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

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

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Caching Game Dec. 9, 2003 Byung-Gon Chun, Marco Barreno 1 Contents Motivation Game Theory Problem Formulation Theoretical Results Simulation Results Extensions 2 Motivation Server Server Server Server

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Caching Game

  • Dec. 9, 2003

Byung-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

s c

n i i

∑

=

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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 α-1 C(WNE) = α + (α-1)n/2 C(SO) = 2α PoA =

α α α 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 + β (#access) → PoA ≤ α/β

  • 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]