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A Network Formation Game for Bipartite Exchange Economies [Even-Dar, K. & Suri]

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Bipartite Exchange Economies Population of N buyers and N sellers 2 abstract commodities, “cash” and “wheat” Buyers have: –An endowment of 1 unit of cash –Utility only for wheat (exact form unimportant) Sellers have: –An endowment of 1 unit of wheat –Utility only for cash Bipartite graph between buyers and sellers –Can only exchange goods with neighbors –Rationality Must always exchange only with neighbors offering best prices (rates) –One-shot game; no resale Previous work: –Network given exogenously –Equilibrium prices (& consumption plans) always exist Including for many commodities, general utility functions, asymmetric endowments, etc. –Prices for the same good may vary depending on network structure! No price/wealth variation in Erdos-Renyi Price variation a root of N in Preferential Attachment Can create networks with any rational price

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A Network Formation Game 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 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 –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) – e(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? –Structural properties? –Price variation?

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Main Result Let NE(N, ) be all NE graphs for a given N and Let NE be the union of NE(N, ) over all N and Let (r,s) denote a connected component with r buyers and s sellers –May be multiple possible topologies 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) Possible price/wealth variation is sharply constrained! –E.g. a wealth of 2/5 is impossible –Contrast with exogenous setting

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Two Important Lemmas 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 – (or > 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, < 1/k. So in any NE graph, 1/k > > 1-w

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Future Work Iterated version of exchange model: –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/~mkearnsmkearns@cis.upenn.edu

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