Caching Game
Dec. 9, 2003
Byung-Gon Chun, Marco Barreno
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Contents
•
•
•
•
•
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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
n
i 1
ci ( s )
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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
-1
n/2 nodes
n/2 nodes
C(WNE) =  + (-1)n/2
C(SO) = 2
PoA =   (  1)n / 2
2
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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) 20
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]
=> 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]
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