Recall that a strategy for a player \(i\) specifies choices at all decision nodes for \(i\), even those nodes that are impossible to reach given earlier moves prescribed by the strategy. Thus, strategies include “counterfactual” information about what players would do if they were given a chance to move at each of their decision nodes. In the single state epistemic model, \(B\) knows that \(A\) is following the strategy \(O_1I_3\). This means that \(B\) knows two things about \(A\)’s choice behavior in the game. The first is that \(A\) is choosing \(O_1\) initially. The second is that if \(A\) where given the opportunity to choose at \(v_3\), then she would choose \(I_3\). Now, given \(B\)’s knowledge about what \(A\) is doing, there is a sense in which whatever \(B\) would choose at \(v_2\), his choice is rational. This follows trivially since \(A\)’s initial choice prescribed by her strategy at \(w\) makes it impossible for \(B\) to move. Say that a player \(i\) is materially rational at a state \(w\) in an epistemic model of a game if \(i\) is rational at all decision nodes \(v\in V_i\) in state \(w\) that are reachable according to the strategy profile \(\sigma(w)\). We have seen that \(B\) is trivially materially rational. Furthermore, \(A\) is materially rational since she knows that \(B\) is choosing \(O_2\) (i.e., \(S_{-A}(w)=\{O_2\}\)). Thus, \(Out_A(v_1, O_1, S_{-i}(w))=\{o_1\}\) and \(Out_A(v_1, O_I, S_{-i}(w))=\{o_2\}\); and so, \(A\)’s choice of \(O_1\) at \(v_1\) makes her materially rational at \(w\). The main point of contention between Aumann and Stalnaker boils down to whether the single state epistemic model includes enough information about what exactly \(B\) thinks about \(A\)’s choice at \(v_3\) when assessing the rationality of \(B\)’s hypothetical choice of \(O_2\) at \(v_2\).