Sleeping bookie

I’ve spent more of this week than is healthy thinking about the Sleeping Beauty puzzle (thanks in large part to this really interesting post by Kenny). I don’t think I’ve got anything terribly novel to say, but I thought I’d set out my current thinking to see if people agree with my take on what the dialectic is on at least one aspect of the puzzle.

Sleeping Beauty is sent to sleep by philosophical experimenters. He (for, in a strike for sexual equality, this Beauty is male) will be woken up on Monday morning, told on Monday afternoon what day it is, and sent to sleep again after being given a drug which will mean that the next time he wakes up, he will have no memories of what transpired. Depending on the result of a fair coin flip, he will either be woken up in exactly similar circumstances on Tuesday morning, or be left to sleep through the day. Beauty is aware of the setup.

How confident should Beauty be on Monday morning that the coin to be flipped in a few hours will land heads (remember, he knows it’s a fair coin). Halfers say: he should have credence 1/2 that it’ll be heads. Thirders say: the credence should be 1/3. (All sides agree that on Sunday his credence should be 1/2).

What I’m interested in is whether there are Dutch book arguments for either view. The very simplest takes the following form. Sell Beauty a [$30,T] bet for $15 on Sunday evening. Then, if Beauty’s a halfer, on Monday and (if awoken) Tuesday mornings, sell him [$20,H] bets on each awakening for $10.

If H obtains, Beauty loses the first bet but wins the sole remaining bet (on Monday morning), for a net loss of $5. If T obtains, Beauty wins the first bet, but loses the next two, for a net loss of $5 again. So Beauty is guaranteed to lose money.

This is in some sense a diachronic dutch book. But as several people note, it’s not a particularly convincing argument that there’s something wrong with Beauty being a halfer. For notice that the information here is asymmetric: the bookie offering the bets needs to have more information than Beauty, since it is crucial to their strategy to offer twice as many bets if the result of the coin flip is tails, than if it is heads.

Hitchcock aims to give a revised Dutch book argument for the same conclusion that avoids this problem. He suggests that the experimenters put the bookie through the same procedure as they put Beauty through, and the bookie’s strategy should then simply be to offer Beauty the bets every time they both wake. That has the net effect of offering the same set of bets as above for a sure loss for Beauty, but the bookie and Beauty are in the same epistemic state. This is the sleeping bookie argument.

What I’d like to claim (inspired by Bradley and Leitgeb) is that if we concentrate too much on the epistemic state of hypothetical bookies, we’ll get led astray. Looking at the overall mechanism whereby bets are offered to Beauty, we initially described this as one where an agent (bookie) is offering bets to Beauty each time they are both awake. But I’d prefer to describe this as a case where a complex agency (the bookie and the experimenters in cahoots) are offering bets to Beauty. The second description seems at least as good as the first: after all, without the compliance of the experimenters, the bookie’s dutch book strategy can’t be implemented. But the system constituted by the experimenters and the bookie clearly has access to the information about the result of the coin toss, and arranges for the bets to be made appropriately, even though the bookie alone lacks this information.

Now dutch book arguments are only as good as the results we can extract from them about what credences are rational to have in given circumstances. And clearly, if Beauty knows that the bets coming at him encode information about the outcome on which the bet turns, then he needn’t (perhaps shouldn’t) simply bet according to his credences, but adjust them to take into account the encoded information. That’s why, to get a fix on what Beauty’s credences are, we put a ban on the bookie having excess information. That’s why the first dutch book argument for thirding looks like a bad way to get a fix on what Beauty’s credences are. But this rationale for forbidding the bookie from having excess information generalizes, so that we shouldn’t trust dutch books in any situation where the mechanism whereby bets are offered (whether in the hands of a single individual, or a system) relies on information about the outcome on which the bet turns. (Equally, if the bookie had extra information, but the system of bets doesn’t exploit this in any way, there’s as yet no case against trusting the dutch book argument, it seems to me.)

