 Limit behavior Rationality is ‘self-fulfilling’ in the limit: if players commit to it in deliberation for long enough, they prune away all irrational moves and, a fortiori, all moves incompatible with common belief of rationality. The final outcome is a model with rational play at every point, a form of common knowledge of rationality. However, iterated announcements can also yield a different type of limit behavior: self-refutation. A prime example for this is the classic Muddy Children puzzle (Gierasimczuk &amp; Szymanik 2011) where repeatedly communicating ignorance leads to knowledge in the end. Also within game theory, a number of situations exist where (credible) announcements of future irrationality can leave some player better off than the Backward Induction solution (Leyton-Brown &amp; Shoham 2008). Limit behavior of announcements in infinite games is studied in Baltag, Bezhanishvili &amp; Fernández-Duque (2022), linking up with Cantor’s derivative in classical topology.