Cable Guy Paradox

[IdleRich]

My girlfriend just said that she is going to a lecture on this and I thought it was quite interesting.
Suppose that I am told that some guy is going to come round and fit me with cable tv sometime tomorrow between 8am and 4pm and I'm going to have to wait in for him. A friend agrees to wait with me to keep me company and to make things more interesting we decide to have a bet on when the guy will come. We divide the interval into two (approximately) equal bits, the first being any time after eight up to twelve and the second part being any time after twelve and before four and each person can pick an interval with the person who picks the interval in which the guy comes collecting ten pounds off the loser. I get first pick and it at first seems that the two intervals are effectively the same and that it shouldn't matter which one I pick as there is absolutely no reason to believe that the guy should come in one interval rather than the other.
Then it occurs to me that, if I choose the first interval, then, after any given period of time in which the cable guy doesn't show it will appear to me that the second interval will now be more attractive as the remaining part of the first interval in which the guy can arrive is now smaller than the second interval. Whatever time the cable guy does come after eight, there is a time before that when he could have come so there will have been a point where the person who picked the first interval, if rational, would have desired to change to the second interval.
In other words, if I pick the first interval I am certain to rationally want to change it at some point and surely, if I know that I am going to want to change it at some point, I should not pick it now. As no similar argument can be made for the second interval (as time goes in one direction) I should surely pick the second interval - even though there is clearly a fifty-fifty chance of the guy coming in each interval. Can anyone explain this?

 

[Mr. Tea]

Well leaving aside the obvious point that no sane person would make a bet and then allow his fellow bet-er to change the conditions of the bet once the test the bet was based on was underway...

...I see it like this. The statement that you are going to want to change your bet to the latter half of the time period once the period is underway is only valid if the guy hasn't turned up already. In other words, let's say it's 10 am, and you want to change your bet because remaining part of the first period is only half the length of the (entire) second part - so clearly he's twice as likely to come after midday than he is to come before. But this is predicated on the assumption that he hasn't alread turned up - in which case you would already have won the bet, of course. Let's say there's a probability of 0.25 of the guy arriving in each two-hour slot; in the situation described, where the guy hasn't come by 10am, he is twice as likely to come after 12pm as before, but this situation is only going happen with a probability of 0.75, since there's a prob of 0.25 that he's come already.
Thus the a priori probability of the guy coming after 12 is 0.5, and the prob of the guy coming before 12 is 0.25 (he's already here by 10) + 0.25 (he gets here between 10 and 12) = 0.5.

Interesting problem to think about, but I'm not sure it's much of a 'paradox' to anyone who knows the first thing about probability.

Edit: this is related to the (more interesting, IMO) "gameshow 'paradox'", you know, the one with the three doors, one of which conceals the star prize, etc. etc....

 

[IdleRich]

"Well leaving aside the obvious point that no sane person would make a bet and then allow his fellow bet-er to change the conditions of the bet once the test the bet was based on was underway..."

No, I'm not saying that you are allowed to change, I'm just pointing out that if you choose the first interval you are guaranteed to want to change - and if you know that you are guaranteed to want to change your choice then you shouldn't make that choice.

"...I see it like this. The statement that you are going to want to change your bet to the latter half of the time period once the period is underway is only valid if the guy hasn't turned up already. In other words, let's say it's 10 am, and you want to change your bet because remaining part of the first period is only half the length of the (entire) second part - so clearly he's twice as likely to come after midday than he is to come before. But this is predicated on the assumption that he hasn't alread turned up - in which case you would already have won the bet, of course. Let's say there's a probability of 0.25 of the guy arriving in each two-hour slot; in the situation described, where the guy hasn't come by 10am, he is twice as likely to come after 12pm as before, but this situation is only going happen with a probability of 0.75, since there's a prob of 0.25 that he's come already.
Thus the a priori probability of the guy coming after 12 is 0.5, and the prob of the guy coming before 12 is 0.25 (he's already here by 10) + 0.25 (he gets here between 10 and 12) = 0.5"

You've explained why the probability of the guy coming before twelve is equal to the probability afterwards - which is correct of course - but you haven't dealt with the paradox.
The point is, supposing that you picked the first interval and it got to ten and you were offered the chance to swap intervals then you should take it - right? The same would be true if it got to nine o'clock with no show and also at 8.01 and in fact it would technically be true at x period after 8 where x is any infinitely small number you care to think of. In other words, if you pick the first interval, you are guaranteed to face a moment where given the option you would be rational to change your bet.
The paradox is; how do you reconcile the fact that if you pick the first interval you are guaranteed to have a moment where you will want to change your bet, with the fact that it seems clear that the two intervals are equally likely for the cable guy to show?

"Edit: this is related to the (more interesting, IMO) "gameshow 'paradox'", you know, the one with the three doors, one of which conceals the star prize, etc. etc....

No it's not. That's to do with the fixing of discrete probabilities, this is to do with continuous probability and the asymmetry of time.

