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Caching Game

Dec. 9, 2003Byung-Gon Chun, Marco

Barreno

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Contents

• Motivation

• Game Theory

• Problem Formulation

• Theoretical Results

• Simulation Results

• Extensions

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Motivation

Server

Server

Server

Server

Wide-area file systems, web caches, p2p caches, distributed computation

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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)

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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)

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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

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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.

•

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Selfish Caching (Single Object)

• Si : 1, when replicating the object

0, otherwise

• Cost of node i

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Socially Optimal Caching

• Optimization of a mini-sum facility location problem

• Solution: configuration that minimizes the total cost

• Integer programming – NP-hard

)(1

scn

ii

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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 ?

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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.

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Existence of Nash Equilibrium

• Proof (Sketch)

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Price of Anarchy – Basic Results

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Inefficiency of a Nash Equilibrium

n/2 nodes n/2 nodes

-1

C(WNE) = + (-1)n/2

C(SO) = 2

PoA =

2

2/)1( n

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Special Network Topology

• For CG, PoA = 1

• For star, PoA 2

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Special Network Topology

• For line, PoA = O(n1/2)

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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

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Varying Placement Cost

(Line topology, n = 10)

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Varying Demand Distribution

(Transit-stub topology, n = 20)

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Different Physical Topology

(Power-law topology (Barabasi-Albert model), n = 20)

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Varying Read-write Ratio

(Transit-stub topology, n = 20)

Percentage of writes

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Questions?

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Different Physical Topology

(Transit-stub topology, n = 20)

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Extensions

• Congestion– d’ = d + (#access) PoA /

• Payment– Access model– Store model

[Kamalika Chaudhuri/Hoeteck Wee]

=> Better price of anarchy from cost sharing?

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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

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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]