## 1. The Basic Dutch Book Argument for Probabilism


### 1.1 The Probability Axioms and the Dutch Book Theorem



The conclusion of the basic DBA is that the degrees of belief, or
credences, that an agent attaches to the members of a set \(X\)
of propositions, or sometimes statements or
sentences, should satisfy the axioms of probability. The basic
axioms of probability are generally taken as requiring that for \(A
\in X\),



(1)\(0 \le \pr(A)\) [Non-Negativity];
(2)if \(A\) is a tautology, then \(\pr(A) = 1\) [Normalization];
(3)if \(A\) and \(B\) are incompatible, then \(\pr(A\vee B) = \pr(A) + \pr(B)\) [Finite Additivity].


Condition 2, which requires only that the truths of propositional
logic are given the value one, is sometimes replaced with



(\(2'\))if \(A\) is a logical truth, then \(\pr(A) =1\),


or even



(\(2''\))if \(A\) is a necessary truth, then \(\pr(A) =1\).


The distinctions between these formulations of the axioms are
connected with the objects that probabilities are appropriately
attached to and on the reasonableness of the argument's
conclusion; but, for the immediate purpose of outlining the basic
argument, the differences are not crucial.



The DBA itself begins with the so-called Dutch Book theorem, which
concerns the conditions under which a set of bets guarantees a net
loss to one side, or a Dutch Book. With de Finetti, it is here assumed
that a bet on a proposition \(H\) is an arrangement
that has the following canonical form:




|  |  |
| --- | --- |
| \(H\) | Payoff |
| True | \(S - qS\) |
| False | \(-qS\) |



The table gives the net payoff to an agent who buys a bet with stake
\(S\) for the price \(qS\), where \(S\)
is won if \(H\) is true. \(S\) is called
the *stake*, as it is the total amount involved in the wager,
that is the payoff in the case that \(H\) is true
together with the amount forfeited if \(H\) is false.
The quantity \(q\) is called the *betting
quotient*, which is the amount lost if \(H\) is
false divided by the stake. The Dutch Book theorem can now be
stated:



Given a set of betting quotients that fails to satisfy the probability
axioms, there is a set of bets with those quotients that guarantees a
net loss to one side.



It is easy to show how it is possible to make book against someone
with betting quotients that violate the probability axioms. Let
\(Q(H)\) be the agent's betting quotient for \(H\).
Assuming that the agent's betting quotients
violate the axioms, a bookie can guarantee himself a profit by placing
bets with the agent as described below. For simplicity the stake is
set here at $1, but the following recipes for constructing a book
against such a person are easily adapted for other stakes.



Axiom 1: Suppose that \(Q(H) \lt 0\). In this case, the bookie buys
the bet that pays $1 if \(H\) is true and 0 otherwise,
for the negative price \(Q(H)\), which means that the agent collects
\(Q(H)\), and pays out $1 if \(H\) is true, and $0
otherwise. Here the agent is betting against \(H\) and
the payoff table for the agent is as follows:




|  |  |
| --- | --- |
| \(H\) | Payoff |
| T | \(-[1 - Q(H)]\) |
| F | \(Q(H)\) |



Since \(Q(H)\) is negative, the agent will suffer a net loss
regardless of the truth-value of \(H\).



Axiom 2: Suppose that an agent's betting quotient for a
tautology (or a logical or necessary truth) \(H\) is
not equal to 1. If \(Q(H) \gt 1\) the bookie sells the bet that pays
$1 if \(H\) is true and 0 otherwise, for \(Q(H)\). In
the case where \(Q(H) \lt 1\), the bookie will buy the bet in which
the agent pays the bookie $1 if \(H\) is true and
nothing if \(H\) is false, for \(Q(H)\). In this case,
the payoff table for the agent will again be as above for axiom 1.
Notice that since \(H\) is a tautology (or logical or
necessary truth) it must be true, which means that at the conclusion
of the bet, the agent will have lost [\(1-Q(H)\)].



Axiom 3 (Additivity): Assume that \(H\_{1}\) and \(H\_{2}\) are mutually
exclusive and that \(Q(H\_{1}\vee H\_{2}) \ne Q(H\_{1}) + Q(H\_{2})\).
There are two cases,

 

\[\begin{align} 
Q(H\_1\vee H\_2) &\gt Q(H\_1) + Q(H\_2), \text{ and} \\ 
Q(H\_1\vee H\_2) &\lt Q(H\_1) + Q(H\_2). 
\end{align}\]


