Mechanisms with Verification
for Any Finite Domain
Carmine Ventre
Università degli Studi di Salerno
Task Scheduling [Nisan&Ronen’99]
…
…
Jj
J1
types
tasks
Jn
M1
…
Mi
…
bt11
…
btii
…
• Optimal Makespan:
minx maxi costi(X)
no VCG!
Mechanism design:
Selfish
payments
machines
Mm

utility
= payment - cost
btmm
• Verification
(observe machine behavior)
Allocation X  costi(X) = ti,j + ti,n
Verification

Give the payment if the results are given “in
time”

1.
2.
Machine i gets job j when reporting bi,j
ti,j  bi,j  just wait and get the payment
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



Even for two machines and
exponential running time
Polynomial time
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]
New lower bounds [Mu’Alem&Shapira’06] [Christodoulou&Koutsoupias&Vidali06]
Setup
holds a resource of type ti
feasible solutions
(how we use resources)

Agent i
X1,…, Xk

costi(X) = ti(X) = time

utility = payment – cost

Goal: minimize m(X,t)

(t1,…,tn)
No payment if
ti(X) > bi(X)
(verification)
Truthful mechanism running an optimal algorithm
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
A() A(, b-i)
P()  P(, b-i)
Truthfulness (single player):
P(a) +
- a(A(a))
P(b) - a(A(b))
(a,b)  P(b)
P(b)
+
- b(A(b))
(b,a) P(a)
P(a) - b(A(a))
truth-telling
a
(a,b)
a(Y)
- a(X)
Algorithm
b
b(X)
- b(Y)
(b,a)
Must be non-negative
X=A(a)
Y=A(b)
Existence of the Payments
Truthful mechanism (A, P)
There is no cycle of negative length
a
b
c
…
k
Can satisfy all P(a) + (a,b)  P(b)
[Malkhov&Vohra’04][MV’05][Saks&Yu’05]
[Bikhchandani&Chatterji&Lavi&Mu'alem&Nisan&Sen’06]……
Why Verification Helps
Some edges may “disappear”
a(Y) - a(X)
a
X
a(Y) > b(Y)
b
Y
True type is “a” but report “b”:
1. a(Y)  b(Y)  can
b” and
get P(b)
P(a)“simulate
- a(X)  P(b)
- a(Y)
2. a(Y) > b(Y)  no
payment
helps)
P(a)
- a(X) (verification
- a(Y)
voluntary participation
0
0
nonnegative costs
Why Verification Helps
Only these edges remain:
a(Y) - a(X)
a
X
a(Y)  b(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(Y)  b(Y)
a
X
b(Z)  c(Z)
b
Y
c(X)  a(X)
c
Z
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))
X is OPT(a,b
m(•,b
-i) -i(Y)) is non-decreasing
Optimal Mechanisms
a(Y)  b(Y)
a
X
b(Z)  c(Z)
b
Y
c(X)  a(X)
c
Z
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))
X Y Z  X
X=Y=Z
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 [Nisan&Ronen‘99]
(In-)Finite Domains?
All vertices in a cycle lead to the same outcome
P(Y)
…
P(Y)
Y
X
P(X)
X
Y
D(X,Y)
D(Y,X)
P(X)
X
Nodes=declarations
Nodes=outcomes
P(X) + (a,b)  P(Y)
P(X) + D(X,Y)  P(Y)
D(X,Y) = sup {(a,b)| (a,b) edge from “X” to “Y”}
(In-)Finite Domains?
m(i,j) = max(i,j), two outcomes X and Y
b-i
10
11
9 X
a
1 Y
a(Y) -8
- a(X)
13 X
b
a(Y)  b(Y) 12 Y
P(a) > P(c) + 7
b(X) 1- a(Y) 13X
c
b(X)  c(X) 14Y
agent i -8
X
1
Y
(In-)Finite Domains?
There exists a class of
social choice functions
(SCFs) s.t. …
SCFs implementable
without verification
… using the allocation graph
SCFs implementable
with verification
Looking for alternative
techniques
Compound Agents
…
…
Jj
J1
types
Jn
M1
…
Mi
…
Mm
bt11
…
btii
…
btmm
agentl
…
agentk
agent1
…
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
a = (a1, a2)
b = (b1, b2)
a(Y) - a(X)
a
X
b
Y
Edge (a,b) exists iff a1(Y)  b1(Y) and a2(Y)  b2(Y)
OPT is implementable w/verification
Compound Agents
…
…
Jj
J1
M1
b1
agent1

…
…
…
Jn
Mi
bi
…
…
agentl
… agentk
Mm
bm
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
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]