Economic Models ofNetwork Formation

Networked LifeCSE 112

Spring 2006Prof. Michael Kearns

Background and Motivation• First half of course:

– identification/quantification of common or “universal” structural properties of “natural” networks

• small diameter, high clustering, heavy-tailed degree distributions,…

– development of statistical models of network formation• Watt’s “Caveman/Solaria”, alpha model, pref. att., Kleinberg’s model…

– analyzed/criticized simple “transmission” dynamics• disease/fad spread, forest fires, PageRank,…

• Second half of course:– examination of “rational” dynamics

• interdependent security games, exchange economies,…

– interaction of rational dynamics with statistical formation models• e.g. when network is formed via pref. att., what will price variation be?

• This lecture: let network formation be “rational” as well• A very recent topic

– thx to Eyal Even-Dar & Sid Suri

A Shortest Paths Game• Let’s consider a simply stated network formation game• Have N players, consider them vertices in the network• Each player has to decide which edges to build or buy• Assume a fixed cost c to build an edge• Player’s goal: be as “central” in the network as possible• Cost to player i:

– Cost(i) = {j <> i} Distance(i,j) + c x (# edges bought by i)

– Distance (i,j) = shortest-path distance between i and j in the network jointly formed by all the players

– Players want to minimize their cost– So need to balance edge costs vs. centrality

Comments and Clarifications

• Are formalizing as a one-shot game– contrast with “gradual” or incremental statistical formation

models– could imagine multi-round or stage game; more complex

• Each player has a huge choice of actions– action for player i: any subset S_i of all the N-1 edges i could buy– number of choices for S_i = 2^(N-1)– cost of choosing S_i = c|S_i|

• Are assuming that if i buys edge to j, j (and all others) can “use” or benefit from this edge

• Joint action for all N players:– choice of edge sets for all players: S_1, S_2, …, S_N

• Let G = G(S_1,S_2,…,S_N) be the result overall graph/NW

From Incentives to Networks

• Q: How can we view this game a NW formation model?• A: View the NWs generated as being the Nash

equilibria• More precisely: say that G can be “formed” by the

game if:– G = G(S_1,S_2,…,S_N) for some choices for the S_i– S_1,S_2,…,S_N form a Nash equilibrium of the shortest-paths

game– so, no player i can improve Cost(i) by unilaterally:

• dropping an edge they bought and saving the cost c• adding an edge they didn’t buy and paying the cost c

Properties of Equilibria• Questions we might ask in NW Life:

– What’s the diameter of the equilibria graphs?– What do their degree distributions look like?– What are their clustering coefficients?– Etc.

• Not much known precisely, but we’ll make some inferences• Another measure of interest: the Price of Anarchy:

– for a given G = G(S_1,S_2,…,S_N), consider sum of all player costs:

• Cost(G) = i Cost(i)• Let Cost* = minimum possible Cost(G) (social

optimum)• Price of Anarchy = Cost(G)/Cost* for G a Nash eq.• Which Nash equilibrium? Pick worst (largest Cost(G))

• Inefficiency or cost of “capitalism” over “socialism”

What Happens?• Note that for a single player, sum of distances is between

– a small constant independent of N (e.g. constant diameter graphs)– ~ N^2 (e.g. a cycle or a line graph)

• Price of Anarchy:– edge cost c < sqrt(N): P.O.A. < some constant (independent of N)– edge cost c > N log(N): P.O.A. < 1.5– in between: unknown whether P.O.A. is bounded

• Structural properties: very little is known, but seems– Nash equilibria very sparse

• often trees, but not always!

– Nash equilibria very “regular” or “structured”• e.g. “star” or “hub” graph

– Small diameter? Sometimes. – Heavy-tailed degree distribution? Don’t know.– High clustering? Seems unlikely.

Kleinberg’s Model• Similar in spirit to the -model• Start with an n by n grid of vertices (so N = n^2)

– add local connections: all vertices within grid distance p (e.g. 2)– add distant connections:

• q additional connections• probability of connection at distance d: ~ 1/d^r

– so full model given by choice of p, q and r– large r: heavy bias towards “more local” long-distance connections– small r: approach uniformly random

• Kleinberg’s question:– what value of r permits effective search?

