News and Notes 4/8
• HW3 due date delayed to Tuesday 4/13
– will hand out HW4 on 4/13 also
• Today: finish up NW economics
• Tuesday 4/13
– another mandatory class exercise
– topic: evolutionary game theory
Market Economies
and Networks
Networked Life
CSE 112
Spring 2004
Prof. Michael Kearns
Market Economies
• Suppose there are a bunch of different goods
– wheat, rice, paper, raccoon pelts, matches, grain alcohol,…
– no differences or distinctions within a good: rice is rice
• We may all have different initial amounts or endowments
– I might have 10 sacks of rice and two raccoon pelts
– you might have 6 bushels of wheat, 2 boxes of matches
– etc. etc. etc.
• Of course, we may want to exchange some of our goods
– I can’t eat 10 sacks of rice, and I need matches to light a fire
– it’s getting cold and you need raccoon mittens
– etc. etc. etc.
• How should we engage in exchange?
• What should be the rates of exchange?
• These are among the oldest questions in economics
Cash and Prices
• Suppose we introduce an abstract good called cash
– no inherent value
– simply meant to facilitate trade, encode exchange rates
• And now suppose we introduce prices in cash
– i.e. rates of exchange between each “real” good and cash
• Then if we all believed in cash and the prices…
– we might try to sell our initial endowments for cash
– then use the cash to buy exactly what we most want
• But will there really be:
– others who want to buy all of our endowments? (demand)
– others who will be selling what we want? (supply)
Mathematical Economics
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Have k abstract goods or commodities g1, g2, … , gk
Have n consumers or players
Each player has an initial endowment e = (e1,e2,…,ek) > 0
Each consumer has their own utility function:
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assigns a personal valuation or utility to any amounts of the k goods
e.g. if k = 4, U(x1,x2,x3,x4) = 0.2*x1 + 0.7*x2 + 0.3*x3 + 0.5*x4
here g2 is my “favorite” good --- but it might be expensive
generally assume utility functions are insatiable
• always some bundle of goods you’d prefer more
– utility functions not necessarily linear, though
Market Equilibrium
• Suppose we post prices p = (p1,p2,…,pk) for the k goods
• Assume consumers are rational:
– they will attempt to sell their endowment e at the prices p (supply)
– if successful, they will get cash e*p = e1*p1 + e2*p2 + … + ek*pk
– with this cash, they will then attempt to purchase x = (x1,x2,…,xk)
that maximizes their utility U(x) subject to their budget (demand)
– example:
• U(x1,x2,x3,x4) = 0.2*x1 + 0.7*x2 + 0.3*x3 + 0.5*x4
• p = (1.0,0.35,0.15,2.0)
• look at “bang for the buck” for each good i, wi/pi:
– g1: 0.2/1.0 = 0.2; g2: 0.7/0.35 = 2.0; g3: 0.3/0.15 = 2.0; g4: 0.5/2.0 = 0.25
– so we will purchase as much of g2 and/or g3 as we can subject to budget
• Say that the prices p are an equilibrium if there are exactly
enough goods to accomplish all supply and demand steps
• That is, supply exactly balances demand --- market clears
The Phone Call from Stockholm
• Arrow and Debreu, 1954:
– There is always a set of equilibrium prices!
– Both won Nobel prizes in Economics
• Intuition: suppose p is not an equilibrium
– if there is excess demand for some good at p, raise its price
– if there is excess supply for some good at p, lower its price
– the “invisible hand” of the market
• The trickiness:
– changing prices can radically alter consumer preferences
• not necessarily a gradual process; see “bang for the buck” argument
– everyone reacting/adjusting simultaneously
– utility functions may be extremely complex
• May also have to specify “consumption plans”:
– who buys exactly what from whom
– example:
• A has Fruit Loops and Lucky Charms, but wants granola
• B and C have only granola, both want either FL or LC (indifferent)
• need to “coordinate” B and C to buy A’s FL and LC
Remarks
• A&D 1954 a mathematical tour-de-force
– resolved and clarified a hundred of years of confusion
– proof related to Nash’s; (n+1)-player game with “price player”
• Actual markets have been around for millennia
– highly structured social systems
– it’s the mathematical formalism and understanding that’s new
• Model abstracts away details of price adjustment process
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modern financial markets
pre-currency bartering and trade
auctions
etc. etc. etc.
• Model can be augmented in various way:
– labor as a commodity
– firms producing goods from raw materials and labor
– etc. etc. etc.
