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

ALGOS TRUTH JUSTICE

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

1

β

THE PROBLEM

• Each player 𝑗 ∈ 𝑂 has a non-

negative valuation 𝑊

𝑗 over

pieces of cake

• Additive: for 𝑌 ∩ 𝑍 = ∅,

𝑊

𝑗 𝑌 + 𝑊 𝑗 𝑍 = 𝑊 𝑗(𝑌 ∪ 𝑍)

• Normalized: For all 𝑗 ∈ 𝑂,

𝑊

𝑗

0,1 = 1

• Divisible: ∀𝜇 ∈ 0,1 can cut

𝐽′ ⊆ 𝐽 s.t. 𝑊

𝑗 𝐽′ = 𝜇𝑊 𝑗(𝐽)

𝛽 β 𝛽 𝜇𝛽

𝛽 + 𝛾

FAIRNESS PROPERTIES

• Our goal is to find an allocation 𝐵1, … , 𝐵𝑜
• Proportionality:

∀𝑗 ∈ 𝑂, 𝑊

𝑗 𝐵𝑗 ≥ 1 𝑜

• Envy-Freeness (EF):

∀𝑗, 𝑘 ∈ 𝑂, 𝑊

𝑗 𝐵𝑗 ≥ 𝑊 𝑗(𝐵𝑘)

For 𝑜 = 2, which is stronger?

• Proportionality
• Equivalent
• Envy-Freeness
• Incomparable

Question

?

FAIRNESS PROPERTIES

• Our goal is to find an allocation 𝐵1, … , 𝐵𝑜
• Proportionality:

∀𝑗 ∈ 𝑂, 𝑊

𝑗 𝐵𝑗 ≥ 1 𝑜

• Envy-Freeness (EF):

∀𝑗, 𝑘 ∈ 𝑂, 𝑊

𝑗 𝐵𝑗 ≥ 𝑊 𝑗(𝐵𝑘)

For 𝑜 ≥ 3, which is stronger?

• Proportionality
• Equivalent
• Envy-Freeness
• Incomparable

Poll 1

?

CUT-AND-CHOOSE

• Algorithm for 𝑜 = 2 [Procaccia

and Procaccia, circa 1985]

• Player 1 divides into two

pieces 𝑌, 𝑍 s.t. 𝑊

1 𝑌 =

Τ 1 2 , 𝑊

1 𝑍 =

Τ 1 2

• Player 2 chooses preferred

piece

• This is EF (hence proportional)

1/2 1/3 1/2 2/3

THE ROBERTSON-WEBB MODEL

• What is the complexity of Cut-and-

Choose?

• Input size is 𝑜
• Two types of operations
• Eval𝑗 𝑦, 𝑧 returns 𝑊

𝑗( 𝑦, 𝑧 )

• Cut𝑗 𝑦, 𝛽 returns 𝑧 such that 𝑊

𝑗

𝑦, 𝑧 = 𝛽

𝑦 𝑧

𝛽

eval output cut output

THE ROBERTSON-WEBB MODEL

• Two types of operations
• Eval𝑗 𝑦, 𝑧 returns 𝑊

𝑗( 𝑦, 𝑧 )

• Cut𝑗 𝑦, 𝛽 returns 𝑧 such that 𝑊

𝑗

𝑦, 𝑧 = 𝛽

#Operations needed to find an EF allocation when 𝑜 = 2?

• One
• Three
• Two
• Four

Question

?

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

What is the complexity of DS?

• Θ(𝑜)
• Θ 𝑜2
• Θ(𝑜 log 𝑜)
• Θ(𝑜2 log 𝑜)

Poll 2

?

DUBINS-SPANIER

1/3

DUBINS-SPANIER

DUBINS-SPANIER

DUBINS-SPANIER

EVEN-PAZ

• Given [𝑦, 𝑧], assume 𝑜 = 2𝑙 for ease of

exposition

• If 𝑜 = 1, give [𝑦, 𝑧] to the single player
• Otherwise, each player 𝑗 makes a mark 𝑨 s.t.

𝑊

𝑗 [𝑦, 𝑨] = 1

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 1/2𝑙 for the piece he’s sharing, then at stage 𝑙 + 1 each player has value at least

1 2𝑙+1

• The number of stages is log 𝑜 ∎

𝑜/2 𝑜/2 𝑜/2

2𝑜

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

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
• ptimal!
• What about envy?

SELFRIDGE-CONWAY

• Stage 0
• Player 1 divides the cake into three equal pieces according to 𝑊1
• Player 2 trims the largest piece s.t. there is a tie between the two

largest pieces according to 𝑊2

• 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 𝑊𝑈′
• Players 𝑈, 1, and 𝑈′ choose the pieces of Cake 2, in that order

• 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(𝑜2) queries in the RW model

THE COMPLEXITY OF EF

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 𝑜𝑜𝑜𝑜𝑜𝑜

• Stay tuned for more next time…

𝑃

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 cut𝑗(0, Τ

1 2)

• Player 𝑗 replies with 𝑧𝑗 = 𝑔(𝑊

𝑗)

• The protocol simulates the Aziz-Mackenzie

protocol ‘in the background’ using 𝑔−1(𝑧𝑗) for all 𝑗 ∈ 𝑂

• Is this a valid EF protocol in the RW model?