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

2. 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

3. 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?

4. 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

5. 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

6. 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