• “Efficient markets” ~ in equilibrium (at least at any given moment)
Network Economics
• All of what we’ve said so far assumes:
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that anyone can trade (buy or sell) with anyone else
wheat bought from Nick is the same as wheat bought from Kilian
equivalently, exchange takes place on a complete network
global prices must emerge due to competition
• But there are many economic settings in which everyone is not free to
trade with everyone else
– geography:
• perishability: you buy groceries from local markets so it won’t spoil
• labor: you purchases services from local residents
– legality:
• if one were to purchase drugs, it is likely to be from an acquaintance (no
centralized market possible)
• peer-to-peer music exchange
– politics:
• there may be trade embargoes between nations
– regulations:
• on Wall Street, certain transactions (within a firm) may be prohibited
Next Up
• A model of network economics
• Analysis of our experiment
• Network economics and preferential attachment
A Network Model of
Market Economies
• Still begin with the same framework:
– k goods or commodities
– n consumers, each with their own endowments and utility functions
• But now assume an undirected network dictating exchange
– 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
• Note: can “encode” network in goods and utilities
– for each raw good g and consumer i, introduce virtual good (g,i)
– think of (g,i) as “good g when sold by consumer i”
– consumer j will have
• zero utility for (g,i) if no edge between i and j
• j’s original utility for g if there is an edge between i and j
Network Equilibrium
• Now prices are for each (g,i), not for just raw goods
– permits the possibility of variation in price for raw goods
– prices of (g,i) and (g,j) may differ
– what would cause such variation at equilibrium?
• Each consumer must still behave rationally
– attempt to sell all of initial endowment, but only to NW neighbors
– attempt to purchase goods maximizing utility within budget
– will only purchase g from those neighbors with minimum price for g
• Market equilibrium still always exists!
– set of prices (and consumptions plans) such that:
• all initial endowments sold (no excess supply)
• no consumer has money left over (no excess demand)
Our Experimental Network
• Four “island economies” --- groups A, B, C, D
– can only engage in trade within group and with group E
• One economic “superpower” --- group E
– can trade with anyone, including internally
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Group A: 6 buyers, 3 sellers (excess demand)
Group B: 3 buyers, 7 sellers (excess supply)
Group C: 5 buyers, 5 sellers (balanced)
Group D: 2 buyers, 5 sellers (excess supply)
Group E: 3 buyers, 3 sellers (internal balance irrelevant)
Overall: 19 buyers, 23 sellers (excess supply)
What should happen at equilibrium?
The Theory Says…
• At equilibrium:
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group E buyers join groups B and D (excess supply)
create merged economy of B, D (all parties) and E’s buyers
group E sellers join groups A (excess demand)
group C remains isolated, only trades internally
price computations:
• B-D-Ebuyers: cash/goods = 8/12 ~ 0.67
• A-Esellers: cash/goods = 6/6 = 1.0
• C: cash/goods = 5/5 = 1.0
– market incentive for E players to equalize opportunity
– but price variation remains
sellers in blue
buyers in red
ignore edge color
D
B
E
A
C
the exchange subgraph at equilibrium
0.67
black: competitive, used
yellow: not competitive, unused
dashed: competitive, unused
1.00
1.00
What Actually Happened?
Esell3
Asell1
Abuy2
Abuy5
CSE112 Perpetrators:
• colluded to inflate profits
of Esell3 and Abuy5
• $0 for 1, $1 for 0
transactions
• agreed to split any prizes
• we’ll come back to their fate
Some Analysis
• Market clearance:
– 18.56 dollars spent (out of 19)
– 21.34 wheat sold (out of 23)
• Group A: 6 buyers, 3 sellers (excess demand)
– average buyer price: 1.14; 1.23 excluding perp Abuy5
– average seller price: 0.72; 1.13 excluding perp Asell1
– versus 1.00 equilibrium
• Group B: 3 buyers, 7 sellers (excess supply)
– average buyer price: 0.5, average seller price: 0.59
– versus 0.67 equilibrium
• Group C: 5 buyers, 5 sellers (balanced)
– average buyer price: 1.10, average seller price: 1.10
– versus 1.00 equilibrium
• Group D: 2 buyers, 5 sellers (excess supply)
– average buyer price: 0.65, average seller price: 0.71
– versus 0.67 equilibrium
• Group E: 3 buyers, 3 sellers (internal balance irrelevant)
– average buyer price: 0.72 (0.67 equilibrium)
– average seller price: 1.47; 1.21 excluding perp Esell3 (1.00 equilibrium)
• Qualitative agreement with equilibrium; higher prices overall
Prizes: $10 Each
• Must have cleared (unloaded all of endowment)
• Lowest avg prices for a buyer compared to equilibrium
– Bbuy1 (Eric Pierce): 0.38, vs 0.67 equilibrium (0.29 differential)
– Abuy5 (Chenxi Jiao): 0.50, vs. 1.00 equilibrium (0.50 differential)
• Highest avg prices for a seller compared to equilibrium
– Esell2 (Sarah Dong): 1.35, vs. 1.00 equilibrium (0.35 differential)
– Esell3 (Scott Brown): 2.00, vs. 1.00 equilibrium (1.00 differential)
• Congratulations!
