Robust multiplicity with (transfinitely) vanishing naiveté

We extend significantly a result in Heifetz and Kets (Theor Econ 18:415–465, 2018) for Carlsson and van Damme’s (Econometrica 61:989–1018, 1993) global games by which even slight deviations from common belief in infinite depth of reasoning restore the robustness of rationalizable actions multiplicity, in contrast with the intriguing findings of Weinstein and Yildiz (Econometrica 75:365–400, 2007) under an idealized lack of such deviations. Here we show that multiplicity of rationalizable actions is a robust phenomenon even if finite depth of reasoning is an ‘extremely remote rumor’, where someone suspects that someone suspects (..) that somebody might have a finite depth of reasoning, and where the dots range over a transfinite range.