Improvements for Truthful Mechanisms with Verifiable One-Parameter Selfish Agents

Carmine Ventre

Joint work with

A. Ferrante, G. Parlato and F. Sorrentino

Selfish agents

An Autonomous System may report false link status to redirect traffic to another AS

Different “components” which have their own goal may not follow the “protocol”

The Internet

source destination

AS1

AS2

Link down

Selfish agents

One-Parameter Selfish Agents

Selfish agents own the links and privately know their speeds (one single number)

How to compute opt(s1,…, sm)?

Routing/Scheduling

s1

sm

s2source destination

J1, …, Jn

Unsplittable traffic (jobs)

b1

b2

bm

GOAL: compute opt(s1,…, sm)

e.g. minimize the makespan (maxi worki/si)

Mechanism design

Mechanism: M=(A, P)

Computes a schedule

X = A(b1, …, bm)

Provides a payment

Pi(b1, …, bm)

Agents’ GOAL: maximize their own utility ui(b1, …, bm) := Pi(b1, …, bm) – costi(X)

Truthful Mechanisms for One-Parameter Selfish Agents

A(b1, …, bm) solution

… …

w1(b1,…,bm) wi(b1,…,bm) wm(b1,…,bm) work

wi(b1,…,bm) ¢ ti cost

ui(b1, …, bm) = Pi(b1,…, bm) – wi(b1,…,bm) ¢ ti utility

Truthfulness: ui(ti,b-i) ¸ ui(bi,b-i) 8 b-i, bi with bi 2 i

ti = 1/si is the i’s type

i’s type set

Prior Works

Concept of mechanism with verification (observe jobs’ release time) [Nisan & Ronen, 99]

Truthful mechanisms for one-parameter selfish agents [Archer & Tardos, 01]

Truthful mechanisms with verification for one-parameter selfish agents [Auletta et al, 04]

Our main contribution

Payments computation depends by i No polynomial-time in general

It works only in some case

Payments computation does not depend on i It does not require finite I Polynomial-time Preserve approximation ratio

Solves for the continuous case

[Prior]

[Ours]

Verifiable One-Parameter Selfish Agents

ti = 1

i underbids1/2

1

3

i’s release time should be 2 but…

… i’s finishing time is 4

i overbids

2

1

1 i can wait 2 time slots delivering the results in the right time

IDEA ([NR99]): No payment for underbidding agents

Verification is impossible!

costi(X, ti) = wi(X) ¢ ti

i={1/2, 1, 2}

Verification = observe jobs’ release time

Weakly-Monotone Algorithms

Truthful mechanism with verification for one-parameter agents must use weakly-monotone algorithms ([ADPP04])

wi(bi, b-i)

bi

An easier and more powerful payment function

speeds are integers

A weakly-monotone

Pi(1)(bi, b-i)= Wmax / bi (= Wmax ¢ s’i)

)(A, P(1)) is truthful

Proof idea:

ti

utility) = (payment) - (cost)

true - false

(payment) ¸ (cost)

Wmax is an upper bound to the work assigned by A

Verification

No payment

si 2 N ) Wmax ¢ si ¸ Wmax ¢ si-1 (*)

¸ 1 · 1

bi bi

ui(bi) · 0, ui(ti)¸ 0 by (*)

Payment ¸ Cost

payment) ¸ (cost)

“Proof”

ti = 1/si & bi = 1/(si-1)

(payment) = Wmax Pi(ti) = Wmax ¢ si

Pi(bi) = Wmax ¢ (si-1)) Pi(ti) - Pi(bi) = Wmax

cost) · Wmax cost) =

· Wmax/si · Wmax

wi(si) – wi(si-1)

si

·wi(si)

si

Generalization of P(1)

Upper-bounded and discrete type sets: Pi

(2)(bi, b-i)= Pi(1)(bi, b-i) ¢ ci

(2)

ci(2) is a suitable constant

Applications CPU speeds are expressed as multiple of (M)Hz

(discrete) It does not exist CPU of 100.83 Mhz

“good” solution don’t use very slow machines (upper-bounded)

P(2) and our results

Continuous type sets: generalization of P(2)

(b1, …, bm) (b1R, …, bm

R)

(-r1, …, -rm)

rounding

si

ri-1 ri

siR

Pi(3)(bi, b-i)= Wmax/ bi

R ¢ ci(3)

ci(3) constant (value depends by rmin and )

rmin depends by the minimum possible speed

“continuous” “discrete”

P(3) and our results

computation time is independent from the chosen smaller ) better apx ratio but larger payments bigger upper bound ) larger payments

Makespan problem in general environments

i’s finite & discrete

[ADPP04] characterization

i’s discrete but not finite?

i’s continuous?

Use P(2)

Use P(3)

Are i’s upper bounded?

don’t care

“good” algorithms induces upper bounded type sets

Conclusions

Power of verification Payments more efficient then the previous ones In (many) real life applications we can preserve

the approximation ratio of existing algorithms Weak-monotonicity suffices (for truthful

mechanisms) in more general settings Open problems

Unbounded continuous type sets in general Running time vs Amount of money (Tradeoff)