CAKE CUTTING How to fairly divide a heterogeneous divisible good - PowerPoint PPT Presentation

T RUTH J USTICE A LGOS Fair Division I: Cake Cutting Basics Teachers: Ariel Procaccia (this time) and Alex Psomas

CAKE CUTTING How to fairly divide a heterogeneous divisible good between players with different preferences?

THE PROBLEM • Cake is interval [0,1] • Set of players N = {1, … , 𝑜} • Piece of cake 𝑌 ⊆ [0,1]: finite union of disjoint intervals 0 1

THE PROBLEM • Each player 𝑗 ∈ 𝑂 has a non- negative valuation 𝑊 𝑗 over 𝛽 β pieces of cake • Additive: for 𝑌 ∩ 𝑍 = ∅ , β 𝛽 + 𝛾 𝑊 𝑗 𝑌 + 𝑊 𝑗 𝑍 = 𝑊 𝑗 (𝑌 ∪ 𝑍) • Normalized: For all 𝑗 ∈ 𝑂 , 𝛽 𝑊 0,1 = 1 𝑗 • Divisible: ∀𝜇 ∈ 0,1 can cut 𝜇𝛽 𝐽 ′ ⊆ 𝐽 s.t. 𝑊 𝑗 𝐽 ′ = 𝜇𝑊 𝑗 (𝐽)

FAIRNESS PROPERTIES • Our goal is to find an allocation 𝐵 1 , … , 𝐵 𝑜 • Proportionality: 1 ∀𝑗 ∈ 𝑂, 𝑊 𝑗 𝐵 𝑗 ≥ 𝑜 • Envy-Freeness (EF): ∀𝑗, 𝑘 ∈ 𝑂, 𝑊 𝑗 𝐵 𝑗 ≥ 𝑊 𝑗 (𝐵 𝑘 ) Question ? For 𝑜 = 2 , which is stronger? • Proportionality • Equivalent • Envy-Freeness • Incomparable

FAIRNESS PROPERTIES • Our goal is to find an allocation 𝐵 1 , … , 𝐵 𝑜 • Proportionality: 1 ∀𝑗 ∈ 𝑂, 𝑊 𝑗 𝐵 𝑗 ≥ 𝑜 • Envy-Freeness (EF): ∀𝑗, 𝑘 ∈ 𝑂, 𝑊 𝑗 𝐵 𝑗 ≥ 𝑊 𝑗 (𝐵 𝑘 ) ? Poll 1 For 𝑜 ≥ 3 , which is stronger? • Proportionality • Equivalent • Envy-Freeness • Incomparable

CUT-AND-CHOOSE • Algorithm for 𝑜 = 2 [Procaccia and Procaccia, circa 1985] 1/2 2/3 • Player 1 divides into two pieces 𝑌, 𝑍 s.t. Τ Τ 𝑊 1 𝑌 = 1 2 , 𝑊 1 𝑍 = 1 2 1/2 1/3 • Player 2 chooses preferred piece • This is EF (hence proportional)

THE ROBERTSON-WEBB MODEL • What is the complexity of Cut-and- Choose? • Input size is 𝑜 • Two types of operations ◦ Eval 𝑗 𝑦, 𝑧 returns 𝑊 𝑗 ( 𝑦, 𝑧 ) ◦ Cut 𝑗 𝑦, 𝛽 returns 𝑧 such that 𝑊 𝑦, 𝑧 = 𝛽 𝑗 𝛽 eval output 𝑦 𝑧 cut output

THE ROBERTSON-WEBB MODEL • Two types of operations ◦ Eval 𝑗 𝑦, 𝑧 returns 𝑊 𝑗 ( 𝑦, 𝑧 ) ◦ Cut 𝑗 𝑦, 𝛽 returns 𝑧 such that 𝑊 𝑦, 𝑧 = 𝛽 𝑗 Question ? #Operations needed to find an EF allocation when 𝑜 = 2 ? • One • Three • Two • Four

DUBINS-SPANIER • Referee continuously moves knife • Repeat: when piece left of knife is worth 1/𝑜 to player, player shouts “ stop ” and gets piece • That player is removed • Last player gets remaining piece ? Poll 2 What is the complexity of DS? • Θ 𝑜 2 • Θ(𝑜) • Θ(𝑜 2 log 𝑜) • Θ(𝑜 log 𝑜)

