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&Ω2) where Ω2 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

–x – y + (γ – 1)z when –B&Y is achieved

– y – z 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-

versity of Chicago Press.

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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