Sharing the cost of multicast transmissions in wireless networksCarmine Ventre

Joint work with Paolo Penna

University of Salerno, WP2

Wireless transmission

Power(i)= d(i,j)α = range(i) α, α>1 (empty space α = 2)

A message sent by station i to j can be also received by every station in transmission range of ii

j

d(i,j)α

Wireless multicast transmission

Who receives Roma-Juventus How to transmit Goal: maximize

Benefit – Cost i.e. the social welfare

Paolo 1€

10€ 1€ 1€ 3€

Carmine 1€ Christos 10€ Andrea 30€

Pino 50€

known

private

source

Selfish agents

COST = 10 + 5 = 15 WORTH = 50 + 30 = 80 NET WORTH = 80 – 15 =

65

source

10

10

5

Pino 50 €

Andrea 30 €

Paolo 9 €

0 €

Pino says 0 € and gets

Roma – Juventus

for free

5.1 €Andrea says 5.1 € and gets

Roma – Juventus

for a lower price

Andrea says 5.1 €

Pino says 0 €Nobody gets

Roma - Juventus

NW’ = 0

WYSWYP (What You Say What You Pay)

Graph model

A complete directed weighted communication graph G=(S,E,w)

w(i,j) = cost of link (i,j) w(1,4) = d(1,4)2.1

w(1,2) = d(1,2)5

w(2,4) = ∞ w(4,2) = d(4,2)2.1

A source node s vi = private valuation of

agent i

21

4 3v4v3

v1 v2

Mechanism design: model

Design a mechanism M=(A,P) Each agent declares bi

Algorithm A selects, based on (b1, …, bn), a set of receivers a subset of connection T E

Agent i must pay Pi(b1, …, bi-1, bi, bi+1 ..., bn) Utility of the agent

ui(bi)=

Goal of agent i: maximize ui(bi)

otherwise.0

ion, transmiss thereceives i if)b,...,b,...,(bPv ni1ii

Mechanism’s desired properties No positive transfer (NPT)

Payments are nonnegative: Pi 0

Voluntary Participation (VP) User i is charged less then his reported valuation

bi (i.e. bi ≥ Pi)

Consumer Sovereignty (CS) Each user can receive the transmission if he is

willing to pay a high price.

Mechanism’s desired properties: Incentive Compatibility Strategyproof (truthful) mechanism

Telling the true vi is a dominant strategy for any agent

Group-strategyproof mechanism No coalition of agents has an incentive to jointly

misreport their true vi

Stronger form of Incentive Compatibility.

Mechanism’s desired properties Budget Balance (BB)

Pi = COST(T) (where T is the solution set)

Efficiency (NW) the mechanism should maximize the

NET WORTH(T) := WORTH(T)-COST(T)

where WORTH(T):= iT vj

Mutually exclusive!!

Efficiency No Group strategy-proof

Previous work

Wireless broadcast 1d: COSTopt in polynomial time [Clementi et al, to appear] 2d: NP-hard, MST is an O(1)-apx [Clementi et al, ‘01] On graphs: (log n)-apx [Guha et al ‘96, Caragiannis et al, ‘02] Many others…

Wired cost sharing (selfish receivers) Distributed polytime truthful, efficient, NPT, VP, and CS mechanism

for trees (no BB) [Feigenbaum et al, ‘99] Budget balance, NPT, VP, CS and group strategy-proof mechanism

(no efficiency) [Jain et al, ‘00] No α-efficiency and β-BB for each α, β > 1 [Feigenbaum et al, ‘02] polytime algorithm no R-efficiency, for each R > 1 [Feigenbaum et

al, ‘99]

Our results

G is a tree NWopt in polytime distributed algorithm Polytime mechanism M=(A,P) truthful, NPT, VP and CS Extensions to “metric trees” graphs

G is not a tree 2d: NP-hard to compute NWopt

1d: Polytime mechanism M=(A,P) truthful, NPT, VP, CS and efficient (i.e. NW is maximized)

Precompute an universal multicast tree T G A polytime truthful, NPT, VP and CS mechanism O(1) or O(n)-efficiency, in some cases

polytime algorithm no R-efficiency, for every R > 1

VCG Trick (marginal cost mechanism) Utilitarian problem:

Xsol, measure(X)=i valuationi(X)

Aopt computes X sol maximizing measure(X)

PVCG: M=(Aopt, PVCG) is truthful

VCG Trick (marginal cost mechanism)Making our problem utilitarian:

measure(X) valuationi(X)

WORTH(X)-COST(X)

= i

iXvi = WORTH(X)

vi

ciInitially, charge to every receiver ithe cost ci of its ingoing connection

- ci

- COST(X)

Pi = ci + PVCG

Free edges on Trees

21

4

3

5

s

graphtree

21

4

3

5

s

RECURSION?

NO! YES!

3 4

4 5 4 5

43

Trees algorithm: recursive equation

jk ccchkoptj

chjopt kNWcNW

),i()i(

i )(max,0maxv)i(

It is easy to see that the best solution has an optimal substructure

It is simple to compute NWopt(s) in distributed bottom-up fashion

O(n) time, 2 msgs per link

k s.t. ck ≤ cj

i

j

cj

vi

Trees with metric free edges

Path(i,4)=w(i,1)+w(1,4) w(i,3) ≥ path(i,4)

(i,4) metric free edge

21

4

3

5

i

1 5

75

6

Tree with metric free edge: idea A node k reached for free gets some credit

i

j

cj

k gets cj-ck units of credit

k

ck

Tree with metric free edge: credit usage k can use its credit to

reach all of its children If there is a child l s.t. cl >

credit(k) and NWopt(l)>0 then credit(k) is useless For each r Є ch(k):

cl – cr > credit(k) – cr

Paying a free edge is not a good solution (i.e. we have a smallest credit and a greater cost)

k

k

lr

credit(r)=cl-cr

r

credit(r) = credit(k)-cr

credit(l)=0

Tree with metric free edge: recursive equations We have two contributions:

the nodes whose ingoing edge is paid

the nodes with credit c whose ingoing edge is free

)cc,k(NWc)(NW kicc),)(p(chk

optipay

ik

i

i

)(max,),(maxv)ci,(),(

),i(i jNWccjNWNW pay

ccichjccchj

joptoptj

j

NOTE: the optimum is NWopt(s,0)

The one dimensional Euclidean case Stations located on a line (linear network)

si j1 n

receivers

Clementi et al algo

(Some) Open problems

2d Euclidean case: O(1)-APX multicast algorithm “Good” universal Euclidean multicast trees Truthful mechanism with O(1)-APX BB truthful mechanisms