I have removalists arriving tomorrow. I don’t know when they will arrive.
It’s times like these that I think about the Cable Guy Paradox by Hajek.
Imagine that you’re like me and you’ve called a removalist (or – in the original – a guy to come and fix your cable television). They’ve told you that they will, with 100% certainty, be at your place between 9am and 3pm. So there are three hours in which they could arrive before noon and three hours in which they could arrive after noon. If you ignore the chance that the removalist will arrive at exactly noon, there is a 50% chance that she will arrive in the morning and an equal chance that she will arrive in the afternoon.
Your housemate decides to have a bit of a wager with you as to when the removalist will arrive: morning or afternoon?
‘[W]e may put the reasoning in terms of a plausible diachronic rationality principle somewhat in the spirit of van Fraassen’s ‘Reflection Principle’ (1984 and 1995). The idea is that you should not knowingly frustrate a rational future self of yours. I will call it 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.‘ — Hajek [Source]
So we’re placing this bet and we’re trying to avoid certain frustration. If we are to place our bet on the removalist arriving in the morning, for each second that passses, the odds increase that the removalist will arrive in the afternoon.
For example, if the removalist has not arrived by 10am, then there are only two morning hours left, but three afternoon hours left.
‘The choice to bet on the morning interval falls squarely under the purview of the Avoid Certain Frustration Principle. It is thus ruled out. Rationality, then, requires you to bet on the afternoon interval (the only choice that is not so ruled out). This is paradoxical, because your initial reasoning that there is nothing to favour one interval over the other seemed impeccable.‘ — Hajek [ibid.]
If you’ve got a 50-50 chance of being correct, it doesn’t matter which you choose because you’ve got equal chance of being correct. And yet, here we are, knowing that, in the future, these even odds will change predictably against us if we choose the morning bet.
Isn’t that cool? Yes, yes it is.
Interestingly (perhaps worryingly), Hajek thinks that the Avoid Certain Frustration Principle should be challenged.
‘Sometimes it is rational to knowingly act against your rationally-formed future preferences, even when you know exactly how to avoid doing so.‘ — Hajek [ibid.]
Over on his blog, Consequentially.org, Greg Restall says:
‘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.‘ — Restall [Source]
For me, it seems that Restall bypasses the point. The point is that you’re given 50-50 odds and yet know that these odds are going to shift against you. Within the context of just the bet, these odds are going to change.
Imagine that you’re sitting an introduction to probability class. You’re asked the question: ‘You’ve got a six-sided fair dice and you roll it. What are the odds that you’ll roll a 6.’ The correct answer is not: ‘Well, it would be one in six but it’s slightly less because you’re not accounting for your pet dog eating the dice when you roll it, or the Large Hadron Collider creates a miniature black hole which swallows your dice, or… &c., &c., &c.’
Sure, you’re correct that these are all possible but they’re not really relevant to the problem at hand. Restall’s objection can be easily bypassed with the following.
‘There is 100% certaintly event S will happen between time t0 and time t1. Time t0.5 is directly between t0 and t1. P(S occurs after t0.5) = 0.5. If S has not happened by any time after t0, P(S occurs after to.5) increase.’
The Avoid Certain Frustration Principle still holds but worries about being mugged and the like no longer apply.
It is interesting that both Hajek and Restall suggest rejecting the ACF Principle and not rejecting the unstated Principle of Making Decisions Based Exclusively on Probability. As we know that the probability will change, we should ignore the odds and go with our rational deliberations. Probability is only one tool in our reasoning toolkit. Just as you don’t use a screwdriver to chop down the mightiest oak in the forest, you wouldn’t exclusively use probability to decide how you’ll place your bet.
Don’t get me wrong. That’s really, really weird. The origins of probability go back to when mathematicians would rip off people in ye ancient pubs by keeping the odds in their (the mathematicians’) favour. To say that there is a class of gamble which doesn’t come down to probability is a bit disturbing.