A Network Formation Game
for Bipartite Exchange Economies
[Even-Dar, K. & Suri]
Bipartite Exchange Economies
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Population of N buyers and N sellers
2 abstract commodities, “cash” and “wheat”
Buyers have:
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Sellers have:
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An endowment of 1 unit of wheat
Utility only for cash
Bipartite graph between buyers and sellers
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An endowment of 1 unit of cash
Utility only for wheat (exact form unimportant)
Can only exchange goods with neighbors
Rationality  Must always exchange only with neighbors offering best prices (rates)
One-shot game; no resale
Previous work:
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Network given exogenously
Equilibrium prices (& consumption plans) always exist
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Including for many commodities, general utility functions, asymmetric endowments, etc.
Prices for the same good may vary depending on network structure!
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No price/wealth variation in Erdos-Renyi
Price variation a root of N in Preferential Attachment
Can create networks with any rational price
A Network Formation Game
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Now endogenize the formation of the network
Buyers and sellers are 2N players in a game
Players can purchase edges to the other side at cost a per edge
Edges represent trading opportunities in an exchange economy
Given any bipartite graph G, let w(G,i) be the exchange equilibrium wealth of player i
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The amount of the opposing commodity obtained by i
Let e(G,i) be the number of edges purchased by i
Overall utility to player i =
w(G,i)
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ae(G,i)
(participation) (capital outlay)
Now edges represent trading opportunities in an exchange economy
What are the (pure) Nash equilibria graphs G of this formation game?
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Structural properties?
Price variation?
Main Result
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Let NE(N,a) be all NE graphs for a given N and a
Let NE be the union of NE(N,a) over all N and a
Let (r,s) denote a connected component with r buyers and s sellers
– May be multiple possible topologies
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The set NE is exactly the union of the following three types:
– Perfect Matchings:
• All wealths are = 1 (no variation)
– Exploitation Graphs:
• For any k and l, graph is a union of (1,k), (1,k+1), (l,1), (l+1,1) components
• Number of buyers must equal number of sellers
• Seller wealths: 1/k, 1/(k+1), l, l+1 (unbounded variation)
– Near-Balanced Graphs:
• For any k, graph is a union of (k-1,k), (k,k+1),(k,k-1),(k+1,k) components
• Number of buyers must equal number of sellers
• Seller wealths: k/(k-1), (k+1)/k, (k-1)/k, k/(k+1) (limited variation)
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Possible price/wealth variation is sharply constrained!
– E.g. a wealth of 2/5 is impossible
– Contrast with exogenous setting
Two Important Lemmas
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Let G be a NE graph of the formation game, and let w be the minimum exchange
equilibrium wealth of any player in G. Then w > 1 – a (or a > 1-w).
Let G be any bipartite graph with an (m,k) component, m > k. Then there is some
seller such that the removal of some edge costs the seller at most 1/k of exchange
equilibrium wealth. Thus if G is a NE of the formation game, a < 1/k.
So in any NE graph, 1/k > a > 1-w
Future Work
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Iterated version of exchange model:
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Start next round with your payoff from last
Will wealth variation amplify with time? By how much?
Small-world model: Provable navigation properties?
General: What properties can be “explained” by economic formation?
Contact: mkearns@cis.upenn.edu, www.cis.upenn.edu/~mkearns