Definition 4.5 (Rationality at a decision node) Player \(i\) is rational at node \ v\in V_i\) in state \(w\) provided, for all strategies \(s_i\) such that \(s_i\ne \sigma_i(w)\), there is an \(o'\in Out_i v, s_i, S_{-i}(w))\) and \(o\in Out_i v, \sigma_i(w), S_{-i}(w))\) such that \ u_i(o)\ge u_i(o')\).
Explanations of human behaviour typically appeal to the subject's beliefs and desires, as well as other facts about him, and proceed on the assumption that the behaviour in question is rational (at least to a minimum degree).
But we also recognize that sometimes it may be necessary to go to war, for instance, in self-defense, and that not everyone is rational, at least to our standards.
Definition 4.6 (Substantive rationality) Player \(i\) is substantively rational at state \(w\) provided for all decision nodes \ v\in V_i\), \(i\) is rational at \ v\) in state \(w\).
Now we turn to those who see nationalist sentiments as being rational, at least in a very wide sense.
It is entirely possible for diligent analysts to award ratings that are rational at the individual level but entirely irrational when viewed collectively.
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 \(\sigma(w)\).
So, a player \(i\) is rational at a decision node \ v\in V_i\) in state \(w\) provided that \(i\) does not know that there is an alternative strategy that would give her a higher payoff.
This means, for example, that for any state \(w\) in an epistemic model for the extensive game in Figure 17, the only strategies that are rational at node \ v_3\) in \(w\) are those that prescribe that \(A\) chooses \(O_3\) at node \ v_3\).
