 K K K u2 K 1 K K · · · K K 1 K un−1 K K K 1 un 1 1 1 1 where 1 < K < ∞ bad a1 · · · aq aq+1 · · · an−1 z1 q/q + 1 · · · q/q + 1 zq q/q + 1 zq+1 1 · · · 1 zn−1 1 zn 1/q + 1 1/q + 1 1/q + 1 is Competitive for any q, 1 ≤ q ≤ n − 1,

a common characterization of the Competitive rule, by means of an incentive property recall

• efficiency =

⇒ need to split at most n − 1 items among n participants

• efficiency =

⇒ at least (n − 1)(p − 1) zero entries in the allocation matrix given p items = ⇒ many lost bids

an easy manipulation under EG: exaggerate lost bids, if they remain lost good a b c u1 6 3 1 u2 1 3 6 → ze = a b c 1 1/2 1/2 1 ; good a b c u

1

6 3 3 u2 1 3 6 → ze = a b c 1 8/11 3/11 1 this always work with EG ! symmetrically with bads, minimizing lost bids is always profitable

Independence of Lost Bids (ILB) for any u, u ∈ RN×A

+

that only differ in coordinate ia and uia > u

ia

(goods) or uia < u

ia (bads) we have

∀z ∈ f(N, A, u) : zia = 0 = ⇒ z ∈ f(N, A, u)

Theorem: i) goods: the Competitive rule is the only single-valued division rule meet- ing Efficiency, Equal Treatment of Equals and/or Fair Share Guaranteed, and Independence of Lost Bids

ii) bads:

any division rule meeting Efficiency, Equal Treatment of Equals and/or Fair Share Guaranteed, and Independence of Lost Bids, contains the (multivalued) Competitive rule Note: in our model ILB is a version of Maskin Monotonicity; the proof is simple and similar to earlier arguments by Gevers (1986) and Nagahisa (1991)

more normative requirements a single-valued division rule is vulnerable to more potential normative objections than a multi-valued one closely watched tests: how does the rule reacts to shocks? → Continuity (CONT): of u → U, from the utility matrix to the final utility profile → Resource Monotonicity (RM): new goods, or more of the same goods (resp. fewer bads, or less of the same bads) is weakly good news for everyone → Population Monotonicity (PM): more people to share the same goods is weakly bad news for everyone common property implies solidarity

take-home point # 2: implementing these three tests is much harder with bads than with goods dividing goods → the Competitive rule meets CONT, RM and PM (true for cake-division as well: Sziklai/Segal-Halevi 2015) → the Egalitarian rule meets CONT and PM, but not RM

dividing bads → no (single-valued) Efficient rule can meet Fair Share Guaranteed and Re- source Monotonicity → no (single-valued) Efficient and Envy-Free rule can be Continuous → the Egalitarian rule still meets CONT and PM, but not RM

an example with goods where EG fails RM good a b c u1 3 1 1 u2 1 3 1 u3 1 1 3

• →

a b c 1 1 1

• → Ueg

1

= Ueg

2

= Ueg

3

= 3 good a b c d u1 3 1 1 u2 1 3 1 4 u3 1 1 3 4

• →

a b c d 55/59 2/59 1 1/2 2/59 1 1/2

• → Ueg

1

< 3

explaining EFF + FSG + RM = ∅ pick F : EFF + FSG ω : a b u1 1 4 u2 4 1 EFF = ⇒ one of Ui is ≤ 1, say U1 ≤ 1 ω :

1 9a

b u1 1/9 4 u2 4/9 1 z

2b ≤ u2 · z 2 ≤ u2 · (1

2eA) = 13 18 = ⇒ z

1b ≥ 5

18 = ⇒ u1 · z

1 = U 1 ≥ 10

9 > U1 contradicting RM.

explaining EFF + NoEnvy + CONT =∅: the EFF + No Envy correspondence may have up to n

2 connected components

an example with two components bad a b u1 1 3 u2 3 1 u3 4 1

first component: from z1 = a b 5/12 5/12 1/6 1 (competitive) to z0 = a b 4/9 4/9 1/9 1 second: from z2 = a b 1 1/2 1/2 (competitive) to z3 = a b 1 1/9 4/9 4/9 and all a b 1 − x x y 1 − y where 4x + 2y ≥ 1 , 2x + 3y ≥ 1 , 3x + 2y ≤ 1

Conclusion 1: the Competitive rule is very compelling for dividing goods under additive utilities, much more so than the Egalitarian one Conclusion 2: it is hard to recommend a single-valued rule for the division of bads: Fair Share Guarantee comes at the cost of Resource Monotonicity; No Envy at the cost of Continuity no single-valued selection of the Competitive correspondence stands out the Egalitarian rule meets all requirements, except No Envy, RM and ILB

further research

• sharing indivisible bads: can we still get Efficiency + No Envy up to one
• bject ?
• how to divide a mix of goods and bads?

Thank You