CSE 522 – Algorithmic and
Economic Aspects of the
Internet
Instructors:
Nicole Immorlica
Mohammad Mahdian
Previously in this class
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Properties of social networks
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Generative models for power law distribution
and power law graphs
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Generative models for small-world networks
This Lecture
Final remarks on small-world networks
Network formation games, and a short
introduction to game theory
Geographic Routing
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Experiments suggest that the first criterion
that people use for forwarding a message is
geographic proximity.
Kleinberg: In a 2-d grid with long-range
contact probability proportional to dist –2,
“geographic routing” works.
However, experiments show that this
probability is closer to dist –1.
Geographic Routing, cont’d

Liben-Nowell et al., PNAS 2005:
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Justification: in Kleinberg’s model, population is
distributed uniformly on a 2-d grid, but in the real
world the distribution is not uniform.
Model: probability of a long-range contact from u
to v proportional to the inverse of the # of people
that are closer to u than v.
Result: In this model, geographic routing works.
Experiments on ~500,000 blogs on LiveJournal
confirms the assumption of the model.
Getting Closer or Drifting Apart?
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Rosenblat and Mobius, QJE 2004:
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Technology has made it less costly to interact with
people across the globe (‘global village’).
As a result, people become more selective in
whom to interact/collaborate with.
Could this fragment the social network into
clusters of like-minded people?
Prominent example: scientific community
Getting Closer or Drifting Apart?
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Model:
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Agents of types A and B are arranged uniformly around a
circle.
Each person collaborates with a fixed # of other people,
and receives a payoff from each collaboration.
The payoff for collaborating with someone of the same type
is higher.
Collaborating with someone who is not close has a cost C.
Results:
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As C decreases, individual separation (diameter)
decreases, but group separation increases.
Experiments on co-authorship among economists (69-99)
Network Formation Games
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Models that use formal game theoretic
reasoning to study network formation
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Individuals in a network face economic incentives
to form or break links with other individuals
Individuals make self-motivated decisions about
which links to form
Applications: professional network, Internet
Incentives in Networks
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Each individual is a source of benefits
(information, resources)
Others can share the benefits of an individual
via formation of links
Link formation is costly (time, effort, money)
Given these incentives, which links will form?
Game Theory Framework
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Set of players (agents)
Each player selects a strategy from the set of
allowed strategies.
A payoff function specifies how much each
player receives given the strategy profile.
An equilibrium is a strategy profile in which
no player can benefit by unilaterally changing
his strategy.
Network Formation Games
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Players {1,…,n} are nodes in the network
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Each player i must simultaneously choose
some subset of {1,…,n} as his strategy si
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A strategy profile defines a (directed) graph G
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Nodes are players
Edge (i,j) is in G if j 2 si
Example: Graph
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Players = {1, 2, 3, 4}
s1 = {4}
s2 = {3}
1
2
4
3
s2 = {3,4}
s3 = {4}
Game Theory Framework
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Let Ni = |si| be number of links i forms
Let Ci be “connectedness” of i (definition
varies depending on model)
Given a strategy profile (i.e., graph G), the
payoff for a player i is a function i(Ni,Ci)
decreasing in Ni and increasing in Ci
Players seek to maximize their payoff
Example: Payoffs
1 = 2 – 1 = 1
3 = 1 – 1 = 0
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2 = 2 – 2 = 0
1
2
4
3
3 = 1 – 1 = 0
E.g., i is number of nodes that i can reach
via a directed path in G minus the number of
links i forms
Equilibrium Networks
When do we expect a graph to be stable?
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A graph G is a Nash equilibrium if no player
has an incentive to unilaterally sever or
create links, i.e. for any other strategy s’i of i,
his payoff ’i in the resulting graph G’ is at
most his payoff i in G
Example: Equilibrium Networks
1 = 2 – 1 = 1
’1 = 3 – 1 = 2
3 = 1 – 1 = 0
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2 = 2 – 2 = 0
1
2
4
3
3 = 1 – 1 = 0
Node 1 has an incentive to sever connection
to 4 and instead form a connection to 2 for a
resulting payoff of ’1 = 3 – 1 = 2
Strict Equilibria
What if there is another strategy for a player
which does not change his payoff?
