Repeated Game Modeling of Multicast Overlays

Mike Afergan (MIT CSAIL/Akamai) Rahul Sami (University of Michigan)

April 25, 2006

Talk Overview Introduction Repeated Games A Repeated Game Model of

Multicast Overlays Results Summary

Application-Layer Multicast

Position in tree can impact QoS. [Mathay et al 04] Users have motive and means to alter tree. In the limit, becomes the unicast tree.

……

Wants to move up tree

Want fewer children

Problem: Selfish users can degrade system performance

A Double ProblemSystem Design Problem

Goal: A protocol which creates efficient trees even with selfish users.

This problem is hard: Real-time and unidirectional Heavyweight solutions (e.g., payments, complicated

trees) are undesirable. NATs make many solutions (e.g., monitoring) challenging.

Modeling Problem On a small scale, these trees exist in practice without

such mechanisms. [Chu et al ’04]Goal: A model that explains observed behavior and

provides practical insight for building robust protocols.

Key Insight

Cheating degrades system efficiency and quality

Can reduce lifespan of system

Even selfish users want the system to exist in the future.

Key Contributions A repeated model of cooperation

Cooperation is endogenous to model Does not require heavyweight

mechanisms

Prescriptive results for building more efficient systems

Talk Overview Introduction Repeated Games A Repeated Game Model of

Multicast Overlays Results Summary

One-Shot Prisoner's Dilemma

P1\P2 C D

C (5,5) (0,9)

D (9,0) (1,1) Static EquilibriumOutcome

In the one-shot game, (D,D) is the outcome of the unique Nash Equilibrium.

Repeated Prisoner's Dilemma

P1\P2 C D

C (5,5) (0,9)

D (9,0) (1,1)

$$$ or $+$+$+ $+ $ + S

Key Takeaway: The equilibrium of the repeated game may differ from the equilibrium of the stage

game.

Example Strategy: 1. Play C 2. If the other player defects, play D forever

Outcome ofthe RepeatedGame

Sample AnalysisP1\P2 C D

C (5,5) (0,9)

D (9,0) (1,1)

$$$ or $+$+$+ $+ $ + S

Parameterized by discount factor () Patience Factor (infinite game) Probability of game ending (finite game with unknown horizon)

Example: is an equilibrium of the RPD iff: (Playing forever) (One-time “cheat”) + (Resulting payoffs)

“Play C forever. If other plays D, play D forever” is an equilibrium iff:

10

)1(95t

t

t

t ½

Talk Overview Introduction Repeated Games A Repeated Game Model of

Multicast Overlays Results Summary

Model Intuition Nodes in a network form an overlay. Per time-period benefit to user dependant on:

Quality of content received Load on user

Network Efficiency: Relative network load of given tree Defines per-period probability of network

continuing Selfish players maximize the (discounted)

series of per-period payoffs.

Formal Game Model Instance

Network: G = (V,E) Nodes to be served: N V Single source: s N, sV Single atomic piece of content

An algorithm constructs a tree (T) which serves all nodes, N.

Load of tree L(T, G) is sum of load on all links.

Players and Actions User Utility Function – ui(di,ci)

Decreasing in d and c as fixed and exogenous Action Space: {Connect to Root, Drop Child,

Stay} Response Function – R(L)

1. R(L(Faithful Tree)) = 1.02. 1.0 > R(L(Unicast Tree)) ≥ 03. R(L) is monotonic

Equilibrium Condition: ''

1

''

0

,)'(,,)( iiit

ttiii

tiii

tt cduLRcducduLR

Talk Overview Introduction Repeated Games A Repeated Game Model of

Multicast Overlays Results Summary

Simulator1. Take inputs (topology, ui(.), N, , A)2. Randomly select source and N end-nodes.3. Each node learns di , ci, and f(L).4. Each node can connect to root, drop child,

or take no action.5. Repeat Step #4 until stable.6. Collect Statistics. All datapoints represent 90 simulator runs.

We prove that stable points of simulator are sub-game perfect equilibria.

Results

1. System efficiency decreases with decreasing .

2. System efficiency decreases with increasing N.

3. Specific insight for particular tree formation protocols.

Goal: A model that explains observed behavior and provides practical insight for building robust protocols.

Benchmark Algorithm:Naïve Min Cost Spanning Tree Inputs:

Nodes Pair-Wise distances

Outputs: Min Cost Spanning Tree

Assumes all reports are truthful

NICE (Banerjee et al ’02)

Nodes create hierarchical tree of clusters of size k Completely distributed NICE has been shown to have good performance

characteristics. [Banerjee et al, ’02]

NICE is more efficient than a Naïve Min-Cost Spanning Tree

NMC better for faithful users

But for even mildly selfish usersNICE performs better.

Utility DistributionNaïve Min Cost NICE

NICE has an inherent tradeoff between depth and load.

050

100

150

200

250

300350

400

450

4 5 6 7 8 9

UtilityCount

050

100150

200

250300

350400

450

4 5 6 7 8 9

Utility

Count

1

1.5

2

2.5

3

3.5

4

10.9

80.9

60.9

40.9

2 0.9 0.88

0.86

NMC

32

8

2

Impact of Cluster Size

Under reasonable assumptions, increasing clustersize can increase efficiency.

Load

Generalizations Core results and intuition apply to

more general cases: Large class of utility functions Large class of response functions Noisy signal of state Noisy understanding of response

function

Exogenous Types vs Endogenous Motivations

Prior models use exogenous types: Cheater/not [Mathy et al] Altruism parameter [Feldman et al ‘04, Chu/Zhang ‘04 ]

A repeated game model captures these factors in an endogenous fashion.

Benefits: Fewer degrees of freedom Behavior is dependant on the system.

This enable practical conclusions.

Summary Users have the means and motive

to alter multicast overlay trees. A repeated model of interactions

can explain user cooperation without heavyweight mechanisms.

Behavior which is endogenous to the model enables practical conclusions.