Mechanisms with Verification for Any Finite Domain

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Mechanisms with Verification for Any Finite Domain

• Carmine Ventre
• Università degli Studi di Salerno

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Task Scheduling NisanRonen99
tasks
Mechanism design payments ? utility payment
- cost
Selfish
machines

• Optimal Makespan
• minx maxi costi(X)

• Verification
• (observe machine behavior)

no VCG!
Allocation X ? costi(X)
ti,n
ti,j
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Verification

• Give the payment if the results are given in
  time
• Machine i gets job j when reporting bi,j
• ti,j ? bi,j ? just wait and get the payment
• ti,j gt bi,j ? no payment (punish agent i)

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Why Verification?
Even for two machines and exponential running
time

• Provably better approximation
• No verification ? No c-APX mechanism
• Makespan on unrelated machines NisanRonen99
• Weighted sum on related machines
  ArcherTardos01
• Verification ? Exact mechanisms
• Makespan on unrelated machines NisanRonen99
• Comparable Types Auletta et al. 06
• Verification ? (1?)-APX mechanism
• Makespan on unrelated machines NisanRonen99
• Weighted sum on related machines Auletta et
  al.06
• Things become simpler
• Can recycle existing algorithms Auletta et
  al.06

Polynomial time
New lower bounds MuAlemShapira06
ChristodoulouKoutsoupiasVidali06
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Setup

• Agent i holds a resource of type ti
• X1,, Xk feasible solutions
• (how we use resources)
• costi(X) ti(X) time
• utility payment cost
• Goal minimize m(X,t)

Truthful mechanism running an optimal algorithm
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Our Contribution

• Can implement the optimum in general
• Minimize any
• m(X,t)m(t1(X),,tn(X))
• non decreasing in the agents costs ti(X)
• Can implement any optimum in general for
  compound agents
• Agents declaring more than a value (e.g., agent
  controlling more than one machine)
• Impossibility results on mechanisms with
  verification for infinite domains

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Existence of the Payments
A(?) ?A(?, b-i) P(?) ? P(?, b-i)
Truthfulness (single player)
P(a) - a(A(a)) ? P(b) - a(A(b))
P(a) ?(a,b) ? P(b)
P(b) - b(A(b)) ? P(a) - b(A(a))
P(b) ?(b,a) ? P(a)
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Existence of the Payments
Truthful mechanism (A, P)
Can satisfy all P(a) ?(a,b) ? P(b)
MalkhovVohra04MV05SaksYu05 Bikhchandan
iChatterjiLaviMu'alemNisanSen06
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Why Verification Helps
Some edges may disappear
X
Y

• True type is a but report b
• a(Y) ? b(Y) ? can simulate b and get P(b)
• a(Y) gt b(Y) ? no payment (verification helps)

P(a) - a(X) ? P(b) - a(Y)
P(a) - a(X) ? - a(Y)
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Why Verification Helps
Only these edges remain
X
Y
a(Y) ? b(Y)
Negative cycles may desappear
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Optimal Mechanisms

• Algorithm OPT
• Fix lexicographic order
• X1 ? X2 ? ? Xk
• Return the lexicographically minimal
• Xj minimizing m(b,Xj)

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Optimal Mechanisms
a(Y) ? b(Y)
b(Z) ? c(Z)
X
Y
Z
c(X) ? a(X)
m(a(X),b-i(X)) ? m(a(Y),b-i(Y))
? m(b(Y),b-i(Y))
? m(b(Z),b-i(Z)) ?
m(c(Z),b-i(Z))
? m(c(X),b-i(X)) ?
m(a(X),b-i(X))
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Optimal Mechanisms
a(Y) ? b(Y)
b(Z) ? c(Z)
X
Y
Z
c(X) ? a(X)
m(a(X),b-i(X)) m(a(Y),b-i(Y))
m(b(Y),b-i(Y))
m(b(Z),b-i(Z))
m(c(Z),b-i(Z))
m(c(X),b-i(X))
m(a(X),b-i(X))
? Z
? X
X ? Y
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Finite Domains
All vertices in a cycle lead to the same outcome
Theorem Truthful OPT mechanism with verification
for any finite domain and any m(X,b)m(b1(X),,bm(
X)) non decreasing in the agents costs bi(X)
Different proof of existence of exact truthful
mechanism w/ verification for makespan on
unrelated machines NisanRonen99
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(In-)Finite Domains?
P(Y)
Y
P(X)
X
P(X) ?(a,b) ? P(Y)
D(X,Y) sup ?(a,b) (a,b) edge from X to Y
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(In-)Finite Domains?
m(i,j) max(i,j), two outcomes X and Y
P(a) gt P(c) 7
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(In-)Finite Domains?
There exists a class of social choice functions
(SCFs) s.t.
SCFs implementable without verification
SCFs implementable with verification
Looking for alternative techniques
using the allocation graph
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Compound Agents
Each agent declares more than a type
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Verification for Compound Agents

• Punish agent i whenever uncovered lying over one
  of its dimensions (e.g., machines)
• Collusion-Resistant mechanisms w/ verification
  w.r.t. known coalitions

a (a1, a2)
b (b1, b2)
Edge (a,b) exists iff a1(Y) ? b1(Y) and a2(Y) ?
b2(Y)
OPT is implementable w/verification
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Compound Agents

• Collusion-Resistant for known coalitions
  mechanisms w/ verification for
• makespan on unrelated machines
• makespan on related machines

Exponential time Exact mechanisms
Polynomial time c (1?) - APX
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Conclusions Further Research

• OPT is always implementable w/ verification for
  finite domains
• Breaking lower bounds for classical mechanisms
  ArcherTardos01BilòGualàProietti06NR99
• Infinite domains and verification?
• Are collusion-resistant (for unknown coalitions)
  mechanisms w/ verification possible?
• Some answers in PennaV, Submitted