 

[Mr. Tea]

Ahh, I see, so it's more of a psychology thing, right?

Well look at it this way: if you chose the latter half, on the basis that you'd invariably want to change to that half if you'd instead chosen the first half, you'd be pretty hacked off it the guy did actually turn up in the first half, wouldn't you? Or is that just another way of stating the paradox, and I'm completely missing the point?

The reason I don't see it as a paradox is that the statement "You'd inevitably want to change your choice to the second half if you'd initially chosen the first half" is only true if the guy hasn't already arrived. So you can talk about this situation during the bet *if* the guy hasn't arrived yet, but at any time after 8am this condition isn't garuanteed.

I think it's related to the gameshow thing because in that situation, there's a better than even chance that the gameshow host gives you some information about the winning door after you've made your initial choice: here, the information you have is that, at some time before 12pm, the cable guy still hasn't turned up. Both 'paradoxes' are, in a sense, about causality and memory, and the arrow of time, namely that we remember things that have happened in the past but not things still to come in the future.

 

[IdleRich]

"Ahh, I see, so it's more of a psychology thing, right?"

NO!

"The reason I don't see it as a paradox is that the statement "You'd inevitably want to change your choice to the second half if you'd initially chosen the first half" is only true if the guy hasn't already arrived. So you can talk about this situation during the bet *if* the guy hasn't arrived yet, but at any time after 8am this condition isn't garuanteed."

Yes, it is only true if the guy hasn't arrived - but (given the continuous nature of time) there is bound to be some point after eight where the guy hasn't arrived because whatever time he arrives he could have come earlier. So, (as you say,) at any given time after eight it isn't guaranteed that the man won't be there - but it is guaranteed that there will be some point in time after eight when he isn't there - in other words it is guaranteed that if you pick the first interval then there will come a time when it would be rational to want to change intervals.
How can it be rational to make a choice knowing that at a future time you will rationally want to change your choice?

 

[vimothy]

This is a bit like the football game last night...?

 

[hucks]

In other words, if you pick the first interval, you are guaranteed to face a moment where given the option you would be rational to change your bet.

Is that true? Surely for any arbitrarily small time after 8am, there is a (arbitrarily small) probability the guy has arrived. So there is no guarantee.

Edit: Or, alternatively, what Mr Tea said here

So you can talk about this situation during the bet *if* the guy hasn't arrived yet, but at any time after 8am this condition isn't garuanteed.

 

[IdleRich]

"This is a bit like the football game last night...?"

How do you mean?

 

[Mr. Tea]

Hmm, ok, I think I see what you're getting at. But when you say "there is bound to be a time after 8 when the guy hasn't come so you're bound to want to change your choice" will only be true for an infinitesimal interval, which is kind of the same thing as saying it last for zero time, right? At 8.00.01 there's an overwhelming chance you'd wish you could change, because it would offer very slightly better odds, but equally there's a very small chance the guy turned up in that first second. As time goes on, the advantage to having chosen the latter half over the first half increases, but so does the chance that the guy has already come, doesn't it?

 

[IdleRich]

"Is that true? Surely for any arbitrarily small time after 8am, there is a (arbitrarily small) probability the guy has arrived. So there is no guarantee."

I see what you're thinking but because time is infinitely subdivisble the guy could have always arrived earlier. Think of it this way, suppose the guy comes at x minutes after 8, then at (x/2) minutes after eight the guy who picked the first interval would have rationally wanted to change to the second interval because it is longer by x/2 minutes.

 

[mixed_biscuits]

but (given the continuous nature of time) there is bound to be some point after eight where the guy hasn't arrived because whatever time he arrives he could have come earlier.

Yeah, but this point would be infinitesimally small = we are then in the realm of the misconception that permits Zeno's paradox (a stretch of time can be divided infinitely into smaller stretches).

 

[Mr. Tea]

I see what you're thinking but because time is infinitely subdivisble the guy could have always arrived earlier. Think of it this way, suppose the guy comes at x minutes after 8, then at (x/2) minutes after eight the guy who picked the first interval would have rationally wanted to change to the second interval because it is longer by x/2 minutes.

So it's more like a Zeno's paradox kinda situation? Very well then, as the interval after 8 shrinks to zero, the ratio P(first half) tends towards P(second half) and the advantage shrinks to zero. Just as for Zeno, as the time interval shrinks to zero so the distance covered by something moving at finite speed shrinks to zero too.

 

[IdleRich]

"At 8.00.01 there's an overwhelming chance you'd wish you could change, because it would offer very slightly better odds, but equally there's a very small chance the guy turned up in that first second. As time goes on, the advantage to having chosen the latter half over the first half increases, but so does the chance that the guy has already come, doesn't it?"

No, you keep getting hung up on "the chance that the guy has already come" - which isn't the best way to think about it. If the guy comes at any given time he hasn't already come before that - by definition. The main thing to focus on is that at any given time after eight the man arrives, he could have arrived earlier and also after eight. At that earlier instant at which he could have (but didn't actually) arrived you would have been in a position of rationally wanting to change your bet.

