Lecture 6 Fair Division 1: Cake-Cutting [Some illustrations due to: - PowerPoint PPT Presentation

CSC2556 Lecture 6 Fair Division 1: Cake-Cutting [Some illustrations due to: Ariel Procaccia] CSC2556 - Nisarg Shah 1

Announcements • Reminder ➢ Project proposal due by March 1 st by 12:59PM ➢ If you want to run your idea by me, this is a good time to approach me. • Remember to use office hours (drop me an email) if you ’ re having any difficulty with homework questions. CSC2556 - Nisarg Shah 2

Fair Division CSC2556 - Nisarg Shah 3

Cake-Cutting • A heterogeneous, divisible good ➢ Heterogeneous: it may be valued differently by different individuals ➢ Divisible: we can share/divide it between individuals • Represented as [0,1] ➢ Almost without loss of generality • Set of players 𝑂 = {1, … , 𝑜} • Piece of cake 𝑌 ⊆ [0,1] ➢ A finite union of disjoint intervals CSC2556 - Nisarg Shah 4

Agent Valuations • Each player 𝑗 has a valuation 𝑊 𝑗 that is very much like a probability distribution over [0,1] 𝛽 β • Additive: For 𝑌 ∩ 𝑍 = ∅ , 𝑊 𝑗 𝑌 + 𝑊 𝑗 𝑍 = 𝑊 𝑗 𝑌 ∪ 𝑍 β 𝛽 + 𝛾 • Normalized: 𝑊 0,1 = 1 𝑗 𝛽 • Divisible: ∀𝜇 ∈ [0,1] and 𝑌 , ∃𝑍 ⊆ 𝑌 s.t. 𝑊 𝑗 𝑍 = 𝜇𝑊 𝑗 (𝑌) 𝜇𝛽 CSC2556 - Nisarg Shah 5

Fairness Goals • An allocation is a disjoint partition 𝐵 = (𝐵 1 , … , 𝐵 𝑜 ) of the cake • We desire the following fairness properties from our allocation 𝐵 : • Proportionality (Prop): 𝑗 𝐵 𝑗 ≥ 1 ∀𝑗 ∈ 𝑂: 𝑊 𝑜 • Envy-Freeness (EF): ∀𝑗, 𝑘 ∈ 𝑂: 𝑊 𝑗 𝐵 𝑗 ≥ 𝑊 𝑗 (𝐵 𝑘 ) CSC2556 - Nisarg Shah 6

Fairness Goals • Prop: ∀𝑗 ∈ 𝑂: 𝑊 Τ 𝑗 𝐵 𝑗 ≥ 1 𝑜 • EF: ∀𝑗, 𝑘 ∈ 𝑂: 𝑊 𝑗 𝐵 𝑗 ≥ 𝑊 𝑗 𝐵 𝑘 • Question: What is the relation between proportionality and EF? Prop ⇒ EF 1. EF ⇒ Prop 2. Equivalent 3. Incomparable 4. CSC2556 - Nisarg Shah 7

C UT - AND -C HOOSE • Algorithm for 𝑜 = 2 players • Player 1 divides the cake into two pieces 𝑌, 𝑍 s.t. Τ 𝑊 1 𝑌 = 𝑊 1 𝑍 = 1 2 • Player 2 chooses the piece she prefers. • This is EF and therefore proportional. ➢ Why? CSC2556 - Nisarg Shah 8

Input Model • How do we measure the “time complexity” of a cake-cutting algorithm for 𝑜 players? • Typically, time complexity is a function of the length of input encoded as binary. • Our input consists of functions 𝑊 𝑗 , which requires infinite bits to encode. • We want running time just as a function of 𝑜 . CSC2556 - Nisarg Shah 9

Robertson-Webb Model • We restrict access to valuations 𝑊 𝑗 ’ s through two types of queries: ➢ Eval 𝑗 (𝑦, 𝑧) returns 𝑊 𝑦, 𝑧 𝑗 ➢ Cut 𝑗 (𝑦, 𝛽) returns 𝑧 such that 𝑊 𝑦, 𝑧 = 𝛽 𝑗 𝛽 eval output 𝑦 𝑧 cut output CSC2556 - Nisarg Shah 10

