1. Computer Science, Economics, and the Effects of Network Structure
Michael Kearns
Computer and Information Science
University of Pennsylvania
Nemmers Prize Conference
in honor of Ariel Rubenstein
May 7, 2005

2. An Economic System… …on a network. Decentralized
Heterogenous preferences
Competition
Cooperation
Free riding
Tragedies of the Commons
Adaptation
…on a network.

3. Economic Thought in Computer Science
The Economics of Spam
a (nearly) free resource
Unlimited usage
Favorable ROI for spammers
even under vanishing uptake
Technology solutions:
spam filters
blacklists & whitelists
Economic policy:
micropayment schemes
Selfish Routing
Internet: a shared resource
Competing traffic flows
Societal goal: max throughput
“Socialist” solution:
centralized route assignment
not realistic
“Capitalist” solution:
let competition flourish!
Formalize as game theory
rational player: minimize latency
very large model
Price of Anarchy:
< 4/3 with linear latencies
Policy: tax congested links

4. Network Models of Strategic and Economic Interaction
Network dictates restrictions on direct (local) interactions:
players in a game
trading partners & embargoes
exchange of information
etc.
Propagation of local interactions  global outcome
traditionally an equilibrium outcome
Long history in economics and game theory
Matt Jackson’s talk

5. The Computer Science Perspective
Computational:
How can we manipulate such network models algorithmically?
A rich computational theory (efficient algorithms & intractability)
Generally interested in large populations
Structural: What are the relationships between:
Structural (topological) properties of the network
Properties of the outcome:
cooperation and correlation
“social value” of the equilibria
price variation and wealth distribution
stability (e.g. ESS)
etc.
What kinds of structural properties?
Generality:
For all or large classes of games
What is implied by network structure alone?

6. Correlated Equilibria and Network Structure
Economic Inequality and Expander Graphs
Evolutionary Stable Strategies and Edge Density
Networks and the Behavioral Price of Anarchy

7. Correlated Equilibria and Network Structure

8. Network Models for Game Theory
Alternative to normal form, which grows exp(n) for n players
assume that action space (pure strategies) is “small”
Undirected graph G capturing local (strategic) interactions
Each player represented by a vertex
N_i(G) : neighbors of i in G (includes i)
Assume: Payoffs expressible as M_i(a) where a over only N_i(G)
Graphical game: (G,{M_i})
“qualitative and “quantitative” components
Compact representation of game
Exponential in max degree (<< # of players)
Must still look for special structure for efficient computations
NashProp algorithm effective for relatively sparse graphs 8 7 3 2 1 5 4 6

9. Advantages of CE Correlated actions a fact of the real world
allows “cooperation via correlation”
modeling of shared exogenous influences
Enlarged solution space: all mixtures of NE, and more
new (non-Nash) outcomes emerge, often natural ones
Natural convergence notion for “greedy” learning
But how do we represent an arbitrary CE?
first, only seek to find CE up to (expected) payoff equivalence
second, look to network models for probabilistic reasoning!

10. Graphical Games and Markov Networks
Let G be the graph of such a network game
arbitrary local payoff functions
Consider the Markov network MN(G):
consider (maximal) cliques of G
introduce potential function f_c on each clique c
joint distribution P(a) = (1/Z) P_c f_c(a)
so G defines a family of joint distributions on actions
Theorem: For any game with graph G, and any CE of this game, there is a CE with the same payoffs that can be represented in MN(G)
thus only need to correlate local collections of players
even though full joint may have long-distance correlations
depends only on G, not payoffs!
Direct link between strategic and probabilistic reasoning in CE
Computation:
LP formulation for trees and sparse networks
fast computation of a single CE in any network [Papadimitriou 05]

11. Economic Inequality and Expander Graphs

12. Network Exchange Markets
The classical framework (Arrow-Debreu):
k goods or commodities
n consumers, each with their own endowments and utility functions
equilibrium prices: all consumers rational  all markets clear
Network version:
each vertex is a consumer
edge between i and j means they are free to engage in trade
no edge between i and j: direct exchange is forbidden
Equilibrium
set of local prices for each good g
price for same good may vary across network!
implies local market clearance

13. What Characterizes Price Variation?
Price variation (max/min price) in arbitrary networks:
Characterized by an expansion property
Connections to eigenvalues of adjacency matrix
Theory of random walks
Economic vs. geographic isolation S
N(S)
Expansion: For all “small” S, |N(S)| >= |S|

14. Application to Social Network Models
small diameter
clustering
Price variation (max/min) at equilibrium:
Root of n in preferential attachment
None in random graphs (Erdos-Renyi)
Wealth distribution at equilibrium:
Power law (heavy-tailed) in networks generated by preferential attachment
Sharply peaked (Poisson) in E-R
Random graphs  “socialist” outcomes
Without a centralized formation process
heavy-tailed connectivity

15. Evolutionary Stable Strategies and Edge Density

16. Network EGT Arbitrary 2-player game
Infinite family of undirected graphs {G_n}
As usual, will examine limit of large n
Individual fitness: average payoffs against neighbors
Incumbent strategy p
Choose mutant strategy q, mutation set M, |M| < en
Not all choices of M are equivalent!
One reasonable ESS definition: all such M “contract”
i.e. some mutant has an incumbent neighbor of higher fitness
Question: What graph families preserve the ESS of the classical setting?

17. The Power of Randomization
I. {G_n} a family of random (Erdos-Renyi) graphs, M arbitrary
II. {G_n} arbitrary family, M random
subject to minimum edge density requirements
#edges superlinear in n, but possibly far less than n^2
In any {G_n} meeting I or II, ESS of any game preserved (w.p. 1 for large n)
Local statistics in {G_n} must approximate global “sufficiently often”