and h(v -i )

Optimal Payments in
Dominant-Strategy Mechanisms
Victor Naroditskiy Maria Polukarov Nick Jennings
University of Southampton
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mechanism
allocation function
who is allocated
fixed
payment function
payment to each agent
optimized
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truthful mechanisms with payments:
groves class
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2
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minimize the
amount burnt
[Moulin 07]
[Guo&Conitzer 07]
individual
rationality
optimize fairness
[Porter et al 04]
no subsidy
weak BB
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single-parameter domains:
characterization of DS mechanisms
if allocated when reporting
x, then allocated when
reporting y ≥ x
if not allocated
when reporting x,
then not allocated
when reporting y ≤ x
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2
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h(9,2) - g(9,2)
h(7,9)
g - price (critical value)
h - rebate
g(v-i) - the minimum
value agent i can
report to be allocated
h(7,2) - g(7,2)
v-i = (v1,...,vi-1,v
xi,vi+1,...,vn)
h(v-i) is the only degree of
freedom
in the payment function
determined by
the allocation
function
g(v-i) =
minx | fi(x,v-i) = 1
optimize h(v-i)
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optimal payment function
constructive characterization
objective
IN
constraints
allocation function
OUT
optimal payment
(rebate) function
e.g., maximize
social welfare
e.g., no subsidy and
voluntary participation
e.g., efficient
AMD
[Conitzer, Sandholm, Guo]
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dominant-strategy implementation
no prior on the agents' values
V = [0,1]n
f: V  {0,1}n
W = [0,1]n-1
g, h: WR
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example MD problem
welfare maximizing allocation
[Moulin 07]
[Guo&Conitzer 07]
maxh(w) r
n agents
m items
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)
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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
objective and constraints are linear in
f(v), g(v-i), and h(v-i)
optimization is over functions
infinite number of constraints
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example
2 agents
1 free item
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allocation regions
f(v) = (0,1)
f(v) = (1,0)
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regions with linear constraints
f(v) = (0,1)
g(v2) = v2
constant allocation and
linear critical value
on each triangle
g(v1) = v1
f(v) = (1,0)
constraints linear in h(w)
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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)
2v1 + v2 ≤ 5
v2
v1
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allocation of free items
V=
[0,1]2
linear f,g,h =>
constraints
are linear in v
constraints(f(v), g(v-1),
g(v-2), h(v-1), h(v-2))
V=
{(0,0) (1,0) (0,1) (1,1)}
W = {(0) (1)}
[Guo&Conitzer 08]
optimal solution
restricted problem
LP with variables
h(0), h(1), objVal - upper bound!
the upper bound (objVal) is achieved
and the constraints hold throughout V
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allocation with costs
hb(v2), hb(v1)
ha(v2), hb(v1)
ha(w1)
hb(w1)
w1
each payment region has
n extreme points
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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
q*
q
q'
PX = {q,q',q*}
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vertex consistency
project
points
0
1,0
v-1
k
w1
1
v-2
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region consistency
v2 · k
lift
regions
w1 · k
0
v1 · k
k
w1
1
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triangulation
each polytope in PW is a simplex
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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
restricted problem
with any subset of value space
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lower bound
(approximate solutions)
not a triangulation:
cannot linearly
interpolate the
extreme points
ha(w1)
0
k*
hb(w1)
k
1
w1
allocate to agent 1 if
v1 ≥ kv2
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examples
V = {v in Rn | 1 ≥ v1 ≥ v2 ≥ ... ≥ vn ≥ 0}
h: WR
W = {w in Rn-1 | 1 ≥ w1 ≥ w2 ≥ ... ≥ wn-1 ≥ 0}
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efficient allocation of free
items
n agents with
private values
m free items/tasks
throughout V
agents 1..m are allocated
f(v) = (1,...1,0,...0)
m
social welfare:
[Moulin 07]
[Guo&Conitzer 07]
fairness: [Porter 04]
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extreme points
restricted problem is
a 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 cost
n agents with
private values
m items with
increasing costs
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4
7
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m+1 possible efficient
allocations depending on
agents' values
tragedy of the commons:
cost of the ith item
measures disutility that i
agents experience from
sharing the resource with
one more user
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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
side in dimension i is of length ai
subdivided via hyperplanes xi/ai = xj/aj
applies to initial subdivisions that can be
obtained with hyperplanes of the form xi = ci
where ci is a constant
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arbitrary initial subdivision
can be approximated with a piecewise
constant function
we know consistent partitions for the modified problem
triangulations of hyperrectangles
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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?
build a bridge if v1 + ... + vm ≤ c
where c is the cost
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...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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