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 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
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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) = 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 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:
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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 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 a-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:
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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:
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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
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Wealth distribution at equilibrium:
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Price variation (max/min) at equilibrium:
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Random graphs result in “socialist” outcomes
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Price variation in arbitrary networks:
– Power law (heavy-tailed) in networks generated by preferential attachment
– Sharply peaked (Poisson) in random graphs
– Grows as a root of n in preferential attachment
– None in random graphs
– Despite lack of centralized formation process
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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?
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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