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Game Theory in Caching Models: Analyzing Nash Equilibria and Price of Anarchy

Explore the application of game theory in caching models, analyzing Nash equilibria, price of anarchy, and social optimal caching. Learn about selfish caching strategies, cost models, and simulation results showing phase transitions in network optimization. Discover major questions and results related to caching games. Ongoing and future work includes theoretical analysis under different constraints and large-scale simulations with realistic workloads.

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Game Theory in Caching Models: Analyzing Nash Equilibria and Price of Anarchy

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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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