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Recent Advances in Fair Resource Allocation Recent Advances in Fair Resource Allocation Rupert Freeman and Nisarg Shah Microsoft Research New York University of Toronto AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 1 Disclaimer • In this tutorial, we will NOT ➢ Assume any prior knowledge of fair division ➢ Walk you through tedious, detailed proofs ➢ Claim to present a complete overview of the entire fair division realm ➢ Present (recent) unpublished results • Instead, we will ➢ Focus mostly on the case of “additive preferences” for coherence o With some results for and pointers to domains with non-additive preferences • If you spot any errors, missing results, or incorrect attributions: ➢ Please email [email protected] or [email protected] AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 2 Outline • Fairness Axioms • Implications of fairness ➢ Proportionality ➢ Price of fairness ➢ Envy-freeness ➢ Interplay with strategyproofness ➢ Maximin share guarantee and Pareto optimality ➢ Groupwise fairness ➢ Restricted cases o Core o Group envy-freeness • Settings o Groupwise MMS ➢ Cake-cutting o Group fairness ➢ Homogeneous divisible goods ➢ Indivisible goods AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 3 A Generic Resource Allocation Framework • A set of agents 푁 = {1,2, … , 푛} • A set of resources 푀 ➢ May be finite or infinite • Valuations 푀 ➢ Valuation of agent 푖 is 푣푖 ∶ 2 → ℝ ➢ Range is ℝ+ when resources are goods, and ℝ− when they are bads • Allocations ➢ 퐴 = 퐴1, … , 퐴푛 ∈ Π푛 푀 is a partition of resources among agents o 퐴푖 ∩ 퐴푗 = ∅, ∀푖, 푗 ∈ 푁 and ∪푖∈푁 퐴푖 = 푀 ➢ A partial allocation 퐴 may have ∪푖∈푁 퐴푖 ≠ 푀 AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 4 Cake Cutting • Formally introduced by Steinhaus [1948] • Agents: 푁 = {1,2, … , 푛} • Resource (cake): 푀 = [0,1] • Constraints on an allocation 퐴 ➢ The entire cake is allocated (full allocation) ➢ Each 퐴푖 ∈ 풜, where 풜 is the set of finite unions of disjoint intervals • Simple allocations ➢ Each agent is allocated a single interval ➢ Cuts cake at 푛 − 1 points AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 5 Agent Valuations • Each agent 푖 has an integrable density 훼 β function 푓푖: 0,1 → ℝ+ 훼 +β훽 For each 푋 ∈ 풜, 푣푖 푋 = ׬푥∈푋 푓푖 푥 푑푥 • 1 훼 For normalization, we require ׬0 푓푖 푥 푑푥 = 1 • ➢ Without loss of generality 휆훼 AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 6 Agent Valuations • In this model, the valuations satisfy the 훼 β following properties • Normalized: 푣 0,1 = 1 푖 훼 +β훽 • Divisible: ∀휆 ∈ [0,1] and 퐼 = [푥, 푦], ∃푧 ∈ [푥, 푦] s.t. 푣푖 [푥, 푧] = 휆푣푖([푥, 푦]) 훼 • Additive: For disjoint intervals 퐼 and 퐼′, 푣 퐼 + 푣 퐼′ = 푣 퐼 ∪ 퐼′ 푖 푖 푖 휆훼 AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 7 Complexity • Inputs are functions eval output ➢ Infinitely many bits may be needed to fully represent the input ➢ Query complexity is more useful 훼 • Robertson-Webb Model ➢ Eval푖(푥, 푦) returns 푣푖 푥, 푦 푥 푦 ➢ Cut푖(푥, 훼) returns 푦 such that 푣푖 푥, 푦 = 훼 cut output AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 8 Three Classic Fairness Desiderata 1 • Proportionality (Prop): ∀푖 ∈ 푁: 푣푖 퐴푖 ≥ Τ푛 ➢ Each agent should receive her “fair share” of the utility. • Envy-Freeness (EF): ∀푖, 푗 ∈ 푁: 푣푖 퐴푖 ≥ 푣푖(퐴푗) ➢ No agent should wish to swap her allocation with another agent. • Equitability (EQ): ∀푖, 푗 ∈ 푁 ∶ 푣푖 퐴푖 = 푣푗 퐴푗 ➢ All agents should have the exact same value for their allocations. ➢ No agent should be jealous of what another agent received. AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 9 Example 1 • Value density functions • Agent 1 wants [0, Τ3] uniformly and does not want anything else 3 • Agent 2 wants the entire cake 2 uniformly 2 1 • Agent 3 wants [ Τ3 , 1] uniformly and does not want anything else 0 1 2 1 ൗ3 ൗ3 AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 10 Example • Value density functions • Consider the following allocation 3 1 1 • 퐴1 = 0, Τ9 ⇒ 푣1 퐴1 = Τ3 1 8 7 • 퐴2 = Τ9 , Τ9 ⇒ 푣2 퐴2 = Τ9 2 8 1 • 퐴3 = Τ9 , 1 ⇒ 푣3 퐴3 = Τ3 1 • The allocation is proportional, but not envy-free or equitable 0 1 2 1 ൗ3 ൗ3 AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 11 Example • Value density functions • Consider the following allocation 3 1 1 • 퐴1 = 0, Τ6 ⇒ 푣1 퐴1 = Τ2 1 5 2 • 퐴2 = Τ6 , Τ6 ⇒ 푣2 퐴2 = Τ3 2 5 1 • 퐴3 = Τ6 , 1 ⇒ 푣3 퐴3 = Τ2 1 • The allocation is proportional