If \(Q(H\_{1}\vee H\_{2}) \lt Q(H\_{1}) + Q(H\_{2})\), then the bookie
will offer the agent the bet that pays $1 if \(H\_{1}\) and 0 otherwise
for \(Q(H\_{1})\) and the bet that pays $1 if \(H\_{2}\) is true and 0
otherwise for \(Q(H\_{2})\). The bookie then buys the bet that will pay
him $1, if \((H\_{1}\vee H\_{2})\) is true and 0 otherwise, for the
price of \(Q(H\_{1}\vee H\_{2})\). The possible payoffs to the agent are
summed up in the following table:




|  |  |  |
| --- | --- | --- |
| \(H\_{1}\) | \(H\_{2}\) | Net Payoff |
| T | F | [\(1 - Q(H\_{1}) - Q(H\_{2}) + Q(H\_{1}\vee H\_{2}) - 1\)] |
| F | T | [\(1 - Q(H\_{1}) - Q(H\_{2}) + Q(H\_{1}\vee H\_{2}) - 1\)] |
| F | F | [\(-Q(H\_{1}) - Q(H\_{2}) + Q(H\_{1}\vee H\_{2})\)] |



Since \(Q(H\_{1}\vee H\_{2}) \lt Q(H\_{1}) + Q(H\_{2})\), the agent loses
in each case and thus the collection of bets assures a loss. If
\(Q(H\_{1}\vee H\_{2}) \gt Q(H\_{1}) + Q(H\_{2})\), then the bookie simply
reverses the direction of the bets.



Letting \(V(H)\) be the payoff if \(H\) is true, the
expected value of a bet on \(H\) is expressed by the
equation:

 

\[ \text{Exp}(H) = V(H)Q(H) + V(-H)(1-Q(H)). \]


Thus, for each axiom, the individual bets involved in making the book
are fair, which is to say they have an expected value of zero, when
calculated using the agent's betting quotients, yet collectively
they will produce a sure loss. The Dutch Book argument assumes that an
agent's degrees of belief are linked with her betting quotients.
This, together with the Dutch Book theorem, is used to establish
that degrees of belief that violate the probability axioms are
associated with bets that are fair in the above sense, yet are also
tied to a sure loss. The argument then concludes that agents ought to
obey the axioms. This leaves open just what the association amounts to
and what sort of problem the prospect of such sure loss is supposed to
be.


### 1.2 A Detailed Version of the Dutch Book Argument



The Dutch Book argument has often been presented as establishing that
degrees of belief that violate the axioms are irrational because they
can (or do) lead to bad consequences. It assumes that an agent's
degrees of belief are associated with her betting quotients such that
(1) for an agent with degree of belief \(q\) in \(M\)
a bet on or against \(M\) at
corresponding odds will be acceptable. With payoffs in dollars and
stake \(S\), this amounts to the claim that the agent
should be willing to take either direction of a bet with a cost of
$\(Sq\) that pays




|  |  |
| --- | --- |
| \($S\) | if \(M\) |
| \($0\) | otherwise |



But then (2) by the Dutch Book theorem a cunning bettor could assure
himself a profit from someone who violates the probability axioms.
Since (3) violating the axioms leaves the bettor open to being Dutch
Booked (that is, being on the losing side of the Dutch Book) because
her degrees of belief make acceptable bets leading to a sure loss, it
is concluded that (4) one ought to satisfy the probability axioms
(i.e. that probabilism is true). In some presentations, the first
premise is strengthened to the claim that the agent actually will
accept those bets on \(M\) for which her degree of
belief matches the betting quotient, in an effort to tighten the
connection between violating the axioms and the sure loss.



A bit of terminology is in need of clarification here. De Finetti
identified degrees of belief with betting quotients and termed degrees
of belief that are susceptible to a Dutch Book *incoherent*;
those that are not so susceptible he called coherent (de Finetti
1937). 'Susceptible' here should be understood in the
sense of the above theorem, namely that bets are specifiable
corresponding to those degrees of belief that will produce a sure loss
to one side. Given the theorem, coherence amounts to satisfaction of
the probability axioms, with incoherence involving their violation,
and accordingly the terms are often used as a shorthand way of
specifying whether the axioms are satisfied. However, both Ramsey and
de Finetti understood incoherence to be a kind of inconsistency, and
some use the term in this sense. There are various questions about
understanding violation of the probability axioms and susceptibility
to a Dutch Book as a kind of inconsistency, as will be
discussed, and so here it will be best to use 'incoherent'
for degrees of belief that violate the probability axioms, and that by
the Dutch Book theorem are associated with susceptibility to a sure
loss, and leave open whether incoherence understood as such involves a
form of inconsistency.


