doi: 10.1111/j.1467-6419.2011.00680.x EQUILIBRIA EXISTENCE AND CHARACTERIZATION IN AUCTIONS: ACHIEVEMENTS AND OPEN QUESTIONS Luciano I. de Castro Northwestern University Daniel H. Karney University of Illinois (Urbana-Champaign) Abstract. Establishing existence and characterizing equilibria are both important achievements in the study of auctions. However, we recognize that equilibria existence results form the basis for well accepted characterizations. In this survey, we review the landmark results and highlight open questions regarding equilibria existence and characterizations in auctions. In addition, we review the standard assumptions underlying these results, and discuss the suitability of the Nash equilibrium solution concept. We focus our review on single-object auctions, but also review results in multi-unit, divisible, combinatorial and double auctions. Keywords. Equilibrium existence; Multi-unit auctions; Single-unit auctions 1. Introduction The widespread use of auctions creates strong demand for results both to inform practitioners and to advance the theoretical understanding of auctions. Although many textbooks and surveys have been written, 1 the literature on equilibria existence and characterization in auctions still lacks a comprehensive overview. We begin by recognizing that answering any question about auctions must begin with an analysis of the equilibrium (or equilibria) of a game-theoretic model. 2 Without an equilibrium, any characterization of players’ strategies or actions, and the final auction outcome, is unstable. Yet, characterizing an auction’s equilibrium is important too because the characterization provides a prediction of players’ behaviour. Our survey provides an overview of the equilibria existence and characterization results in auctions, highlights the opportunities for research to economists and mathematicians and proceeds as follows. Section 2 describes the standard model of single-object auctions, and discusses the concepts of types, values, formats, as well as the suitability of the (Bayesian-)Nash equilibrium concept. (A reader familiar with game theory and basic auction theory can safely skip this section, but Journal of Economic Surveys (2011) C  2011 Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.