Robertson-Webb Model • Two types of queries: ➢ Eval 𝑗 𝑦, 𝑧 = 𝑊 𝑦, 𝑧 𝑗 ➢ Cut 𝑗 𝑦, 𝛽 = 𝑧 s.t. 𝑊 𝑦, 𝑧 = 𝛽 𝑗 • Question: How many queries are needed to find an EF allocation when 𝑜 = 2 ? • Answer: 2 ➢ Why? CSC2556 - Nisarg Shah 11

D UBINS -S PANIER • Protocol for finding a proportional allocation for 𝑜 players • Referee starts at 0 , and continuously moves knife to the right. • Repeat: when piece to the left of knife is worth 1 /𝑜 to a player, the player shouts “ stop ” , gets the piece, and exits. • The last player gets the remaining piece. CSC2556 - Nisarg Shah 12

D UBINS -S PANIER 1/3 1/3 ≥ 1/3 CSC2556 - Nisarg Shah 13

D UBINS -S PANIER • Moving knife is not really needed. • At each stage, we can ask each remaining player a cut query to mark his 1/𝑜 point in the remaining cake. • Move the knife to the leftmost mark. CSC2556 - Nisarg Shah 14

D UBINS -S PANIER CSC2556 - Nisarg Shah 15

D UBINS -S PANIER 1 3 Τ CSC2556 - Nisarg Shah 16

D UBINS -S PANIER 1 3 Τ 1 3 Τ CSC2556 - Nisarg Shah 17

D UBINS -S PANIER 1 3 Τ 1 3 Τ ≥ 1 3 Τ CSC2556 - Nisarg Shah 18

D UBINS -S PANIER • Question: What is the complexity of the Dubins- Spanier protocol in the Robertson-Webb model? Θ 𝑜 1. Θ 𝑜 log 𝑜 2. Θ 𝑜 2 3. Θ 𝑜 2 log 𝑜 4. CSC2556 - Nisarg Shah 19

E VEN -P AZ • Input: Interval [𝑦, 𝑧] , number of players 𝑜 ➢ Assume 𝑜 = 2 𝑙 for some 𝑙 • If 𝑜 = 1 , give [𝑦, 𝑧] to the single player. • Otherwise, let each player 𝑗 mark 𝑨 𝑗 s.t. = 1 𝑊 𝑦, 𝑨 𝑗 2 𝑊 𝑦, 𝑧 𝑗 𝑗 • Let 𝑨 ∗ be the 𝑜/2 mark from the left. • Recurse on [𝑦, 𝑨 ∗ ] with the left 𝑜/2 players, and on [𝑨 ∗ , 𝑧] with the right 𝑜/2 players. CSC2556 - Nisarg Shah 20

E VEN -P AZ CSC2556 - Nisarg Shah 21

E VEN -P AZ • Theorem: E VEN -P AZ returns a Prop allocation. • Proof: ➢ Inductive proof. We want to prove that if player 𝑗 is allocated piece 𝐵 𝑗 when [𝑦, 𝑧] is divided between 𝑜 Τ players, 𝑊 𝑗 𝐵 𝑗 ≥ 1 𝑜 𝑊 𝑦, 𝑧 𝑗 o Then Prop follows because initially 𝑊 𝑦, 𝑧 = 𝑊 0,1 = 1 𝑗 𝑗 ➢ Base case: 𝑜 = 1 is trivial. ➢ Suppose it holds for 𝑜 = 2 𝑙−1 . We prove for 𝑜 = 2 𝑙 . ➢ Take the 2 𝑙−1 left players. 𝑦, 𝑨 ∗ o Every left player 𝑗 has 𝑊 Τ ≥ 1 2 𝑊 𝑦, 𝑧 𝑗 𝑗 1 1 𝑦, 𝑨 ∗ o If it gets 𝐵 𝑗 , by induction, 𝑊 𝑗 𝐵 𝑗 ≥ 2 𝑙−1 𝑊 ≥ 2 𝑙 𝑊 𝑦, 𝑧 𝑗 𝑗 CSC2556 - Nisarg Shah 22

