In a Bayes-Nash Equilibrium of a private information game players engage in an iterative beliefs formation process of the form "I believe that you believe that I believe...", and so on, ad infinitum. However, in reality beliefs might extend only a few steps. We propose a non-equilibrium concept in which a player is Lk (i.e., her depth of reasoning is k) if she correctly forms up to k^{th}-order belief. Thus, an L0 does not process the reciprocal belief formation process of the game at all, in that she simplistically believes that her rival is of her same type.
We propose a simple game to test our level-k concept, and we show its prediction in standard games. In some games there is a sharp discontinuity between the infinite unraveling of reciprocal beliefs formation process and the Bayes-Nash Equilibrium.