The moral I take from all this is that what’s going on in the head of some individual we deign to call “bookie” is neither here nor there: what matters is the pattern of bets and whether that pattern exploits information about the outcomes on which the bet turns. This is effectively what I take Bradley and Leitgeb to argue for in their very nice article. What they suggest (roughly) is that a necessary condition on taking a dutch book argument to give a fix on rational credences, is that the pattern of bets be uncorrelated with the outcomes on which the bets turn. I conjecture (tentatively), that this is really what the ban on bookie’s having extra information was trying to get at all along. The upshot is that Hitchcock’s sleeping bookie argument is problematic in the same way as the initial dutch book argument against halfers.

But more than this. If we refocus attention on the issue of the goodstanding of the pattern of bets, rather than the epistemic states of hypothetic bookies, we can put together a dutch book argument against thirders. For suppose that the experimenters offer Beauty a [$30,H] bet for $15 on Sunday, and then a genuine bet of [$30,T] for $20 on Monday morning no matter what happens, and (so he can’t tell what’s going on) a fake bet where he’ll automatically get his stake returned, apparently of [$30,T] for $20 on Tuesday. Then he’ll be guaranteed a loss of $5 no matter what happens. Of course, the experimenters here have knowledge of the outcomes. But (arguably) that doesn’t matter, because the bets they offer are uncorrelated with the outcomes of the event on which the bets turn: the system of bets offered is the same no matter what the outcome is, so (it seems to me) the information that the experimenters have isn’t implicit in the pattern of bets in any sense. So I think there’s a better dutch book argument against thirding, than there is against halfing. (Or at least, I’d be interested in seeing the case against this in detail).

All this is not to say that the halfer is out of the woods. A quite different dutch book argument is given in a paper by Draper and Pust, which exploits the standard halfer’s story (Lewis’s) about what happens on Monday afternoon, once Beauty has been told what day it is. The Lewisian halfer thinks that once Beauty realizes its Monday, he should have credence 2/3 that Heads is the result. And that, it appears, is a dutch-bookable situation.

Notice that this isn’t directly an argument against the thesis that Beauty should have credence 1/2 in Heads on Monday morning. It is, in effect, an argument that he should also have credence 1/2 in Heads on Tuesday. And, with a few other widely accepted assumptions, these combine to give rise to a contradiction (see for example, Cian Dorr’s presentation of the Beauty case as a paradox).

If this is all we say, then we should conclude that we really do have here a puzzling argument for a contradiction, where all the premises look pretty plausible and the two crucial ones both seem prima facie defensible via dutch book strategies. Maybe, as some suggest, we should revise our claims about updating of credences to make halfing in both circumstances appropriate: or maybe there’s something unavoidably irrational in Beauty’s predicament. What will finally come out in the wash as the best response to the puzzle is one matter; whether the dutch book considerations support halfing or thirding on Monday morning is another; and it is only on this narrow point that I’m claiming that there is a pro tanto case to be a halfer.


7 responses to “Sleeping bookie

  1. Hi Robbie,

    Your Dutch book argument against the thirders is interesting. As you know I think SB should be a thirder. I agree that SB should take the Sunday bet, and the Monday bet looks tempting, but I think SB should refuse it.

    Presumably SB should be aware of the betting set up as you have described it (otherwise the bookie has extra information). In particular SB should know that he will be offered a dummy bet on the Tuesday, and this should be factored into his calculations. If SB is a thirder then when he wakes up during the experiment he has credence of [H&Monday] of 1/3, but his credence in [T&Monday] is not 2/3 but 1/3. His expected payout is the money value of the winning payout ($30) multiplied by his credence in [T&Monday] (1/3) i.e. $10. He shouldn’t pay $20 for such a bet. If there is going to be a genuine bet on tails on the Monday and an equal dummy bet on the Tuesday then the payout needs to be at least 3 times the stake for SB to accept it, and there is no way to set up a diachronic Dutch book with the kind of betting pattern you present.

    The reason I am not persuaded by the Dutch book argument against the thirder is that when SB bets on Monday he is not betting on Heads alone he is betting on Heads being thrown and today being Monday. Just as the odds of Heads being thrown are relevant to him when working out whether or not to accept the bet so are the odds of today being Monday. Granted he doesn’t lose anything if today isn’t Monday, but equally he doesn’t win anything either. Whether or not today is Monday is germane to his expected change in utility if the bet is accepted consequently he should factor the odds of this into his calculations. I think something like this is the case for the examples that Bradley and Leitgreb give in their paper – their paper doesn’t convince me that betting behaviour and credence should diverge in the way they suggest, instead the bettor needs to think a little more carefully about what it is they are betting on.

