Computer Science, Economics, and the Effects of Network Structure

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… • Decentralized • Heterogenous preferences • Competition • Cooperation • Free riding • Tragedies of the Commons • Adaptation …on a network.

3. Economic Thought in Computer Science • 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 • The Economics of Spam • Email: a (nearly) free resource • Unlimited usage • Favorable ROI for spammers • even under vanishing uptake • Technology solutions: • spam filters • blacklists & whitelists • Economic policy: • email micropayment schemes

4. Network Models ofStrategic 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 onlyN_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 structurefor 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 networkMN(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 localmarket 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. 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 small diameter clustering heavy-tailed connectivity Application to Social Network Models

15. Evolutionary Stable Strategiesand 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 edgedensity 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”