Have you ever waited for a tradesperson to come to your place? Alan Hájek uses this example in an article entitled “The Cable Guy Paradox.” The issue is straightforward. You and I are waiting for the cable guy who has given you a window between 8am and 4pm for his arrival. He will arrive after 8am, and before 4pm. We while away our time with a bet. Before 8am, we bet on whether he’ll come in the morning or the afternoon. It seems that there’s no reason to choose between the interval (8am,12noon) and (12noon,4pm). (Let’s assume, with Hájek that it’s probability zero that the moment of arrival is 12noon exactly.) It’s equally likely that he arrive in the morning and in the afternoon.
Now, there seems to be an asymmetry. Hájek cites the ‘Avoid Certain Frustration Principle‘:
Suppose you now have a choice between two options. You should not choose one of these options if you are certain that a rational future self of yours will prefer that you had chosen the other one – unless both your options have this property.
Hájek seems that using this principle, we should bet on the afternoon and not the morning, since whenever the cable guy arrives, it will be after 8am (we specified that in the setup of the problem), so the will be some interval of time (8 am, 8+ε am) during which I will take the probability of the morning arrival to be something smaller than the probability of the afternoon arrival, and I will regret my choice for the bet if I had chosen morning.
In the paper, Hájek uses this example to motivate a rejection and revision of the Avoid Certain Frustration Principle. I think that rejecting this principle seems sound, but I don’t think that this example shows it. Here’s why:
I don’t think that in this case I am certain to be frustrated. For I’m not certain that there is, in fact, an interval of time where I will regret making the bet. First, the bet might be cancelled for some reason – the guy might arrive early, contrary to his promise. More interestingly, we might decide to go out and leave someone else to mind the house, only to return after 4pm. In that case, at any time after 8am, I don’t know that the guy hasn’t arrived, so I don’t have the same grounds for regret. So, I’m not certain that I’ll have my regret.
My favourite example, however, is this one: I don’t know that you won’t mug me, put me in the cupboard and mind the house without my company. Furthermore, you might record the front door with your camera, and then, after 4pm, let me out, and play the recording backwards. In that case, there’s some interval of time in which I’ll prefer having chosen the morning rather than the afternoon, by completely symmetric reasoning.
As a matter of fact, I don’t see how I can be certain that I’ll be frustrated with my morning bet in the sense required by Hájek. (He’s very careful to distinguish certain frustration from very likely frustration).
Now, I’ve mentioned this to a number of people. I mentioned it to Alan Hájek at the 2004 AAP conference at which he presented the paper. I’ve talked about it with some students here at Melbourne who’ve been worrying about this issue. My line doesn’t seem to convince anyone accept me. Why is that? If you think it’s irrelevant to the considerations in the paper, or if there’s a way to reconfigure the example to make my worries not apply, please let me know.
Oh, and on other counterexamples to the Avoid Certain Frustration Principle? Lloyd Humberstone mentioned this scenario to me today, which is in his “You’ll Regret It.” The scenario goes like this. You’re going to bet $n on a horse in the Melbourne Cup. If the horse loses, you’ll regret having made the bet. If the horse wins, you’ll regret not betting more than $n. In either case, you’ll regret having bet $n on the horse. Isn’t that cute? I’m not sure that this is a completely convincing counterexample to the principle, but it bears thinking about.