Trading in Networks
Networked Life
CIS 112
Spring 2010
Prof. Michael Kearns
strategic games
trade economies
Nash equilibrium
price equilibrium
networked games
networked trade
behavior
behavior
Trade Economies
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Suppose there are a bunch of different goods orcommodities
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We may all have different initial amounts or endowments
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Of course, we may want to trade or exchange some of our goods
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How should we engage in trade?
What should be the rates of trade?
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These are among the oldest questions in markets and economics
Obviously can be specialized to “modern” markets (e.g. stocks)
– wheat, milk, rice, paper, raccoon pelts, matches, grain alcohol,…
– commodity = no differences or distinctions within a good: rice is rice
– 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.
– 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 many sacks of rice per box of matches?
Cash and Prices
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Suppose we introduce an abstract resource called cash
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And now suppose we introduce prices in cash (from where?)
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Then if we all believed in cash and the prices…
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But will there really be:
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A complex, distributed market coordination problem
– no inherent value
– simply meant to facilitate trade; “encode” pairwise exchange rates
– 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
– we might try to sell our initial endowments for cash
– then use the cash to buy exactly what we most want
– others who want to buy all of our endowments? (demand)
– others who will be selling what we want? (supply)
– how might we find them?
Mathematical Microeconomics
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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:
– 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
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– here g2 is my “favorite” good --- but it might be expensive
Rationality = utility maximization subject to budget
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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
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Suppose we “announce” prices p = (p1,p2,…,pk) for the k goods
Assume consumers are rational:
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A specific mechanism:
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What could go wrong?
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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
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– 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
– 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
– 1) stuff left on stage 2) not enough stuff on stage
Examples
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Example 1: 3 consumers, 2 goods
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Claim: equilibrium prices = (1.0,1.0)
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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)
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all three consumers receive 2.0 from sale of endowments
3 units of Good 1:
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3 units of Good 2:
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1 unit remains of each good
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Consumer B buys as much as he can  2 units
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Consumer C buys as much as he can  2 units
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Consumer A is indifferent, buys both
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Example of non-equilibrium prices: (1.0, 2.0)
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Example 2:
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Claim: equilibrium prices = (2.0,1.0)
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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
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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)
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All three consumers receive 2+1 = 3.0 from sale of endowments
3 units of Good 1:
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3 units of Good 2
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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
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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
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Arrow and Debreu, 1954:
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Intuition: suppose p is not an equilibrium
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The problems with this intuition:
– there is always a set of equilibrium prices!
– no matter how many consumers & goods, any utility functions, etc.
– both won Nobel prizes in Economics
– 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
– changing prices can radically alter consumer preferences
• not necessarily a gradual process; see “bang for the buck” argument
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– 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
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A&D 1954 a mathematical tour-de-force
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Actual markets have been around for millennia
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Model abstracts away details of price adjustment/formation process
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Model can be augmented in various way:
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“Efficient markets” ~ in equilibrium (at least at any given moment)
– resolved and clarified a hundred of years of confusion
– proof related to Nash’s; (n+1)-player game with “price player”
– highly structured social systems
– it’s the mathematical formalism and understanding that’s new
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does not specify any particular “mechanism”
modern financial markets
pre-currency bartering and trade
auctions
etc. etc. etc.
– labor as a commodity
– firms producing goods from raw materials and labor
– etc. etc. etc.
Networked Trade: Motivation
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All of what we’ve said so far assumes:
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But there are many economic settings in which everyone is not free
to directly trade with everyone else
– 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
– 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:
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• 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
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Still begin with the same framework:
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But now assume an undirected network dictating exchange
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Note: can “encode” network in goods and utilities
– k goods or commodities
– n consumers, each with their own endowments and utility functions
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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
– 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
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Now prices are for each (g,i), not for just raw goods
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Each consumer must still behave rationally
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Market equilibrium still always exists!
– 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?
– 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
– 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
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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:
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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?
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“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
An Example
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2
a
2/3
b
2/3
c
2/3
d
w
x
y
z
1/2
3/2
1/2
3/2
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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:
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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
1
a
1
b
1
c
1
d
w
x
y
z
1
1
1
1
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Suppose we add the single green edge
Now equilibrium has no wealth variation!
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
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Consider any bipartite “Milk-Wheat” network economy
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Necessary and sufficient condition for all equilibrium prices and wealths to be equal:
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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)
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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…
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again, all endowments equal to 1.0, equal numbers of Milks and Wheats
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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
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Can actually iterate: remove S and N(S) from the network, find S’ maximizing |S’|/|N(S’)|,…
A Bipartite Network Formation Model
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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 n > 1 links back to extant graph
Distribution of new buyer’s links:
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So (a,n) characterizes distribution of generative model
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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
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 (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
Full Network
wealth
sorted equilibrium wealth
vertex degree
USA: 4.42
Germany: 4.01
Italy: 3.67
France: 3.16
Japan: 2.27
European Union Network
Full Network
EU network
price
sorted equilibrium prices
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
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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
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Exchange mechanism:
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payoffs proportional to amount obtained (10 units = $2)
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)
demo
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  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”