Caching GameDec. 9, 2003Byung-Gon Chun, Marco Barreno
ContentsMotivationGame TheoryProblem FormulationTheoretical ResultsSimulation ResultsExtensions
MotivationWide-area file systems, web caches, p2p caches, distributed computation
Game TheoryGamePlayersStrategies S = (S1, S2, , SN)Preference relation of S represented by a payoff function (or a cost function)Nash equilibrium Meets one deviation propertyPure strategy and mixed strategy equilibriumQuantification of the lack of coordinationPrice of anarchy : C(WNE)/C(SO) Optimistic price of anarchy : C(BNE)/C(SO)
Caching Modeln nodes (servers) (N)m objects (M)distance matrix that models a underlying network (D)demand matrix (W)placement cost matrix (P)(uncapacitated)
Selfish CachingN: the set of nodes, M: the set of objectsSi: the set of objects player i places S = (S1, S2, , Sn)Ci: the cost of node i
Separability for uncapacitated versionwe can look at individual object placement separatelyNash equilibria of the game is the crossproduct of nash equilibria of single object caching game.
Selfish Caching (Single Object)Si : 1, when replicating the object 0, otherwiseCost of node i
Socially Optimal CachingOptimization of a mini-sum facility location problemSolution: configuration that minimizes the total cost Integer programming NP-hard
Major QuestionsDoes 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 ?
Major ResultsPure strategy Nash equilibria exist.The price of anarchy can be bad. It is O(n).The distribution of distances is important.Undersupply (freeriding) problemConstrained 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.
Existence of Nash EquilibriumProof (Sketch)
Price of Anarchy Basic Results
Inefficiency of a Nash Equilibriumn/2 nodesn/2 nodes-1
Special Network TopologyFor CG, PoA = 1For star, PoA 2
Special Network TopologyFor line, PoA = O(n1/2)
Simulation MethodologyGame simulations to compute Nash equilibriaInteger programming to compute social optima
Underlying topology transit-stub (1000 physical nodes), power-law (1000 physical nodes), random graph, line, and treeDemand distribution Bernoulli(p)Different placement cost and read-write ratioDifferent number of servers
Metrics PoA, Latency, Number of replicas
Varying Placement Cost(Line topology, n = 10)
Varying Demand Distribution(Transit-stub topology, n = 20)
Different Physical Topology(Power-law topology (Barabasi-Albert model), n = 20)
Varying Read-write Ratio(Transit-stub topology, n = 20)Percentage of writes
Different Physical Topology(Transit-stub topology, n = 20)
Extensions Congestiond = d + (#access) PoA /PaymentAccess modelStore model [Kamalika Chaudhuri/Hoeteck Wee]=> Better price of anarchy from cost sharing?
Ongoing and future workTheoretical analysis underDifferent distance constraintsHeterogeneous placement costCapacitated versionDemand random variables
Large-scale simulations with realistic workload traces
Related WorkNash 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]