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Network Creation Game

A. Fabrikant, A. Luthra, E. Maneva, C. H. Papadimitriou, and S. Shenker,

PODC 2003

(Part of the Slides are taken from Alex Fabrikant’s presentation)

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Context The internet has over 20,000 Autonomous Systems (AS)

Every AS picks their own peers to speed-up routing or minimize cost

Question:

What is the performance penalty in terms of the poor network structure resulting from selfish users creating

the network, without centralized control?

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Goal of the paper

Introduces a simple model of network creation by selfish agents

Briefly reviews game-theoretic concepts Computes the price of anarchy for different cost functions

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A Simple Model for constructing G

N agents, each represented by a vertex and can buy (undirected) links to a set of others (si)

One agent buys a link, but anyone can use it

Cost to agent:

Pay $ for each link you buy

Pay $1 for every hop to every node( may depend on n)

Distance from i to j

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Example

(Convention: arrow from the node buying the link)

+

1

1

2

2

3

4

-1

-3 c(i)=+13c(i)=2+9

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Definitions V={1..n} set of players A strategy for v is a set of vertices Sv V\{v}, such that v creates an edge to every w Sv. G(S)=(V,E) is the resulting graph given a combination of strategies S=(S1,..,Sn), V set of players / nodes and E the laid edges.

Social optimum: A central administrator’s approach to combining strategies and minimizing the the total cost (social cost) It may not be liked by every node.

Social cost:

),()(,

jidEcGCji

Gi

i ∑∑ +==

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Definitions: Nash Equilibria

Nash equilibrium: a situation such that no single player can unilaterally modify its strategy and lower its cost Presumes complete rationality and knowledge on behalf of each agent

Nash Equilibrium is not guaranteed to exist, but they do for our model

The private cost of player i under s:

),(),( jidSSiCj

Gv ∑+=

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Definitions: Nash Equilibria

A combination of strategies S forms Nash equilibrium, if for any player i and every other strategy U (such that U differs from S only in i’s component)

G(S) is the equilibrium graph.

),(),( UiCSiC ≤

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+5

-2

Example Set =5, and consider:

-5

+2

+4

-1

-5

+5-1

+1

+5

+1

+1

-1

-1

-5

-5

?

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Definitions: Price of Anarchy

Price of Anarchy (Koutsoupias & Papadimitriou, 1999): the ratio between the worst-case social cost of a Nash equilibrium network and the optimum social cost over all Nash equilibria.

We bound the worst-case price of anarchy to limit “the price we pay” for operating without centralized control

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Social optima for < 2

When < 2, the social optima is a clique. Any missing edge can be added adding to the social cost and subtracting at least 2 from social cost.

A clique

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Nash Equilibrium for 1<<2

When 1<<2, the worst-case equilibrium configuration is a star. The total cost here is

(n-1) + 2n(n-1) - 2In a Nash Equilibrium, no single node can unilaterally add or delete an edge to bring down its cost.

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Social optima for > 2

When > 2, the social optima is a star. Any extra edges are too expensive.

))1(2)(1()( −+−= nnstarC

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Complexity issues

Theorem. Computing the best response of a given peer is NP-hard.

Proof hint. When 1 < < 2, for a given node k, if there are no incoming edges, then the problem can be reduced to the Dominating Set problem.

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Equilibria: very small (<2) For <1, the clique is the only N.E.

For 1<<2, clique no longer N.E., but the diameter is at most 2

Then, the star is the worst N.E., can be seen to yield P.o.A. of at most 4/3

+

-2

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P.O.A for very small (<2)

The star is also a Nash equilibrium, but there may be worse Nash equilibrium.

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P.O.A for very small (<2)

Proof.

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The case of > n2

The Nash equilibrium is a tree, and the price of anarchy is 1.

Why?