 One can draw several morals from the e-mail game of Example 5.1. Rubinstein (1987) argues that his conclusion seems paradoxical for the same reason the backwards induction solution of Alan’s and Fiona’s perfect information game might seem paradoxical: Mathematical induction does not appear to be part of our “everyday” reasoning. This game also shows that in order for A to be a common truism for a set of agents, they ordinarily must perceive an event which implies A simultaneously in each others’ presence. A third moral is that in some cases, it may make sense for the agents to employ some solution concept weaker than Nash or correlated equilibrium. In their analysis of the e-mail game, Monderer and Samet (1989) introduce the notions of ex ante and ex post \(\varepsilon\)-equilibrium. An ex ante equilibrium \(h\) is a system of strategy profiles such that no agent \(i\) expects to gain more than \(\varepsilon\)-utiles if \(i\) deviates from \(h\). An ex post equilibrium \(h'\) is a system of strategy profiles such that no agent \(i\) expects to gain more than \(\varepsilon\)-utiles by deviating from \(h'\) given \(i\)’s private information. When \(\varepsilon = 0\), these concepts coincide, and \(h\) is a Nash equilibrium. Monderer and Samet show that, while the agents in the e-mail game can never achieve common knowledge of the world \(\omega\), if they have common \(p\)-belief of \(\omega\) for sufficiently high \(p\), then there is an ex ante equilibrium at which they follow \((A,A)\) if \(\omega = \omega_1\) and \((B,B)\), if \(\omega = \omega_2\). This equilibrium turns out not to be ex post. However, if the situation is changed so that there are no replies, then Lizzi and Joanna could have at most first order mutual knowledge that \(\omega = \omega_2\). Monderer and Samet show that in this situation, given sufficiently high common \(p\)-belief that \(\omega = \omega_2\), there is an ex post equilibrium at which Joanna and Lizzi choose \((B,B)\) if \(\omega = \omega_2\)! So another way one might view this third moral of the e-mail game is that agents’ prospects for coordination can sometimes improve dramatically if they rely on their common beliefs as well as their mutual knowledge. More recently, the notion of \(p\)-belief and \(p\)-common belief proved useful (Paternotte, 2011) to analyze and formalize Lewis’s account of common knowledge, while Paternotte (2017), establishing a link between “ordinary” common knowledge and \(p\)-common belief, uses the latter to show that only a limited number of exchanges in the e-mail game or coordinated attack paradox would be sufficient to determine coordination. The result, building on foundations provided by Leitgeb (2014), is used to show that our “ordinary” understanding of common knowledge is captured by probabilistic common belief, although at the price of decreased robustness relative to the number of individuals sharing common belief and their awareness.