Free-Riders in Steiner TreeCost-Sharing Games

Paolo Penna and Carmine Ventre

Università di Salerno

Cost-Sharing Games

UQ

1. Which customers to service?2. At which price?

S

Service provider Customers

user i wants to pay at most vi

Cost-Sharing Games

UQS

Service provider Customers

S

0.9 0.9

1

Multicast:S

0.9 0.9

1

wired wireless

Cost-Sharing Games

UQS

Service provider Customers

1. Budget balance: Cost(Q) = Pi

2. Users can form coalitions Group strategyproof mechanisms

Group Strategyproof Mechanisms

user i wants to pay at most viPrivate knowledge

Service i and charge Pi Don’t service i

0vi - Pi

Utility ui

Pi’

Pi’

ui’

ri

Group Strategyproof Mechanisms

Customers

UCuiTruth-telling

Lie ui’

None gains

i

At least one looses

Coalition is uselessui’ < ui

Breaks off the coalition

Requirements

1. Budget Balance2. Group Strategyproof3. Voluntary Participation: Pi vi (unless i lies)4. Consumer Sovereignity: ri “large enough” service i5. No Positive Transfer: Pi 0 (do not pay customers)

Steiner Tree GameGiven network G = (U s, E, w), with s source node

s

1

1 1

1.6 1.6

0.9 0.9

0

Cost(Q) = cost of opt Steiner tree connecting s to Q

Related Work

Polytime mechanisms:• 2-APX budget balance [Jain & Vazirani, STOC01]• Budget balance [Penna &Ventre, WAOA04]

Mechanisms and free-riders:• No free-riders [Immorlica et al, SODA05]

Relax budget balance Ignores free-ridersIgnores computational issues

Budget balance + polytime + free-riders issue ?

Our Contribution

N-L free-riders

Less free-riders? Polytime?

Polytime, budget balance, N-L free-riders (L = #leaf nodes)

Budget balance, no free-riders NP-hard

No free-riders

[IMM04]

N-1 free-ridersfairness

[PV04]

-APX budget balance, no free-riders NP-hard for some > 1

(1+)-APX BB

Wireless case: similar results (BB 6-APX BB)

How to Build Mechanisms

UQ

(Q,i) = Cost(Q)

Cost-sharing methods: distribute Cost(Q) among users in Q

(Q,i) 0

(Q,i) = 0, i Q

Idea: associate prices to service set

How to Build MechanismsCost-sharing method (•,•) Mechanism M( )

(Q,i) > ri U Drop iQ

UQ1=U

How to Build MechanismsCost-sharing method (•,•) Mechanism M( )

Q3

Qk…

Q2 Prices do not decrease

Group Strategyproof

(Qk,i)

(Q2,i)

(Q3,i)

Pi = (Qk,i)

Changes

Monotonicity

[Moulin & Shenker ’97] [PV04]

Our Method

no Steiner nodes MST(U) is optimal

s

1

1 1

1.6 1.6

G = (U s, E, w)

Easy case: Q = U

Hard case: Q “any”

U

sOur Method

MST

pay

s

prune

Q MST(Q)

opt Steiner tree

T + = opt

s

uQ

v

s

uQ

T

s

T*>

Qu

v

v

+ +

Our MethodMonotonicity: Prices do no lower

u

v

/ (Lv-1)

0 / (Lv-1)

s

/ Lv

Lv

QConsumer Sovergnity

Hardness

Any U

Budget balance, no free-riders NP-hard

vi

0

Voluntary Participation

Pi vi = 0

Free-riders

Consumer Sovergnity

Hardness

U

Budget balance, no free-riders NP-hard-APX budget balance, no free-riders NP-hard for some > 1

Voluntary Participation

AnyQ =

Budget balance Must compute Cost(Any)

Open Questions

Trade-offs (Polytime mechanisms)a-APX Budget balance + bN-free-riders

Complete Metric GraphsBudget balance + (N/2) -free-riders?

Thank You