Network Economics Networked Life CIS 112 Spring 2008 Prof. Michael Kearns

Exchange Economies

• Suppose there are a bunch of different – etc. etc. etc.

• Of course, we may want to – etc. etc. etc.

• What should be the

rates exchange

• How should we engage in exchange?

of exchange?

goods

• We may all have different initial amounts or – it’s getting cold and you need raccoon mittens – how many sacks of rice per box of matches?

or – I might have 10 sacks of rice and two raccoon pelts

commodities

– wheat, milk, rice, paper, raccoon pelts, matches, grain alcohol,… – no differences or distinctions within a good: rice is rice

endowments

– you might have 6 bushels of wheat, 2 boxes of matches some of our goods – I can’t eat 10 sacks of rice, and I need matches to light a fire • These are among the oldest questions in markets and economics

Cash and Prices

• Suppose we introduce an abstract resource called

cash

– no inherent value – simply meant to facilitate trade, “encode” pairwise exchange rates • And now suppose we introduce • Then if we all believed in cash

prices

in cash – i.e. rates of exchange between each “real” good and cash – e.g. a racoon pelt is worth $5.25, a box of matches $1.10

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 Microeconomics

• Have

k abstract goods

• Have

n consumers

or

commodities

g1, g2, … , gk or “players” • Each player has an

initial endowment

e = (e1,e2,…,ek) > 0 • Each consumer has their own • this is an example of a

linear utility function:

– assigns a subjective “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 (* = multiplication) utility function • lots of other possibilities; e.g. diminishing utility as amount becomes large – 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

Market Equilibrium

• Suppose we announce 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 C = e1*p1 + e2*p2 + … + ek*pk (* = times) – with this cash, they will then attempt to purchase x = (x1,x2,…,xk) that

maximizes their utility U(x) subject to their budget C

(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 • A specific mechanism: – bring your endowments to the stage – I act as banker, distribute cash for endowments – return to stage, use cash to buy optimal bundle of goods • What could go wrong? – 1) stuff left on stage 2) not enough stuff on stage • Say that the prices p are an

equilibrium

if there are exactly enough goods to accomplish

all

supply and demand constraints • That is, supply exactly balances demand --- market

clears

• • • • • •

Examples

Example 1: 3 consumers, 2 goods – Consumer A: utility 0.5*x1 + 0.5*x2 (indifferent) – – – Claim: equilibrium prices = (1.0,1.0) – All three consumers receive 2.0 from sale of endowments – – – Consumer B: utility 0.75*x1 + 0.25*x2 (prefers Good 1) Consumer C: utility 0.25*x1 + 0.75*x2 (prefers Good 2) All endowments = (1,1) 3 units of Good 1: • Consumer B buys as much as he can  3 units of Good 2: • Consumer C buys as much as he can  1 unit remains of each good • Consumer A is indifferent, buys both 2 units 2 units Example 2: – Consumer A: utility 0.5*x1 + 0.5*x2 (indifferent) – – Claim: equilibrium prices = (2.0,1.0) – All three consumers receive 2+1 = 3.0 from sale of endowments – – Consumer B: 1.0*x1 (prefers Good 1) Consumer C: 1.0*x1 (also prefers Good 1) 3 units of Good 1: • Consumer B buys as much as he can  • • Consumer C buys as much as he can  Supply of Good 1 is exhausted 3 units of Good 2 • Consumer A can exactly purchase all 3 1.5 units 1.5 units How did I figure this out? Guess that B and C must split Good 1 Note: even for

centralized

 1.5*p1 = p1+p2 computation, finding equilibrium is challenging (but tractable)

Another 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

– if there is excess

supply

for some good at p,

raise

for some good at p,

lower

its price its price – the famed “invisible hand” of the market • The trickiness: – changing prices can – example:

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, and from whom – in previous example, may have to specify how much of g2 and g3 to buy • 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/formation process – 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: – that anyone can trade (buy or sell) with anyone else – equivalently, exchange takes place on a complete network – at equilibrium,

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

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 – will only purchase g from those neighbors with

neighbors

– attempt to purchase goods maximizing utility within budget

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)

Network Structure and Outcome

• Q: How does the structure of a network influence the prices/wealths at equilibrium?

• Need to separate asymmetries of endowments & utilities from those of NW structure • We will thus consider

bipartite

economies • Only two kinds of players/consumers: – “Milks”: start with 1 unit of milk, but have utility only for wheat – “Wheats” start with 1 unit of wheat, but have utility only for milk – exact form of utility functions irrelevant • Equal numbers of Milks and Wheats • Network is bipartite --- only have edges between Milks and Wheats • When will such a network have variation in prices?

