Aggregating the Single Crossing Property


  • John K.-H. Quah,

    1. Dept. of Economics, Oxford University, Manor Road, OX1 3UL, United Kingdom;
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  • Bruno Strulovici

    1. Dept. of Economics, Northwestern University, Evanston, IL 60208, U.S.A.;
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    • We are grateful to Ian Jewitt for stimulating conversations and to David McAdams, Paul Milgrom, Phil Reny, and Shmuel Zamir for helpful comments. Part of this research was carried out while John Quah was visiting professor at the National University of Singapore and he would like to thank the Economics Department at NUS for its hospitality and support. The financial support of the ESRC to this research project through Grant RES-000-22-3187 (for John Quah) is gratefully acknowledged.


The single crossing property plays a crucial role in economic theory, yet there are important instances where the property cannot be directly assumed or easily derived. Difficulties often arise because the property cannot be aggregated: the sum or convex combination of two functions with the single crossing property need not have that property. We introduce a new condition characterizing when the single crossing property is stable under aggregation, and also identify sufficient conditions for the preservation of the single crossing property under multidimensional aggregation. We use our results to establish properties of objective functions (convexity, logsupermodularity), the monotonicity of optimal decisions under uncertainty, and the existence of monotone equilibria in Bayesian games.