Trading in Networks

Networked LifeCIS 112

Spring 2010Prof. Michael Kearns

strategic games

Nash equilibrium

networked games

behavior

trade economies

price equilibrium

networked trade

behavior

Trade Economies• Suppose there are a bunch of different goods orcommodities

– wheat, milk, rice, paper, raccoon pelts, matches, grain alcohol,…– commodity = 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 trade or 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 trade?• What should be the rates of trade?

– how many sacks of rice per box of matches?• These are among the oldest questions in markets and economics• Obviously can be specialized to “modern” markets (e.g. stocks)

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 prices in cash (from where?)

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

• 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)– how might we find them?

• A complex, distributed market coordination problem

Mathematical Microeconomics

• 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:

– 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)• this is an example of a linear utility function, with weights/coefficients w_i• lots of other possibilities; e.g. diminishing utility as amount becomes large

– here g2 is my “favorite” good --- but it might be expensive• Rationality = utility maximization subject to budget

– e.g. suppose U as above and prices p = (1.0,0.35,0.15,2.0) per unit– look at “bang for the buck” for each good i, w_i/p_i:

• 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 our budget• we are indifferent between g2 and g3

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 posted 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), as per the last slide

• 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)– 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)

• Claim: equilibrium prices = (1.0,1.0)– all three consumers receive 2.0 from sale of endowments– 3 units of Good 1:

• Consumer B buys as much as he can 2 units– 3 units of Good 2:

• Consumer C buys as much as he can 2 units– 1 unit remains of each good

• Consumer A is indifferent, buys both

• Example of non-equilibrium prices: (1.0, 2.0)– all three consumers get 3.0 from endowments– Consumers A and B each want to buy 3 units of Good 1, but there’s not enough– Consumer C can buy 1.5 units of Good 2, but that leaves 1.5 units of surplus– important: consumers don’t have to worry about balancing supply demand --- prices should do

that• Example 2:

– Consumer A: utility 0.5*x1 + 0.5*x2 (indifferent)– Consumer B: 1.0*x1 (prefers Good 1)– Consumer C: 1.0*x1 (also prefers Good 1)– all endowments = (1,1)

• Claim: equilibrium prices = (2.0,1.0)– All three consumers receive 2+1 = 3.0 from sale of endowments– 3 units of Good 1:

• Consumer B buys as much as he can 1.5 units• Consumer C buys as much as he can 1.5 units• supply of Good 1 is exhausted

– 3 units of Good 2• Consumer A can exactly purchase all 3

• How did I figure this out? Guess that B and C must split Good 1 1.5*p1 = 3*p2• Note: even for centralized computation, finding equilibrium is challenging (but

tractable)

Another Phone Call from Stockholm• Arrow and Debreu, 1954:

– there is always a set of equilibrium prices!– no matter how many consumers & goods, any utility functions, etc.– 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 famed “invisible hand” of the market

• The problems with this intuition:– 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, and from whom– in previous example, may have to specify how much of g2 and g3 to buy– 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/formation process– does not specify any particular “mechanism”– 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)

Networked Trade: Motivation• 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 directly 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

• Nice real-world example of a market with strong network constraints: electricity markets

• e.g. PJM Interconnect• challenges of electricity storage, regional generation & consumption

Networked Trade: A Model• 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 represents a consumer– edge between i and j means they are free to engage in trade– no edge between i and j: direct trade is forbidden– simplest case: no “resale” allowed --- one “round” of trading

• 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– Q: 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 --- from

neighbors– 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)• no trades except between network neighbors!

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?

2 2/3 2/32/3

1/2 1/2 3/23/2

• Price = amount of the other good received = wealth

• Wealths at opposite ends of any used edge always reciprocal: p and 1/p

• 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• x sends 2/3 to b, 1/3 to c• y sends 1 to a• z sends 1/3 to c, 2/3 to d

• Some edges unused at equilibrium– exchange subgraph

a dcb

w x y z

An Example

1 1 1 1

1 1 1 1

• Suppose we add the single green edge

• Now equilibrium has no wealth variation!

a dcb

w x y z

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 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• To find wealth inequality, look for S much larger than N(S)• Find the S such that |S|/|N(S)| = p is 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 much larger than N(S), 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 Network Formation Model

• 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 a Wheat and a Milk• Each new party will have links back to extant graph• Distribution of new buyer’s links:

– with prob. 1 – : extant seller chosen w.r.t. preferential attachment– with prob. : extant seller chosen uniformly at random– is pure pref. att.; is “like” Erdos-Renyi model

• So () characterizes distribution of generative model

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 more “socialist” outcomes– Despite lack of centralized formation process

An Amusing Case Study

U.N. Comtrade Data Network

USA: 4.42Germany: 4.01Italy: 3.67France: 3.16Japan: 2.27

Full Network

sorted equilibrium wealth

vertex degree

wealt

h

European Union Network

USA: 4.42Germany: 4.01Italy: 3.67France: 3.16Japan: 2.27

Full Network EU network

sorted equilibrium prices

vertex degree

pri

ce

EU: 7.18USA: 4.50Japan: 2.96

Behavioral Experimentsin 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

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)

demo

Collective Performance and Topology

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

overall mean ~ 0.88

Equilibrium and Collective Performance

correlation ~ -0.8 (p < 0.001) correlation ~ 0.96 (p < 0.001)

• greater equilibrium variation behavioral performance degrades• greater equilibrium variation 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”