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]