optimal payments in dominant-strategy mechanisms victor naroditskiy maria polukarov nick jennings...
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Optimal Payments in Dominant-Strategy
Mechanisms
Victor Naroditskiy Maria Polukarov Nick JenningsUniversity of Southampton
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allocation function
who is allocated
payment function payment to each agent
fixedfixed optimizedoptimized
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mechanism
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truthful mechanisms with payments: groves class
minimize the amount burnt
optimize fairness
no subsidyweak BB
individual rationality
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[Moulin 07] [Guo&Conitzer
07]
[Porter et al 04]
single-parameter domains: characterization of DS
mechanisms
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if allocated when reporting x, then allocated when reporting y ≥ x
if not allocated when reporting x, then not allocated when reporting y
≤ x
h(7,9) h(7,2) - g(7,2)
h(v-i) is the only degree of freedom
in the payment function optimize h(v-i) 5
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if allocated when reporting x, then allocated when reporting y ≥ x
h(9,2) - g(9,2)
g(v-i) - the minimum value agent i can
report to be allocated
v-i = (v1,...,vi-
1,vi,vi+1,...,vn)x
determined by the
allocation functiong(v-i) =
minx | fi(x,v-i) = 1
g - price (critical value)h - rebate
optimal payment functionconstructive characterization
optimal payment (rebate) function
IN
OUT
objective
e.g., maximize social welfare
constraints
e.g., no subsidy andvoluntary
participation
allocation functione.g., efficient
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AMD [Conitzer, Sandholm,
Guo]
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dominant-strategy implementation
no prior on the agents' valuesV = [0,1]n
f: V {0,1}n
W = [0,1]n-1
g, h: WR
example MD problemwelfare maximizing
allocation
maxh(w) r s.t. for all v in V
(social welfare within r of the efficient surplus v1 + ... + vm)
v1 + ... + vm + i h(v-i) - mvm+1 ≤ r(v1 + ... + vm)
i h(v-i) - mvm+1 ≤ 0 (weak BB)
h(v-i) ≥ 0 (IR)
maxh(w) r s.t. for all v in V
(social welfare within r of the efficient surplus v1 + ... + vm)
v1 + ... + vm + i h(v-i) - mvm+1 ≤ r(v1 + ... + vm)
i h(v-i) - mvm+1 ≤ 0 (weak BB)
h(v-i) ≥ 0 (IR) 8
[Moulin 07] [Guo&Conitzer
07]
n agentsm items
generic MD problem
maxh(w),objVal objVal s.t. for all v in V
objective(f(v), g(v-i), h(v-i)) ≥ objVal
constraints(f(v), g(v-i), h(v-i)) ≥ 0
maxh(w),objVal objVal s.t. for all v in V
objective(f(v), g(v-i), h(v-i)) ≥ objVal
constraints(f(v), g(v-i), h(v-i)) ≥ 0
objective and constraints are linear in f(v), g(v-i), and h(v-i)
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optimization is over functionsinfinite number of constraints
example
2 agents
1 free item
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allocation regions
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f(v) = (0,1)
f(v) = (1,0)
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f(v) = (0,1)
f(v) = (1,0)
g(v1) = v1
g(v2) = v2
regions with linear constraints
constant allocation and linear critical
value on each triangle
constraints linear in h(w)
linear constraints on a polytope
a linear constraint c1v1 + ... + cnvn ≤ cn+1
holds at all points v in P of a polytope P iff it holds at the extreme points v in extremePoints(P)
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v2
v1
2v1 + v2 ≤ 5
allocation of free items
restricted problemLP with variables
h(0), h(1), objVal - upper bound!
