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Aggregating the single crossing property: theory and applications to comparative statics and Bayesian games

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  • John Quah
  • Bruno Strulovici

Abstract

The single crossing property plays a crucial role in monotone comparative statics (Milgrom and Shannon (1994)), yet in some important applications the property cannot be directly assumed or easily derived. Difficulties often arise because the property cannot be aggregated: the sum of two functions with the single crossing property need not have the same property. We obtain the precise conditions under which functions with the single crossing property add up to functions with this property. We apply our results to certain Bayesian games when establishing the monotonicity of strategies is an important step in proving equilibrium existence. In particular, we find conditions under which first-price auctions have monotone equilibria, generalizing the result of Reny and Zamir (2004).

Suggested Citation

  • John Quah & Bruno Strulovici, 2010. "Aggregating the single crossing property: theory and applications to comparative statics and Bayesian games," Economics Series Working Papers 493, University of Oxford, Department of Economics.
    • Handle: RePEc:oxf:wpaper:493
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    File URL: https://ora.ox.ac.uk/objects/uuid:f68b7498-2c7f-48de-895c-89dc3758e551
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    References listed on IDEAS

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    1. Van Zandt, Timothy & Vives, Xavier, 2007. "Monotone equilibria in Bayesian games of strategic complementarities," Journal of Economic Theory, Elsevier, vol. 134(1), pages 339-360, May.
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    Cited by:

    1. Charles Z. Zheng, 2014. "Existence of Monotone Equilibria in First-Price Auctions with Resale," University of Western Ontario, Economic Policy Research Institute Working Papers 20141, University of Western Ontario, Economic Policy Research Institute.

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    More about this item

    Keywords

    Monotone comparative statics; Single crossing property; Bayesian games; Monotone strategies; First-price auctions; Logsupermodularity;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • D42 - Microeconomics - - Market Structure, Pricing, and Design - - - Monopoly

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