The rationality of a strategy at a decision node depends both on what actions the strategy prescribes at all future decision nodes and what the players know about the strategies that their opponents are following. Let \(S_{-i}(w)=\{\sigma_{-i}(w') \mid w'\in \Pi_i(w)\}\) be the set of strategy profiles of player \(i\)’s opponents that \(i\) thinks are possible at state \(w\). Then, \(Out_i(v, s_i, S_{-i}(w))\) is the set of outcomes that player \(i\) thinks are possible starting at node \(v\) if she follows strategy \(s_i\).