[PPT] - Congestion Games and Selfish Routing Maria Serna Fall 2016 PowerPoint Presentation

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion Games and Selfish Routing

Maria Serna Fall 2016

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

1 Congestion games and variants 2 Selfish Routing 3 Price of Anarchy/Stability

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games

A congestion game (E, N, (de)e∈E, (ci)i∈N) is defined on a finite set E of resources and has n players using a delay function de mapping N to the integers, for each resource e. The actions for each player are subsets of E. The cost functions are the following: ci(a1, . . . , an) =

e∈ai

de(fe(a1, . . . , an)) being fe(a1, . . . , an) = |{i | e ∈ ai}|.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Weighted congestion games

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Weighted congestion games

A weighted congestion game (E, N, (de)e∈E, (ci)i∈N, (wi)i∈N) is defined on a finite set E of resources and has n players. Player i has an associated natural weight wi. Using a delay function de mapping N to the integers, for each resource e. The actions for each player are subsets of E. The cost functions are the following: ci(a1, . . . , an) =

e∈ai

de(fe(a1, . . . , an)) being fe(a1, . . . , an) =

{i|e∈ai} wi.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Network weighted congestion games

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Network weighted congestion games

A network weighted congestion game is defined on a directed graph G = (V .E), (N, G, (de)e∈E, (ci)i∈N, (wi)i∈N, (si)i∈N, (ti)i∈N). The resources are the arcs in G. The game has n players. Player i has an associated natural weight wi. Using a delay function de mapping N to the integers, for each arc e ∈ E. The action set for player i is the set of (si, ti)-paths in G. The cost functions are the following: ci(a1, . . . , an) =

e∈ai

de(fe(a1, . . . , an)) being fe(a1, . . . , an) =

{i|e∈ai} wi.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted)

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted): wi = 1.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted): wi = 1. symmetric (vs. non-symmetric) strategies:

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted): wi = 1. symmetric (vs. non-symmetric) strategies: all the players have the same set of actions.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted): wi = 1. symmetric (vs. non-symmetric) strategies: all the players have the same set of actions. symmetric congestion games:

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted): wi = 1. symmetric (vs. non-symmetric) strategies: all the players have the same set of actions. symmetric congestion games: unweighted with symmetric strategies.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted): wi = 1. symmetric (vs. non-symmetric) strategies: all the players have the same set of actions. symmetric congestion games: unweighted with symmetric strategies. singleton congestion games:

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted): wi = 1. symmetric (vs. non-symmetric) strategies: all the players have the same set of actions. symmetric congestion games: unweighted with symmetric strategies. singleton congestion games: all possible actions have only one resource.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted): wi = 1. symmetric (vs. non-symmetric) strategies: all the players have the same set of actions. symmetric congestion games: unweighted with symmetric strategies. singleton congestion games: all possible actions have only one resource. nonatomic network congestion games (vs. atomic)

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted): wi = 1. symmetric (vs. non-symmetric) strategies: all the players have the same set of actions. symmetric congestion games: unweighted with symmetric strategies. singleton congestion games: all possible actions have only one resource. nonatomic network congestion games (vs. atomic) In nonatomic congestion games the number of players is infinite and each player controls an infinitesimal weight of the total traffic.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Congestion games terminology

unweighted (vs. weighted): wi = 1. symmetric (vs. non-symmetric) strategies: all the players have the same set of actions. symmetric congestion games: unweighted with symmetric strategies. singleton congestion games: all possible actions have only one resource. nonatomic network congestion games (vs. atomic) In nonatomic congestion games the number of players is infinite and each player controls an infinitesimal weight of the total traffic. Named also Selfish routing games.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

An example of a network congestion game

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

An example of a network congestion game

There are three players. and a network (with a delay function on arcs)

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

An example of a network congestion game

There are three players. and a network (with a delay function on arcs) 1/2/4 4/5/9 2/3/7 0/2/9 A U R B

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

An example of a network congestion game

There are three players. and a network (with a delay function on arcs) 1/2/4 4/5/9 2/3/7 0/2/9 A U R B Player’s objective: going from s = A to t = B as fast as possible.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

An example of a network congestion game

There are three players. and a network (with a delay function on arcs) 1/2/4 4/5/9 2/3/7 0/2/9 A U R B Player’s objective: going from s = A to t = B as fast as possible. Strategy profiles: paths from A to B. A NE?

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

An example of a weighted network congestion game

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

An example of a weighted network congestion game

There are three players with weights 1,1,2

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

An example of a weighted network congestion game

There are three players with weights 1,1,2 and a network (with a delay function on arcs)

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

An example of a weighted network congestion game

There are three players with weights 1,1,2 and a network (with a delay function on arcs) 1/2/4/5 4/5/9/9 2/3/7/8 0/2/9/10 A U R B

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

An example of a weighted network congestion game

There are three players with weights 1,1,2 and a network (with a delay function on arcs) 1/2/4/5 4/5/9/9 2/3/7/8 0/2/9/10 A U R B Player’s objective: send wi units from s = A to t = B as fast as possible. Strategy profiles: paths from A to B. A NE?