### 1.3 The Converse Dutch Book Theorem



There is reason to accept the conclusion of the DBA that one ought to
satisfy the probability axioms so as to avoid a Dutch Book only
insofar as satisfying the axioms affords at least the possibility of
avoiding exposure to a sure loss (Hajek 2005; 2008a,b). Here, the
Converse Dutch Book theorem, proven independently by Lehman (1955) and
Kemeny (1955), plays an important role. Roughly stated the result is
that



> 
> For a set of betting quotients that obeys the probability axioms,
> there is no set of bets (with those quotients) that guarantees a sure
> loss (win) to one side.
> 



It assures that the agent would not be exploitable in any event, and
thus establishes an advantage to obeying the axioms. Notice that the
theorem states that having betting quotients adhering to the axioms
assures that one will not be susceptible to a sure loss, rather than
making just the minimal claim required to establish an advantage to
coherence that such vulnerability may be avoided.



Both the Dutch Book theorem and its converse are sensitive to the
formulation of the axioms, as well as to the understanding of
'bet', 'sure loss' and what it means for such
a loss to be guaranteed. Special care must be taken with the
characterization of the probability axioms when it comes to the
Converse Dutch Book theorem. Given that the axioms are formulated such
that the second axiom only requires that tautologies receive
probability one, it is possible to satisfy the axioms, yet still be
open to a sure loss. Consider the claim that if \(b\)
has \(F\) then something has \(F\). This
is not tautological, but betting against it would leave one vulnerable
to a sure loss. Sometimes the second axiom is instead formulated as
requiring that all logical truths receive probability one, but
satisfying this constraint leaves open the possibility of a sure loss
by betting against a necessary truth such as 'nothing is both
red and green all over'. Even with strengthening the second
axiom to require that all necessary truths receive probability one,
there is still a reading upon which the Converse Dutch Book theorem is
false, since an agent will be vulnerable to a sure loss if she
attaches a probability less than one to a known truth (or a
probability greater than zero to a known falsehood). One response to
this is to restrict 'sure loss' to those losses that do
not depend on contingent facts. Instead the restriction could be made
to losses that are 'sure' in the sense that there is a
mechanical formula for inflicting the loss, thus removing the sort of
counterexample to the Converse Dutch Book theorem with which we began,
and the need to strengthen the axioms. A related move involves
restricting the allowable bets, rather than limiting what counts as a
sure loss.



In observing that the Dutch Book theorem and the DBA are sensitive to
the formulation of the probability axioms, it should be noted that
whereas classical logic is generally presumed, probability axioms can
be formulated, with suitable adjustments, for non-classical logics.
Weatherson argues that this is appropriate in representations of
rational belief, and formulates axioms of probability that are built
upon an intuitionistic logic, for which he then gives a Dutch Book
argument (Weatherson 2003).



It is clear that in order for the Dutch Book theorem to hold,
'sure loss' must be taken to mean loss if the bets are in
fact placed and settled. Violation of the probability axioms under any
of their formulations does not guarantee an actual loss. Still more
must be said here about the meaning of 'sure loss', which
suggests some sort of necessity but does not tell us whether this is
supposed to be a particular form of logical necessity, metaphysical
necessity, or perhaps some other relative necessity. If 'sure
loss' were taken as foreseeable loss, then an agent could
violate the axioms by attaching positive probability to a necessary
falsehood, where there is no foreseeable loss given the current state
of knowledge concerning the proposition in question. There is also an
issue concerning whether it is the agent who must be able to foresee
the loss. If 'sure' means decidable, then neither the
formulation that logical nor that necessary truths receive probability
one will do, since there is no decision procedure for determining in
general whether a given sentence is a logical truth, let alone a
necessary one. Clearly, there is a delicate balance that must be
struck between the formulation of the probability axioms, and the
understanding of 'bets' and 'sure loss' for
the Dutch Book theorem and its converse to hold.



There is a further issue with the Converse Dutch Book theorem, because
there are books that can be made against agents who violate other
probabilistic norms, such as Reflection, countable additivity and
others (see sections 3, 4, and 5). Satisfaction of the basic axioms is
no guarantee that one will not be open to a book due to violation of
some other norm. A correct formulation of the theorem must accordingly
restrict the form and number of the allowable bets. Kemeny and Lehman
put restrictions on the allowable bets (for example their betting
systems are restricted to a finite number of bets), and these seem to
rule out such counterexamples. However, it remains to be shown that
avoiding book with such a restricted set of bets suffices to justify
adherence to the axioms. The problem here becomes especially pressing
with the observation, discussed in the following section, that
incoherent agents need not be generally vulnerable to a sure loss, and
thus that both coherent and incoherent agents alike may be subject to
some books and not others. What is needed in arguing for adherence to
the probability axioms is the further claim that the bets which lead
to sure losses and that are associated with incoherence pose a special
problem, although this threatens the use that many proponents of the
DBA have wanted to make of Dutch Book arguments in defending other
norms.



