Evidence shows that people fail to account for informational differences and instead project their information onto others in that they too often act as if others had access to the same information they did. In this study, we find that while people naively project their information onto others, they also anticipate the projection of their differentially informed opponents onto them. Specifically, we find not only that better-informed principals exaggerate the extent to which lesser informed agents should act as if they were better-informed, but that lesser-informed agents anticipate such misperceptions as revealed by their choice of incentive scheme and elicited second-order estimates. Crucially, we estimate directly a parameter of projection equilibrium, Madarasz (2014), that characterizes simultaneously the extent to which principals project and the extent to which agents fail to recognize the principal's projection - because of their own projection - and find remarkably consistent estimates.