DUBINS-SPANIER

DUBINS-SPANIER 1/3

DUBINS-SPANIER

DUBINS-SPANIER

EVEN-PAZ • Given [𝑦, 𝑧] , assume 𝑜 = 2 𝑙 for ease of exposition • If 𝑜 = 1 , give [𝑦, 𝑧] to the single player • Otherwise, each player 𝑗 makes a mark 𝑨 s.t. 𝑗 [𝑦, 𝑨] = 1 𝑊 2 𝑊 𝑗 ([𝑦, 𝑧]) • Let 𝑨 ∗ be the Τ 𝑜 2 mark from the left • Recurse on [𝑦, 𝑨 ∗ ] with the left Τ 𝑜 2 players, and on [𝑨 ∗ , 𝑧] with the right Τ 𝑜 2 players

EVEN-PAZ

EVEN-PAZ • Claim: The Even-Paz protocol produces a proportional allocation • Proof: • At stage 0 , each of the 𝑜 players values the whole cake at 1 • At each stage the players who share a piece of cake value it at least at 𝑊 𝑗 ( 𝑦, 𝑧 )/2 • Hence, if at stage 𝑙 each player has value at least 1/2 𝑙 for the piece he ’ s sharing, then at 1 stage 𝑙 + 1 each player has value at least 2 𝑙+1 • The number of stages is log 𝑜 ∎

𝑈 1 = 0, 𝑈 𝑜 = 2𝑜 + 2𝑈 𝑜 2 2𝑜 𝑜 𝑜 = 2𝑜 log 𝑜 𝑜/2 𝑜/2 𝑜/2 𝑜/2 = 2𝑜 𝑜/2 pairs 4 4 4 4 = 2𝑜 Overall: 2𝑜 log 𝑜

COMPLEXITY OF PROPORTIONALITY • Theorem [Edmonds and Pruhs 2006]: Any proportional protocol needs Ω(𝑜 log𝑜) operations in the RW model • The Even-Paz protocol is provably optimal! • What about envy?

SELFRIDGE-CONWAY • Stage 0 ◦ Player 1 divides the cake into three equal pieces according to 𝑊 1 ◦ Player 2 trims the largest piece s.t. there is a tie between the two largest pieces according to 𝑊 2 ◦ Cake 1 = cake w/o trimmings, Cake 2 = trimmings • Stage 1 (division of Cake 1) ◦ Player 3 chooses one of the three pieces of Cake 1 ◦ If player 3 did not choose the trimmed piece, player 2 is allocated the trimmed piece ◦ Otherwise, player 2 chooses one of the two remaining pieces ◦ Player 1 gets the remaining piece ◦ Denote the player 𝑗 ∈ {2, 3} that received the trimmed piece by 𝑈 , and the other by 𝑈′ • Stage 2 (division of Cake 2) ◦ 𝑈′ divides Cake 2 into three equal pieces according to 𝑊 𝑈 ′ ◦ Players 𝑈 , 1, and 𝑈′ choose the pieces of Cake 2, in that order

THE COMPLEXITY OF EF • Theorem [Brams and Taylor 1995]: There is an EF cake cutting algorithm in the RW model • But it is unbounded • Theorem [P 2009]: Any EF algorithm requires Ω(𝑜 2 ) queries in the RW model

THE COMPLEXITY OF EF • Theorem [Aziz and Mackenzie 2016a]: There is a bounded EF algorithm for four players • Theorem [Aziz and Mackenzie 2016b]: There is a bounded EF algorithm for any 𝑜 , whose complexity is 𝑜 𝑜 𝑜𝑜𝑜𝑜 𝑃 • Stay tuned for more next time …

A SUBTLETY • EF protocol that uses 𝑜 queries • 𝑔 = encoding of the information needed by the Aziz-Mackenzie protocol into [0,1] • The protocol asks each player cut 𝑗 (0, Τ 1 2) • Player 𝑗 replies with 𝑧 𝑗 = 𝑔(𝑊 𝑗 ) • The protocol simulates the Aziz-Mackenzie protocol ‘ in the background ’ using 𝑔 −1 (𝑧 𝑗 ) for all 𝑗 ∈ 𝑂 • Is this a valid EF protocol in the RW model?