A related problem concerns the assumption made in the version of the
DBA presented in section 1.2 that agents will accept, or at least find
acceptable, those 'fair bets' on or against \(P\)
where their degree of belief matches the betting
quotient for the bet. As discussed in the next section, it is not true
that all such bets will be accepted, or even be regarded as
acceptable, and additional bets may be accepted (or found acceptable).
For various reasons someone might accept bets in cases where her
credences do not correspond to the betting quotients of those bets,
including where the bets constitute a Dutch Book. Thus, we need to be
concerned more generally with the array of bets that agents will
accept (or find acceptable). But since this widens the class of bets
that can be used to produce a book, if anything it makes the problem
for the DBA that a coherent agent may be subject to a Dutch Book even
worse. The version of the DBA in section 1.2 could be modified with
the claim that agents accept (or find acceptable) only bets that are
fair or favorable, that is those with a non-negative expected value as
calculated using their degrees of belief. This rules out some
additional bets that would produce a Dutch Book for a coherent agent,
but not those that enter into the DBAs for other norms. However, this
assumption does not hold generally, and fails to address other issues
confronting the argument.


### 1.4 Can it be Rational to Violate the Probability Axioms?



As noted above, the sure loss guaranteed by the Dutch Book theorem
need not amount to an actual loss. Indeed a bookie could reverse the
direction of the bets that would guarantee a loss for an incoherent
agent so as to guarantee her a win, or there could be some sort of
other prize that might be awarded for being incoherent. An incoherent
agent might not be confronted by a clever bookie who could, or would,
take advantage of her, perhaps because she can take effective measures
to avoid such folk. Even if so confronted, the agent can always
prevent a sure loss by simply refusing to bet. It won't do to
respond to such a possibility by insisting on the stronger assumption
that an agent will always take bets where the expected value of each
is non-negative, because this simply gives the argument a false
premise. In any case, it is clear that the trouble with being
incoherent cannot be that it will lead to losses. Rather, the idea
behind the version of the argument in section 1.2 should be taken to
be that incoherence opens one up to sure losses, in a way that having
coherent beliefs does not (given an appropriate version of the
Converse Dutch Book theorem), and that such potential vulnerability
demands adherence to the axioms.



We may question whether either the conclusion that one ought to
satisfy the axioms or that incoherence is irrational follows
from simply being open to a sure loss. If the threat of such a loss is
seen as unlikely, say, if the agent thinks she won't face a
clever bookie, has confidence that her charms will prevent her from
ending up on the losing side of a Dutch Book, or just thinks she
won't take bets leading to a Dutch Book if they are offered,
then it is hard to see why the mere potential of a sure loss should
demand coherence. Another problem here is the special status accorded
to sure losses. In fact the potential sure loss is symmetric with a
possible sure gain. As Hajek emphasizes, there is a
corresponding "Czech Book Argument", which parallels the
DBA, with the conclusion that one ought to violate the probability
calculus (Hajek 2005, 2008a). Construed as above, the DBA
appears to be canceled out by the "Czech Book Argument",
although Hajek shows that the potential canceling out can be
avoided by reformulating the DBA with the assumption that the agent
should accept bets that are either fair or favorable (Hajek
2008a). Still, it is compatible with the Dutch Book theorem that an
incoherent agent could end up on the side of a sure gain, and so
incoherence cannot be condemned as irrational simply by citing the
possibility of sure losses.



Although the threat of a sure loss may be remote, and an agent could
face a sure gain, there are some situations in which incoherence is
more closely tied to actual losses. Some versions of the DBA appeal to
such scenarios as a preliminary to attempting to establish the general
irrationality of being incoherent. One tactic has been to argue that
incoherence is irrational in so-called forced betting situations, and
then that the prohibition against being incoherent in such situations
can be universalized (Jackson and Pargetter 1976). In a forced betting
situation the agent is required to post betting quotients, and then to
place bets with those quotients, where the stake and direction of the
bets are dictated by an opponent. Assuming that the only source of
value is the money (valued linearly) involved in the wagers, it is
claimed that it is irrational to be incoherent in such a situation.
But this doesn't follow if the irrationality is taken as arising
from actual losses that would result. Even in a forced betting
scenario, there may be reasons to suppose that incoherent betting
quotients will not be taken advantage of, as may occur if making book
would require knowledge of some very complicated logical truth
(Kennedy and Chihara 1979). Here too, the direction of the bets could
be chosen so that an incoherent agent will have a sure gain. This
would be an odd forced betting situation, but if the agent had reason
to believe that this would occur, it could actually be advisable for
her to post incoherent odds.



