A Note on Reduced Strategies and Cognitive Hierarchies in the Extensive and Normal Form

In a recent paper, Lin & Palfrey (2022, revised 2023) developed a theory of cognitive hierarchies (CH) in sequential games and observed that this solution concept in not reduced-normal-form invariant. In this note I qualify this observation by showing that the CH model is normal-form invariant, and that the diferences arising from the application of the CH model to the reduced normal form depend only on how randomization by level-0 types is modeled. Indeed, while the uniform behavior strategy in the extensive form yields the uniform mixed strategy in the normal form, the latter does not correspond to the uniform randomization in the reduced normal form, because different reduced strategies may correspond to sets of equivalent strategies with different cardinalities. I also note that results in the literature on transformations of sequential games imply that the sequential CH model of Lin & Palfrey is in variant to the interchanging of essentially simultaneous moves, but it is not invariant to coalescing of moves(and, of course, its inverse, sequential agents splitting).Finally, I note that the independence of ex ante beliefs about the level-types of co-players is preserved by updated beliefs conditional on information sets in all games with observable deviators.