Convergence of a Dynamic Matching and Bargaining Market with Two-sided Incomplete Information to Perfect Competition

Consider a decentralized, dynamic market with an infinite horizon in which both buyers and sellers have private information concerning their values for the indivisible traded good. Time is discrete, each period has length δ, and each unit of time a large number of new buyers and sellers enter the market to trade. Within a period each buyer is matched with a seller and each seller is matched with zero, one, or more buyers. Every seller runs a first price auction with a reservation price and, if trade occurs, both the seller and winning buyer exit the market with their realized utility. Traders who fail to trade either continue in the market to be rematched or become discouraged with probability δμ (μ is the discouragement rate) and exit with zero utility. We characterize the steady-state, perfect Bayesian equilibria as δ becomes small and the market–in effect– becomes large. We show that, as δ converges to zero, equilibrium prices at which trades occur converge to the Walrasian price and the realized allocations converge to the competitive allocation. ∗Kellogg School of Management, Northwestern University and Department of Economics, University of British Columbia respectively. †We owe special thanks to Zvika Neeman who originaly devised a proof showing the strict monotonicity of strategies. We also thank Hector Chade, Patrick Francois, Paul Milgrom, Dale Mortensen, Asher Wolinsky, and Jianjun Wu; the participants at the “Electronic Market Design Meeting” (June 2002, Schloss Dagstuhl), the 13th Annual International Conference on Game Theory at Stony Brook, the 2003 NSF Decentralization Conference held at Purdue University, the 2003 General Equilibrium Conference held at Washington University, and the 2003 Summer Econometric Society Meeting held at Northwestern University; seminar participants at Carnegie-Mellon, Washington University, Northwestern University, University of Michigan, Harvard and MIT, UBC, and Stanford; and the members of the collaborative research group on “Foundations of Electronic Marketplaces” for their constructive comments. Finally, both of us acknowledge gratefully that this material is based on work supported by the National Science Foundation under Grant IIS-0121541. Artyom Shneyerov also acknowledges support from the Canadian SSHRC Grant 410-2003-1366.