Fair Bets.pdf

When Betting Odds and Credences Come Apart: More Worries for Dutch Book Arguments

Darren BRADLEY and Hannes LEITGEB

If an agent believes that the probability of E being true is 1/2, should she accept a bet on E at

even odds or better? Yes, but only given certain conditions. This paper is about what those

conditions are. In particular, we think that there is a condition that has been overlooked so far in

the literature. We discovered it in response to a paper by Hitchcock (2004) in which he argues

for the 1/3 answer to the Sleeping Beauty problem. Hitchcock argues that this credence follows

from calculating her fair betting odds, plus the assumption that Sleeping Beauty’s credences

should track her fair betting odds. We will show that this last assumption is false. Sleeping

Beauty’s credences should not follow her fair betting odds due to a peculiar feature of her

epistemic situation.

1. Dutch Books

Suppose that rational agents bet in line with their beliefs. This means that if an agent believes

proposition E with certainty, he will bet in favour of the truth of E at any odds, no matter how

long. If he believes E with 50% certainty, he will accept a bet on E that pays twice the stake (or

more). If he believes E with 33% certainty, he will accept a bet on E that pays 3 times the stake

(or more). Some writers defined partial beliefs in terms of betting behaviour, making the link

constitutive. We have no need for such a strong link. All we need is for there to be a normative

link between the belief and the bet. Something like “Other things being equal (risk-neutral, utility

linear with money,...), an agent who accepts E with 50% certainty is rationally permitted to

accept a bet on E that pays twice the stake or better“. This link is broadly accepted, and will be

all we need. The issue that we are interested in within this paper is the “other things”.

Assuming agents bet in line with their beliefs, can we say anything about the beliefs an

agent may rationally have by looking at the bets they will make? Dutch book arguments say that

we can. A Dutch book is a series of bets such that anyone who accepts the bets will end up losing

money however the world turns out. A Dutch book argument says that any set of beliefs that

justifies an agent’s accepting a Dutch book is irrational. The beliefs lead to the bets; the bets

leads to a guaranteed loss; therefore the beliefs were irrational. Dutch book arguments have been

the main arguments given for probabilism – the doctrine that one’s beliefs should conform to the

probability calculus (Ramsey 1927; cf. Skyrms 1987). Given the importance of this idea, the

argument deserves careful scrutiny.

A Dutch book argument is also used by Chris Hitchcock (2004); not for probabilism, but

in arguing that an agent should have a particular set of beliefs. In the Sleeping Beauty problem

(as explained below), the disagreement is about the degree of belief Sleeping Beauty should have

that a coin landed Heads. Some argue for 1/3, others for 1/2. Hitchcock points out that 1/3 is the

only degree of belief that avoids a Dutch book. We agree with him on this point. Hitchcock

concludes that 1/3 is the only rationally permissible belief. We disagree. Various examples have

already been given in the literature where correct betting behaviour comes apart from rational

degree of belief. Hitchcock is careful to make sure that his example avoids being like any of

these cases. But we think he has highlighted a new case, not previously noticed, where betting

behaviour should come apart from rational degrees of belief. Thus betting considerations in

Sleeping Beauty, as in other cases, are inconclusive.

2. Sleeping Beauty

Sleeping Beauty is about to be put to sleep. She will be woken on Monday then put back to

sleep. If a fair coin lands Heads, she will not be awoken again. If it lands Tails, she will also be

woken on Tuesday. But the drug is such that on Tuesday she will have no memory of the

Monday awakening. So she will not know, when awoken, whether it is Monday or Tuesday. And

of course this will be true of the Monday awakening as well, as she is not told how the coin

lands. When Beauty finds herself awake, what credence should she have in the proposition that

the coin landed Heads?

There are two compelling, mutually exclusive arguments.

Half: Her credence should be 1/2, because she has learnt no new evidence that is relevant to the

coin landing Heads.

Third: Her credence should be 1/3, because if the experiment is repeated there will be twice as

many awakenings due to Tails.

Admittedly this last one is not a very good argument. Hitchcock has a better one. He imagines a

bookie who offers Sleeping Beauty various bets, but each on the outcomes of the single coin toss

that was described in the story above. The bookie has no more information than Sleeping Beauty

(otherwise Dutch books are not a sign of irrationality), so we can imagine him being subject to

the same druggings and awakenings as Beauty. Nevertheless, Hitchcock shows that Beauty can

avoid being Dutch booked if and only if she assigns Tails a credence of 1/3 on being awoken.

Let us review the betting situations that occur in the 1/2 and in the 1/3 case:

P ( E ) = 1/2:

Suppose Beauty refuses to follow Hitchcock’s advice, and stubbornly assigns P ( E ) = 1/2 on

being awoken. The bookie then offers the following set of bets: On Sunday, Beauty is offered a

bet of £15 that wins £15 if Tails lands; on each awakening, Beauty is offered a bet of £10 that

wins £10 if Heads lands; i.e.:

Sunday

Monday

Tuesday

Net

Heads

-15

-5

Tails

-10

-5

Suppose the coin lands Heads. The first bet loses £15. The second bet, on Monday, wins £10.

Beauty and the bookie sleep through Tuesday. Overall, Beauty loses £5. Suppose the coin lands

Tails. The first bet wins £15. The second bet, on Monday, loses £10. The third bet, on Tuesday,

loses £10. Overall, Beauty loses £5. Either way, Beauty loses £5. She has accepted a Dutch book.

She did so because she bet in accordance with her 50% credence that the coin landed Heads.