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the upper bound (objVal) is achieved and the constraints
hold throughout V
V = [0,1]2
V = {(0,0) (1,0) (0,1) (1,1)}
W = {(0) (1)}
constraints(f(v), g(v-1), g(v-2), h(v-1), h(v-2))
[Guo&Conitzer 08]
linear f,g,h =>
constraints are linear in
v
optimal solution
ha(v2), hb(v1)
hb(v2), hb(v1)
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ha(w1) hb(w1)
w1
ha (w
1 ) hb(w1
)
allocation with costs
each payment region has n extreme points
overview of the approach
• find consistent V and W space subdivisions
• solve the restricted problem– extreme points of the value space
subdivision
• payments at the extreme points of W region x define a linear function hx
• optimal rebate function is h(w) = {hx(w) if w in x}
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subdivisions
• PX - subdivision (partition) of polytope X
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q q'
q*
PX = {q,q',q*}
vertex consistency
w1
0 1k
1,0 v-1v-2
projectpoints
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region consistency
w1
0 1k
w1 · kliftregions
v2 · k
v1 · k
v2 · k
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triangulation
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each polytope in PW is a simplex
characterization• if there exist PV and PW satisfying
– PV refine the initial subdivision• allocation constant on q in PV
• critical value linear on q in PV
– vertex consistency– region consistency
– PW is a triangulation
• then an optimal rebate function is given by– interpolation of optimal rebate values from the
restricted problem– by construction, the optimal rebates are piecewise
linear
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upper bound
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restricted problemwith any subset of value space
lower bound(approximate solutions)
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not a triangulation:cannot linearly interpolate the extreme points
allocate to agent 1 if v1 ≥ kv2
ha(w1) hb(w1)
w1
k* k 10
ha(w1) hb(w1)
w1
k 10
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examples
V = {v in Rn | 1 ≥ v1 ≥ v2 ≥ ... ≥ vn ≥ 0}
h: WRW = {w in Rn-1 | 1 ≥ w1 ≥ w2 ≥ ... ≥ wn-1
≥ 0}
efficient allocation of free items
n agents with private values
m free items/tasks
social welfare: [Moulin 07]
[Guo&Conitzer 07]
fairness: [Porter 04]
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throughout V agents 1..m are allocated
m
f(v) = (1,...1,0,...0)
extreme points
restricted problem isa linear program with constraints for n+1
points(0...0) (10...0) (1110...0) ... (1...1)
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fairness: [Porter 04]
results follow immediately from the restricted problem
the feasible region is empty for k<m+1
=> impossibility result
unique linear (m+2)-fair mechanism
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efficient allocation of items with increasing marginal
costn agents with private values
m items with increasing costs
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m+1 possible efficient allocations depending on
agents' values
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tragedy of the commons:
cost of the ith item measures disutility that
i agents experience from sharing the
resource with one more user
algorithmic solution
input: n, cost profile
output: percentage of efficient surplus
optimal payment function
piecewise linear on each region
number of regions is exponential in the
number of agents/costs
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hypercube triangulation
• a hypercube [0,1]n can be subdivided into n! simplices with hyperplanes xi = xj comparing each pair of coordinates
• each simplex corresponds to a permutation σ(1)... σ(n) of 1...n
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hyperrectangle triangulation
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applies to initial subdivisions that can be obtained with hyperplanes of the form xi =
ci
where ci is a constant
side in dimension i is of length ai
subdivided via hyperplanes xi/ai = xj/aj
arbitrary initial subdivisioncan be approximated with a piecewise constant function
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we know consistent partitions for the modified problem
triangulations of hyperrectangles
contribution• characterized linearity of mechanism
design problems– consistent partitions
• piecewise linear payments are optimal• interpolate values at the extreme points
• approach for finding optimal payments– unified technique for old and new
problems
• algorithm for finding approximate payments and an upper bound
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open questions
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consistent partitions for public good?
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build a bridge if v1 + ... + vm ≤ cwhere c is the cost
...open questions
• full characterization of allocation functions that have consistent partitions
• is a consistent partition necessary for the existence of (piecewise) linear optimal payments
• approximations: simple payment functions that are close to optimal
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