I was just reading the interesting paper When Betting Odds and Credences Come Apart, by Darren Bradley and Hannes Leitgeb, at least in part because of some issues that are coming up in my dissertation about the relations between bets and credences. Their paper is a response to a paper by Chris Hitchcock arguing for the 1/3 answer in the Sleeping Beauty problem, where he shows that if Beauty bets as if her credences were anything other than 1/3, then she is susceptible to a Dutch book.

They end up agreeing that she should bet as if her credences were 1/3, but they argue that this doesn’t mean that her credences should actually be 1/3, because of some similarities this case has to other cases where betting odds and credences come apart. I know at least Darren supports (or has supported) the 1/2 answer in the Sleeping Beauty case, so he’s got a reason to argue for this position.

I think in the end though, their paper has convinced me of the opposite – the correct thing to do in this situation is to bet as if one’s credence is 1/2, even though one’s credence should actually be 1/3! I get the 1/3 credence argument from a bunch of sources (especially Mike Titelbaum’s work on the topic). But for the betting as if one’s credence is 1/2, I might be using the term “bet” in a somewhat non-standard way. However, I think my usage is inspired by my attempt to resist some of the claims of Bradley and Leitgeb.

They give some examples of other cases in which it might look as if one should bet at different odds than one’s credences. For instance, if one is offered a bet on a coin coming up heads, but knows that this bet will only be offered if the coin has actually come up tails, then it looks as if one should bet at odds different from one’s credences. However, they agree that in this case one’s credences change as soon as the bet is offered, and one should bet at odds equal to the new credences.

Their next example is very similar, but without the shift in credences. One is offered a bet on a coin coming up heads, but knows that if the coin actually came up heads then the bet is carried out with fake money (indistinguishably replacing the real money in your and the bookie’s pockets) and is real if the coin actually came up tails. In this case, it looks like one should bet at odds different from one’s credences, which should still be 1/2.

However, I think that in this case what’s going on is that one isn’t really being offered a proper bet on heads at odds of 1/2. Functionally speaking, the money transfer involved will be like a bet on heads at odds of 1. It might be described as a bet at different odds, but I think bets should be individuated in some sort of functionalist way here, rather than according to their description in this sense. Thus, since one’s credence in heads is less than 1, one shouldn’t accept this bet.

Bradley and Leitgeb then say that what goes on in Hitchcock’s set-up of the Sleeping Beauty bets is similar. The bet will be repeated twice if the coin comes up tails (because Beauty and the bookie both forget the Monday bet), and thus this is a situation like the one with the bet that might turn out to be with pretend money, but in the opposite direction. Thus, this bet ends up being one that costs the agent $20 if the coin comes up heads, and wins her $20 if it comes up tails, so it’s functionally a bet at odds of 1/2. I think this is the set of bets she should be willing to accept, but that her credence in heads should be 1/3, so her betting odds and credences should come apart.

Of course, there may be a slight difference between the situations. In this version of the Sleeping Beauty bets, the bet gets made twice if the coin comes up tails, rather than paying off double. Perhaps the fact that it’s agreed to multiple times doesn’t make the same difference that having money replaced by something twice as valuable would. If so, then this bet really was properly described as a bet at odds of 1/3, so that I would no longer think that this is an example where betting odds and credences should come apart.

So I think I don’t really accept the particular claims that Bradley and Leitgeb make in this paper, but it’s only because I’m trying to do something subtle about how to individuate bets in functional terms. I’m sure there are good examples out there on which betting odds and credences could rationally come apart, but I’m not convinced whether the Sleeping Beauty case is one of them.

Lukas Biewald(10:37:59) :You would really take an even bet in the sleeping beauty situation??? Really?

Kenny(10:43:26) :No, I’d take the bet where I lose $10 if it’s tails and gain $20 if it’s heads – but since I’d accidentally take this bet twice if it were tails and only once if it’s heads, it would in some sense effectively be an even bet.

Franco(17:03:35) :The Sleeping Beauty prolem is easily solved when transformed from a statistical problem into a probability one.