P ( E ) = 1/3:

Disaster can be avoided if Beauty follows Hitchcock’s advice: while she first assigns P ( E ) = 1/2

on Sunday, she changes her assignment to P ( E ) = 1/3 when awoken. She will not accept the

evens bet on Heads when awoken. The deal would have to be sweetened. A layout of £10 would

have to be rewarded with winnings of £20 (instead of £10), as Heads has fallen in probability:

Sunday

Monday

Tuesday

Net

Heads

-15

Tails

-10

-5

Now the bet looks as it should. She loses £5 if Tails, and wins £5 if Heads. It can be shown that

no Dutch book can be made against Beauty in this new setting. In order to see this, note that the

bookie is not able to distinguish the Monday awakening from the Tuesday awakening himself –

otherwise he would have more background information than Beauty has, which we want to avoid

– so he is not able to come up with two distinct bets on Monday and Tuesday in any systematic

manner. Thus, we may assume that he actually offers the same bet twice: now let the money

which Beauty would lose on Monday and Tuesday given Tails, respectively, be of amount x, and

let what she would win on Monday given Heads be of amount 2 x or more; if y is what she would

lose on Sunday given Heads while winning y or more given Tails, then it is impossible that both

the total Heads outcome - y + 2 x and the total Tails outcome y - 2 x are negative, so there is no

way Beauty is bound to lose. We can also see that 1/3 is the only probability that leads to “fair

bets” on awakening, in the sense that Beauty is equally happy to take either side of the bet. Each

waking bet costs x , so 2 x is lost if Tails. Whence the fair payoff given Heads must be 2 x . This bet

will be considered fair if and only if Beauty’s credence in Heads on waking is 1/3.

Hitchcock concludes that P ( E ) = 1/3 is the rationally required answer, which tells us

Beauty really ought to believe with 1/3 probability that the coin landed Tails. We think this is

incorrect. It is true that 1/3 is the only credence that avoids a Dutch book, but we think the

example is one in which the agent should not bet in line with her credences. The only way to

avoid a Dutch book is to bet as if one believed Heads landed with 1/3 certainty. But from this it

does not follow that the agent really is rationally required nor even permitted to believe that

Heads landed with 1/3 probability. Let us take a look at a similar example, where betting as if

one believed a proposition to 1/3 certainty will avoid a Dutch book.

3. Separating Credences From Betting Odds

What we need is a case where it is clear that the probability of a coin landing Heads is 1/2, but

nevertheless, one should bet as if the probability was something other than 1/2. This would

happen if the bet were only “actually” offered if the coin landed Tails. We propose two ways of

getting this result.

Forgery

Imagine that you knew a fair coin was about to be flipped. If the coin lands Heads, no bet will be

made. If the coin lands Tails, you are offered a bet on Heads, but not Heads of a new coin flip

but of the flip that has just taken place. Should you accept this bet? Of course not. You should

not take a bet, no matter how generous the odds, on the proposition that the coin landed Heads.

So perhaps we have a case where your betting odds have come apart from your credences? Not

yet; this is no good as it stands, because the fact that you have been offered the bet might tell you

that the coin landed Tails. You have received extra information that shifts your credences. So in

fact your credence in Tails is close to 1. And it is therefore in line with your credences not to

accept bets on Heads. Credences and betting odds are still aligned.

What we need is a way of making sure that offering the bet does not inform the agent that

the coin landed Tails. And we can do that by offering a fake bet. Imagine that instead of no bet

being offered if the coin lands Heads, a bet will indeed be offered, except with fake money. Your

notes, and the bookie’s, have been switched for excellent, but worthless, forgeries. Neither you

nor the bookie can tell the difference. If the coin lands Tails, you will be offered a bet (on Heads)

with real money. If it lands Heads, you will be offered a bet (on Heads) with fake money. Should

you take the bet? Of course not. Either the coin lands Tails and you lose real money, or it lands

Heads and you win fake money. You are much better off holding onto your real money.

Nevertheless, your credence that the coin landed Heads should remain at 1/2. Why should your

subjective degree of belief in the outcome of the coin landing event be affected by the existence

of a fake bet that you are not even aware of as being fake? So we have a case where your

credence that the (fair) coin landed Heads (1/2) should not guide your betting behaviour.

Hallucination

The point can be made even more vividly by making the example such that the fake bet does not

exist at all; it will just be in your head. As in the Sleeping Beauty case, we also add a second

time period. Suppose that if the coin lands Tails, you will be offered two real bets on Heads (of

the same flip), one after the other. There is no funny business here. But if the coin lands Heads,

you will be offered a real bet on Heads and you will also hallucinate being offered a bet on

Heads. You won’t know whether the hallucination occurs at the first stage or the second stage.

You do know that one of the bets will be real and one will be a hallucination. Whether or not you

accept the hallucinatory bet, you will later wake up and find your wallet untouched. So we have:

Stage 1

Stage 2

Tails

Real Bet

Heads

Real Bet or Hallucinatory Bet Real if the first bet was

hallucinatory, hallucinatory if

the first bet was real.

Should you accept any of these bets? No. Your credence in Tails should remain at 50%, but you

should not accept either (evens) bet on Heads. Again, we have found a case where your

credences and betting odds come apart. Hopefully this is already intuitively correct, but let us go

carefully through the reasoning: It is straightforward why the credence should stay the same.

You have the same experiences given either Heads or Tails, so you have learnt nothing that

could give you relevant information. What about the bets? We can sum over the possible bets to

find the expected utility is negative. There are 4 possible bets:

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