 The moral that Pettigrew draws from this example is the following. Not all dominated options are irrational. Whether or not a dominated option is irrational depends on the status of the options that dominate it. If all of the options that dominate a given option are themselves dominated, then being dominated does not rule out the given option as irrational. Thus, in  Name Your Fortune\(^*\),  none of the options are ruled irrational because they are dominated; after all, all of the dominated options are only dominated by other options that are also themselves dominated. Thus, Pettigrew instead suggests a decision-theoretic principle to replace  Naive Dominance.  To state it, we must distinguish between two notions of dominance: a strong notion and a weak notion. Suppose \(o^*\) and \(o'\) are options. We say that \(o^*\) strongly dominates \(o'\) if \(o^*\) has greater utility than \(o'\) at all worlds. We say that \(o^*\) weakly dominates \(o'\) if \(o^*\) has at least as great utility as \(o'\) at all worlds and greater utility at some world.