The space of events from the point of view of the Sleeping Beauty is the set {A,B,C} where the atomic events are:

A = “the coin toss landed heads”

B = “the coin toss landed tails and the day is Monday”

C = “the coin toss landed tails and the day is Tuesday”

Clearly p(A)=p(B) and, since C follows surely B, p(B)=p(C). Hence p(A)=p(B)=p(C)=1/3. Thus the credence of the Sleeping Beauty that the coin landed heads must equal p(A), that is 1/3.

This is obviusly different from the point of view of the researcher whose space of events is {heads, tails} and hence his correct credence must be 1/2.

On this basis, I disagree with your opinion that there exists examples in which credences and betting odds can rationally come apart. Every time that the betting odds can be computed, they must equal the rational credence of the agents involved: every other different behavior will eventually produce a loss, and this is hardly rational.

Franco(17:33:04) :Incidentally, the Sleeping Beauty problem can be shown to be equivalent to the Monty Hall problem. This becomes apparent if we rephrase the Monty Hall problem in the following way: first the prize is put at random behind one of the doors and then the player tosses a coin. If heads lands, he must choose the first door, whereas if tails lands, he chooses both the other two doors. Since his winning probability is 1/3 if heads lands and 2/3 if tails lands, the player’s credence must assign 1/3 of probability of getting the prize to heads, exactly as the Sleeping Beauty must do.

Kenny(00:26:09) :I’m not sure what you mean by transforming it from a statistical problem to a probability one – I didn’t think there was anything statistical at all in what I was talking about, since I’m thinking of it all in terms of subjective probabilities and not frequencies or anything of the sort.

I agree with you that the answer is 1/3, and that this is fairly clear, but there are some arguments people have given for the 1/2 answer. I was just saying that there’s some sense in which the proper way to bet in this case behaves more like a bet at 1:1 odds than like a bet at 1:2 odds, even though the credence is 1/3. Of course, that’s only if you accept my (admittedly strange) way of describing a bet.

Franco(07:52:28) :When I spoke of transforming a statistical argument into a probabilistic one (“probability one” was a mispelling) I had in mind Wikipedia’s explanation of the Sleeping Beauty problem, where the 1/3 answer is justified by a sample counting argument.

My point, however, is that when the probability space is fully known, as in this case, then it is possible to compute the subjective probability by means of an “objective” probability space.

Indeed, the arguments given in favor of the 1/2 answer refer to a different probability space with different atomic events, which is fit to describe the state of knowledge of an external oberver, but it is unfit to describe the state of knowledge of the Sleeping Beauty, and vice versa. Whence the different subjective credences.

It seems to me that the attempts to separate credence from betting strategy are necessarily based on the construction of two or more probability spaces, but fail to recognize that each space pertains to a different (and separate) agent. My feeling is that once this is properly taken in account, all differences between betting strategy and credence disappear.

horus(08:59:11) :The use of dutch book arguments as a consistency check on probability assignments was developed in a synchronic context. As with Bayesianism in general, the extension to the diachronic case is far from trivial. In particular, it’s essential that the hypothetical bookie be in the same epistemic state as the individual whose probability assignment is being checked for consistency.

In the case of Hitchcock, it is not at all clear to me that his extension of the dutch book argument to the diachronic case preserves this property. Beauty is explicitly prevented in the setup of the problem from having access to any external memory aids for distinguishing her path through the causal structure of the problem. In Hitchcock’s setup, however, the exchange of money adds such a memory aid – without some antecedent criterion for identity of problem, here, I’m not certain how analysis of this problem will reflect upon the sleeping beauty problem as stated by Elga.

To put it another way, Hitchcock has returned us to the Absent Minded Driver problem. There, the paradox was not one of inconsistency of beliefs, but rather one of apparent inconsistency between beliefs and optimal action. But in the absent minded driver case, this inconsistency depends upon the asymmetry between agent and world: the agent cannot remember whether or not he has passed through the intersection, but the world “remembers” in the sense that the event has happened or it has not. If Hitchcock’s case is really parallel to this, then he simply hasn’t extended the notion of a dutch book argument to the diachronic case in an appropriate manner.

Thus, I think Bradley and Leitgeb are confused by Hitchcock’s inappropriate analysis when they claim it demonstrates a problem for the Dutch Book method. Rather, their problem of credences and betting “coming apart” is just equivalent to the divergence of beliefs and actions in the absent minded driver case, which is not a problem for consistency of beliefs as the “epistemic” situation of the world is asymmetrical to that of the agent.