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A graph G is a strict Nash equilibrium if each
player’s strategy is his unique best-response,
i.e. for any other strategy s’i of i, his payoff ’i
in the resulting graph G’ is strictly less than
his payoff i in G
Example: Strict Equilibria
1 = 3 – 1 = 2
3 = 3 – 1 = 2

2 = 3 – 1 = 2
1
2
4
3
3 = 3 – 1 = 2
Any unilateral deviation by a node strictly
decreases his payoff
Models: Bala and Goyal
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Two models (Bala and Goyal, Econometrica 2000)
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One-way flow: A link can be used only by the
person who formed it to send information
Two-way flow: A link between two people can be
used by either person
Model is frictionless if value of information does
not decay with distance: Ci is number of nodes i
can reach in G by a path of any length
Equilibria in Bala and Goyal
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For any payoff function
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In both models, every Nash equilibrium is either
connected or empty
In the one-way flow model, the only strict Nash
equilibria are the directed cycle and/or the empty
network
In the two-way flow model, the only strict Nash
equilibria are the center-sponsored star (one node
connects to all others) and/or the empty network
Experimentation: Falk and Kosfeld
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Implemented game with 4 players
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Players were offered 10 points (worth 65 cents
each) for each player they had a direct or indirect
connection to (including themselves)
Players were charged C points for each link they
formed
There were five treatments: C = 5, 15, and 25 in
one-way model and C = 5 and 15 in two-way
model
Predictions vs Results
Freq. of Strict
NE
Treatment
Strict NE
C=5, 1-way
Circle
Circle, ;
48%
41%
52%
52% (all circ.)
59%
59% (83% circ)
Star
;
31%
0%
9%
0%
C=15, 1-way
C=25, 1-way
C=5, 2-way
C=15, 2-way
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Freq. of
NE
Circle, ;
Explanations:
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Symmetry of strategies/coordination issue
Inequity aversion (people prefer equal payoffs)
Concern for efficiency (empty graph gives no payoffs)
Dynamics in Bala and Goyal
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Does not imply that equilibria are unique! For
example, there are n possible stars. Can
players find an equilibria?
Consider following best-response dynamic
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Start from an arbitrary initial graph
In each period, each player independently decides
to “move” with probability p
If a player decides to move, he picks a new
strategy randomly from his set of best responses
to graph in previous period
Dynamics in Bala and Goyal
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Theorem: In either model, the dynamic
process converges to a strict Nash
equilibrium network with probability one.
Simulations show that rate of convergence is
quite rapid.
Accounting for Distances
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Bala and Goyal
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Value of information decays by a factor of  for
each link traversed (model with “friction”)
Results similar to frictionless models still hold
Fabrikant et al. (PODC 2003)
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Value of connection to j for a node i is -d(i,j) (and
payoff function is linear)
Nash equilibria become slightly more complex
(e.g., trees are Nash equilibria in some cases)
Model: Fabrikant et al.
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The payoff incurred by player i is
i = -  Ni – jd(i,j)
where Ni is the number of links formed by i
and d(i,j) is the distance between i and j in
the underlying undirected network (two-way
flow model).
Equilibria: Fabrikant et al.
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For  < 1, complete graph is only Nash
equilibrium
For  > n2, all Nash equilibria are trees
Conjecture: For  > some constant, all strict
Nash equilibria are trees.
Upcoming paper in SODA 2006 disproves this.
Efficiency of the Equilibria
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The social welfare or efficiency of a strategy
profile in a game is defined as the sum of
payoffs of all players
The price of anarchy of a game is the ratio of
least-efficient Nash equilibrium to the mostefficient strategy profile (which need not be
an equilibrium)
Theorem [Fabrikant et al.]: For any tree Nash
equilibrium T, the welfare of T to the optimum
is at most 5.
Other Network Formation Games
What if agents cooperate to form links?
Cooperative Game Theory
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Players cooperate to achieve a common goal
(e.g., building a network).
Achieving goal has a value for each agent.
In a transferable utility game, agents must
additionally decide how to share this value
among each other.
As in non-cooperative game theory, analyze
stable situations, but now must consider
coalitions as well as individuals.
Cooperative Network Formation
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Jackson and Wolinsky
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Studied network formation as a cooperative game
with transferable utilities, in particular individuals
can share cost of links.
Showed there are natural situations in which no
efficient network is pairwise stable for any utilitytransfer rule.