Monotone Equilibrium in Multi-Unit Auctions

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) non-decreasing 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.

[1]  Paul R. Milgrom,et al.  A theory of auctions and competitive bidding , 1982 .

[2]  Robert J. Weber,et al.  Distributional Strategies for Games with Incomplete Information , 1985, Math. Oper. Res..

[3]  Kerry Back,et al.  Auctions of divisible goods with endogenous supply , 2001 .

[4]  Michal Bresky.  Equilibria in Multi-unit Auctions , 1999 .

[5]  Jeroen M. Swinkels,et al.  Efficiency and Information Aggregation in Auctions , 2000 .

[6]  Roy Radner,et al.  Private Information and Pure-Strategy Equilibria , 1982, Math. Oper. Res..

[7]  Philip J. Reny,et al.  On the Existence of Pure Strategy Monotone Equilibria in Asymmetric First-Price Auctions , 2004 .

[8]  Bidding Lower with Higher Values in Multi-Object Auctions , 2002 .

[9]  M. Whinston,et al.  Menu Auctions, Resource Allocation, and Economic Influence , 1986 .

[10]  Jaime F. Zender,et al.  Auctions of Divisible Goods: On the Rationale for the Treasury Experiment , 1993 .

[11]  E. Maskin,et al.  Equilibrium in Sealed High Bid Auctions , 2000 .

[12]  Jeroen M. Swinkels,et al.  Communication and equilibrium in discontinuous games of incomplete information , 2002 .

[13]  Susan Athey,et al.  Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information , 1997 .

[14]  E. Stacchetti,et al.  How (not) to sell nuclear weapons , 1996 .

[15]  Paul R. Milgrom,et al.  Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities , 1990 .

[16]  X. Vives.  Nash equilibrium with strategic complementarities , 1990 .

[17]  Gustavo E. Rodriguez.  First Price Auctions: Monotonicity and Uniqueness , 1997 .

[18]  E. Maskin,et al.  The Existence of Equilibrium in Discontinuous Economic Games, I: Theory , 1986 .

[19]  Robert B. Wilson.  Auctions of Shares , 1979 .

[20]  Steven R. Williams,et al.  Convergence to Efficiency in a Simple Market with Incomplete Information , 1994 .

[21]  David McAdams.  Isotone Equilibrium in Multi-Unit Auctions , 2000 .

[22]  P. Reny.  On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games , 1999 .

[23]  Alessandro Lizzeri,et al.  Uniqueness and Existence of Equilibrium in Auctions with a Reserve Price , 2000, Games Econ. Behav..

[24]  David McAdams,et al.  Isotone Equilibrium in Games of Incomplete Information , 2002 .

[25]  Jeroen M. Swinkels,et al.  Existence of Equilibrium in Single and Double Private Value Auctions , 2004 .

[26]  Guoqiang Tian,et al.  Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs , 1993 .

[27]  Paul R. Milgrom,et al.  Monotone Comparative Statics , 1994 .