• Open Access

Finding all pure-strategy equilibria in games with continuous strategies


  • Kenneth L. Judd,

    1. Hoover Institution, Stanford University; kennethjudd@mac.com
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  • Philipp Renner,

    1. Department of Business Administration, Universität Zürich; philipp.renner@business.uzh.ch
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  • Karl Schmedders

    1. DBA—Universität Zürich, and Swiss Finance Institute; karl.schmedders@business.uzh.ch
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    • We are indebted to five anonymous referees and the co-editor for very helpful comments on an earlier version of the paper. We thank Guy Arie, Paul Grieco, Felix Kubler, Andrew McLennan, Walt Pohl, Mark Satterthwaite, Andrew Sommese, Jan Verschelde, and Layne Watson for helpful discussions on the subject. We are very grateful to Jonathan Hauenstein for providing us with many details on the software package Bertini and his patient explanations of the underlying mathematical theory. We are also grateful for comments from seminar audiences at the University of Chicago ICE workshops 2009–2011, the University of Zurich, ESWC 2010 in Shanghai, the University of Fribourg, and the Zurich ICE workshop 2011. Karl Schmedders gratefully acknowledges financial support from the Swiss Finance Institute.


Static and dynamic games are important tools for the analysis of strategic interactions among economic agents and have found many applications in economics. In many games, equilibria can be described as solutions of polynomial equations. In this paper, we describe state-of-the-art techniques for finding all solutions of polynomial systems of equations, and illustrate these techniques by computing all equilibria of both static and dynamic games with continuous strategies. We compute the equilibrium manifold for a Bertrand pricing game in which the number of equilibria changes with the market size. Moreover, we apply these techniques to two stochastic dynamic games of industry competition and check for equilibrium uniqueness.