An Example

2 a 2/3 b 2/3 c 2/3 d w 1/2 x 3/2 y 1/2 z 3/2 • Price = amount of the • Prices at opposite ends of an edge always

reciprocal:

• Checking equilibrium conditions: – only “cheapest” edges used – supply and demand balance: • a sends 1/2 each to w and y • b sends 1 to x • c sends 1/2 each to x and z • d sends 1 to z • w sends 1 to a • y sends 1 to a – exchange subgraph

other

p and 1/p • x sends 2/3 to b, 1/3 to c • z sends 1/3 to c, 2/3 to d good • Some edges unused at equilibrium

1 a 1 b c 1 1 d • Suppose we add the single green edge • Now equilibrium has

no

wealth variation!

w 1 x 1 y 1 z 1

A More Complex Example

• 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

Characterizing Price Variation

• • • • • • • • • • Consider any bipartite “Milk-Wheat” network economy – again, all endowments equal to 1.0, equal numbers of Milks and Wheats

Necessary and sufficient

– – condition for Find the S such that |S|/|N(S)| = p is

all equilibrium prices and wealths to be equal:

network has a

perfect matching

as a subgraph a pairing of Milks and Wheats such that everyone has

exactly one

trading partner on the other side What if there is no perfect matching subgraph? How large can the price variation be?

For any set of vertices S on one side (e.g. Milks), let N(S) be its set of neighbors on the other side maximized (here |S| is the number of vertices in S) Then the largest price/wealth in the network will be p, and the smallest 1/p Intuition: When S is very large but N(S) is small, consumers in S are “captives” of their neighbors N(S) – Can actually iterate: remove S and N(S) from the network, find S’ maximizing |S’|/|N(S’)|,… Note: When network has a perfect matching, N(S) is always at least as large as S Note: Finding the maximizing set S may involve some computation… Now let’s examine price variation in a statistical network formation model…

A Bipartite Economy Network Formation Model

• Consider economies with only two goods: milk and wheat… • …and only two kinds of players/consumers: – Milks: start with 1 unit of milk, have utility only for wheat – Wheats: start with 1 unit of wheat, have utility only for milk – exact form of utility functions irrelevant • Wheats and Milks added incrementally in pairs at each time step • Goal: bipartite network formation model interpolating between P.A. and E-R • Probabilistically generates a bipartite graph • All edges between buyers and sellers • Each new party will have n > 1 links back to extant graph – note: n = 1 – larger n generates bipartite trees 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

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

An Amusing Case Study

U.N. Comtrade Data Network

Full Network

sorted equilibrium prices vertex degree USA: 4.42

Germany: 4.01

Italy: 3.67

France: 3.16

Japan: 2.27

European Union Network

Full Network

sorted equilibrium prices

EU network

vertex degree USA: 4.42

Germany: 4.01

Italy: 3.67

France: 3.16

Japan: 2.27

EU: 7.18

USA: 4.50

Japan: 2.96

Behavioral Experiments in Networked Trade

Game Overview

• • • • • • Simplified version of classic exchange economies (Arrow-Debreu) Players divided into two equal populations; all graphs bipartite Start with 10 divisible units endowment of either “Milk” or “Wheat” Only value the

other

good – payoffs proportional to amount obtained (10 units = $2) Exchange mechanism: –

can only trade with network neighbors

– simple limit orders (e.g. offer 2 units Milk for 3 units Wheat) – no price discrimination in a neighborhood: prices on vertices, not edges – partial executions possible – no resale Only source of asymmetry is network position

• • • •

Equilibrium Theory and Network Structure

Equilibrium: set of prices (exchange rates) at which market

clears

– no

local

supply/demand imbalances – accompanied by

exchange subgraph;

– a static notion; does not specify a trading mechanism – network structure may give rise to only trade with neighbors offering best prices

different prices and wealths

throughout the graph – centralized computation uses linear programming as a subroutine

Theorem:

[Kakade, K., Ortiz, Pemantle, Suri] – No wealth variation at equilibrium  network contains a perfect matching – Max/min wealth correspond to maximum

contraction

: large set with few neighbors – degree alone does

not

determine wealth Preferential attachment: wealth imbalance grows with network size Random (Erdos-Renyi) networks: no wealth variation

Pairs (1 trial) 2-Cycle (3) 4-Cycle (3) Clan (3) Clan + 5% (3 samples) Clan + 10% (3) Erdos-Renyi, p=0.2 (3) E-R, p=0.4 (3) Pref. Att. Tree (3) Pref. Att. Dense (3)

Collective Performance and Topology

overall mean ~ 0.88

• overall behavioral performance is strong • topology matters; many (but not all) pairs distinguished

Equilibrium and Collective Performance

correlation ~ -0.8 (p < 0.001) correlation ~ 0.96 (p < 0.001) • greater equilibrium variation  • greater equilibrium variation  behavioral performance degrades greater behavioral variation

Equilibrium and Collective Performance

• equilibrium theory relevant: beats degree, uniform, centrality • but best model (so far) tilts towards equality • “network inequality aversion”

Behavioral Dynamics: Prices and Volumes

mean in first 30s ~ 1.05; last 90s ~ 1.71 (highly sig.) • preponderance of early 1-for-1 trading • may contribute largely to inequality aversion • no rush of trading at the closing

Fragmentation of Liquidity

Conditional Equilibrium Wealth (CEW): actual earnings so far + equilibrium wealth given (global) trades so far Almost all topology pairs are distinguished by individual CEW variation Cumulative CEW: decreases are structural “traumas” that isolate goods [demo]