A level-k theory for private information games

In a Bayesian Nash Equilibrium (BNE) of a private information game players form reciprocal beliefs over types (i.e., payoff-relevant parameters) of the form "I believe that you believe that I believe...", and so on, ad infinitum. We propose a level-k theory for private information games where a player of level of reasoning k forms "equilibrium beliefs" up to the k{th}-order, and "non-equilibrium beliefs" from the (k 1){th} -order onwards. Equilibrium beliefs follow the distribution of types, as in a BNE. Non-equilibrium beliefs ignore the distribution of types and are rather heuristic projections of one own's type onto the rival, of the form "my rival is of my type." As a result, k→∞ coincides with the definition of a BNE, and k=0 coincides with the Nash equilibrium of the symmetric-type complete information version of the game. Finally, we illustrate our belief-based level-k theory through a simple game