• The SEC may be contacting some of you.
Markets on Natural Networks
• NW for class experiment highly artificial
– very regular and simple structure
– facilitated an easy experiment
• What happens on the NW types from SNT?
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e.g. Erdos-Renyi, a model, preferential attachment,…
we’ll take a quick look at preferential attachment
will have only buyers ($1) and sellers (1 wheat)
introduce a bipartite version of pref. att.
• Some personal comments on:
– serving up only the freshest fare 
– the research-teaching-research cycle
– MK talk tomorrow at CMU; (optional) paper on web site
A Preferential Attachment Model
for Buyer-Seller Networks
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Probabilistically generates a bipartite graph
Buyers and sellers added in pairs at each time step
All edges between buyers and sellers
Each new party will have n > 1 links back to extant graph
– note: n = 1 generates bipartite trees
– larger n generates cyclical graphs
• Distribution of new buyer’s links:
– with prob. 1 – a: extant seller chosen w.r.t. preferential attachment
– with prob. a: extant seller chosen uniformly at random
– a = 0 is pure pref. att.; a = 1 is “like” Erdos-Renyi model
• So (a,n) characterizes distribution of generative model
A Sample Network and Equilibrium
• Solid edges:
– exchange at equilibrium
• Dashed edges:
– competitive but unused
• Dotted edges:
– non-competitive prices
• Note price variation
– 0.33 to 2.00
• Degree alone does not
determine price!
– e.g. B2 vs. B11
– e.g. S5 vs. S14
Basic Theory
Statistics of the Network
• Can generate standard range of degree distributions
• Define b = (1-a)n/(1+n)
– varies from (1-a)/2 to 1-a
• Theorem: in the (a,n) model, for x = o(n^(1/b)), the fraction
of sellers at time n with degree > x is Q(x^(-1/b)).
– closely follows standard techniques
– (a = 0, n = 1) yields cumulative distribution Q(x^(-2))
– as a  1, tails become lighter; exponential decay at a = 1
Economics of the Network
• Now not just interested in structural
properties of NW
• Examine the properties of global
(equilibrium) computation
• A useful monotonicity lemma:
– take a subgraph with a buyer
(respectively, seller) frontier
– let p be the global equilibrium price of
some good in this subgraph
– let p’ be its equilibrium price when
computed just on the subgraph
– then p’ > p (respectively, p’ < p)
• For example, seller degree is an upper
bound on seller wealth
• This lemma has algorithmic applications
– compute controlled, local approximations
to global equilibrium prices
– how well will this work?
Economics of the Network
• Theorem (Wealth Distribution): For w = o(n^(1/b)), the
fraction of sellers with wealth > w is O(w^(-1/b)).
– no corresponding lower bound yet
– power laws seen empirically
– not explained by degree distribution!
• Theorem (Price Variation): If (e.g.) a = 0 (pure pref. att.),
then (max price)/(min price) = W(n^(2/(1 + n)))
– = W(n) for n = 1
– = W(n^(2/3)) for n = 1
– variation generally scaling as a root of population size
Simulation Studies
Degree and Wealth Distributions
degree
wealth
Model: (a = 0.4, n = 1)
n = 250
average of 25 trials
Power law wealth distribution at (rational) economic equilibrium
Degree and Wealth Distribution versus
n
Model: (a = 0, n = 2)
n = 250
average of 25 trials
Increased n lightens wealth tail, separates wealth and degree
Price Variation vs. NW Size &
n
Model: (a = 0, n varying)
average of 25 trials
Power of network size (matches theory); decreasing with n
Price Variation vs.
a and n
n=1
n = 250, scatter plot
n=2
Exponential decrease with a; rapid decrease with n
Quality of Local Approximations
Model: (a = 0, n = 1)
n = 50 to 250 (five plots)
each plot averages 5 trials
Very good approximations in small neighborhoods
Error decays exponentially with k
Quality of Local Approximation II
Model: (a = 0, n = 1)
n = 50 to 250 (five plots)
each plot averages 5 trials
Very mild dependence on n (Chung & Lu on loglog(n) core)
k = 5 gives exact solution; k = 3 is 60% faster (n = 250)