Sharing the cost of
multicast transmissions in
wireless networks
Carmine Ventre
Joint work with
Paolo Penna
University of Salerno, WP2
Wireless transmission


d(i,j)α
i
j
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
Wireless multicast transmission
known
10€
1€
1€
3€
source
Paolo 1€
Carmine 1€
Christos 10€
Pino 50€
private
Who receives Roma-Juventus
 How to transmit
 Goal: maximize
Benefit – Cost
i.e. the social welfare

Andrea 30€
Selfish agents
WYSWYP (What You
Say What You Pay)
source

10
0€

Pino 50 €
5.1 €
5
Andrea 30 €
10
Paolo 9 €

COST = 10 + 5 = 15
WORTH = 50 + 30 = 80
NET WORTH = 80 – 15 =
65
Pino
says
0 €5.1
and€ gets
Andrea
says
Andrea says 5.1 € and gets
Roma
– Juventus
Pino
says
0€
Roma – Juventus
Nobody
for free gets
for a lower price
Roma - Juventus
NW’ = 0
Graph model

A complete directed
weighted communication
graph G=(S,E,w)

w(i,j) = cost of link (i,j)


v1
1
2
v2


v4
4
3 v3


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
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
vi  Pi (b1 ,..., bi ,..., b n ) if i receives the transmiss ion,
ui(bi)= 
0
otherwise.


Goal of agent i: maximize ui(bi)
Mechanism’s desired properties

No positive transfer (NPT)


Voluntary Participation (VP)


Payments are nonnegative:
Pi  0
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):= iT 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:
  Xsol, 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)
WORTH(X)-COST(X)
= i valuationi(X)
iX
vi - ci = WORTH(X)
- COST(X)
ci
vi
Initially, charge to every receiver i
the cost ci of its ingoing connection
Pi = ci + PVCG
Free edges on Trees
RECURSION?
tree
graph
s
s
1
4
3
2
1
3
5
4
4
5
NO!
2
5
4
4
3
5
3
4
YES!
Trees algorithm: recursive equation







NWopt (i)  vi  max 0, max  c j   NWopt (k )
jch ( i )


kch ( i ),ck c j






vi
i

cj
j

k s.t. ck ≤ cj
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
Trees with metric free edges


Path(i,4)=w(i,1)+w(1,4)
w(i,3) ≥ path(i,4)
i
7
5
6
1

2
(i,4) metric free edge
1
4
5
5
3
Tree with metric free edge: idea

A node k reached for free gets some credit
i
cj
j
k gets cj-ck
units of credit
ck
k
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


credit(r) = credit(k)-cr
r
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
r
l
credit(l)=0
credit(r)=cl-cr
Tree with metric free edge: recursive
equations

We have two contributions:

the nodes whose ingoing edge is paid
NWpay (i )  c i 
NW
opt
(k, c i  c k )
kch( p( i) ),c k c i

the nodes with credit c whose ingoing edge is free




NWopt (i, c)  v i  max   NWopt ( j , c  c j ), max NWpay ( j )
jch ( i ),c  c j


 jch (i ),cc j

NOTE: the optimum is NWopt(s,0)
The one dimensional Euclidean case

Stations located on a line (linear network)
1
i
s
receivers
Clementi et al algo
j
n
(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