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
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The internet has over 20,000
Autonomous Systems (AS)
Every AS picks their own peers to
speed-up routing or minimize cost
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Question:
What is the performance penalty in
terms of the poor network structure
resulting from selfish users creating
the network, without centralized
control?
Goal of the paper
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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
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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:
Distance from i to j
Pay $a for each
link you buy
Pay $1 for every
hop to every node
(a may depend on n)
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Example
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2
-1 3
-3
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1
+a
2
1
a
c(i)=a+13
c(i)=2a+9
(Convention: arrow from the node buying the link)
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Definitions
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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: C(G)   ci  a E   dG (i, j )
i
i, j
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Definitions: Nash Equilibria
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Nash equilibrium: a situation such that no
single player can unilaterally modify its
strategy and lower its cost
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L
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:
C (i, S )  a Sv   dG (i, j )
j
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Definitions: Nash Equilibria
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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)
C (i, S )  C (i,U )
G(S) is the equilibrium graph.
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Example ?
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Set a=5, and consider:
+1
-2
-1
-5
-1
+2
-1
+5
+5
+5
-5
+4
+1
-1
-5
-5
+1
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Definitions: Price of Anarchy
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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 a < 2
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When a < 2, the social optima
is a clique. Any missing edge
can be added adding a to the
social cost and subtracting at
least 2 from social cost.
A clique
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Nash Equilibrium for 1<a<2
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When 1<a<2, the worst-case
equilibrium configuration is a
star. The total cost here is
(n-1)a + 2n(n-1) - 2
In 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 a > 2
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When a > 2, the social
optima is a star. Any
extra edges are too
expensive.
C (star)  (n  1)(a  2(n  1))
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Complexity issues
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Theorem. Computing the best response of
a given peer is NP-hard.
Proof hint. When 1 < a < 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 a (<2)
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For a<1, the clique is the only N.E.
For 1<a<2, clique no longer N.E., but
the diameter is at most 2
-2
+a
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Then, the star is the worst N.E., can be
seen to yield P.o.A. of at most 4/3
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P.O.A for very small a (<2)
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The star is also a Nash
equilibrium, but there may be
worse Nash equilibrium.
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P.O.A for very small a (<2)
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Proof.
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The case of a > n2
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The Nash equilibrium is a tree, and the
price of anarchy is 1.
Why?
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