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 u i ∈ R A + the linear utilities or disutilities ordinal content only: u i � λu i if λ > 0 feasible allocations: z = ( z i ) i ∈ N , z i ∈ [0 , 1] A , � N z i = e A (convention: one unit of each good) final utilities/disutilities: U i = u i · z i

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 = ( U i , i ∈ N ) or a set of feasible utility vectors { U } = ⇒ requires downstream negotiation

the oldest and most compelling test of fairness Fair Share Guarantee (FSG) u i · z i ≥ u i · (1 ne A ) , resp. u i · z i ≤ u i · (1 ne A ) → 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 u i · e A = u j · z j u i · z i u j · e A if some u ia = 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 + such that � allocation z ; price p ∈ R A A p a = n and goods : z i ∈ arg max { u i · y i | p · y i ≤ 1 } for all i y i ∈ R A + bads : z i ∈ arg min { u i · y i | p · y i ≥ 1 } for all i y i ∈ R A + 2 . = ⇒ No Envy: u i · z i ≥ u i · z j 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: ⇒ u ia = p a } and u ib goods : { z ia > 0 = ≤ p b , for all i, a, b U i U i ⇒ u ia = p a } and u ib bads : { z ia > 0 = ≥ p b , for all i, a, b U i U i recall U i = u i · z i

two alternative formulations: U � i : � • (Kelly) for any feasible allocation z � with U � i = u i · z � i U i ≤ n (goods) N U � ; � i U i ≥ n (bads) N • z is a critical point of the Nash product of utilities Π N u i · z i in the set of feasible allocations

a b c example: u 1 2 1 4 u 2 1 1 5 good a b c good a b c Competitive u 1 1 1 1 / 8 Egalitarian u 1 1 1 2 / 9 u 2 0 0 7 / 8 u 2 0 0 7 / 9 bad bad a b c a b c Competitive u 1 0 0 7 / 10 Egalitarian u 1 1 1 7 / 9 u 2 1 1 3 / 10 u 2 0 0 2 / 9

U 2 7 6 5 1 U U 1 U U 2 1 U 1 2 3 4 5 7 Figure 1:

take-home point #1 : first key difference between goods and bads dividing goods • (Eisenberg Gale) the Competitive rule maximizes the Nash Product Π N u i · z i in the feasible set = ⇒ single-valued dividing bads • the Competitive rule is multivalued

an example with five CEEI allocations bad a b c u 1 3 2 8 u 2 6 3 2

bad a b c bad a b c price 12 / 11 6 / 11 4 / 11 price 6 / 13 4 / 13 16 / 13 z 1 = z 5 = z 1 11 / 12 0 0 z 1 1 1 3 / 16 z 2 1 / 12 1 1 z 2 0 0 13 / 16 bad a b c price 18 / 19 12 / 19 8 / 19 z 3 = z 1 1 1 / 12 0 0 11 / 12 1 z 2 bad a b c bad a b c price 1 3 / 5 2 / 5 price 3 / 5 2 / 5 1 z 2 = z 4 = z 1 1 0 0 z 1 1 1 0 z 2 0 1 1 z 2 0 0 1

U 2 11 9 1 U 1 1 6 U 2 U 1 1 U 3 U 2 5 U 3 1 U 4 1 U 5 2 U 4 U 5 U 1 3 5 8 10 13 Figure 2:

what is the largest possible number of different Competitive allocations ? • if n = 2 it is 2 p − 1 • if p = 2 it is 2 n − 1 • for general n, p it is no less than 2 min { n,p } − 1

bad a 1 a 2 · · · a n − 1 u 1 1 K K K u 2 K 1 K K where 1 < K < ∞ · · · K K 1 K u n − 1 K K K 1 u n 1 1 1 1 bad a 1 · · · a q a q +1 · · · a n − 1 z 1 q/q + 1 0 0 · · · 0 q/q + 1 0 0 0 0 q/q + 1 z q 1 0 0 z q +1 · · · 0 0 1 0 0 0 1 z n − 1 1 /q + 1 1 /q + 1 1 /q + 1 0 0 0 z n 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 a b c good a b c a b c → z e = → z � e = u � u 1 6 3 1 1 1 / 2 0 ; 6 3 3 1 8 / 11 0 1 u 2 1 3 6 0 1 / 2 1 u 2 1 3 6 0 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 � ∈ R N × A that only differ in coordinate ia and u ia > u � + ia (goods) or u ia < u � ia (bads) we have ⇒ z ∈ f ( N, A, u � ) ∀ z ∈ f ( N, A, u ) : z ia = 0 =

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 a b c u 1 3 1 1 1 0 0 → U eg = U eg = U eg � → � = 3 u 2 1 2 3 1 3 1 0 1 0 u 3 0 0 1 1 1 3 good a b c d a b c d u 1 3 1 1 0 55 / 59 0 0 0 → U eg � → � < 3 u 2 1 2 / 59 1 0 1 / 2 1 3 1 4 u 3 2 / 59 0 1 1 / 2 1 1 3 4

explaining EFF + FSG + RM = ∅ pick F : EFF + FSG a b ω : u 1 1 4 EFF = ⇒ one of U i is ≤ 1 , say U 1 ≤ 1 u 2 4 1 1 9 a b ω � : u 1 1 / 9 4 4 / 9 1 u 2 2 ≤ u 2 · (1 2 e A � ) = 13 z � 2 b ≤ u 2 · z � 18 1 b ≥ 5 1 ≥ 10 ⇒ z � ⇒ u 1 · z � 1 = U � = 18 = 9 > U 1 contradicting RM.