PowerPoint Presentationarielpro/15251/Lectures/lecture05.pdfΒ Β· Bit representation β’ 221
PowerPoint Presentationarielpro/15896s15/docs/896s15-11.pdfΒ Β· 15896 Spring 2015: Lecture 11 A...
Transcript of PowerPoint Presentationarielpro/15896s15/docs/896s15-11.pdfΒ Β· 15896 Spring 2015: Lecture 11 A...
CMU 15-896 Fair division 3: Rent division Teacher: Ariel Procaccia
15896 Spring 2015: Lecture 11
Strategyproof Cake cutting
β’ We discussed strategyproofness (SP) in voting β’ All the cake cutting algorithms we discussed are
not SP: agents can gain from manipulation o Cut and choose: player 1 can manipulate o Dubins-Spanier: shout later
β’ Assumption: agents report full valuations β’ Deterministic EF and SP algs exist in some
special cases, but they are rather involved [Chen et al. 2010]
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15896 Spring 2015: Lecture 11
A randomized algorithm β’ π1, β¦ ,ππ is a perfect partition if ππ ππ = 1 πβ for all π, π β’ Algorithm
o Compute a perfect partition o Draw a random permutation π over {1, β¦ ,π} o Allocate to agent π the piece ππ π
β’ Theorem [Chen et al. 2010; Mossel and Tamuz 2010]: the algorithm is SP in expectation and always produces an EF allocation
β’ Proof: if an agent lies the algorithm may compute a different partition, but for any partition:
οΏ½1πππ ππβ² =
1ποΏ½ππ ππβ² =
1π
βπβππβπ
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15896 Spring 2015: Lecture 11
Computing a perfect partition
β’ Theorem [Alon, 1986]: a perfect partition always exists, needs polynomially many cuts
β’ Proof is nonconstructive β’ Can find perfect
partitions for special valuation functions
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15896 Spring 2015: Lecture 11
A true story
β’ In 2001 I moved into an apartment in Jerusalem with Naomi and Nir
β’ One larger bedroom, two smaller bedrooms β’ Naomi and I searched for the apartment, Nir
was having fun in South America β’ Nirβs argument: I should have the large room
because I had no say in choosing apartment o Made sense at the time!
β’ How to fairly divide the rent?
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15896 Spring 2015: Lecture 11
Spernerβs Lemma β’ Triangle π partitioned into
elementary triangles β’ Label vertices by {1,2,3} using
Sperner labeling: o Main vertices are different o Label of vertex on an edge
(π, π) of π is π or π β’ Lemma: Any Sperner
labeling contains at least one fully labeled elementary triangle
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1 2 2 1 1 2
1 1 2 1 2
3 3 2 2
1 2 3
1 2
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15896 Spring 2015: Lecture 11
Proof of Lemma
β’ Doors are 12 edges β’ Rooms are elementary
triangles β’ #doors on the boundary
of π is odd β’ Every room has β€ 2
doors; one door iff the room is 123
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3 3
3
3
2 1 1 2
1 1 2 2
2 2
1 2
1 2
1
1 2
15896 Spring 2015: Lecture 11
Proof of Lemma β’ Start at door on boundary
and walk through it β’ Room is fully labeled or it
has another door... β’ No room visited twice β’ Eventually walk into fully
labeled room or back to boundary
β’ But #doors on boundary is odd β
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3 3
3
3
2 1 1 2
1
2 2
1 2
1 2
2 2 1 1
1 2
15896 Spring 2015: Lecture 11
Fair rent division β’ Assume there are three housemates
A, B, C β’ Goal is to divide rent so that each person
wants different room β’ Sum of prices for three rooms is 1 β’ Can represent possible partitions as
triangle
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15896 Spring 2015: Lecture 11 10
(0,0,1)
(1,0,0) (0,1,0)
0,12
,12
13
,13
,13
15896 Spring 2015: Lecture 11
Fair rent division
β’ βTriangulateβ and assign βownershipβ of each vertex to each of A, B, and C ...
