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
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:
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 =
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
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
Bet 1: Mary bets stake $20 on TT but conditionally upon the
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
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.
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