and envy-free, but not equitable 0 1 2 1 ൗ3 ൗ3 AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 12 Example • Value density functions • Consider the following allocation 3 1 3 • 퐴1 = 0, Τ5 ⇒ 푣1 퐴1 = Τ5 1 4 3 • 퐴2 = Τ5 , Τ5 ⇒ 푣2 퐴2 = Τ5 2 4 3 • 퐴3 = Τ5 , 1 ⇒ 푣3 퐴3 = Τ5 1 • The allocation is proportional, envy-free, and equitable 0 1 2 1 ൗ3 ൗ3 AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 13 Relations Between Fairness Desiderata • Envy-freeness implies proportionality ➢ Summing 푣푖 퐴푖 ≥ 푣푖 퐴푗 over all 푗 gives proportionality • For 2 agents, proportionality also implies envy-freeness ➢ Hence, they are equivalent. • Equitability is incomparable to proportionality and envy-freeness ➢ E.g. if each agent has value 0 for her own allocation and 1 for the other agent’s allocation, it is equitable but not proportional or envy-free. AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 14 Existence • Theorem [Alon, 1987] Suppose the value density function 푓푖 of each agent valuation 푣푖 is continuous. Then, we can cut the cake at 푛2 − 푛 places and rearrange 2 the 푛 − 푛 + 1 intervals into 푛 pieces 퐴1, … , 퐴푛 such that 1 푣푖 퐴푗 = ൗ푛 , ∀푖, 푗 ∈ 푁 • This is called a “perfect partition” ➢ It is trivially envy-free (thus proportional) and equitable • As we will later see, this cannot be found with finitely many queries in Robertson-Webb model AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 15 Proportionality AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 16 PROPORTIONALITY : 푛 = 2 AGENTS • CUT-AND-CHOOSE ➢ Agent 1 cuts the cake at 푥 such that 푣1 0, 푥 = 푣1 푥, 1 = 1Τ2 ➢ Agent 2 chooses the piece that she prefers. • Elegant protocol ➢ Proportional (equivalent to envy-freeness for 2 agents) ➢ Needs only one cut and one eval query (optimal) • More agents? AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 17 PROPORTIONALITY: DUBINS-SPANIER • DUBINS-SPANIER ➢ Referee starts a knife at 0 and moves the knife to the right. ➢ Repeat: When the piece to the left of the knife is worth 1/푛 to an agent, the agent shouts “stop”, receives the piece, and exits. ➢ When only one agent remains, she gets the remaining piece. • Can be implemented easily in Robertson-Webb model ➢ When [푥, 1] is left, ask each remaining agent 푖 to cut at 푦푖 so that 푣푖 푥, 푦푖 = ∗ 1/푛, and give agent 푖 ∈ arg min푖 푦푖 the piece [푥, 푦푖∗]. • Query complexity: Θ(푛2) AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 18 PROPORTIONALITY: EVEN-PAZ • EVEN-PAZ • Input: ➢ Interval [푥, 푦], number of agents 푛 (assume a power of 2 for simplicity) • Recursive procedure: ➢ If 푛 = 1, give [푥, 푦] to the single agent. ➢ Otherwise: o Each agent 푖 marks 푧푖 such that 푣푖 푥, 푧푖 = 푣푖 푧푖, 푦 o 푧∗ = 푛Τ2 th mark from the left. o Recurse on [푥, 푧∗] with the left 푛/2 agents, and on [푧∗, 푦] with the right 푛/2 agents. • Query complexity: Θ(푛 log 푛) AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 19 Complexity of Proportionality • Theorem [Edmonds and Pruhs, 2006]: ➢ Any protocol returning a proportional allocation needs Ω(푛 log 푛) queries in the Robertson-Webb model. • Hence, EVEN-PAZ is provably (asymptotically) optimal! AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 20 Envy-Freeness AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 21 Envy-Freeness : Few Agents • 푛 = 2 agents : CUT-AND-CHOOSE (2 queries) • 푛 = 3 agents : SELFRIDGE-CONWAY (14 queries) Gets complex pretty quickly! AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 22 Envy-Freeness : Few Agents • [Brams and Taylor, 1995] ➢ The first finite (but unbounded) protocol for any number of agents • [Aziz and Mackenzie, 2016a] ➢ The first bounded protocol for 4 agents (at most 203 queries) • [Amanatidis et al., 2018] ➢ A simplified version of the above protocol for 4 agents (at most 171 queries) AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 23 Envy-Freeness • Theorem [Aziz and Mackenzie, 2016b] ➢ There exists a bounded protocol for computing an envy-free allocation with 푛 푛푛 푛푛 agents, which requires 푂(푛푛 ) queries 2푛+3 ➢ After 푂 푛 queries, the protocol can output a partial allocation that is both proportional and envy-free • What about lower bounds? AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 24 Complexity of Envy-Freeness • Theorem [Procaccia, 2009] Any protocol for finding an envy-free allocation requires Ω(푛2) queries.
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