Private capacities in mechanism design Vincenzo Auletta Paolo Penna Giuseppe Persiano Università di Salerno, Italy.

Private capacities in mechanism design

Vincenzo Auletta Paolo Penna Giuseppe Persiano

Università di Salerno, Italy

Ideal World

In practice…

The Internet

Self organized, no central authority, anarchic

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

Selfish agents

Example: BGP Routing

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

AS1

AS2source destination

Link down

Toy problem

source destination

Routing cost (time/latency)

t1

t2

0

0

Payment (large enough)

Algorithm: smallest cost

Toy problem

source destination

Routing cost (time/latency)

C1

C2

Payment: 2nd smallest cost

Algorithm: smallest cost

Toy problem

source destination

Routing cost (time/latency)

7

10

Payment: 2nd smallest cost

Algorithm: smallest cost

A truthtelling agent maximizes his/her utility

Truthful mechanism

7 10 – 7 8 10 – 7 11 0 – 0

Mechanism design

When are truthful mechanisms possible?

How to design them?

Mechanism = Algorithm + Payments

Focus on the Algorithm!!

truthful mechanism monotone algorithm

One-parameter setting

Mechanism = Algorithm + Payments

t1 ti tn

utilityi = paymenti – worki ti

r1 ri tn

workn work1 worki

… …t1 ti tn

paymentipayment1 paymentn

per-unit cost

One-parameter setting

Mechanism = Algorithm + Payments

utilityi = paymenti – worki ti

• P2P, Grid computing, scheduling• Routing• Wireless communications

speed

One-parameter setting

Mechanism = Algorithm + Payments

utilityi = paymenti – worki ti

• P2P, Grid computing, scheduling• Routing• Wireless communications

Well understood…but assumes infinite capacities!

battery

worki

ti

truthful [Myerson81][Archer&Tardos01]

Our contribution

• New model (one-parameter + capacities)

• Characterization of truthfulness

• Few applications (optimal mechanisms)

• Open questions

ALGORITHMIC!!

Private capacities

… …t1 ti tnc1 ci cn

per-unit cost

max work that can effort

utilityi = paymenti – worki ti

when “work capacity”

utilityi = – when “work > capacity”

Characterization of truthfulness

Weakly monotone algorithm: ti > ti

worki worki or worki > ci

truthful mechanism weakly monotone algorithm

… …t1 ti tnc1 ci cn

algorithm

worki

… …t1 ti tnc1 ci cn

algorithm

worki

per-unit cost increasesless work

too much work

Example: Greedy Algorithm

5

610

Weakly monotone algorithm: ti > ti

worki worki or worki > ci

fixed capacities

x2 x

106

5

NO!

1 x

10 6 5

Characterization of truthfulness: proof ingredients

truthfulness cycle monotonicity [Rochet 87]

weak monotonicity

Hard to understand/us

e

simple

…

nonnegative length nonnegative length

k different inputs for agent i

how worki changes

ti ci

worki

t i c i

work i

worki worki or worki > ci

ti•(work i worki )

t i•(worki work i)

(t i ti ) •(worki work i)

Characterization of truthfulness: proof ingredients

truthfulness cycle monotonicity [Rochet 87]

weak monotonicity

Hard to understand/us

e

simple

…

nonnegative length nonnegative length

obvioushard

How to design the mechanism

Algorithm + Payments

Truthful mechanism

Algorithm

polytimeweakly

monotone optimal cost

not always[Archer&Tardos 01]

Can we have all of them?

min-max problems (max congestion, makespan, fairness)

yes[this work]

Without capacities[AT 01]

How to design the mechanism

min-max problems (max congestion, makespan, fairness)

Scheduling: truthful PTAS for O(1) machines [this work]

exact + rounding

Algorithm

polytimeweakly

monotone optimal cost

Without capacities[Andelmann et al

07]

Open questions

• PTAS for any number of machines?– Without capacities: Yes [Christodoulou & Kovacs 09]

• Complexity of “truthful” algorithms– BGP routing– Network lifetime

• Multi-parameter settings? – No simple characterization [this work]

Private capacities

Thank You