caching game
Post on 31-Dec-2015
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2
Contents
• Motivation
• Game Theory
• Problem Formulation
• Theoretical Results
• Simulation Results
• Extensions
3
Motivation
Server
Server
Server
Server
Wide-area file systems, web caches, p2p caches, distributed computation
4
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)
5
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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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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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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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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