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At risk of being obtuse, why isn’t Lloyd’s example a clear refutation of the principle? It’s easy to come up with models where the utility maximising thing to do is bet exactly n dollars on the horse, but you know you’ll regret doing just that later. What you don’t know is whether you’ll regret betting more or betting less.

Of course, Lloyd’s case doesn’t rule out a principle that says you shouldn’t do A if there is some action B such that whatever happens, you’ll regret doing A rather than B, but it does seem to rule out the principle that Al states.

Brian Weatherson , April 21, 2006 07:41 AM

I don’t think you’re being obtuse. I suppose that if you can come up with a model in which betting n is the utility maximising thing (no more, no less), and according to which you know you’ll regret it later, (but, as you point out, you don’t know you’ll regret having bet something because you lost, or that you’ll regret having bet only n, because you won), then you’ve got a proof that utility maximisation does not satisfy the principle as Al states it.

But I suppose I don’t know enough about how preferences about past selves are meant to work here. (This is what makes it difficult for me to say anything without hedging.) After all, can’t I be glad (on your account) that I did the utility maximising thing and therefore not prefer to have done something else? It’s hard to see how the details work out…

Greg Restall , April 21, 2006 10:00 AM

Perhaps the relevant principle is supposed to be a kind of dominance principle - if I am right now choosing between A and B (say, betting on the morning and betting on the afternoon), and I am certain that there will be some point at which I will prefer having bet on B to having bet on A, then I should already prefer betting on B to betting on A. With the Humberstone example, we don’t have this situation. The principle sounds plausible when “regret” is a relation one bears to a single action (as opposed to its negation), but the Humberstone example requires us to read it as a relation one bears to one action as contrasted with another - and the other one isn’t fixed.

Perhaps Alan should have phrased this paper with the stronger principle that Brian mentions, but the Cable Guy example looks like a counterexample to the stronger one. I’ve always taken the case to show that we really want a principle with the existential quantifier first and the universal second - the Humberstone example shows that this is the case when we’re comparing the existential over alternatives and the universal over times. We can avoid the Cable Guy problem if we state the principle as:

One should avoid the situation where one has chosen A, and yet there is some B such that for some future time t one is certain that at t one will prefer B to A.

The open interval at 8 am means that one is certain there will be some time at which one is frustrated, but there is no time such that one is certain to be frustrated at that time.

Another relevant quantifier switch occurs if we cash out preference in terms of being willing to pay e to switch. The principle as stated suggests that one should avoid the situation in which one knows that at some future time there is some e such that one will be willing to pay e to switch from morning to afternoon. However, more plausibly, one should just avoid the situation in which there is some e such that one knows that at some future time one will be willing to pay e to switch.

Kenny Easwaran , April 21, 2006 11:29 AM

Maybe I shouldn’t regret doing something that was utility maximising. But I think I regret doing something if I now know there was something else I could have done that increases utility now. I think.

Anyway, just for completeness (and because it works out surprisingly well) here’s the model I had in mind.

Let the utility function u(d), where d is the wealth in dollars, equal log(d).

Assume the agent starts with $1,000,000.

Assume that horse h is at odds of 2-1.

Assume also that the agent’s credence that horse h will win is 0.4.

I should say at this stage that I just made those numbers up for the sake an example. I then sat down with a giant Excel spreadsheet to work out what the utility maximising thing to do is.

It turns out the utility maximising thing to do is to be exactly one hundred thousand dollars. I’m a little befuddled as to why, given how the model was drawn, we should get such a nice round number as the amount to be bet, we don’t if the numbers are different, for instance, but that’s where the spreadsheet tells me.

So our agent does that, the horse loses, and they later regret betting. Is that irrational? It doesn’t seem so to me.

I’ll leave it as an exercise for the reader to work out what the agent should do if the credence that h will win is 0.5. (This is the first time I’ve left an exercise for the reader in a blog comment. First time for everything I guess!)

Brian Weatherson , April 22, 2006 03:25 AM

Here is a thought: what is it that is being regretted in the horseracing example? There seems to be two ways of describing one action; which may lead to differences in regret.

Action 1: betting $100,000 Action 2: betting rationally by choosing an amount that is calculated to maximise utility

Is the suggestion that we should regret action 1 but not action 2? My gut feeling is that we should not regret not having decided to bet more, but maybe we should regret that we didn’t make a typo and put in an extra 0 when filling out the online betting form.

Presumably the question about whether or not we should regret is one about whether or not it is rational to regret. In which case for it to be rational to regret that we did not bet a greater amount is for it to be rational to regret that we did not behave irrationally. Perhaps this is one of those odd cases where it is (or at least was!) rational to be irrational?

