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