Monotone Equilibrium in Multi-Unit Auctions

31 Pages Posted: 22 Jul 2002

See all articles by David McAdams

David McAdams

Massachusetts Institute of Technology (MIT) - Economics, Finance, Accounting (EFA)

Date Written: July 2002


In a large class of multi-unit auctions of identical objects that includes the uniform-price, as-bid (or discriminatory), and Vickrey auctions, a Bayesian Nash equilibrium exists in monotone pure strategies whenever there is a finite price / quantity grid and each bidder's interim expected payoff function satisfies single-crossing in own bid and type. A stronger condition, non-decreasing differences in own bid and type, is satisfied in this class of auctions given (a) independent types and (b) risk-neutral bidders with marginal values that are (c) nondecreasing in own type and have (d) non-increasing differences in own type and others' quantities. A key observation behind this analysis is that each bidder's valuation for what he wins is always modular in own bid in any multi-unit auction in which the allocation is determined by market-clearing. This paper also provides the first proof of pure strategy equilibrium existence in the uniform-price auction when bidders have multi-unit demand and values that are not private.

Keywords: Autions, Multi-unit Auctions, Uniform-price Auction, Vickrey Auctions Bidders, Bayesian Nash Equilibrium

Suggested Citation

McAdams, David, Monotone Equilibrium in Multi-Unit Auctions (July 2002). Available at SSRN: or

David McAdams (Contact Author)

Massachusetts Institute of Technology (MIT) - Economics, Finance, Accounting (EFA) ( email )

77 Massachusetts Avenue
Cambridge, MA 02139-4307
United States
617-253-1306 (Phone)

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