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Another family: Fair Cost Sharing Games

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Another family: Fair Cost Sharing Games

A fair cost sharing game (E, N, (ce)e∈E) is defined on a finite set E of resources and has n players a fixed cost ce, for each resource e. The actions for each player are subsets of E. The cost functions are the following: ci(a1, . . . , an) =

e∈ai

ce fe(a1, . . . , an) being fe(a1, . . . , an) = |{i | e ∈ ai}|.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

1 Congestion games and variants 2 Selfish Routing 3 Price of Anarchy/Stability

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish Routing: an Example

Total traffic is r = 1. Network (with delay functions on arcs)

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish Routing: an Example

Total traffic is r = 1. Network (with delay functions on arcs) x 1 1 x A U R B

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish Routing: an Example

Total traffic is r = 1. Network (with delay functions on arcs) x 1 1 x A U R B Player’s objective going from s = A to s = B with minimum delay.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish Routing: an Example

Total traffic is r = 1. Network (with delay functions on arcs) x 1 1 x A U R B Player’s objective going from s = A to s = B with minimum delay. Strategy profiles: flows from A to B with total flow r = 1

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: strategy profiles

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: strategy profiles

Traffic as Flows:

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: strategy profiles

Traffic as Flows:

A flow vector f giving the routing of traffic. fP = amount of traffic routed on (si − ti) path P.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: strategy profiles

Traffic as Flows:

A flow vector f giving the routing of traffic. fP = amount of traffic routed on (si − ti) path P.

x f 1 1 − f 1 x 1 A U R B

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: strategy profiles

Traffic as Flows:

A flow vector f giving the routing of traffic. fP = amount of traffic routed on (si − ti) path P.

x f 1 1 − f 1 x 1 A U R B Notation: for a path P and a feasible flow f , C P(f ) denotes the cost corresponding to the traffic routed through P by f .

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria

A flow is a Nash equilibrium (or is a Nash flow) if all flow is routed on min-latency paths (given current edge congestion)

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria

A flow is a Nash equilibrium (or is a Nash flow) if all flow is routed on min-latency paths (given current edge congestion) x f 1 1 − f 1 f x 1 − f A U R B

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria

A flow is a Nash equilibrium (or is a Nash flow) if all flow is routed on min-latency paths (given current edge congestion) x f 1 1 − f 1 f x 1 − f A U R B This flow is not a Nash flow unless f = 1/2.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria

Theorem A feasible flow x is an equilibrium flow iff for any feasible flow y

e∈E

de(x[e])x[e]

e∈E

de(y[e])x[e].

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria

Theorem A feasible flow x is an equilibrium flow iff for any feasible flow y

e∈E

de(x[e])x[e]

e∈E

de(y[e])x[e]. Called Variational Inequality (Smith 79 and Dafermos 80)

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria

Theorem A feasible flow x is an equilibrium flow iff for any feasible flow y

e∈E

de(x[e])x[e]

e∈E

de(y[e])x[e]. Called Variational Inequality (Smith 79 and Dafermos 80) As a consequence all Nash flows have the same cost per edge.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria

Theorem A feasible flow x is an equilibrium flow iff for any feasible flow y

e∈E

de(x[e])x[e]

e∈E

de(y[e])x[e]. Called Variational Inequality (Smith 79 and Dafermos 80) As a consequence all Nash flows have the same cost per edge. Do PNE exist?

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria existence

As for the atomic case we can consider a potential, for a given flow x Ψ(x) =

e∈E

x[e] de(u)du

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria existence

As for the atomic case we can consider a potential, for a given flow x Ψ(x) =

e∈E

x[e] de(u)du Theorem A feasible flow x is an equilibrium flow iff x is a minimum of Ψ

ver the set of feasible flows.

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Another example

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Another example

x 1 1 x A U R B

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Another example

A feasible flow: x f 1 1 − f 1 x 1 f A U R B

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Another example

A feasible flow: x f 1 1 − f 1 x 1 f A U R B Is this a NE?

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Another example

A feasible flow: x f 1 1 − f 1 x 1 f A U R B Is this a NE? No unless f = 1

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria

A Nash flow

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Selfish routing: equilibria

A Nash flow x 1 1 1 x 1 1 A U R B

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Braess’ paradox: Nash flows

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Braess’ paradox: Nash flows

x .5 1 .5 1 .5 x .5 A U R B

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Braess’ paradox: Nash flows

x .5 1 .5 1 .5 x .5 A U R B Delay is 1.5. We add a fast connection

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Braess’ paradox: Nash flows

x .5 1 .5 1 .5 x .5 A U R B Delay is 1.5. We add a fast connection x 1 1 1 x 1 1 A U R B

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

Braess’ paradox: Nash flows

x .5 1 .5 1 .5 x .5 A U R B Delay is 1.5. We add a fast connection x 1 1 1 x 1 1 A U R B and delay is 2!

AGT-MIRI, FIB-UPC Congestion Games

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Contents Congestion games and variants Selfish Routing Price of Anarchy/Stability

1 Congestion games and variants 2 Selfish Routing 3 Price of Anarchy/Stability

AGT-MIRI, FIB-UPC Congestion Games