One might begin instead with a competitive betting situation, in which
it is given that both sides attempt to maximize their gain. Here there
is a closer connection between incoherence and loss, since the bookie
will be trying to maximize his profit, and so a sure gain for an
incoherent agent could only be the result of a mistake. If it is
assumed that the agent is also forced to bet and that the bookie will
act optimally, the agent should post coherent odds, for otherwise she
will be subject to a bankrupting Dutch Book. Of course if the
competitive situation is one in which the agent can refuse to bet,
having incoherent betting quotients need not leave her open to an
actual loss.



Even in those forced betting situations and competitive betting
situations in which it is irrational to post incoherent betting
quotients, it still need not be irrational to have incoherent degrees
of belief (Kennedy and Chihara 1979; Adams and Rosenkrantz 1980). For
example, if the agent must post odds on what she knows to be either a
logical truth or a logical falsehood in a forced betting situation,
she may be better off with a betting quotient of either 1 or 0, rather
than some intermediate value, for she is susceptible to a Dutch Book
in the latter event, but she could end up with the objectively correct
value in the former case and thus would avoid a loss. But it seems
that she can have reasons for an intermediate level of confidence,
making such an evaluation more rational, at least in the sense of
reflecting her available evidence, than an extreme one. Alternatively
the agent may simply have no idea whatsoever about the logical status
of what is in fact a logical truth upon which she must post odds, in
which case a confidence of .5, or perhaps remaining entirely agnostic
by not adopting any particular level of confidence, seems more
reasonable than her being fully confident despite her ignorance.



At the least, such examples raise questions about the connection that
the DBA presupposes between partial beliefs and favorable/unfavorable
betting odds. In the case where an agent must give her betting
quotient for what is in fact a logical truth, any value less than one
leaves her vulnerable to a sure loss, possibly a bankrupting one, yet
in such cases not all such credences are rationally on a par. The
issue here seems to be more than simply one about how well the
underlying theory of action and belief works for extreme values, but
also about the relevant conception of rational belief. Rational
action-guiding credences should generally reflect the agent's
evidence, yet at least in the cases above, probabilism seems sometimes
to require that they do not. One may question whether the appropriate
or intended sense of 'rational' is invoked in these cases
in which it seems right to say that rational beliefs can come apart
from betting quotients, but such objections just highlight the lack of
clarity in most presentations of the argument over what sort of
rationality is supposed to be at stake. It has been said, for
instance, that it is ideal rationality that demands adherence to the
axioms, but this notion too is unclear. What can be said though is
that there are cases in which degrees of belief do satisfy an
important ideal of rationality but where they seem to come apart from
betting quotients, and this puts additional pressure on the premise
that agents should be willing to accept bets where their degrees of
belief match the betting quotient.



Neither a forced nor a competitive betting situation per se guarantees
that incoherence will result in actual losses, and even in the special
cases in which it would bring them about, it appears that having
incoherent degrees of belief need not be irrational. Indeed, whereas
degrees of belief may generally be thought of as guides to action, and
as well-modeled by betting quotients in many cases, forced and
competitive betting situations in fact reinforce the point that they
are not always straightforwardly linked. Thus, arguments for the claim
that incoherence is generally irrational that begin with the claim
that it is irrational in forced betting situations fail to get off the
ground, let alone show that it is irrational outside of such
situations, where the connection between incoherence and the
possibility of losses is even more tenuous. Still, forced and
competitive betting situations are helpful in identifying rational
constraints under idealized conditions, and can serve as useful models
of action in some situations. Under highly circumscribed conditions,
where the agent's goals are limited, they show that it is
prudent to post coherent betting quotients. It is a bad feature of
incoherent betting quotients, whether or not they correspond to such
degrees of belief, that in a forced betting situation they provide the
opportunity for a clever bookie to inflict a sure loss, and guarantee
it in competitive ones with a utility-maximizing opponent. But, the
special conditions that make it rational to have coherent betting
quotients, so as to avoid a sure loss, are not met in general. In some
circumstances, a small sure loss can be better than the chance of a
greater loss. Additionally, where there is something of value that is
linked to a wager besides the stake, it can also be rational to act so
as to bring about a sure loss on a bet.



Aside from the reasons given above for the fact that it can be
rational to leave oneself open to, or act so as to guarantee, a sure
loss, situations in which the value of a bet is not fully represented
by its monetary payoff conditions present challenges for versions of
the DBA that take the problem with incoherence as stemming from the
threat of losing money. An agent may not wish to risk losing money on
a large bet, due to additional bad consequences that would or might
ensue, or the agent may suffer anxiety over the prospect of loss. De
Finetti and others insist on limiting the stakes in an effort to get
around this problem, but Ramsey already observed that one might have a
"reluctance to bother with trifles", which is to say that
there is a negative value to placing bets with little potential gain.
Perhaps there is some range for which the stakes are neither too high
nor too low, such that the monetary payoffs may be taken as at least a
reasonable approximation of the value of the bets, but this further
limits the situations in which a sure loss may be inflicted.
 



