a game theoretic framework for incentives in p2p systems --- cs. uni. california
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A Game Theoretic Framework for Incentives in P2P Systems--- CS. Uni. California
Jun Cai
Advisor: Jens Graupmann
Outline
Introduction (problem, motivation) Incentive model Nash Equilibrium in Homogeneous
Systems of Peers Nash Equilibrium in Heterogeneous
Systems of Peers Simulation result Summary
Introduction Democratic nature, no central authority
mandate resource Distributed resources are highly variable
and unpredictableMost of users are “free riders” (In Gnutella,
25% users share nothing)User session are relative short, 50% of
sessions are shorter than 1 hour
How to build a reliable P2P system
Require: Contribution should be predictable Peers can be motivated using economic
principleMonetary payment (one pays to consume
resources and paid to contribute resource)Differential service (peers that contributes more
get better quality of service) eg: reputation index (participation level in KaZaA)
KaZaA: Participation level = upload in MB / download in MB x 100
Modeling interaction of peers by Game Theory
Peers are strategic and rational player Non-cooperative game Each player wants to maximize his utility
Utility depends on benefit and cost Utility depends not only on his own strategy but
everybody else’s strategy Find equilibrium (a locally optimum set of
strategies) where no peer can improve his utility --- Nash equilibrium
Level of contributionUptime or shared disk space, bandwidth
Incentive model (measure contribution)
P1,P2,P3…PN as peers
Utility function for Pi is Ui
Contribution of Pi is Di (D0 is absolute measure of contribution)
Dimensionless contribution: Unit cost ci
Total cost:ciDi
0/i id D D
Incentive model (Benefit matrix)
NxN benefit matrix B Bij denote how much the
contribution made by Pj is worth to Pi
bi is the total benefit that Pi can get from the system
/
1
ij ij i
i ijj
av ii
b B c
b b
b bN
There exists a critical value bc.
Incentive model (A peer reward other peers in proportion to their contribution)
Pj accepts a request for a file from peer Pi with probability p(di) and rejects it with probability 1-p(di)
Each request is tagged with di as metadata
( ) , 01
(0) 0
lim ( ) 1d
dp d
dp
p d
Incentive model (Utility function)
Utility function
Dimensionless utility function
0
, ( ) , 0ii i i i ij j ii
ji
Uu u d p d b d b
c D
( ) , 0i i i i ij j iij
U c D p d B D B
0(0) 0 lim 0
( ) 1 limi
i
id
id
p u
p u
cost
benefit
worth
Be able to download?
Utility vs. contribution (different benefit)
So far…
Incentive model Now find equilibrium…
Homogeneous (simple)Heterogeneous (by analogy of Homogeneous
system)
Homogeneous System of Peers (1)
All peers derive equal benefit form everybody else (bij=b for )
By symmetry, reduce the problem to Two player game
Best response function
1 1 12 2 1
2 2 21 1 2
( )
( )
u d b d p d
u d b d p d
( ) , ( 1)
(1 )
dp d
d
( 1) ( )u d N bdp d
1 2 1 12 2
2 1 2 21 1
( ) 1
( ) 1
r d d b d
r d d b d
i j
Differentiate w.r.t. d1
Differentiate w.r.t. d2
P1:
P2:
Nash Equilibrium in Homogeneous System of Peers (2)
Best response function
Nash equilibrium exists if forms a fix point for above equation
1 2 1 12 2
2 1 2 21 1
( ) 1
( ) 1
r d d b d
r d d b d
* *1 2( , )d d
* * *1 2
* 2( 1) ( 1) 12 2
d d d
b bd
Solution exists only if
4 cb b
Utility
contribution
Critical benefit value bc
b=bc
Nash Equilibrium in Homogeneous System of Peers (3)
N player game
* *
* 2
( 1) 1
( 1) ( 1)( 1) ( 1) 1
2 2
d b N d
b N b Nd
Replace b(N-1) to b, this formula is two player game.
Courtnot learning & convergence process
*1 1 1 1 1 2
*2 2 2 2 2 1
( ( ( (...( )))))
( ( ( (...( )))))
d r r r r d
d r r r r d
High
Low
Nash Equilibrium in Heterogeneous System In Homogeneous system, fix point equation:
In Heterogeneous system, fix point equation:
* *1 12 2
* *2 21 1
1
1
d b d
d b d
* * 1i ij jj i
d b d
By analogy of
Homogeneous system
Iterative learning model
Algorithms: iterative learning model
di = random contribution
While (converge == false){
new_di = computeContribution (d, b);
if (new_di == di) {
converge = true;
}
di = new_di;
}
Convergence of learning algorithms
How fast it converge?
High benefit
Low benefit
Simulation: dav vs. (bav/bc-1)
Eq
uilib
rium
av
erag
e co
ntrib
utio
n
1. Monotonically
2. Peer size independent
3. If bav < bc, d ---> 0
Simulation: leave system
bav/bc-1=2.0
Summary
Differential service based incentive model for p2p system that eliminating free riding and increasing availability of the system
Critical benefit bc
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