Reeling about the room, I say to the man: 'Go ahead. I love my children. So please kill them.' The man tries to get the gold by torturing me. I cry out: 'This is agony. So please go on.'
Given the state that I am in, the man is now powerless. He can do nothing that would force me to open the safe. Threats and torture cannot force concessions from someone is so irrational. The man can only flee, hoping to escape the police. (p.13)
Threats of all sorts depend on exploiting another person's rationality. We can thus neutralize threats by making ourselves irrational. Since we have good reason to want to neutralize threats, it can be rational to make ourselves irrational!
This can work just as well for offence as defence. Parfit later describes a society of perfectly rational individuals who are also transparent (others can tell whether they are being honest). Suppose one of these individuals could turn themselves into a threat-fulfiller - someone who always carries out their threats, no matter the cost to themselves. Would it be rational to make yourself irrational in such a way? In this case it would. Since you are transparent, whenever you make a threat others would know that you would carry it out if not appeased. And the others are all perfectly rational, so they would always placate you if the threat was serious enough. So if you strapped a bomb to yourself, you could get others to do whatever you wanted simply by threatening to blow everyone up if they didn't do as you say!
Everyone else should then (rationally speaking) impair their own rationality by becoming threat-ignorers and advertising this fact. You would no longer threaten them if you knew that they would ignore it, since then you would have to blow yourself up and of course you would rather not have to do that!
It's been a couple of years since I read Steven Pinker's How the Mind Works, but I think he explained the evolution of overpowering emotions in terms of the advantages such irrationality affords us. The best way to win a game of 'chicken' is to conspicuously remove your steering wheel and throw it out the window. When others can no longer rely on your rationality to compel you to compromise, their own rationality then forces them to surrender. The craziest man is the most dangerous, and the most dangerous man wins. Overpowering emotions such as jealous rage thus make one very powerful. Pinker argues that the jealous man's wife wouldn't dare have an affair if she knew that he'd kill her if he found out.
I'm not entirely convinced of the evolutionary tale, but the game-theoretical issue is certainly an interesting one. And it doesn't only apply to rationality, but indeed to any goal-directed activity which may sometimes be best achieved through indirect means.
Thus utilitarians could argue that morality requires us to form our character in such ways that we are disposed to sometimes act wrongly (e.g. saving your own child at the cost of many others' lives). Such favouritism is still wrong, just like ignoring the threat-fulfiller's threat is still irrational. But it is justified in an indirect sense, since it results from dispositions that are dictated by the goal in question. It is a case of what Parfit calls blameless wrongdoing.