Strongly Polynomial-Time Truthful Mechanisms in

One ShotPaolo Penna1, Guido Proietti2, Peter Widmayer3

1 Università di Salerno2 Università de l’Aquila3 ETH Zurich

Example: BGP Routing

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

AS1

AS2source destination

Link down

Networks, Protocols, Mechanisms

Network

Protocol “efficient communication”

Private Costs

Selfish agent

selected not selected

ti

cost ti cost 0

“bids” bi

Payments that incentivize agents to be truthfulTruthtelling is a dominant strategyReporting bi = ti maximizes the utility of agent i, always

utility = payment - cost

Networks, Protocols, Mechanisms

Network

Protocol “efficient communication”

Private Costs

Selfish agent

selected not selected

ti

cost ti cost 0

“bids” bi

“Efficient protocol” “Incentive compatible efficient protocol”

?“Efficient” Alg “Efficient” truthful mechanism (Alg,Pay)

? Algorithmic Mechanism Design [Nisan&Ronen’99]

< 3

Truthful Mechanisms

1 2

xShortest Path Tree

cheap expensivex

3

selected not selected

3 <

Depends on the “1” and “2”

agent bid

Monotone Algorithm

Truthful Mechanisms

Monotone algorithm Truthful mechanism

(Vickrey’61, Myerson’81)

Pay

0Tcheap expensivex

T

selected not selected

bid of agent ibids of other agents Alg

Truthful Mechanisms

Monotone algorithm Truthful mechanism

(Vickrey’61, Myerson’81)

Payments Thresholds

Truthful Mechanisms

(Alg,Pay)

Monotonicity Compute the payments

Techniques to solve both?

Efficient mechanism in one shot

Two algorithmic problemsnaive approach: weakly polynomial-time

Our (and Prior) Work

General technique for obtaining efficient mechanisms is one-shot

• write Alg as a “combination” of simpler algorithms

• compute the payments from the simpler algorithms

Monotonicity (easy to prove)Payment computations (efficient)

Prior work:• Monotone “combinations” [Mu’Alem&Nisan’02] • Fast payment computations for several “combinations” [Kao&Li&Wang’05]

compute payment no min-max problems

Limitations

Our (and Prior) Work

General technique for obtaining efficient mechanisms is one-shot

• write Alg as a “combination” of simpler algorithms

• compute the payments from the simpler algorithms

Mechanism for the Minimum Diameter Spanning Tree

max latency

Our (and Prior) Work

General technique for obtaining efficient mechanisms is one-shot

• write Alg as a “combination” of simpler algorithms

• compute the payments from the simpler algorithms

Mechanism for the Minimum Diameter Spanning Tree

• first mechanism, strongly polynomial-time (close to “best algorithm”)

MIN(A1,A2) : • run A1 and A2 independently;• choose the solution whose cost is smaller.

MIN(A1,A2):• compute X1 := A1(b) and X2:=A2(b)• if cost(X1,b) cost(X2,b) then return X1 else return X2

“Min combinations”

Objective function

MIN (A1,…,Ak) := MIN(A1,MIN(A2,…,Ak))

[Mu’Alem&Nisan’02]

Agents’ bids

Example: Minimum Radius Spanning Tree

Rooted tree minimizing the longest path to the root(locate a server and minimize the maximum latency)

x

1 20

1

Cost = height

1 1

x

SPT2

T2

SPT1

MIN(SPT1,SPT2)

MRST

Example: Minimum Radius Spanning Tree

1 1

x

x

T1

SPT2

T2

1 20

1

SPT1

MIN(SPT1,SPT2)

SPT2SPT1 SPT2

MRST

MIN(SPT2,SPT1)

Order matters!!

Different payments!Cost = height

Example: Minimum Radius Spanning Tree

Payment Computations (Idea)

x

A1

A2

MIN (A1,A2)MIN (A2,A1)

A1A2

The order

matters

T1R1L1TT

non-decreasing constant

Ah

Aj

Payment Computations (Idea)

x

The order

matters

RhLh

lowest

MIN(A1,…, Aj, …Ak)

Ah Ah

Threshold T = “leftmost ’’

General Technique

Alg is MIN-reducible in time

x

Ajnon-decreasing constant

Tj

Alg = MIN(A1,…,Ak)

Lj Rj

General Technique

Alg is MIN-reducible in time

Truthful (Alg,Pay) running in O(timeAlg + ) time

Main Application: MDST

MDST

Truthful mechnism for MDST running inO(n•(n,m)•timeMDST) time

Easy

Hard

Min Diam. Spanning Tree (Idea)

MDST = MIN(SPTe1,…,SPTem)

SPT

e1

Cost = height (max dist to every other node)

x

L R[Assin&Tamir’95]

Min Diam. Spanning Tree (Idea)

MDST = MIN(SPTe1,…,SPTem)

x

u

e1

SPT

Min Diam. Spanning Tree (Idea)

MDST = MIN(SPTe1,…,SPTem)

x

u

e1

SPTSPTSPT

x

Min Diam. Spanning Tree (Idea)

MDST = MIN(SPTe1,…,SPTem)

x

u

e1

SPT max x

Min Diam. Spanning Tree (Idea)

MDST = MIN(SPTe1,…,SPTem)

x

u

e1

SPT max x

u’ u’’

Min Diam. Spanning Tree (Idea)

MDST = MIN(SPTe1,…,SPTem)

x

u

e1

SPT max x

u’ u’’ xL R

limit

= “at least limit”

LR

= “larger than limit”

Extensions• Technique: Every “Binary Game” (selected/not selected) • Minimum Radius Spanning Tree (better running time)• p-center (1-center MRST)

Open

• Approximation: NP-hard problems• MIN-reducible APX Alg?

• More general Agents (e.g. two edges per agent)

Thank You