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tedious, detailed proofs ➢ Claim to present a complete overview of the entire fair division realm ➢ Present (recent) unpublished results • Instead, we will ➢ Focus mostly on the case of “additive preferences” for coherence o With some results for and pointers to domains with non-additive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.