A Revealed Preference Theory Of Monotone Choice And Strategic Complementarity

Milgrom and Shannon's (1994) fundamental result on monotone comparative statics says that an agent's optimal action increases with the parameter for every constraint set if, and only if, her preference over actions changes with the parameter in a way that obeys single crossing differences. Our objective in this paper is to conduct a revealed preference analysis of monotone comparative statics. Suppose we could observe an agent's choice at different feasible sets and parameter values, but not the agent's preference. How can we ascertain whether the observations are consistent with the agent having a preference that obeys single crossing differences? We show that a data set is consistent with such a hypothesis if, and only if, it obeys an intuitive and easy-to-check property we call the axiom of revealed complementarity. We can extend our single agent results to a game-theoretic setting where strategy sets, parameters, and players' chosen actions are observed. This leads to a characterization of data sets that are consistent with pure strategy Nash equilibrium play in games with strategic complementarity.

It is not always possible in an empirical setting to observe the same agent behaving under different conditions. Instead, one observes the distribution of actions taken by many agents with heterogenous preferences under different conditions. The issue then is whether changes to this distribution (as feasible sets or parameters vary) are consistent with agents in the population having preferences obeying single crossing differences. We show that this issue can be addressed with a stochastic version of the axiom of revealed complementarity. We then apply our tests to data on cigarette smoking amongst couples. Thinking of these interdependent decisions as a two-by-two game between partners, we investigate whether the distribution of smoking behavior amongst couples is consistent with strategic complementarity.

Keywords: Monotone Comparative Statics, Single Crossing Differences, Interval Dominance, Super-Modular Games, Lattices, Unobserved Heterogeneity

JEL Classification Numbers: C5, C6, C7, D01, D7