Aumann's Theorem states that common knowledge of substantive rationality implies play of the backward induction profile

In the above game the backward induction profile is \((I_1I_3, I_2)\) leading to the outcome \(o_4\) with both players receiving a payoff of \(3\). Consider an epistemic model with a single state \(w\) where \(\sigma(w)=(O_1I_3,O_2)\). This is not the backward induction profile, and so, by Aumann’s Theorem (Theorem 4.7) it cannot be common knowledge among \(A\) and \(B\) at state \(w\) that both \(A\) and \(B\) are substantively rational.