    If this is right then it has the happy result that Hitchcock’s paper still stands, which given that it has the best title by a significant margin of any paper on the Sleeping Beauty problem is only fair.

  2. …I used to play a game called sleeping beauty with my first girlfriend… it wasn’t like this… but it was a bit more fun…

  3. I think my maths was out my earlier post. SB’s expected payout is 16 and 2/3 dollars, the $10 as calculated plus of course there is a 1/3 chance that he will pay his $20 and get $20 back (this happens if it is a Tuesday, and he should set his credence in [today is Tuesday] to 1/3 if he is a thirder). The upshot is still the same though- SB isn’t rationally obliged to take the bet and so he avoids the Dutch book.

  4. Hi Duncan,

    I think that’s an interesting line of defense, and raises the intriguing question of who gets to say what bets are being offered!

    First thought: it’s my example, and I *stipulate* that the bet is simply on Heads.

    But of course, there’s room for “hostile interpretations” of my example: saying that offering those returns for that price in those circs *should* be described as something other than “a bet on H”.

    So what hostile interpretations of the anti-thirder might be available? The one that seems most plausible to me is to claim that the bet offered in those circs is really a conditional bet: a bet at 1/3 odds, but made conditionally on it being monday. And one should bet on this according to one’s conditional credences: and of course the standard thirder will say that SB’s credence in heads conditionally on it being monday, is 1/2. So she isn’t obligated to take the bet, and won’t be dutch booked.

    (Your hostile interpretation of the example may be slightly different: construing the bets unconditional but as turning on a conjunctive outcome. I’m not sure much turns on this, but I find the conditional-bet construal easier to think about…).

    Obviously, hostile interpretations aren’t automatically successful. I’d personally be prepared to concede that the genuine-bet-on-monday/fake-bet-on-tuesday pair is *equivalent* in some reasonable sense to two (genuine) conditional bets being offered on monday and tuesday (that’s definitely not something that other halfers, e.g. Lewis, would concede). But the hostile interpreter needs more than this equivalence: she needs to say that the bets are *not* what I say they are.

    Stand off threatens. One way to sort out the stand off is to figure out who has the burden of proof, and who would win by default. But unfortunately, I don’t think that issue is clear at all. There’s a case for the hostile interpreter having the burden: after all, the thought experiment seems pretty coherent, and they’re saying it is misdescribed. But there’s also a case that, in order to have a good argument against thirders, I incur the burden of showing that no reasonable hostile interpretation works.

    Speculation: we’ll only make progress in resolving the stand off once we’re clear how the argument from dutchbookability to epistemic irrationality is supposed to go. My hope would be that we don’t have to give SB full info about the setup: just the info that bets will be “unbiased”, in the sense that the bets offered will be uncorrelated with the outcomes on which they turn. I hope that’ll start to undermine potentially hostile interpretations. End Speculation.


    p.s. the thirder has got to do some thinking about hostile interpretations too, if they’re going to use Hitchcock-style dutch-booking arguments against halfers. That I take to be the core idea of Kenny’s post on Antimeta: he thinks (based on a functional conception of what a bet is) that the Hitchcock SB scenario is really one where Beauty is offered a bet at 1:1, despite the scenario being one where Beauty *appears* to be accepting a bet at 2:1.

  5. I think that the conditional-bet construal is the way to think about the bets- it was the appreciation of this that led me to realise I had not calculated the expected payout correctly. I want to think of it as a pair of conditional bets though – if it is Monday then Tails pays $30, if it is Tuesday then Tails pays $20.

    I wouldn’t be persuaded by a stipulation argument. If I stipulate that I am offering you a bet on Red Rum winning and only pay out if Shergar wins then it looks like you were betting on Shergar. You still would have been rational to bet in accordance with your credence on Red Rum winning despite the fact that the odds of you winning that bet were not correlated with the odds of Red Rum winning.

