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Caching Game

Caching Game. Dec. 9, 2003 Byung-Gon Chun, Marco Barreno. Contents. Motivation Game Theory Problem Formulation Theoretical Results Simulation Results Extensions. Motivation. Wide-area file systems, web caches, p2p caches, distributed computation. Game Theory. Game Players

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Caching Game

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  1. Caching Game Dec. 9, 2003 Byung-Gon Chun, Marco Barreno

  2. Contents • Motivation • Game Theory • Problem Formulation • Theoretical Results • Simulation Results • Extensions

  3. Motivation Wide-area file systems, web caches, p2p caches, distributed computation

  4. Game Theory • Game • Players • Strategies S = (S1, S2, …, SN) • Preference relation of S represented by a payoff function (or a cost function) • Nash equilibrium • Meets one deviation property • Pure strategy and mixed strategy equilibrium • Quantification of the lack of coordination • Price of anarchy : C(WNE)/C(SO) • Optimistic price of anarchy : C(BNE)/C(SO)

  5. Caching Model • n nodes (servers) (N) • m objects (M) • distance matrix that models a underlying network (D) • demand matrix (W) • placement cost matrix (P) • (uncapacitated)

  6. Selfish Caching • N: the set of nodes, M: the set of objects • Si: the set of objects player i places S = (S1, S2, …, Sn) • Ci: the cost of node i

  7. Cost Model • Separability for uncapacitated version • we can look at individual object placement separately • Nash equilibria of the game is the crossproduct of nash equilibria of single object caching game. •

  8. Selfish Caching (Single Object) • Si : 1, when replicating the object 0, otherwise • Cost of node i

  9. Socially Optimal Caching • Optimization of a mini-sum facility location problem • Solution: configuration that minimizes the total cost • Integer programming – NP-hard

  10. Major Questions • Does a pure strategy Nash equilibrium exist? • What is the price of anarchy in general or under special distance constraints? • What is the price of anarchy under different demand distribution, underlying physical topology, and placement cost ?

  11. Major Results • Pure strategy Nash equilibria exist. • The price of anarchy can be bad. It is O(n). • The distribution of distances is important. • Undersupply (freeriding) problem • Constrained distances (unit edge distance) • For CG, PoA = 1. For star, PoA  2. • For line, PoA is O(n1/2 ) • For D-dimensional grid, PoA is O(n1-1/(D+1)) • Simulation results show phase transitions, for example, when the placement cost exceeds the network diameter.

  12. Existence of Nash Equilibrium • Proof (Sketch)

  13. Price of Anarchy – Basic Results

  14. C(WNE) =  + (-1)n/2 C(SO) = 2 PoA = Inefficiency of a Nash Equilibrium -1 n/2 nodes n/2 nodes

  15. Special Network Topology • For CG, PoA = 1 • For star, PoA  2

  16. Special Network Topology • For line, PoA = O(n1/2)

  17. Simulation Methodology • Game simulations to compute Nash equilibria • Integer programming to compute social optima • Underlying topology – transit-stub (1000 physical nodes), power-law (1000 physical nodes), random graph, line, and tree • Demand distribution – Bernoulli(p) • Different placement cost and read-write ratio • Different number of servers • Metrics – PoA, Latency, Number of replicas

  18. Varying Placement Cost (Line topology, n = 10)

  19. Varying Demand Distribution (Transit-stub topology, n = 20)

  20. Different Physical Topology (Power-law topology (Barabasi-Albert model), n = 20)

  21. Varying Read-write Ratio Percentage of writes (Transit-stub topology, n = 20)

  22. Questions?

  23. Different Physical Topology (Transit-stub topology, n = 20)

  24. Extensions • Congestion • d’ = d +  (#access)  PoA / • Payment • Access model • Store model [Kamalika Chaudhuri/Hoeteck Wee] => Better price of anarchy from cost sharing?

  25. Ongoing and future work • Theoretical analysis under • Different distance constraints • Heterogeneous placement cost • Capacitated version • Demand random variables • Large-scale simulations with realistic workload traces

  26. Related Work • Nash Equilibria in Competitive Societies, with Applications to Facility Location, Traffic Routing and Auctions [Vetta 02] • Cooperative Facility Location Games [Goemans/Skutella 00] • Strategyproof Cost-sharing Mechanisms for Set Cover and Facility Location Games [Devanur/Mihail/Vazirani 03] • Strategy Proof Mechanisms via Primal-dual Algorithms [Pal/Tardos 03]

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