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Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information


  • Athey, Susan


This paper analyzes a class of games of incomplete information where each agent has private information about her own type, and the types are drawn from an atomless joint probability distribution. The main result establishes existence of pure strategy Nash equilibria (PSNE) under a condition we call the single crossing condition (SCC), roughly described as follows: whenever each opponent uses a nondecreasing strategy (in the sense that higher types choose higher actions), a player's best response strategy is also nondecreasing. When the SCC holds, a PSNE exists in every finite-action game. Further, for games with continuous payoffs and a continuum of actions, there exists a sequence of PSNE to finite-action games that converges to a PSNE of the continuum-action game. These convergence and existence results also extend to some classes of games with discontinuous payoffs, such as first-price auctions, where bidders may be heterogeneous and reserve prices are permitted. Finally, the paper characterizes the SCC based on properties of utility functions and probability distributions over types. Applications include first-price, multi-unit, and all-pay auctions; pricing games with incomplete information about costs; and noisy signaling games.

Suggested Citation

  • Athey, Susan, 2001. "Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information," Econometrica, Econometric Society, vol. 69(4), pages 861-889, July.
  • Handle: RePEc:ecm:emetrp:v:69:y:2001:i:4:p:861-89

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    References listed on IDEAS

    1. Myerson, Roger B, 1983. "Mechanism Design by an Informed Principal," Econometrica, Econometric Society, vol. 51(6), pages 1767-1797, November.
    2. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
    3. Moscarini, Giuseppe & Ottaviani, Marco, 2001. "Price Competition for an Informed Buyer," Journal of Economic Theory, Elsevier, vol. 101(2), pages 457-493, December.
    4. Roger B. Myerson, 1978. "Optimal Auction Design," Discussion Papers 362, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    5. Maskin, Eric & Tirole, Jean, 1990. "The Principal-Agent Relationship with an Informed Principal: The Case of Private Values," Econometrica, Econometric Society, vol. 58(2), pages 379-409, March.
    6. Milgrom, Paul R & Weber, Robert J, 1982. "A Theory of Auctions and Competitive Bidding," Econometrica, Econometric Society, vol. 50(5), pages 1089-1122, September.
    7. Eric Maskin & John Riley, 1984. "Monopoly with Incomplete Information," RAND Journal of Economics, The RAND Corporation, vol. 15(2), pages 171-196, Summer.
    8. Kenneth L. Judd & Michael H. Riordan, 1994. "Price and Quality in a New Product Monopoly," Review of Economic Studies, Oxford University Press, vol. 61(4), pages 773-789.
    9. Lewis, Tracy R & Sappington, David E M, 1994. "Supplying Information to Facilitate Price Discrimination," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 35(2), pages 309-327, May.
    10. Manelli, Alejandro M & Vincent, Daniel R, 1995. "Optimal Procurement Mechanisms," Econometrica, Econometric Society, vol. 63(3), pages 591-620, May.
    11. Cremer, Jacques & McLean, Richard P, 1985. "Optimal Selling Strategies under Uncertainty for a Discriminating Monopolist When Demands Are Interdependent," Econometrica, Econometric Society, vol. 53(2), pages 345-361, March.
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    15. Maskin, Eric & Tirole, Jean, 1992. "The Principal-Agent Relationship with an Informed Principal, II: Common Values," Econometrica, Econometric Society, vol. 60(1), pages 1-42, January.
    16. Paul R. Milgrom, 1981. "Good News and Bad News: Representation Theorems and Applications," Bell Journal of Economics, The RAND Corporation, vol. 12(2), pages 380-391, Autumn.
    17. George A. Akerlof, 1970. "The Market for "Lemons": Quality Uncertainty and the Market Mechanism," The Quarterly Journal of Economics, Oxford University Press, vol. 84(3), pages 488-500.
    18. McAfee, R Preston & Reny, Philip J, 1992. "Correlated Information and Mechanism Design," Econometrica, Econometric Society, vol. 60(2), pages 395-421, March.
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    More about this item

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design


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