    I’m not sure where the burden of proof lies with a hostile interpretation argument (I would prefer to use the less loaded title ‘unbenevolent interpretation argument’) , one thought is that this might just turn on the exoticness of the bet. The bet you set up looks pretty exotic, so I say the burden of proof lies with you!

    A better argument is that you can discover what it is you are betting on by looking at what your stake buys you. In this case it buys you a bet that pays out $30 on Tails if today is Monday, or it buys you a bet that pays out $20 on Tails if today is Tuesday. And by standard manipulations of conditional credences and the usual assumptions about utility increasing linearly with money this is a bet that a thirder should not take. I don’t see how a case could be made that on the Monday this should be taken as a straightforward bet on Tails – SB does not know that today is Monday so he is gambling on whether or not today is Monday as well as whether or not Tails was thrown. It is fairly uncontroversial that a thirder thinks that if SB is awake he should set his credence in [T&Monday] at 1/3 and his credence in [T&Tuesday] at 1/3 also. These are the only ways that SB gets a payout, so multiply the credences by the respective payouts and add together to get expected payout, if this is less than the stake why take the bet?

    I need to think more about Kenny’s post.

  6. Exotic? Well, y’know, setting aside the amnesia pills and the implausibility of getting it past the ethics committee…

    But seriously, why do you think the case (or my favoured take on it) is exotic?

    Here’s something that strikes me about your version. Suppose we’re on monday, handing a betting contract to SB. Take this very contract, in these very words. No matter what happens, that contract, in those very words, will be honoured. And yet, according to you, depending on what *other* bets are or are not offered tomorrow, this could turn out to be an unconditional bet on Tails; or a conditional one on Tails|Monday (and this makes a difference as to how you should place your bets). Doesn’t that just strike you as a really weird way for bets to behave?

    I’m interested in your final arg, though not sure I get all of it. I’m wondering whether, quite apart from the issues we’ve been discussing till now, some stuff about SB’s uncertainty about which bet she’s being offered might do some work for the thirder.

  7. I think that the exoticness charge, if valid, can be used against Hitchcock too to push the burden of proof to him that the bet he describes is as described. It’s been a while since I read his paper but I think that it convinced me that the bet was as he described, but having read Kenny’s post I am slightly less certain. I suppose the exoticness charge is supposed to be that there is some pretty weird conditional bets being offered, and it just doesn’t look obvious (to me at least) that it is a simple bet on Tails. For that matter it isn’t obvious that it is a simple bet on [Tails and Monday], but if the exoticness charge is correct that needn’t worry the thirder. In addition I think a pretty good case can be made (as I try to in previous posts) that [Tails and Monday] is the bet being offered.

    I think the bet on Tuesday is a red herring to a certain extent. It effects SB’s expected payout because she is does not know whether today is Monday or Tuesday. I think what is doing the work in making it the case that SB ought to reject the bet is that his credence that he is going to get a dividend (as opposed to just getting his stake back like he does on Tuesday) is only 1/3. That dividend doesn’t sufficiently exceed his stake to make the bet worthwhile, no number of bets that simply return his stake are going to alter that, but if he is uncertain about what bet he is participating in these fake bets will effect his expected payout. If I roll a six sided die and offer [1-2, $30] for a $20 stake then you shouldn’t take it since the dividend on a win is only $10 but would need to be $30 to make it worthwhile. If I add a fake bet into the mix and furthermore you don’t know which bet is being offered your expected payout changes from $10 to $16 2/3. No matter how many fake bets you add you will not get a bet worth taking. The situation isn’t exactly parallel with the SB case because of course his being awake counts as evidence, but the point remains, that what is doing the work is simply the unpalatable first bet that he is offered.

    I’m not sure if this really addresses your concerns about the betting contract (which I’m not sure I fully understand). I don’t think that the bet offered will ever turn out to be an unconditional bet on Tails, the bet (singular) offered is that if Today is Monday then Tails will win you $30, if Today is Tuesday Tails will win you $20. (Of course SB may or may not be offered this single bet more than once). As a quick speculation, it may be that this compound bet comes down to the same thing as your thought in the final paragraph about SB being uncertain about which bet he’s being offered (or at least perhaps both are equally acceptable ways of modeling what is going on).

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