Nonequilibrium models of choice (e.g., level-k reasoning) have significantly different, sometimes more accurate, predictions in games than does Nash equilibrium. When it comes to the maximal set of functions that are implementable in mechanism design, however, they turn out to have similar implications. Focusing on single-valued rules, we discuss the role and implications of different behavioral anchors (arbitrary level-0 play), and prove a level-k revelation principle. If a function is level-k implementable given any level-0 play, it must obey a slight weakening of standard strict incentive constraints. Further, the same condition is also sufficient for level-k implementability, although the role of specific level-0 anchors is more controversial for the sufficiency argument. Nonetheless, our results provide tight characterizations of level-k implementable functions under a variety of level-0 play, including truthful, uniform, and atomless anchors.