algorithms and economics of networks abraham flaxman and vahab mirrokni, microsoft research
Post on 21-Dec-2015
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Outline Network Congestion Games Congestion Games
Rosenthal’s Theorem: Congestion games are potential Games:
PoA for Congestion Games. Market Sharing Games. Network Design Games.
Network Congestion Games
A directed graph G=(V,E) with n users, Each edge e in E(G) has a delay function fe,
Strategy of user i is to choose a path Aj from a source si to a destination ti,
The delay of a path is the sum of delays of edges on the path,
Each user wants to minimize his own delay by choosing the best path.
Congestion Games
n players, a set of facilities E, Strategy of player i is to choose a subset of
facilities (from a given family of subsets Ti), Facility i have a cost (delay) function fe
which depends on the number of players playing this facility,
Each player minimizes its total cost,
Example: Congestion Games Picture from
Kapelushnik Lior
6,5,4,22 ffff
f1 f2 f3 f4 F5 F6
4,3,2,3,11 fffff
4,13 ff
Congestion Games: Pure NE
Rosenthal’s Theorem (1979): Any congestion game is an exact potential game.
Proof is based on the following Potential Function
Ee
An
te
e
tf1
Classes of Congestion Games
Every network congestion game is a congestion game
Symmetric and Asymmetric Players Network Design Games. Maximizing Congestion Games: Each player
wants to maximize his payoff (instead of minimizing his delay) Market Sharing Games.
Generalizations: Weighted Congestion Games Player-specific Congestion Games
Congestion Games: Social Cost Two social Cost functions: Consider a pure Strategy A = (A1, A2,
…, An).
Defintion 1:
AcAMax iNimax
Ni
i AcASumDefintion 2:
Congestion Games: PoA PoA for two social Cost functions: Defintion 1:
Defintion 2:
Ni
i AcASum
We Prove bounds for
opt
AMaxNEaisAmax
opt
ASumNEaisAmax
Congestion Games: PoA
PoA for congestion game with linear delay functions is at most 5/2.
Proof: Lemma 1: for a pair of nonnegative
integers a,b:
Proof: …
22
3
5
3
11 baab
Congestion Games: PoA for mixed NE Theorem: PoA for mixed Nash equilibria in
congestion games with linear latency function is 2.61.
Theorem: PoA for mixed Nash equilibria of weighted congestion games with linear latency function is 2.61.
Theorem: PoA for polynomial delay functions of constant degree is a constant.
Other Variants
Atomic Congestion Games: Many infinitesimal users. The load of each user is very small.
Theorem: PoA for non-atomic congestion games with linear latency functions is 4/3.
Splittable Network Congestion Games
Market Sharing Games
Congestion Game Facilities are Markets. Cost function Profit Function. Players share the profit of markets (equally). Each player has some packing constraint for
the set of markets he can satisfy.
PoA: 1/2.
Network Design Games Players want to construct a network. They share the cost of buying links in the
network.
Known Results: Price of Stability, Convergence…