CSE 522 – Algorithmic and Economic Aspects of the Internet

Instructors: Nicole Immorlica

Mohammad Mahdian

Previously in this class

Properties of social networks

Generative models for power law distribution and power law graphs

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

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: 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? Rosenblat and Mobius, QJE 2004:

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? Model:

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:

As C decreases, individual separation (diameter) decreases, but group separation increases.

Experiments on co-authorship among economists (69-99)

Network Formation Games

Models that use formal game theoretic reasoning to study network formation 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

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

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

Players {1,…,n} are nodes in the network

Each player i must simultaneously choose some subset of {1,…,n} as his strategy si

A strategy profile defines a (directed) graph G Nodes are players Edge (i,j) is in G if j 2 si

Example: Graph

Players = {1, 2, 3, 4}

1

4 3

2s1 = {4} s2 = {3,4}

s3 = {4}s2 = {3}

Game Theory Framework

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

E.g., i is number of nodes that i can reach via a directed path in G minus the number of links i forms

1

4 3

21 = 2 – 1 = 1 2 = 2 – 2 = 0

3 = 1 – 1 = 03 = 1 – 1 = 0

Equilibrium Networks

When do we expect a graph to be stable?

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

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

1

4 3

21 = 2 – 1 = 1’1 = 3 – 1 = 2

2 = 2 – 2 = 0

3 = 1 – 1 = 03 = 1 – 1 = 0

Strict Equilibria

What if there is another strategy for a player which does not change his payoff?

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

Any unilateral deviation by a node strictly decreases his payoff

1

4 3

21 = 3 – 1 = 2 2 = 3 – 1 = 2

3 = 3 – 1 = 23 = 3 – 1 = 2

Models: Bala and Goyal

Two models (Bala and Goyal, Econometrica 2000)

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

For any payoff function 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 Implemented game with 4 players

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

Treatment Strict NEFreq. of

NEFreq. of Strict

NE

C=5, 1-way Circle 48% 41%

C=15, 1-way Circle, ; 52% 52% (all circ.)

C=25, 1-way Circle, ; 59% 59% (83% circ)

C=5, 2-way Star 31% 0%

C=15, 2-way ; 9% 0%

Explanations: Symmetry of strategies/coordination issue Inequity aversion (people prefer equal payoffs) Concern for efficiency (empty graph gives no payoffs)

Dynamics in Bala and Goyal

Does not imply that equilibria are unique! For example, there are n possible stars. Can players find an equilibria?

Consider following best-response dynamic 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

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

Bala and Goyal 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) 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.

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.

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

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 most-efficient 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

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 Jackson and Wolinsky

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 utility-transfer rule.