It
is
standardly assumed, as in the version of the DBA given in section
1.2, that an agent will be willing to buy or sell a bet on \(S\) for which
her credence matches the betting quotient. But in practice,
buying a bet may have a different value for an agent than selling that
bet, because there is typically a lag between the time that a bet is
placed and when it is settled, as Rowbottom (2007) points out.
 Selling a bet involves collecting a sum up front and potentially
paying out latter, which often has greater value than paying the same
price up front and with the potential payoff obtained later.
 This asymmetry between the value of buying and selling bets also
constrains the situations in which a Dutch Book can be
made.



Another problem involving the value of bets arises in establishing the
additivity axiom, for even if each in a set of bets is acceptable, it
does not follow that they would be jointly acceptable. Consider a
person with 4 dollars, who needs 2 dollars for bus fare. He might be
willing to take either one of two wagers that cost 2 dollars for a
chance at winning enough money to buy a newspaper to read on the ride,
but be unwilling to risk all 4 dollars, and the possibility of having
to walk home. One response here is to insist that the bets should be
presented in terms of utilities rather than money, although this
raises difficulties of its own, since these are not objective
commodities. Such an approach involves assuming either directly or
indirectly that such utilities are additive, but even this is not
enough. While it may plausibly be argued that given a set of credences
and utilities, pragmatic rationality requires maximizing expected
value with respect to those credences and utilities, so that an agent
with incoherent degrees of belief is committed to accepting bets
(paying off in utility) that lead to a sure loss, the problem for the
DBA remains that being so committed to a sure loss is still not
necessarily irrational. See (Maher 1993).



It is clear that the version of the DBA presented in section 1.2 is
far from cogent. There is a gap between the agent's degrees of
belief and the acceptance of bets that would produce a sure loss, and
the possibility of such a loss need not entail irrationality. One can
specify situations in which the connections are tighter, but this
fails to establish the general claim of probabilism.


## 2. The Dutch Book Argument and Probabilistic Consistency


### 2.1 The Dutch Book Argument as Revealing Inconsistency



A popular response to the objections to the DBA considered in section
one suggests that incoherence is not a pragmatic defect per se, and
that the Dutch Book is merely a dramatic device for illustrating a
kind of logical defect. Skyrms (1987) attributes this reading of the
DBA to Ramsey, finding support in his remarks that



> 
> Any definite set of degrees of belief which broke them [the laws of
> probability] would be inconsistent in the sense that it violated the
> laws of preference between options...If anyone's mental
> condition violated these laws, his choice would depend on the precise
> form in which the options were offered him, which would be absurd. He
> could have a book made against him by a cunning bettor and would stand
> to lose in any event. (Ramsey 1926, p. 41)
> 



Additional support for this interpretation can be found in
Ramsey's assertion that the subject of his paper is the
*logic* of partial belief.



Many authors, including Armendt (1993), Christensen (1996, 2004),
Hellman (1997), Howson and Urbach (1993), and more recently, Briggs
(2009) and Mahtani (2015), have endorsed and elaborated upon the
idea that violation of the probability axioms amounts to a kind of
inconsistency. For example, Armendt (1993) tells us that it involves
an inconsistency that is revealed by the fact that degrees of belief
guide action, and that Dutch Book vulnerability amounts to giving
conflicting evaluations to the same option(s). He calls this
"divided-mind" inconsistency, which he says is a flaw of
rationality. This fits the typical sort of case where an agent
violates additivity, and sometimes applies to violations of the other
axioms, but incoherence does not always entail having two different
evaluations of the same option. Although degrees of belief often serve
as guides to action, they need not be tied to an evaluation of options
at all. However, it is a fundamental assumption of most versions of
the DBA that credences do have such a role, as in the theories of
Ramsey and de Finetti, so it is of interest that even with this
assumption, incoherence need not involve giving conflicting
evaluations (Vineberg 2001). Consider for instance a person who is
less than fully confident of Fermat's Last Theorem, perhaps
because they don't know that it has been proven. On at least
some ways of understanding propositions, this need not entail
attaching two different judgments to the same proposition. One could
also violate coherence by only attaching a probability of less than
one to a single proposition that is a logical truth, or one could
avoid different evaluations by having the same level of confidence in
every proposition. These examples aside, even the additivity axiom is
problematic on Armendt's interpretation, for it requires, for
mutually exclusive propositions \(p\) and \(q\),
that joint bets on \(p\) and on \(q\)
amount to the same option as a bet on \(p\)
or \(q\). This in turn presumes that
values are additive, which goes beyond simply having degrees of
confidence in \(p, q\), and their disjunction, although Armendt (1993)
suggests that the assumption is generally met and appropriate for the
purposes of the DBA.