• Assume parties know only:– grid address of target– addresses of their own direct links

• Algorithm: pass message to neighbor closest to target

An Economic Variation on Kleinberg• Again have N players/vertices, but arrange them in a grid

• Grid connections provide free connectivity• Instead of variable probabilities for long-distance edges, introduce variable costs:

– Let cost to i to purchase edge to j = g(i,j)^a– g(i,j) = grid or “Manhattan” distance from i to j– a = some constant value– so cost grows with distance on grid, at a rate determined by value a

• So now just have another network formation game• Another striking “tipping point”:

– for any a <= 2, all Nash equilibria have constant diameter• i.e. diameter does not grow with N!

– for any a > 2, all Nash equilibria have unbounded diameter• i.e. diameter grows with N

– again, Nash equilibria seem to be “regular”, but we don’t know much at this point…

• Cost to player i:

– Cost(i) = {j <> i} Distance(i,j) + c x (# edges bought by i)

– Distance (i,j) = shortest-path distance between i and j in the network jointly formed by all the players

– Players want to minimize their cost– So need to balance edge costs vs. centrality

Economic Formation + Economic Dynamics

• Recall our simple 2-good exchange economy model:– start with a bipartite network between “buyers” and “sellers”– buyers start with $1 but value only wheat– sellers start with 1 unit wheat but value only dollars– prices = proposed rates of exchange– price p means party is willing to exchange their $1/1u for p of other– equilibrium prices: prices for each party such that

• all parties behave “rationally” = trade only with “best price” neighbor(s)

• everyone is able to trade away their initial endowment

– at equilibrium, party charging p only trades with parties charging 1/p

– equilibrium prices = equilibrium wealths

• Before we examined wealth distribution for given networks

Price Variation vs. and

n = 250, scatter plot

Exponential decrease with rapid decrease with

(Statistical) Structure and Outcome

• Wealth distribution at equilibrium:– Power law (heavy-tailed) in networks generated by preferential attachment – Sharply peaked (Poisson) in random graphs

• Price variation (max/min) at equilibrium: – Grows as a root of n in preferential attachment– None in random graphs

• Random graphs result in “socialist” outcomes– Despite lack of centralized formation process

• Price variation in arbitrary networks:– Characterized by presence/absence of a perfect matching– Alternately: an expansion property– Theory of random walks– Economic vs. geographic isolation

Economic Formation + Economic Dynamics

• Now imagine that network is not given, but must be bought• Each player i (buyer or seller) chooses a set S_i of trading

partners on the opposing side (sellers or buyers)– as before, assume each edge costs c to purchase, where c can be in

dollars (buyers) or wheat (sellers)– edge purchased by one party can be used “for free” by other party– once all parties have decided what edges to buy, have some graph G– at price equilibrium of G, player i receives wealth W_i– overall payoff to player i:

• W_i – c x (#edges purchased by i)

• Can again view this as a network formation model:– possible networks = Nash equilibria of the above game– that is, a choice of edge sets S_i bought for each player i such that

no party can unilaterally improve their overall payoff by dropping or adding an edge

Price Variation?

• Can price/wealth variation still be present? How much?• What do the equilibrium networks look like?• Suppose G = G(S_1,S_2,…,S_N) is a Nash equil. of this

game• Let W_min < 1 be smallest wealth (w/o edge costs)• Let c be the cost of an edge• Then must have W_min > 1 – c (same as c > 1 –

W_min)• This inequality is tight

– can construct networks where W_min = 1-c

• So rational NW formation eradicates inequality…– …up to the cost to buy an edge

Network Structure

• If G is some Nash equil. of this game, then– G equals its “exchange subgraph” --- no “unused” edges– G consists of (possibly multiple) connected components

• each component has uniform prices p, 1/p

– don’t know much yet about structure within components• some components may have a range of degrees