  Decomposition methods are often used for producing counterfactual predictions
in non-strategic settings. When the outcome of interest arises from a
game-theoretic setting where agents are better off by deviating from their
strategies after a new policy, such predictions, despite their practical
simplicity, are hard to justify. We present conditions in Bayesian games under
which the decomposition-based predictions coincide with the equilibrium-based
ones. In many games, such coincidence follows from an invariance condition for
equilibrium selection rules. To illustrate our message, we revisit an empirical
analysis in Ciliberto and Tamer (2009) on firms' entry decisions in the airline
industry.
