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Sharing the cost of multicast transmissions in wireless networks

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### Sharing the cost of multicast transmissions in wireless networks

Carmine 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 i

d(i,j)α

i

j

Wireless multicast transmission

known

10€

1€

1€

3€

source

Paolo 1€

Carmine 1€

Christos 10€

Andrea 30€

Pino 50€

- Who receives Roma-Juventus
- How to transmit
- Goal: maximize

Benefit – Cost

i.e. the social welfare

private

Selfish agents

WYSWYP (What You Say What You Pay)

source

- COST = 10 + 5 = 15
- WORTH = 50 + 30 = 80
- NET WORTH = 80 – 15 = 65

10

0 €

Pino 50 €

5

5.1 €

Pino says 0 € and gets

Roma – Juventus

for free

Andrea says 5.1 €

Pino says 0 €

Andrea says 5.1 € and gets

Roma – Juventus

for a lower price

Andrea 30 €

Nobody gets

Roma - Juventus

10

Paolo 9 €

NW’ = 0

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

v1

1

2

v2

v3

v4

4

3

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)

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 viStronger 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)

whereWORTH(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)
- Aoptcomputes Xsol maximizing measure(X)
- PVCG: M=(Aopt, PVCG) is truthful

VCG Trick (marginal cost mechanism)

Making our problem utilitarian:

= i

measure(X)

valuationi(X)

iX

WORTH(X)-COST(X)

vi

- ci

= WORTH(X)

- COST(X)

Initially, charge to every receiver i

the cost ci of its ingoing connection

ci

Pi = ci + PVCG

vi

Trees algorithm: recursive equation

- 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

vi

i

cj

j

k s.t. ck≤ cj

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

i

7

5

6

1

2

3

1

5

4

5

Tree with metric free edge: idea

- A node k reached for free gets some credit

i

k gets cj-ck

units of credit

ck

cj

k

j

Tree with metric free edge: credit usage

k

- 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)

credit(r) = credit(k)-cr

r

k

r

l

credit(r)=cl-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

NOTE: the optimum is NWopt(s,0)

The one dimensional Euclidean case

- Stations located on a line (linear network)

1

i

j

n

s

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

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