1. Repeated Game Modeling of Multicast Overlays Mike Afergan (MIT CSAIL/Akamai) Rahul Sami (University of Michigan) April 25, 2006

2. Talk Overview Introduction Repeated Games A Repeated Game Model of Multicast Overlays Results Summary

3. 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

4. A Double Problem System 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.

5. Key Insight Cheating degrades system efficiency and quality Can reduce lifespan of system Even selfish users want the system to exist in the future.

6. Key Contributions A repeated model of cooperation Cooperation is endogenous to model Does not require heavyweight mechanisms Prescriptive results for building more efficient systems

7. Talk Overview Introduction Repeated Games A Repeated Game Model of Multicast Overlays Results Summary

8. One-Shot Prisoner's Dilemma P1\P2CD C(5,5)(0,9) D(9,0)(1,1) Static Equilibrium Outcome In the one-shot game, (D,D) is the outcome of the unique Nash Equilibrium.

9. Repeated Prisoner's Dilemma P1\P2CD 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 of the Repeated Game

10. Sample Analysis P1\P2CD 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:    ½

11. Talk Overview Introduction Repeated Games A Repeated Game Model of Multicast Overlays Results Summary

12. 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.

13. Formal Game Model Instance Network: G = (V,E) Nodes to be served: N  V Single source: s  N, s  V 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.

14. Players and Actions User Utility Function – u i (d i,c i ) 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)) = 1.0 2. 1.0 > R(L(Unicast Tree)) ≥ 0 3. R(L) is monotonic Equilibrium Condition:

15. Talk Overview Introduction Repeated Games A Repeated Game Model of Multicast Overlays Results Summary

16. Simulator 1. Take inputs (topology, u i (.), N, , A) 2. Randomly select source and N end-nodes. 3. Each node learns d i, c i, 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.

17. 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.

18. Benchmark Algorithm: Naïve Min Cost Spanning Tree Inputs: Nodes Pair-Wise distances Outputs: Min Cost Spanning Tree Assumes all reports are truthful

19. 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]

20. NICE is more efficient than a Naïve Min-Cost Spanning Tree NMC better for faithful users But for even mildly selfish users NICE performs better.

21. Utility Distribution Naïve Min Cost NICE NICE has an inherent tradeoff between depth and load.

22. Impact of Cluster Size  Under reasonable assumptions, increasing cluster size can increase efficiency. Load

23. 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

24. 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.

25. 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.