Abstract
We discuss the cable guy paradox, both as an object of interest in its own right and as something which can be used to illuminate certain issues in the theories of rational choice and belief. We argue that a crucial principle—The Avoid Certain Frustration (ACF) principle—which is used in stating the paradox is false, thus resolving the paradox. We also explain how the paradox gives us new insight into issues related to the Reflection principle. Our general thesis is that principles that base your current opinions on your current opinions about your future opinions need not make reference to the particular times in the future at which you believe you will have those opinions, but they do need to make reference to the particular degrees of belief you believe you will have in the future.
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Notes
There are other ways to see that the ANCF principle is false, but we focus on the way in the main text because it is related to the cable guy paradox. One might worry about such a “companions in guilt” argument on the grounds that, e.g., the difference between certainty and near certainty can make all the difference in determining whether it is rational to buy a lottery ticket (so long as the prize is large enough). But in such a situation, there is no special discontinuity at probability 1; instead, there is a threshold at a probability slightly less than 1, determined by the number of tickets, the size of the prize, and the utility one gets out of money. What our example in the main text shows is that, if the ACF principle is true, then, given the falsehood of the ANCF principle, there is a special discontinuity at probability 1; note that, in the example, one can make the value of y as small as one likes.
Cf. Hájek (2005, p. 114), who makes the parallel point about his talk of regret.
We are setting aside situations where the outcome of the bet is (causally or counterfactually) dependent on the choice one makes.
That is, any such point will do so long as you are alive then; if not, the time of maximal relevant information can be taken as the last time at which you are alive.
It might be said: what you, in the relevant sense, should now consider is that future time such that you believe, or find it probable, that you will be relevantly maximally informed at that time. Strictly, this is correct. An easy way around the point would just be always to take the last time at which you are alive as your end-of-the-day time. Or at least it would be, if there were no issues about forgetting. We are only interested in rational agents, so we are making the standard Bayesian idealization that an agent never forgets.
If there is a continuum of degrees of gladness, as we think there is, then this definition will strictly have to be modified to make use of integration instead of mere summation. But since the idea should be clear enough, we choose to avoid the complexities involved in such modification.
Although we think it most natural, choosing end-of-the-day times is not necessary; choosing other times (or even aggregating across many different times) leads to principles equivalent to the EFG principle. We leave this for the reader to work out.
Presumably there is a similar non-comparative principle (one for which there is no reason of simplicity for restricting to two-option situations), but we stick with the language of “gladness” and “regret” for continuity with Hájek. Note that Hájek (2005, p. 115) himself restricts his ACF principle to two-option situations for a different reason; this reason is inapplicable once we have moved from certain future regret to expected future gladness. Hájek (2005, pp. 115–116) also restricts his ACF principle so that it does not apply when each option has the feature that one is certain to rationally regret having made it in the future. Given the comparative nature of regret, such a situation may seem impossible, but Hájek cites the situation in the two envelope paradox as a possible case. In case such situations are possible (which we doubt), the reader should feel free to read a similar restriction into the EFG principle; but for simplicity, our formulations will leave out such a restriction.
Since expected utility is not comparative in the way that expected future gladness is (see second point above after formulation of the EFG principle), formulation of this version of the Expected Utility principle requires explicitly building in the comparative aspect.
Hájek (2005, 118n) cites Schervish, Seidenfeld, and Kadane (2004) on the idea of stopping times, and maintains that it is reference to particular times which prevents the cable guy example from being a counter-example to the Reflection principle. Cf. our discussion of the Reflection principle in Section 4.
See Section 5 for a more detailed discussion of the implications of Reflection for present opinion given uncertainty about future opinion; it turns out in fact that Reflection requires the probability assignment of 0.5 to MORNING.
Recall that we are restricting our discussion to ideally rational agents. We think it is legitimate to specify that an ideally rational agent is always aware, for any time t, whether or not that time is in the future.
The sense in which you could go to far to an arbitrarily large extent consists in the fact that the following proportion could be arbitrarily large: (p future−p current)/(0.5−p future), where p future is the smallest probability you in fact attach to MORNING in the future and p current is the present probability you would in fact opt for if influenced by your knowledge of (largely) unspecified future probability. In other words, what you gain by at least getting to p future could be swamped, to an arbitrarily large extent, by how far past p future you go.
Note that this reasoning might not generalize to all cases of Type 3. For example, it would not apply to a situation where an agent does not have defined expectation values, if such a situation were indeed possible for ideally rational agents.
One could also give a principle that deals with the case where the interval is open on the left and closed on the right, and a principle that deals with the case the interval is open on both sides. For reasons similar to the ones we will give in the main text, these other principles that involve open intervals are also false. Similar remarks hold for more complicated ranges, e.g., (x, y] ∪ [w, z).
More generally, whenever the OQGR principle applies, the CQGR also applies. This is because, whenever a certain probability lies in the range [x, y), it also lies in the range [x, y], as the former is a proper subset of the latter.
Why not require that the probability assignment for B be in the interval [0, 0.4)? We think that considerations analogous to those which show that the OQGR principle is false will show that restricting one’s probability assignment for B to the interval [0, 0.4) is too stringent.
We thank Alan Hájek for many helpful comments.
References
Hájek A. (2005). The cable guy paradox. Analysis, 65, 112–119.
Schervish M. J., Seidenfeld T., & Kadane J. B. (2004). Stopping to reflect. Journal of Philosophy, 101, 315–322.
van Fraassen B. (1984). Belief and the will. Journal of Philosophy, 81, 235–256.
van Fraassen B. (1995). Belief and the problem of Ulysses and the Sirens. Philosophical Studies, 77, 7–37.
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Kierland, B., Monton, B. & Ruhmkorff, S. Avoiding certain frustration, reflection, and the cable guy paradox. Philos Stud 138, 317–333 (2008). https://doi.org/10.1007/s11098-006-9044-1
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DOI: https://doi.org/10.1007/s11098-006-9044-1