Caching Game

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Caching Game

• Dec. 9, 2003
• Byung-Gon Chun, Marco Barreno

2
Contents

• Motivation
• Game Theory
• Problem Formulation
• Theoretical Results
• Simulation Results
• Extensions

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Motivation
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

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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/(D1))
• 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
?-1
n/2 nodes
n/2 nodes
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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
Percentage of writes
(Transit-stub topology, n 20)
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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
• gt 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