Selfish Caching in Distributed Systems: A Game-Theoretic Analysis

Transcription

1 Selfish Caching in Distributed Systems: A Game-Theoretic Analysis Symposium on Principles of Distributed Computing July 5, 4 Byung-Gon Chun, Kamalika Chaudhuri, Hoeteck Wee, Marco Barreno, Christos Papadimitriou, and John Kubiatowicz University of California, Berkeley 1

2 Caching in Distributed Systems D Server Server A Access cost C Server Placement cost Server B Wide-area file systems, distributed web caches, pp caches, federated database, distributed computation

3 Socially Selfish Optimal Caching Caching Placement cost Access cost node link 3

4 Game Theory Game Servers (nodes) Strategies Cost function Nash equilibrium No node can benefit by unilaterally changing its strategy. Quantification of lack of coordination Price of anarchy (PoA) Optimistic price of anarchy (OPoA) 4

5 Outline Introduction and Background Basic Game How efficient are Nash equilibria under different placement costs and network topologies? Payment Game What is the effect of adopting payments? Simulation Results Conclusion and Future Work 5

6 Basic Game (Single Object) Strategy of node i - to cache or not to cache Cost of node i c (S) = { i α, when i caches the object wi d closest, otherwise α: placement cost of the object w i : demand that server i has for the object d closest : distance between i and the closest server to i that caches the object 6

7 Price of Anarchy is O(n) α 1 A B α 1 A B n n nodes nodes C SO = α n n nodes nodes C WNE = α + (α-1)n/ PoA = C WNE C SO = α + (α 1)n / α O(n) OPoA = O(n) 7

8 Specific Network Topology Graph with unit edge distance Complete graph Star Line D-dimensional grid PoA 1 O(n 1/ ) O(n D/(D+1) ) 8

9 Outline Introduction and Background Basic Game How efficient are Nash equilibria under different placement costs and network topologies? Payment Game What is the effect of adopting payments? Simulation Results Conclusion and Future Work 9

10 Payment Game (Single Object) Strategy of node i - bid (v i, b i ), threshold (t i ) Cost of node i i t i b j j v j = i R i : total amount of bids i receives Node i caches the object iff R i t i Node j pays b j to i when i caches. α R, when i caches the object c (S) = { i i w d + b, when i does not cache i closest i w d i closest, when i and v i do not cache and v i caches the object 1

11 Payment does not improve PoA The PoA of the payment game is at least PoA of the basic game. Any Nash equilibrium configuration in the basic game can be implemented in the payment game. In fact, given a graph, the PoA of the payment game can be worse. 11

12 Optimistic Price of Anarchy with Payment α 1 A B n n n n n A α 1 B n n n n n n n nodes nodes n n nodes nodes Observation: Both i and j replicate the object in a Nash equilibrium and d ij < α, then node i should have a threshold (t i ) at least (α - d ij ). 1

13 OPoA of the payment game is always one. The OPoA of the payment game is always one. Consider any socially optimal configuration. Find a set of bids and thresholds so that the Nash equilibrium implements the social optimal configuration. Let i be the node replicating the object in social optimum. Distribute the payment needed to replicate the object at i among the nodes accessing the object from i. Ensure that the threshold of i satisfies the requirement in the observation. Ensure the node i collects bids at least the threshold. 13

14 Simulation Game simulations to compute Nash equilibria Integer programming to compute social optima 1 servers, 1 runs Varying placement cost (α) Underlying topology line, transit-stub (35 physical nodes), power-law (35 physical nodes) Demand distribution Bernoulli(p), default p = 1 Metrics PoA, OPoA, and Average Ratio 14

15 Effects of Placement Cost and Payments (a) Basic game (Transit-Stub topology, n = 1) (b) Payment game 15

16 Conclusion Introduce novel game-theoretic models to characterize the caching among selfish servers. The price of anarchy in the basic game is O(n) in general. With specific topologies it has better bounds. With payments, the price of anarchy does not decrease, but the optimistic price of anarchy is always one. 16

17 Future Work Incentive protocols to set bids and thresholds in a distributed fashion Analysis of more realistic system models What s the effect of capacity limit? What s the effect of non-uniform demands? Distance matrices achieved from the Internet system measurements What s the effect of congestion? 17

18 Selfish Caching in Distributed Systems: A Game-Theoretic Analysis Symposium on Principles of Distributed Computing July 5, 4 Byung-Gon Chun, Kamalika Chaudhuri, Hoeteck Wee, Marco Barreno, Christos Papadimitriou, and John Kubiatowicz University of California, Berkeley 18