s="ws39">/H)9. Though Jef-
frey articulates his revised rule in contexts that focus primarily
on diachronic processes (where gamblers place different bets at
different times), the rule makes sense in our more modest con-
text. So we can ask an obvious question: is it wise for Peter to
use Jeffrey-conditionalization for his conditional bets in the cir-
cumstances sketched above, where the information released by
Paul at time τ + 3 no longer provides knowledge that an outcome
has been achieved? Such conditional bets are no longer “epistemic”
then—in the sense contrasted with ontic bets above—and we will
here call them “evidential” since they continue if and only if an
agreed piece of evidence is obtained. If he wishes to avoid a
Dutch-Book, is it sufficient for Peter to use Jeffrey-conditionaliza-
tion when placing an evidentially conditional bet?
The answer to this question should be immediately clear: it is
definitely “NO”! For if the Jeffrey-conditionalization was truly
reliable here, it would have to work in the extreme cases, when
one of μ or λ was zero. But we have already seen that this is not
so. For in such circumstances, Jeffrey’s extended conditional
probability (at least that version of it articulated above) simply
collapses to the standard conditional probability. And by now,
we have repeatedly seen that the latter is enough to allow Mary
to construct Dutch-Books against Peter.
Furthermore, this form of Jeffrey’s extended rule becomes
totally redundant—for, as I will soon show, the conditional
probability Π(X&Y*)/Π(Y*) can in fact accommodate the situa-
tions envisaged by Jeffrey, while the Jeffrey rule allows a
Dutch-Book even when both μ and λ are non-zero. Indeed,
while it is true that the focus of this essay has until now been
articulated in terms of conditions that involved Peter being
supplied with knowledge (with some outcome of the primary
chance-process being reliably verified), that has in fact been
something of a red herring. It was there primarily because it
was required to make sense of the negative claim—the claim
about standard conditional probabilities that we have rejected
here. But if we focus on the alternative positive result—the
claim that Π(X&Y*)/Π(Y*) is required to avoid a Dutch-Book—
then a little reflection will indicate that there is no necessity at
all for Y* to be restricted to the narrow meanings given it above.
It can readily embrace evidential betting as well.
Indeed, we can easily generate an extremely general condi-
9See, e.g., Jeffrey, 1983: pp. 165-166.
Copyright © 2012 SciRes. 199
K. HUTCHISON
tional betting situation by imagining two different chance proc-
esses: process 1 that produces outcomes referred to here using
early letters of the alphabet, A, B, C,... etc.; and process 2 that
produces outcomes referred to here using late letters, Z, Y, X,...
etc. Peter does not presume these processes to be statistically
independent, so allocates joint probabilities Π(A&Z) to the
outcomes A and Z..., etc.; and we use Π(A) as an abbreviation
for Π(A&2) where 2 is the universal outcome for process 2;
Π(Z) is similarly defined in the obvious analogous fashion.
We can now ask the question what probabilities should be
used for bets on the outcomes of process 1, if those bets are
conditional, and proceed if and only if process 2 produces some
outcome Y? Suppose that Peter is willing to accept the betting
ratio 1/α for a conditional bet on B that proceeds if and only if Y
is produced. We show that if α is not Π(Y)/Π(B&Y), then again
a Dutch-Book can be constructed.
We ignore the cases when Peter makes either Π(B&Y) or Π(Y)
zero, for the relevant bets will not be placed if Peter does not
believe the condition will be met, or believes that B is incom-
patible with the condition. So we can make β 1/Π(B&Y) and
γ≡ 1/Π(Y), and consider the following book:
Bet 1: Mary bets stake x on B but conditionally upon Y.
i.e. she pays Peter x, and in return:
Peter pays her αx if both B and Y are achieved.
Peter pays her nothing if Y is achieved but not B.
Peter returns her x if Y is not achieved.
Bet 2: Mary bets stake y on B&Y, unconditionally.
i.e. she pays Peter y, and in return:
Peter pays her βy if both B and Y are achieved.
Peter pays her nothing if either B or Y is not
achieved.
Bet 3: Mary bets stake z on outcome Y, unconditionally.
i.e. she pays Peter z, and in return:
Peter pays her γz if Y is achieved.
Peter pays her nothing otherwise.
Then Mary’s net winnings, in the four possible situations, viz.
B&Y, –B&Y, B&–Y, –B&–Y, (using –B for “not B”…) are:
(α – 1)x + (β – 1)y + (γ – 1)z when B&Y is achieved
xy + (γ – 1)z when –B&Y is achieved
yz for either B&–Y or –B&–Y.
Now, if β αγ, Mary can choose:
x
y and
z , in which case all
these net winnings turn out to be + 1. (This is easy to check.)
So Mary has a Dutch-Book whenever α . But α =
if
and only if Π(B&Y)/Π(Y) is the rate accepted by Peter for the con-
ditional bet. In other words, Peter can be Dutch-Booked if he does
not use this rate. This is the very general result promised above.
Superficially, this result might seem to be little more than the
standard result for ontic conditional bets, but, despite the mis-
leading similarity in expression, the result here is very different,
and far broader. The standard result is in fact an extremely par-
ticular case of this general result, the particular case that pre-
sumes process 1 to be identical to process 2. The result pursued
in this paper for epistemic conditional bets is similarly a par-
ticular case of this more general result, that particular case
when a) process 2 takes place after process 1, and b) consists of
the release of knowledge about the outcome of process 1. The
evidential case is the case where again process 2 takes place
after process 1, where process 2 again involves the inspection
of the outcome of process 1, but where that inspection produces
outcomes that are experiences which fall short of supplying
knowledge about the outcomes of process 1.
