Uncertain rationality, depth of reasoning and robustness in games with incomplete information

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 Yildiz's discontinuity. We weaken ICR to ICRλ, where λ is a sequence whose term λn is the probability players attach to (n- 1)th-order belief in rationality. We find that Weinstein and Yildiz's discontinuity remains when λn is above an appropriate threshold for all n, but fails when λn 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.Germano acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (grant ECO2017-89240-P, AEI/FEDER, UE, and through the Severo Ochoa Program for Centers of Excellence in R&D, grant SEV-2015-0563). Zuazo-Garin acknowledges financial help from the Spanish Ministry of Economy and Competitiveness, from the Department of Language Policy and Culture of the Basque Government (grants ECO2012-31346 and POS-2015-1-0022 and IT568-13, respectively) and the ERC Starting Grant #759424, and thanks Northwestern University (both the Department of Economics and the Math Center at MEDS Department) for generous hospitality