 

[mixed_biscuits]

What if you had made this bet with an infinite number of friends, each backing an infinite number of slots (which the infinite subdivision of time would allow)? Choosing to back a different slot would do nothing to increase your chances of being right.

In any case, neither the moment of the cable guy's arrival would occur nor that immediately before it, in a Zeno's paradox style.

Using the minimum number of waiting periods gives the paradox more force than it might otherwise have.

 

[IdleRich]

"Yeah, but this point would be infinitesimally small = we are then in the realm of the misconception that permits Zeno's paradox (a stretch of time can be divided infinitely into smaller stretches)."

Whichever point the man turns up will be infinitesimally small won't it? A point is infinitesimal. That's what I mean by time being continuous.
(I don't think it's clear that that is the misconception Zeno's Paradox rests on.)

 

[IdleRich]

Not really sure what you're getting at here Andrei

"What if you had made this bet with an infinite number of friends, each backing an infinite number of slots (which the infinite subdivision of time would allow)? Choosing to back a different slot would do nothing to increase your chances of being right."

Yes, but that's obvious and I don't see how it's relevant.

"In any case, neither the moment of the cable guy's arrival would occur nor that immediately before it, in a Zeno's paradox style."

Why not? And if not so what? Either I'm missing the point or you're barking up totally the wrong tree here.

I don't see how Zeno's Paradox is relevant here although it looks superficially similar. I think you're saying that it's not true that for any given time the guy arrived he could have arrived earlier. Yes, the interval between eight and the time the man arrives can be conceived of as infinitely small but as we're assuming that the man arrives at a single point in time anyway that's not a problem (although the paradox doesn't arise if you say that the man can arrive at eight by the way).

 

[Mr. Tea]

No, you keep getting hung up on "the chance that the guy has already come" - which isn't the best way to think about it. If the guy comes at any given time he hasn't already come before that - by definition. The main thing to focus on is that at any given time after eight the man arrives, he could have arrived earlier and also after eight. At that earlier instant at which he could have (but didn't actually) arrived you would have been in a position of rationally wanting to change your bet.

Sure - but if you'd somehow been granted your wish and could have changed it to the second half, there's a chance (call it P') that he arrived after that time but still before 12 - in which case you'd wish you could have changed your bet back to the first half! And as the time interval is brought back to 8am, P' tends to 0.5, which is the chance you stand of winning by picking the first half anyway.

This situation is psychological because we know, rationally, that whether the guy comes before or after 12 is not affected by our choice of bet, so while you might 'rationally' wish you could have changed your bet from first half to second half for as long as the guy hasn't come in the first half, you could just as well 'rationally' wished you'd picked the first half if you didn't and the guy does indeed turn up before 12.

 

[mixed_biscuits]

Yes, the interval between eight and the time the man arrives can be conceived of as infinitely small but as we're assuming that the man arrives at a single point in time anyway that's not a problem (although the paradox doesn't arise if you say that the man can arrive at eight by the way).

I think what I'm getting at is that a point can't not have duration.

 

[IdleRich]

"Sure - but if you'd somehow been granted your wish and could have changed it to the second half, there's a chance (call it P') that he arrived after that time but still before 12 - in which case you'd wish you could have changed your bet back to the first half! And as the time interval is brought back to 8am, P' tends to 0.5, which is the chance you stand of winning by picking the first half anyway."

Well, obviously, there is a chance that he will arrive you change (if you are allowed to change) and you will wish that you had the chance to change back - but that's not guaranteed to happen. I don't see what you're getting at here really, there is no irrationality in making a sensible bet that happens to not pay off. There does however appear to be an irrationality in picking a bet for which there is guaranteed to exist a point at which you will wish you hadn't picked that bet.

"This situation is psychological because we know, rationally, that whether the guy comes before or after 12 is not affected by our choice of bet, so while you might 'rationally' wish you could have changed your bet from first half to second half for as long as the guy hasn't come in the first half, you could just as well 'rationally' wished you'd picked the first half if you didn't and the guy does indeed turn up before 12."

No, it's not psychological at all. The thing doesn't rest on whether the man's arrival is affected by your bet. Again, I'm not sure what you're getting at here, you're saying that if you pick the second half and the guy turns up in the first half you will (rationally) wish that you had picked the first half - but as that is not guaranteed to happen it's not equivalent. That's where the asymmetry occurs.

"I think what I'm getting at is that a point can't not have duration."

Well, if it can't then I think you're right it would be problematic (for the paradox) but I think it can.

 

[mixed_biscuits]

if I know that I am going to want to change it at some point, I should not pick it now.

You might assume that you would want to change your bet at every point after 8am, but you cannot be certain, as there is a point infinitesimally soon after 8am that you could not be reasonably aware of being 8am+t, for obvious reasons.