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Mechanisms with Verification for Any Finite Domain
Carmine Ventre
Università degli Studi di Salerno
Task Scheduling [Nisan&Ronen’99]
Allocation X costi(X) + ti,n= ti,j
Selfish
• Optimal Makespan:
minx maxi costi(X)
• Verification (observe machine behavior)
no VCG!
J1 Jj Jn
… …
M1 Mi Mm… …
b1 bi bm… …
tasks
machines
t1 ti tm… …types
Mechanism design: payments
utility = payment - cost
Verification
Give the payment if the results are given “in time”
Machine i gets job j when reporting bi,j
1. ti,j bi,j just wait and get the payment
2. ti,j > bi,j no payment (punish agent i)
Why Verification?
Provably better approximation No verification No c-APX mechanism
Makespan on unrelated machines [Nisan&Ronen’99] Weighted sum on related machines [Archer&Tardos’01]
Verification Exact mechanisms Makespan on unrelated machines [Nisan&Ronen’99] Comparable Types [Auletta et al. ‘06]
Verification (1+)-APX mechanism Makespan on unrelated machines [Nisan&Ronen’99] Weighted sum on related machines [Auletta et al.’06]
Things become simpler Can “recycle” existing algorithms [Auletta et al.’06]
Even for two machines andexponential running time
Polynomial time
New lower bounds [Mu’Alem&Shapira’06] [Christodoulou&Koutsoupias&Vidali06]
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)
No payment ifti(X) > bi(X) (verification)
Truthful mechanism running an optimal algorithm
(t1,…,tn)
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
Existence of the Payments
Truthfulness (single player):
P(a) - a(A(a)) P(b) - a(A(b))
a b
truth-tellingP(b) - b(A(b)) P(a) - b(A(a))
X=A(a)Y=A(b)
a(Y) - a(X)
b(X) - b(Y)
Must be non-negative
(a,b)
(b,a)
P(a) + (a,b) P(b)
P(b) + (b,a) P(a)
A() A(, b-i)
P() P(, b-i)
Algorithm
Existence of the Payments
Truthful mechanism (A, P)
Can satisfy all P(a) + (a,b) P(b)
There is no cycle of negative length
a b kc …
[Malkhov&Vohra’04][MV’05][Saks&Yu’05]
[Bikhchandani&Chatterji&Lavi&Mu'alem&Nisan&Sen’06]……
Why Verification Helps
a bX
a(Y) - a(X)
Some edges may “disappear”
Y
True type is “a” but report “b”:1. a(Y) b(Y) can “simulate b” and get P(b)2. a(Y) > b(Y) no payment (verification helps)
P(a) - a(X) P(b) - a(Y)P(a) - a(X) - a(Y)
0voluntary participation
0nonnegative costs
a(Y) > b(Y)
Why Verification Helps
a bX
a(Y) - a(X)
Only these edges remain:
Ya(Y) b(Y)
Negative cycles may desappear
Optimal Mechanisms
Algorithm OPT:
• Fix lexicographic order X1 X2 … Xk• Return the lexicographically minimal Xj minimizing m(b,Xj)
Optimal Mechanisms
a bX Y
a(Y) b(Y)
m(a(X),b-i(X)) m(a(Y),b-i(Y))
cZ
b(Z) c(Z)
X is OPT(a,b-i)
c(X) a(X)
m(•,b-i(Y)) is non-decreasing
m(b(Z),b-i(Z)) m(c(Z),b-i(Z)) m(b(Y),b-i(Y))
m(c(X),b-i(X)) m(a(X),b-i(X))
Optimal Mechanisms
a bX Y
a(Y) b(Y)
m(a(X),b-i(X)) = m(a(Y),b-i(Y))
cZ
b(Z) c(Z)
c(X) a(X)
= m(b(Z),b-i(Z)) = m(c(Z),b-i(Z))= m(b(Y),b-i(Y))
= m(c(X),b-i(X)) = m(a(X),b-i(X))
Z XX Y X=Y=Z
Finite Domains
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)
All vertices in a cycle lead to the same outcome
Different proof of existence of exact truthful mechanism w/ verification for makespan on unrelated machines [Nisan&Ronen‘99]
(In-)Finite Domains?
Nodes=declarations
All vertices in a cycle lead to the same outcome
Y
…
Nodes=outcomes
X
Y
P(X) + (a,b) P(Y)
D(X,Y)
P(X) + D(X,Y) P(Y)
D(X,Y) = sup {(a,b)| (a,b) edge from “X” to “Y”}
P(X)
P(Y)
P(X)
P(Y)
X
XD(Y,X)
(In-)Finite Domains?
m(i,j) = max(i,j), two outcomes X and Y
a(Y) b(Y)a b c
b(X) c(X) Y
X
Y
X
Y
X
b-i
11
10a(Y) - a(X) b(X) - a(Y)-8 1
1
9
14
13
12
13
agent i
Y Y
X
P(a) > P(c) + 7 X Y
-8
1
(In-)Finite Domains?
SCFs implementable without verification
SCFs implementable with verification
There exists a class of social choice functions (SCFs) s.t. …
… using the allocation graph
Looking for alternative techniques
Compound Agents
J1 Jj Jn
… …
M1 Mi Mm… …
agent1 agentlagentk… …
t1 ti tm… …types b1 bi bm… …
Each agent declares more than a type
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
aX
a(Y) - a(X)bY
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
Compound Agents
Collusion-Resistant for known coalitions mechanisms w/ verification for makespan on unrelated machines makespan on related machines
J1 Jj Jn
… …
M1 Mi Mm… …
agent1 agentlagentk… …
b1bi bm… …
Polynomial timec (1+) - APX
Exponential time Exact mechanisms
Conclusions & Further Research OPT is “always” implementable w/ verification
for finite domains Breaking lower bounds for classical mechanisms
[Archer&Tardos‘01][Bilò&Gualà&Proietti’06][NR‘99] Infinite domains and verification? Are collusion-resistant (for unknown coalitions)
mechanisms w/ verification possible? Some answers in [Penna&V, Submitted]