Aggregating the single crossing property

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.

[1]  A. Dixit.  Entry and Exit Decisions under Uncertainty , 1989, Journal of Political Economy.

[2]  Iñaki Aguirre,et al.  Monopoly price discrimination with constant elasticity demand , 2015 .

[3]  Paul Milgrom,et al.  Putting Auction Theory to Work , 2004 .

[4]  Paul Milgrom,et al.  Comparing Optima: Do Simplifying Assumptions Affect Conclusions? , 1994, Journal of Political Economy.

[5]  K. Eliaz,et al.  Competing for Consumer Inattention , 2013, Journal of Political Economy.

[6]  David McAdams,et al.  Isotone Equilibrium in Games of Incomplete Information , 2002 .

[7]  Susan Athey,et al.  Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information , 1997 .

[8]  Paul R. Milgrom,et al.  Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities , 1990 .

[9]  Paul R. Milgrom,et al.  Monotone Comparative Statics , 1994 .

[10]  Ying Chen,et al.  Information Transmission in Nested Sender-Receiver Games , 2013 .

[11]  Susan Athey.  Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information , 1997 .

[12]  Ian Jewitt,et al.  Applications of likelihood ratio orderings in economics , 1991 .

[13]  X. Vives.  Nash equilibrium with strategic complementarities , 1990 .

[14]  C. Gollier.  The economics of risk and time , 2001 .

[15]  P. Reny,et al.  On the Existence of Monotone Pure Strategy Equilibria in Bayesian Games , 2009 .

[16]  Frédéric Koessler,et al.  Certifiable Pre‐Play Communication: Full Disclosure , 2014 .

[17]  Bruno H. Strulovici,et al.  On the Smoothness of Value Functions , 2012 .

[18]  L. Booth.  On the theory of the competitive firm under uncertainty , 1978 .

[19]  R. Amir.  Cournot Oligopoly and the Theory of Supermodular Games , 1996 .

[20]  S. Karlin,et al.  Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions , 1980 .