β’ ... in a way that each elementary triangle is an ABC triangle
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15896 Spring 2015: Lecture 11 12
A C B C A B
B C A B C
A B C A
C A B
B C
A
15896 Spring 2015: Lecture 11
Fair rent division
β’ Ask the owner of each vertex to tell us which room he prefers
β’ This gives a new labeling by 1, 2, 3 β’ Assume that a person wants a free room if
one is offered to him
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15896 Spring 2015: Lecture 11 14
A C B C A B
B C A B C
A B C A
C A B
B C
A
3 only
2 or 3
1 or 3
1 or 2
β’ Choice of rooms on edges is constrained by the free room assumption
15896 Spring 2015: Lecture 11 15
A C B C A B
B C A B C
A B C A
C A B
B C
A
3 only
2 or 3
1 or 3
1 or 2
1
2 3
β’ Spernerβs lemma (variant): such a labeling must have a 123 triangle
15896 Spring 2015: Lecture 11
Fair rent division
β’ Such a triangle is nothing but an approximately envy free allocation!
β’ By making the triangulation finer, we can increase accuracy
β’ In the limit we obtain a completely envy free allocation
β’ Same techniques generalize to more housemates [Su 1999]
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15896 Spring 2015: Lecture 11
Computational resources
β’ Setting: allocating multiple homogeneous resources to agents with different requirements
β’ Running example: shared cluster β’ Assumption: agents have proportional demands
for their resources (Leontief preferences) β’ Example:
o Agent has requirement (2 CPU,1 RAM) for each copy of task
o Indifferent between allocations (4,2) and (5,2)
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15896 Spring 2015: Lecture 11
Model
β’ Set of players π = {1, β¦ ,π} and set of resources π , π = π
β’ Demand of player π is π π = πππ, β¦ ,πππ , 0 < πππ β€ 1; βπ s.t. πππ = 1
β’ Allocation π¨π = π΄ππ, β¦ ,π΄ππ where π΄ππ is the fraction of π allocated to π
β’ Preferences induced by the utility function π’π π¨π = minπβπ π΄ππ/πππ
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15896 Spring 2015: Lecture 11
Dominant resource fairness
β’ Dominant resource of π = π s.t. πππ = 1 β’ Dominant share of π = π΄ππ for dominant π β’ Mechanism: allocate proportionally to
demands and equalize dominant shares
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Agent 1 alloc. Agent 2 alloc. Total alloc.
15896 Spring 2015: Lecture 11
Formally... β’ DRF finds π₯ and allocates to π an π₯πππ
fraction of resource r:
max π₯ s.t. βπ β π ,οΏ½π₯ β πππ β€ 1πβπ
β’ Equivalently, π₯ = πmaxπβπ β ππππβπ
β’ Example: πππ = π2
;ππ2 = 1;π2π = 1;π22 = π6
then π₯ = π12+π
= 23
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15896 Spring 2015: Lecture 11
Axiomatic properties
β’ Pareto optimality (PO) β’ Envy-freeness (EF) β’ Proportionality (a.k.a. sharing incentives,
individual rationality):
βπ β π,π’π π¨π β₯ π’π1π
, β¦ ,1π
β’ Strategyproofness (SP)
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15896 Spring 2015: Lecture 11
Properties of DRF
β’ An allocation π¨π is non-wasteful if βπ₯ s.t. π΄ππ = π₯πππ for all π
β’ If π¨π is non-wasteful and π’π π¨π < π’π π¨πβ² then π΄ππ < π΄ππβ² for all π
β’ Theorem [Ghodsi et al. 2011]: DRF is PO, EF, proportional, and SP
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15896 Spring 2015: Lecture 11
Proof of theorem
β’ PO: obvious β’ EF:
o Let π be the dominant resource of π o π΄ππ = π₯ β πππ = π₯ β₯ π₯ β πππ = π΄ππ
β’ Proportionality: o For every π, β πππ β€ ππβπ
o Therefore, π₯ = πmaxπ
β ππππβπβ₯ π
π
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15896 Spring 2015: Lecture 11
Proof of theorem β’ Strategyproofness:
o πππβ² are the manipulated demands; πππβ² = πππ for all π β π
o Allocation is π΄ππβ² = π₯β²πππβ² o If π₯β² β€ π₯, π is the dominant resource of π,
then π΄ππβ² = π₯β²πππβ² β€ π₯πππβ² β€ π₯πππ = π΄ππ o If π₯β² > π₯, let π be the resource saturated by
π¨ (β π₯πππ = 1)πβπ , then
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π΄ππ = 1 βοΏ½π΄πππβ π
= 1 βοΏ½π₯πππ > 1 βοΏ½π₯β²πππ =πβ ππβ π
1 βοΏ½π΄ππβ²
πβ π
β₯ π΄ππβ²