Online Appendix to “Finite Depth of Reasoning and Equilibrium Play in Games with Incomplete Information”

Example 8. Suppose there are three players, Ann, Bob, and Carol. Each player i = a, b, c has four types, labeled as ti , t 2 i , t 3 i , t 4 i . The σ-algebras in Si are the trivial σ-algebra F i and the σ-algebra F i generated by the pairs {ti , ti }, {ti , ti }. We now endow each type for Ann with a product σ-algebra, describing the beliefs it can have about the types for Bob and Carol, and similarly for the types for Bob and Carol. For each player i, the types ti and t 2 i believe that the state of nature is H; the other types believe that the state of nature is L. Suppose that type ta has the product σ-algebra Σa(t 1 a) = F b × F c , and that it assigns probability 1 to the event {h} × {tb , tb} × {tc , tc , tc , tc}. Then, the type believes that Bob believes that the state of nature is H (as both tb and t 2 b assign probability 1 to that event), but it does not have a well-articulated belief about Carol’s belief about nature. That is, Ann can reason about Bob’s first-order beliefs, but not about Carol’s, meaning that her “depth” of reasoning depends on the identity of the player she reasons about. / W. Kets (2014), Finite Depth of Reasoning and Equilibrium Play in Games with Incomplete Information, Working paper, Northwestern University.