Evidential conditional bets do not then require a special new
rule, but fit exactly the same generalized pattern as the ontic
and epistemic bets. Worse, we have seen that the particular rule
proposed by Jeffrey does in fact allow Mary to construct some
Dutch-Books. Whatever (dubious) virtues it might offer for
diachronic betting, it has nothing to offer for the synchronic
bets at issue here.
To give a concrete illustration of this claim that evidential
betting requires no special treatment, let us return to the exam-
ple introduced above (at the beginning of this discussion of
Jeffrey-conditioning), when the need for some expansion of the
rule for Peter’s choice of betting ratio was suggested by the
introduction of poor lighting. Paul’s “Tail”-utterance then ceased
to guarantee that there was a tail on the coin that had been drawn
at time τ, so bets conditional upon the “Tail”-utterance no longer
qualify as “epistemic” (in the sense contrasted with ontic bets
above). But we now know that Peter can be Dutch-Booked if he
accepts a bet on TT that is conditional upon the “Tail”-utterance,
at a rate that is not p(TT&“Tail”)/p(“Tail”).
Peter can calculate this safe betting ratio from his assess-
ments of the probabilities that each of a tail and a head will be
identified as a Tail. Suppose (to be concrete) that he assesses
these as 75% and 15% respectively. Then he will assess
p(“Tail”), the probability of the “Tail”-utterance, as 12(75%
+ 15%) or 45%. Similarly p(TT&“Tail”) will be 75% of 13
,
or 25%. So his safe betting ratio (for the conditional bet on TT,
that proceeds if and only if “Tail” is uttered) in these circum-
stances is 25%45% or 59
55.555...%.
To perform this calculation, Peter needed slightly more data
than is required to apply Jeffrey’s rule, which only used the
probability Peter would allocated to a tail if Paul made the
“Tail”-utterance. The data required for Jeffrey’s rule can be
calculated from that required for our rule, but not vice-versa, so
Jeffrey’s rule is more frugal than ours. Ours however is more
secure: every departure from it generates Dutch-Books.
In the example just given, a “Tail”-utterance leaves Peter
7590 certain that the coin has a tail on it, while a “Head”-
utterance leaves him 85110 certain that the coin has a head
on it. So according to Jeffrey’s rule, the updated probability for
two tails after hearing the “Tail”-utterance, will be 75 180 =
512 41.666...%. This differs considerably from our
55.555...%, but, if used by Peter, allows Mary to set up the
following Dutch-Book:
Bet 1: Mary bets stake $20 on TT but conditionally upon the
“Tail”-utterance.
i.e. she pays Peter $20, and in return:
Peter pays her $48 if “Tail” is uttered and TT is re-
vealed at time τ + 5, for by hypothesis the betting
ratio here is the Jeffrey-conditional 512.
Peter pays her $0 if “Tail” is uttered but TT is not
revealed.
Peter returns her $20 otherwise.
Bet 2: Peter bets stake $12 on TT&“Tail”, unconditionally.
i.e. he pays Mary $12; and in return:
Mary pays Peter $48 if “Tail” is uttered and TT is
revealed at time τ + 5, for Peter’s rate here is 14.
Mary pays Peter nothing if neither “Tail” is uttered
nor TT is revealed.
Bet 3: Mary bets stake $9 on the “Tail”-utterance.
Copyright © 2012 SciRes.
200
K. HUTCHISON
Copyright © 2012 SciRes. 201
the urn or Paul’s actual epistemic procedure, he becomes ex-
posed to (weak) Dutch books. So avoidance of a sure loss re-
quires Peter to do more than just allocate an appropriate condi-
tional probability. And it is surely impossible to establish that
Peter’s avoidance of one particular irrational bet protects him
from all others, right through his life.
i.e. she pays Peter $9; and in return:
Peter pays her $20 if “Tail” is uttered, for Peter’s
betting ratio here is now 45% or 9/20.
Peter pays her nothing otherwise.
Mary will now make a profit of $3 no matter what outcome
was produced in the original draw, or what utterance Paul
makes at τ + 3. This is clear in the tabulation of her earnings in
Table 3 below.
Yet some partial converses must surely hold, since we all do
believe that Peter can sometimes make conditional bets without
being Dutch-Booked. But it remains unclear what restrictions
need to be placed on such converses to make them valid. A
converse claim restricted to stronger books could well survive,
but any such claim would have to be carefully formulated, and I
do not know of any claims that are clearly valid.
An Inconclusive Conclusion: The Search for
a Converse
Having concluded, in the most general of our cases above, that
Peter can be Dutch-Booked if he accepts bets on an outcome B
of process 1 that are conditional on the outcome Y of process 2
at any rate other than Π(B&Y)/Π(Y), one is tempted to ask if
some reasonable converse of this result exists. Can we argue
that Peter is safe if he does use such a betting rate for each of
his conditional bets?
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Table 3.
Mary’s winnings: counter-example to Jeffrey rule.
Jeffrey, R. (1983). The logic of decision (2nd ed.). Chicago, IL: Uni-
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Bet 1: TT/“Tail”
(512)
Bet 2: TT&“Tail”
(14)
Bet 3: “Tail”
(920)
Outcome Loss Gain Loss Gain Loss Gain Net Gain
HH&“Tail” $20 $0 $0 $12 $9 $20 $3
HH&“Head” $20 $20 $0 $12 $9 $0 $3
TT&“Tail” $20 $48 $48 $12 $9 $20 $3
TT&“Head” $20 $20 $0 $12 $9 $0 $3
HT&“Tail” $20 $0 $0 $12 $9 $20 $3
HT&“Head” $20 $20 $0 $12 $9 $0 $3
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