As an aside the following might shed some light on why the numbers came out so well.

Let:

c= credence horse will win m= multiple of original bet that you will receive if you win x= amount bet u= expected utility

so u= c.log(1,000,000+mx) + (1-c)log(1,000,000-x)

The graph of u plotted against x ascends steeply and then descends gradually. The maximum value for u is at that point where x changes from increasing to decreasing, i.e. when the gradient is 0.

So the maximum value of u is given by:

du/dx=0

du/dx is:

[1,000,000(cm+c-1) - mx] / [(1,000,000+mx)(1,000,000-x)ln10]

So is 0 when the numerator is 0 (nasty things happen when x=1,000,000 though). I.e.:

1,000,000(cm+c-1) - mx = 0

so x= [1,000,000(cm+c-1)] / m

All the log function unpleasantness falls out so you can get nice round figures as results.

Incidentally at odds of 2:1 you would get a payout of three times your stake if you won in the UK (the original stake plus twice the original stake being the rationale). I guess the way you have calculated the 2:1 odds is that a win pays just twice the payout (that m=2)?

From this it can be seen that if as c tends to 0.5 the amount bet should tend to 0 when m=2. Which is intuitively sensible, don’t bet if you think the bookmaker is offering unfair odds. The global success of the gambling industry is then either evidence of mass irrationality or that factors have been left of the model, or more likely both.

Duncan Watson , April 24, 2006 01:02 AM

Greg,

Does this argument work for any cable guy? Suppose your cable guy is indecisive. For any time in the interval at which he decides to arrive, he changes his mind and decides to arrive earlier. The indecisive cable guy cannot arrive in the afternoon. So you should not bet on that. And since the cable guy is indecisive over an infinite series of earlier and earlier arrival times, it is not true that there is some arrival time at which you will regret betting on the morning interval.

Mike Almeida , April 28, 2006 02:52 AM

Hmmm… There’s a difference between what you ought to do for an indecisive cable guy if you believe it’s an indecisive cable guy coming, and if you have no expectation about the cable guy’s arrival except for the uniform probability distribution on the entire interval.

But I suppose an indecisive cable guy is an odd beast indeed. Given that the possible interval of arrival is (8am,4pm) (not including endpoints) then if the cable guy arrives at a time, doesn’t it follow that he has decided to arrive at some other time than this? I don’t see how I can prove from the assumption of indecisiveness that he arrives in the morning. Don’t I prove that for any time at which he arrives, either he didn’t decide to arrive at that time, or he did decide to arrive at that time and he later changed his mind after making that decision? As far as I can tell, he can still arrive in the afternoon.

Maybe I’ve misunderstood. This is certainly a weird problem! Thanks for bringing it up.

Greg Restall , May 1, 2006 07:14 AM

The Cable Guy Paradox may eventually turn out to be quite interesting. At my own blog http://ttjohn.blogspot.com/2006/05/qualia-problem-cable-guy-paradox-and.html I add a few more comments and refer to your interpretation as well. I am very keen on sharing this aspect and furthering discussions on the matter.

uv , May 1, 2006 09:56 PM

Greg,

The glb on the large interval is outside the set of possible arrival times, right? So the cable guy cannot arrive at 8am. But the indecisive cable guy is such that for any time t at which he decides to arrive, he changes his mind and decides to arrive earlier (t-n)> 8am (if there is such a (t-n)). Your point is taken that the indecisive cable guy might arrive without deciding to do so. So stipulate further that the indecisive cable guy never arrives without deciding to do so first. I don’t think there is a violation of the equiprobability assumption, since he will have decided the day before to arrive at no time during the interval (every arrival time in the interval is such that there is an earlier arrival time, and so he cannot arrive at any time). I don’t think there is a violation of the assumption that the cable guy will arrive since things change after 8am. At that point, there is a time such that there is no earlier arrival time. And so he can arrive at that time. I’m not assuming that you know your cable guy is indecisive. Rather I’m saying that such a cable guy is consistent with the assumptions of the CG-Paradox. To keep things paradoxical we might have to stipulate that the cable guy is not indecisive.

Mike Almeida , May 3, 2006 02:55 AM

Mike: I don’t get the point where you say:

I don’t think there is a violation of the assumption that the cable guy will arrive since things change after 8am. At that point, there is a time such that there is no earlier arrival time.

