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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