We consider extensive-form games in which the information structure is not known and ask how much of that structure can be inferred from the distribution on action profiles generated by player strategies. One game is said to observationally imitate another when the distribution on action profiles generated by every behavior strategy in the latter can also be generated by an appropriately chosen behavior strategy in the former. The first part of the paper develops analytical methods for testing observational imitation. The central idea is to relate a game's information structure to the conditional independencies in the distributions it generates on action profiles. We present a new analytical device, the influence opportunity diagram of a game, describe how such a diagram is constructed for a given finite-length extensive-form game, and demonstrate that it provides, for a large class of economically interesting games, a simple test for observational imitation. The second part of the paper shifts the focus to the influence assessments of players within a game. A new equilibrium concept, causal Nash equilibrium, is presented and compared to several other well-known alternatives. Cases in which causal Nash equilibrium seems especially well-suited are explored.
