Predictions under common knowledge of payoffs may differ from those under arbitrarily, but finitely, many orders of mutual knowledge; Rubinstein's (1989) Email game is a seminal example. Weinstein and Yildiz (2007) showed that the discontinuity in the example generalizes: for all types with multiple rationalizable (ICR) actions, there exist similar types with unique rationalizable action. This paper studies how a wide class of departures from common belief in rationality impact Weinstein and

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... diz's discontinuity. We weaken ICR to ICR λ , where λ is a sequence whose term λ n is the probability players attach to ( n − 1)th‐order belief in rationality. We find that Weinstein and Yildiz's discontinuity remains when λ n is above an appropriate threshold for all n, but fails when λ n converges to 0. That is, if players' confidence in mutual rationality persists at high orders, the discontinuity persists, but if confidence vanishes at high orders, the discontinuity vanishes.