Economic Models of Network Formation Networked Life CSE 112 Spring 2006 Prof. 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

• This lecture: let dynamics with

statistical network formation

formation models • e.g. when network is formed via pref. att., what will price variation be?

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 • Assume a fixed • Cost to player i:

cost c

to build an edge

game

• Have N players, consider them vertices in the network • Each player has to decide which edges to build or buy • Player’s goal: be as “central” in the network as possible – Cost(i) = S {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 • Each player has a

one-shot

game – contrast with “gradual” or incremental statistical formation models – could imagine multi-round or stage game; more complex

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) =

S

i Cost(i) • Let Cost* =

minimum possible

• Price of Anarchy = Cost(G)/Cost* for G a Nash eq.

• Which Nash equilibrium? Pick Cost(G) (social optimum)

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 a -model • Start with an n by n

grid

of vertices (so N = n^2) – add

local

connections: all vertices within grid distance

p

– add distant connections: • q additional connections • probability of connection at distance d: ~

1/d^r

(e.g. 2) – 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…

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.

a

and

n

n = 1

n = 250, scatter plot

n = 2

Exponential decrease with

a;

rapid decrease with

n

(Statistical) Structure and Outcome

• Wealth distribution at equilibrium: – Power law (heavy-tailed) in networks generated by preferential attachment – Sharply peaked – Grows as a (Poisson) in random graphs • Price variation (max/min) at equilibrium: 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 partners on the opposing side (sellers or buyers) dollars (buyers) or wheat (sellers) – at

price

– overall equilibrium of G, player i receives wealth W_i payoff to player i: • W_i – c x (#edges purchased by i) • Can again view this as a network adding an edge

formation

model: – possible networks = Nash equilibria of the above game

bought

• Each player i (buyer or seller) chooses a set S_i of trading – as before, assume each edge costs c to purchase, where c can be in – 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 – 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

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 • So

must have rational

W_min > 1 – c (same as c > 1 – W_min) • This inequality is

tight

– can construct networks where W_min = 1-c 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