E VEN -P AZ • Question: What is the complexity of the Even-Paz protocol in the Robertson-Webb model? Θ 𝑜 1. Θ 𝑜 log 𝑜 2. Θ 𝑜 2 3. Θ 𝑜 2 log 𝑜 4. CSC2556 - Nisarg Shah 23

Complexity of Proportionality • Theorem [Edmonds and Pruhs, 2006]: Any proportional protocol needs Ω(𝑜 log 𝑜) operations in the Robertson-Webb model. • Thus, the E VEN -P AZ protocol is (asymptotically) provably optimal! CSC2556 - Nisarg Shah 24

Envy-Freeness? • “I suppose you are also going to give such cute algorithms for finding envy- free allocations?” • Bad luck. For 𝑜 -player EF cake-cutting: ➢ [Brams and Taylor, 1995] give an unbounded EF protocol. ➢ [Procaccia 2009] shows Ω 𝑜 2 lower bound for EF. ➢ Last year, the long-standing major open question of “bounded EF protocol” was resolved! ➢ [Aziz and Mackenzie, 2016]: 𝑃(𝑜 𝑜 𝑜 𝑜𝑜𝑜 ) protocol! o Not a typo! CSC2556 - Nisarg Shah 25

Other Desiderata • There are two more properties that we often desire from an allocation. • Pareto optimality (PO) ➢ Notion of efficiency ➢ Informally, it says that there should be no “ obviously better ” allocation • Strategyproofness (SP) ➢ No player should be able to gain by misreporting her valuation CSC2556 - Nisarg Shah 26

Strategyproofness (SP) • For deterministic mechanisms ➢ “ Strategyproof ”: No player should be able to increase her utility by misreporting her valuation, irrespective of what other players report. • For randomized mechanisms ➢ “ Strategyproof-in-expectation ”: No player should be able to increase her expected utility by misreporting. ➢ For simplicity, we’ll call this strategyproofness, and assume we mean “in expectation” if the mechanism is randomized. CSC2556 - Nisarg Shah 27

Strategyproofness (SP) • Deterministic ➢ Bad news! ➢ Theorem [Menon & Larson ‘ 17] : No deterministic SP mechanism is (even approximately) proportional. • Randomized ➢ Good news! ➢ Theorem [Chen et al. ‘ 13, Mossel & Tamuz ‘ 10]: There is a randomized SP mechanism that always returns an envy- free allocation. CSC2556 - Nisarg Shah 28

Perfect Partition • Theorem [Lyapunov ’40]: ➢ There always exists a “perfect partition” (𝐶 1 , … , 𝐶 𝑜 ) of 1 𝑜 for every 𝑗, 𝑘 ∈ [𝑜] . 𝑘 = Τ the cake such that 𝑊 𝑗 𝐶 ➢ Every agent values every bundle equally. • Theorem [Alon ‘87]: ➢ There exists a perfect partition that only cuts the cake at 𝑞𝑝𝑚𝑧(𝑜) points. ➢ In contrast, Lyapunov’s proof is non -constructive, and might need an unbounded number of cuts. CSC2556 - Nisarg Shah 29

Perfect Partition • Q: Can you use an algorithm for computing a perfect partition as a black-box to design a randomized SP+EF mechanism? ➢ Yes! Compute a perfect partition, and assign the 𝑜 bundles to the 𝑜 players uniformly at random. ➢ Why is this EF? o Every agent values every bundle at Τ 1 𝑜 . ➢ Why is this SP-in-expectation? o Because an agent is assigned a random bundle, her expected 1 𝑜 , irrespective of what she reports. utility is Τ CSC2556 - Nisarg Shah 30