The Algorithmic Structure of Group Strategyproof Budget-Balanced Cost-Sharing Mechanisms

Paolo Penna & Carmine Ventre

Università di Salerno

Italy

Why Cost-Sharing Methods?

Town A needs a water distribution system A’s cost is € 11 millions

Town B needs a water distribution system B’s cost is € 7 millions

A and B construct a unique water distribution system for both cities The total cost is € 15

millions Why not collaborate and

save € 3 millions? How to share the cost?

Town A

Town B

11

7

15

Multicast and Cost-Sharing

A service provider s Selfish customers U Who is getting the service? How to share the cost?

real worth is 7

is worth 5 ( 7)

Pi

Accept or reject the service?

Selfish Agents

Each customer/agent has a private valuation vi for the service

declares a (potentially different) valuation bi

pays Pi for the service

Agents’ goal is to maximize their own utility: ui(b1, …, bn) := vi – Pi(b1, …, bn)

Accept iff my

utility ¸ 0!

Coping with Selfishness: Mechanism Design

Algorithm A Who gets serviced (Q(b)) How to reach Q(b)

Payment P How much each user pay

M = (A, P)

bi

bj

P1

P4

P3

P2

M’s Strategyproofness

For all others players’ declarations b-i it holds

ui = ui(vi, b-i) ¸ ui(bi, b-i) = ui

for all bi (ie, truthtelling is a dominant strategy)

M = (A, P)vi

M’s Group Strategyproofness

U

Coalition C

No one gainsAt least one looses (ie, ui < ui)

C is uselessBreaks off C

Mechanism’s Requirements

Budget Balance (BB) i2Q(b) Pi(b) = CA(Q(b))

Cost Optimality (CO) CA(¢) is minimum

No positive transfer (NPT) Payments are nonnegative: Pi 0

Voluntary Participation (VP) User i is charged less then his reported valuation b i (i.e. bi ≥

Pi) Consumer Sovereignty (CS)

Each user can receive the transmission if he is willing to pay a high price

Beyond CS Property

M is not upper continuous E.g., serve i for all bids strictly greater than 1

bii(b-i)

ServicedNot Serviced

M SP

Fix i, b-i

CS

M is upper continuous E.g., serve i for all bids greater or equal than 1

Characterizing GSP, BB, … Mechanisms

M = (A, P)

Cost function is submodular P is cross monotonic

[MS99]

Sufficient condition too

[MS99]

M UC & with no free-riders P is cross monotonic

[IMM05]

And the algorithm?

Extant Approach & Algorithms

A is able to reach any set in 2U Cost hard to compute for

some subset (e.g. Steiner tree) Polynomial-time

mechanisms Relax BB condition

Switched beam wireless antenna Generalized cost-sharing

games

U

Sequential Algorithms

A is sequential if for some bid vectors reaches a chain of sets Q1, …, Q|U|, ;

Sequential algorithm leads to GSP, BB, … mechanisms ([PV04], [IMM05]) Steiner tree game BB

mechanisms ([PV04, PV05]) NP-hard problem

UQ1=U

Q3

Q|U|…

Q2

.

.

.

… Q|U|+1 = ;

Our Results

M = (A, P)

M for 2 users A is sequential

M GSP & UC A is sequential

M is SP, BB, …

9 M for 3 users SP, BB with A not sequential

The Two Users Case

No singleton is reached by A

A cannot reach U

A is not sequential

M=(A,P) SP, BB, …

Users compete for the resource

SP ) P2 must be at least 7

1 2

510 712

Unbounded payments

1

2

b1

b2

SP, VP ) P2(b1,0)=0

SP, VP ) P1(0, b2)=0

SP ) P1(b1,b2)= P2(b1,b2)=0

Users not separable

No payments

Three Users: Working Mechanism

1 2 3

Sets Cost

U 3

{1,2} 1

{1,3} 1

{2,3} 1

A is not sequential (no singleton in the sets)

Mechanism M = (A,P)

Serve U if b1>1, b2>1 and b3 > 1

Serve {i, j} if bi > 1 and bj > 1

Serve {i, i+1 mod 3} if bi > 1

{1,2,3} ) P1=P2=P3=1

{1,2} ) P1=1, P2=0

{1,3} ) P1=0, P3=1

{2,3} ) P2=1, P3=0

M is not UC nor GSP (user 3 can help user 1)

Hints for the General Case

Full Coverage (U reachable) Weak Separation (a singleton reachable) Using Upper Continuity & GSP Working in P P

P

Bids only in {0,B}

A user bidding B is serviced no matter what b-i is

Conclusions Introduction of generalized cost-sharing games (modeling

many real-life applications) Simple technique of [PV04, IMM05] is not less powerful than

more complex one for UC mechanisms Relaxing BB does not allow to solve more problems

Are sequential algorithms necessary for not UC & GSP mechanisms too?

GamesUpper Continuous Mechanism

any (non polytime) poly-time

With Sequential Algorithms

P=2U 1 [PV04, IMM05]· (2U) [PV04]

¸ ({U}) [this work]

P has a sequence 1 [PV04, IMM05]· () [PV04]

¸ ({U}) [this work]

With No Sequential Algorithm

P has no sequence unbounded [this work]

unbounded [this work]