In "Truth and Probability", Ramsey assumes that, at least
under idealized conditions, degrees of belief manifest themselves in
preferences for options. This allows Ramsey to characterize the
inconsistency involved in violating the probability axioms in terms of
violating the axioms of rational preference. While he never claims
that degrees of belief are necessarily linked to preferences, the
model of belief and preference that he offers assumes such an
association, and indeed a great achievement of the paper is what
amounts to a representation theorem establishing that an agent
satisfying the axioms that he specifies for rational preference can be
represented as having degrees of belief that satisfy the probability
axioms. In the model he offers, degrees of belief serve as guides to
action through their connection with preferences, so that therein at
least incoherence does manifest itself in preference
inconsistency. But the way that Ramsey casts the inconsistency
involved in violating the axioms in terms of preference inconsistency
leaves it unclear that his goal of characterizing the logic of partial
belief has been satisfactorily met. One trouble is that it does not
show us that the preference inconsistency to which incoherent beliefs
are tied is appropriately analogous to inconsistency for simple (or
full) belief, which involves believing inconsistent propositions, i.e.
a set of propositions that cannot all be true. Another worry is that
in Ramsey's model degrees of belief are linked with preferences,
and it seems that having a degree of belief does not by its nature
require any such connection with preference and action. One can surely
have finely graded beliefs in propositions without those beliefs being
connected to preferences. Moreover, consistency for full belief
derives from the concept of belief as holding propositions to be true
and from the logic of propositions, without any such assumed tie to
action, and although Ramsey thought that such a connection was
necessary to clarify the idea of partial belief, it seems that a fully
analogous characterization of consistency for partial belief should do
without it. 


### 2.2 Depragmatized Arguments



Several philosophers have sought to establish coherence as a
consistency constraint on degrees of belief through a version of the
DBA that does not presuppose any definite connection between degrees
of belief and preference (Christensen 1996, 2004; Howson and Urbach
1993; Hellman 1997). These so-called *depragmatized* arguments
purport to improve upon Ramsey's treatment by making clear how
coherence is supposed to mirror the ordinary notion of consistency.
Howson and Urbach attempt this by identifying an agent's degree
of belief in \(M\) with the bets she regards as fair,
such that for credence \(q\) in \(M\), a
bet with cost $\(Sq\) that pays




|  |  |
| --- | --- |
| \($S\) | if \(M\) |
| \($0\) | otherwise |



is taken as fair. They then employ the Dutch Book theorem to argue
that incoherent degrees of belief cannot in fact be fair betting
quotients, and thus that incoherence involves taking a set of bets as
fair that cannot be fair, which is supposed to parallel having
inconsistent beliefs.



There are several problems that undermine Howson and Urbach's
approach. The most basic is that an agent with degree of belief \(q\)
in \(M\) need not regard bets of the
above form as fair for a variety of reasons. To begin with she may
simply lack the concept of a fair bet. Secondly, she could have
credence \(q\), yet acknowledge the bet above as
explicitly unfair. For instance, if she knows \(M\) to
be either a logical truth or a logical falsehood, but has no idea
which, her confidence that it is true could be 1/2, though she
may well know that this would not be a fair betting quotient for it.
Casting credences in terms of the bets that are regarded as fair is
also problematic in that it appears to associate degrees of belief
with having certain full beliefs, which some take to be problematic.
The argument also clearly falters over the additivity axiom, since
even if an agent regards each of a set of bets as individually fair,
they may not be collectively fair by her lights, so she need not
evaluate the bets required to produce a book as fair bets. Here, and
in the basic assumption about the individual bets that an agent finds
fair, it is in effect assumed that money acts as a measure of value.
Not only is this not generally true, it calls attention to the fact
that the argument has not really been depragmatized. While Howson and
Urbach do not rely on either the assumption that agents will (or
should) act in accordance with their degrees of belief or that they
should be willing to accept either direction of a fair bet, they do
assume that agents make evaluations of bets, which are tied to the
concept of pragmatic value. Maher (1997) argues that their way of
spelling out fairness in terms of a primitive concept of advantage, so
as to avoid the notions of preference and utility, fails. Indeed, it
is hard to see that the latter notions can be avoided in explicating
the concept of fairness, upon which their argument depends.



