CMU 15-896 Fair division: Incentives and Leontief Teachers: Avrim Blum Ariel Procaccia (this time)
Strategyproof Cake cutting β’ We discussed strategyproofness (SP) in social choice and
auctions β’ All the cake cutting algorithms that 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 their full valuation functions (which are typically assumed to be concisely representable)
β’ Deterministic EF and SP algs exist in some special cases, but they are rather involved [Chen et al. 2010]
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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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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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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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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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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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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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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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(0,0,1)
(1,0,0) (0,1,0)
(0, Β½, Β½)
(β , β , β )
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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Fair rent division
β’ Ask the owner of each vertex to tell us which rooms 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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Fair rent division
β’ Spernerβs lemma (variant): such a labeling must have a 123 triangle
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1
3 2
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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Applications to the cloud
β’ Setting: allocating multiple homogeneous resources to agents with different requirements
β’ Running example: cloud computing β’ State-of-the-art systems employ a single resource
abstraction β’ Assumption: agents have proportional demands for their
resources 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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Model
β’ Set of agents π = {1, β¦ ,π} and set of resources π , π = π
β’ Demand of agent 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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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.
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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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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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
β’ We prove the theorem on the board
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