What’s the “that point” that you mention here? Is it 8am, or some point later than 8am? If it’s 8am, then for any later time t, there’s an time earlier than t but later that 8am. So let’s suppose that it’s 8:01 and the Cable Guy hasn’t yet arrived. How does the fact that he’s indecisive work out now? I would have thought that if he arrives at 8:02, say, we can argue that since he decided to arrive at 8:02 (by your assumption that he doesn’t arrive without deciding to arrive), since he’s indecisive, he decided to arrive at an earlier time. Am I confused?

Greg Restall , May 3, 2006 12:16 PM

Greg,

I don’t assume that the indecisive cable guy (ICG) must decide before he goes. Let 8+am be some time after 8am. He can decide to arrive and arrive (simultaneously) at 8+am. At that very time there is no earlier time at which he could arrive. But as you said it would have to be after 8am.

Another worry for the CG paradox might be worth mentioning. The paradox disappears when you’ve contracted with the (say) Infinite CG Company. This company sends out a cable guy to arrive at time t (in the morning) only if it sends another to arrive at (t-n)> 8am. If the Infinite CG Company decides to send someone to arrive at t in the morning there is no time after 8am at which some cable guy does not arrive. Of course, you’ll have a lot of company on that day, but then no time elapses after 8am without a cable guy present. So you cannot have time to regret your choice of the morning interval if you contract with the Infinite CG Company.

Mike Almeida , May 4, 2006 05:57 AM

The paradox disappears when you?ve contracted with the (say) Infinite CG Company. This company sends out a cable guy to arrive at time t (in the morning) only if it sends another to arrive at (t-n)> 8am.

That’s not really clear. I want to stipulate everything that Hajek assumes about the case (equiprobability of arrival times, the CG will arrive at some time). I add that if the CG arrives during the morning interval at t, another CG arrives earlier than t (but of course after 8am), and so on. Let there be as many CG’s as there are arrival times earlier than t and map the CG’s onto the arrival times. In that case, we have met all of the stipulations of the case, and there is no time during the morning interval when (i) you don’t know whether you have won or lost and (ii) you regret having bet on the morning interval. If you bet on the morning interval, you will know whether you have lost immediately after 8am. I think that works.

Mike Almeida , May 5, 2006 11:56 PM

To avoid Greg’s worries, we could stipulate that the bet is repudiated if you aren’t in your house at precisely 8 AM ready to await the arrival of the cable guy. You now know for sure that there will be a half-open interval (8 AM, 8 AM

• n] in which the cable guy won’t have arrived, you will be aware of this, and will have an opportunity to regret it. If necessary we can strengthen the above stipulation to ensure that you aren’t unconscious or otherwise incapacitated at 8 AM and that you are in an epistemically strong position to observe his arrival.

Of course, none of this guarantees that you will in fact regret the bet, but only that it would be rational for you to do so. There is a small probability that you won’t think about it during the requisite time interval, and I doubt’t this can be ruled out beforehand. Thus the case seems to fall short of certain frustration, and to necessitate only probable frustration, e.g., there is a chance that you will be momentarily distracted at 8 AM, not sufficiently to prevent you from observing the arrival of the cable guy, but enough to preclude the regret from forming.

As to the horse race, apart from what has already been said, there is also the problem of diminishing marginal utility if $n is taken large enough. If there is a threshold below which you won’t feel any regret then the paradox doesn’t arise.

Jason White , May 27, 2006 06:33 PM

It’s been nearly a year since Greg posted his original comment on “The Cable Guy Paradox” - I really ought to get more plugged in to the blogosphere! Thanks for all your comments, guys.

With regard to the Avoid Future Frustration Principle: I can’t resist replying to the Lloyd’s betting case, which Greg mentions, and which Brian suggests is “a clear refutation of the principle”. I don’t see that at all. Again, the principle says:

“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.” What are the TWO options in the betting case, such that if you choose one of them, you are certain that a rational future self of yours will prefer that you had chosen the other one? Certainly not: ‘bet at $n’/’abstain from betting’; you are not certain of one of these that it will be dispreferred to its alternative. Certainly not: ‘bet at $n’/’bet at $(n+1)’; likewise. You may be tempted to say: ‘bet at $n’/’do not bet at $n’, where the latter includes ‘abstain from betting’ and ‘bet at $(n+1)’. But that’s way too disjunctive to count as an ‘option’. After all, it includes much more: betting at every other price besides $n, going for a swim, committing suicide, … ‘Options’ are specific courses of action that you are deciding among, just as decision theory or game theory would have them. Betting on Morning and betting on Afternoon were clearly options in this sense.

So I don’t see how ordinary betting situations provide quick counterexamples to the Principle. Betting on the Morning in the Cable Guy scenario does provide one - and therein was supposed to lay its interest.

Alan

Alan Hajek , April 12, 2007 03:12 PM