Routing Games

By Javad Ghareh Chamani

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Outlines

•Introduction

•Some Definitions

•Non-atomic Selfish Routing

•Atomic Selfish Routing

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Introduction2 Routing models

◦Nonatomic selfish routing

Reasons of explain◦Simple but general◦Understandability of POA◦Different techniques needed for

analyze

◦Atomic selfish routing

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Some DefinitionsPigou’s example

◦ Selfish players◦ Each player wants to minimize cost

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Multicommodity Flow Network

Directed graph G = (V, E), may have parallel edgesCommodities:

set of source/sink pairs: (s1, t1),…,(sk , tk )Assign each player to a\some commoditynonnegative, continuous, non-decreasing cost

function ce

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Multicommodity Flow Network (cnt)

Paths

Routes◦ Traffic described with flow f

◦ fP: amount of traffic of commodity i that chooses path to travel from si to ti

◦ Prescribed amount of traffic for commodity i : ri

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Feasibility of FlowA flow f is feasible for a vector r if it

routes all of the traffic

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Cost DefinitionCost of a path P with respect to a flow

f

fe amount of traffic using paths that contain the edge e.

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Nonatomic EquilibriumA flow (f) is equilibrium if:

no commodity can increase its gain by changing its traffic

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Braess’s Paradox

One commoditywants to route 1

unit of trafficfrom s to t

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Braess’s Paradox(cont)

New equilibrium

Old cost: 1.5New cost: 2

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Existence & Uniqueness of Equilibrium

There exists at least one equilibrium

All equilibriums are equivalent (of equal cost)

Using Potential Function

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Potential FunctionsLet ce(x) be the cost of transporting x

over some edge eThe total amount spent on that edge

is x.ce(x)

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Marginal Cost and Optimality

Follows from convex optimization problem nonnegativity constraints

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Marginal Cost and Optimality(cnt)

How to minimize the total marginal cost?

Create a new network with the marginal cost as cost function, and find an equilibrium.

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Back to Finding EquilibriumRemaining questions?

◦ Can we find equilibrium by finding optimal flows?

◦ What function should we minimize?

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Potential FunctionWe want to minimize the functions

he(x) of all edges. The sum of those is called the potential function and should be minimized.

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ConsequencesThe potential function is convex:

◦ there exists a minimum, and therefore an equilibrium

◦ and all equilibriums form a set◦ with the same cost

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Atomic Selfish RoutingAlmost the same as the nonatomic

instance.Differences:

◦ Each commodity represents ONE player instead of a large number of players.

◦ Player i must route traffic amount ri on a

SINGLE path.Result: Each player must route a

significant amount of traffic instead of a negligible amount.

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Feasibility of a flowA flow f is feasible if:

◦it routes all traffic◦it uses one path per player i to route

ri units

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Atomic equilibrium flow

Also note:Different equilibrium flows can have different

costs

no uniqueness

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Nonexistence of equilibria

Traffic amount: r1 = 1; r2 = 2 P1 s -> t P2 s -> v -> t P3 s -> w -> t P4 s -> v -> w -> t

1- If player 2 takes P1 or P2, player 1 takes P4.

2- If player 1 takes P4, player 2 takes P3. 3- If player 2 takes P3 or P4, player 1

takes P1. 4- If player 1 takes P1, player 2 takes P2. This does never end. . .

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Difference of Atomic and NonDifferent EQ of an atomic

instance can have different costsNonatomic EQ have same costPOA in atomic instance can be

larger than nonatomic

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Overcome nonexistence of equilibrium

To guarantee an equilibrium, additional restrictions are placed on atomic instances. For instance:

Give all players an equal amount of traffic which has to be routed.

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Overcome nonexistence of equilibrium (cnt)

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ProofIn atomic instances we can

discretize the potential function that was used for nonatomic instances:

#Possible flows is finite. One flow f is a global minimum of

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Proof (cnt)Claim: flow f is an equilibrium

flow for instance (G,r,c).Suppose: Player i could strictly decrease cost

in equilibrium flow f by deviating from path P to P’ , yielding the new flow f’ :