A similar argument is offered by Christensen (1996), except that his
depragmatized DBA further weakens the connection between degrees of
belief and bets. Where Howson and Urbach specify that an agent will
regard a bet as fair, Christensen assumes only that the bet is
sanctioned as fair or justified by the agent's beliefs, thus
avoiding the first objection to Howson and Urbach's treatment.
He then invokes the Dutch Book theorem to argue that incoherent
degrees of belief sanction as fair a set of bets that cannot be fair,
and claims this reveals that incoherence is fundamentally an
epistemic, and indeed a logical, defect. But his argument clearly
faces the previous difficulty over the additivity axiom since, even if
two bets are sanctioned individually, it does not follow that they are
sanctioned jointly. In response to this problem, Christensen (2004)
modified his argument by restricting the assumption about the
sanctioning of bets to "simple agents" who value only
money, value it linearly, etc., so that monetary payoffs function as
utilities for them. He then argued that degrees of belief that violate
the probability axioms in a simple agent are rationally defective
because they sanction bets that would be guaranteed to lose money.
From this he claimed that since it is the incoherent beliefs
themselves that have been shown to be defective, such beliefs are
rationally defective in all agents.



Here again, the pragmatic dimension of the argument seems to have been
merely submerged. Degrees of belief do not sanction bets in isolation
from preference, and so the alleged defect in the simple agent who
violates the probability axioms cannot be pinned merely on those
beliefs. Since it is not the simple agent's beliefs themselves
that have been shown to be defective, but rather those beliefs in
combination with her preferences, the inference that incoherent
degrees of belief are rationally defective in all agents is invalid.
For all that has been shown, such beliefs might be fine in combination
with a different preference structure. Not only is it premature to
conclude that incoherence is rationally defective generally,
Christensen has not argued convincingly that incoherence is rationally
or logically defective even in simple agents. It could be rational for
a simple agent to be incoherent if she were surrounded by "Czech
bookies", who will offer bets so that she is on the winning,
rather than the losing, side of the book, because in this situation
incoherence will lead to satisfaction of her one goal as a simple
agent of increasing her monetary gain. Of course it is incoherence as
a logical defect, rather than one of means/ends rationality, that is
Christensen's focus, but since sanctioning bets as fair that
cannot be fair is not a property of incoherent credences alone, he has
not shown incoherence to be on a par with inconsistency with full
belief, where the defect lies with the beliefs themselves.


### 2.3 Coherence as a Pragmatic Consistency Constraint



The depragmatized DBAs attempt to use the Dutch Book theorem to show
that incoherent degrees of belief involve a kind of inconsistency on
their own, irrespective of the way in which they link up with
particular preferences. But the claim of inconsistency here requires
that credences be tied to evaluations of fairness, which in turn
invokes a concept of valuation that goes beyond merely having degrees
of belief. If an agent's degrees of belief are incoherent, and
he makes evaluations of bets using the standard expectation rule, then
there will be bets (with payoffs in some measure of utility) such that
calculated individually these will have expected value zero, and will
in this sense be fair by his lights, but which lead to a net loss, and
so may be said to be unfair. The unfairness can be deduced from the
agent's beliefs and their connection with the agent's
utilities, which establishes the defect brought out by the Dutch Book
as internal to the agent, marking a crucial difference between the
loss so inflicted and the sort requiring superior factual knowledge on
the part of the bookie. While the linkage between the agent's
credences and his evaluation of bets is crucial to this result, the
needed connection is independent of the agent's particular
preference for goods, and so it could be said that the failure to
correspond to fair betting quotients is a property of the
agent's beliefs, as tied to evaluations of acts, yielding the
result that incoherent credences, appropriately connected with
preference, display a property that is analogous to inconsistency for
full belief.



There is then a version of the DBA that seems close to what Ramsey had
in mind, which claims that insofar as degrees of belief link up
suitably with preferences, incoherence is associated with a property
mirroring inconsistency, although this still need not amount to
divided-mind inconsistency in Armendt's sense, nor is it
entirely akin to inconsistency for full beliefs, which are
characterized directly without any assumed connection to preference
and action. The agent by whose lights each of a set of bets looks fair
(or favorable), that is, should be evaluated as fair (or
favorable) although they lead collectively to a Dutch Book, which can
be set up by examining simply the agent's credences, has a
system of evaluation that is self-defeating, and as such may be said
to exhibit a form of irrationality. On this reading, the vulnerability
associated with incoherence is theoretical, and linked closely with
the agent's utilities, which means that actual vulnerability
depends not only upon an appropriately situated bookie, but also upon
the availability of suitable commodities that measure those utilities.
With this in place, bets can be fixed that constitute a book on the
basis of the agent's credences, but which are fair (or
favorable) by the agent's lights. Notice that the
previous points, that an agent might avoid book by refusing to bet and
that one might end up on the side of a sure gain, are now otiose, for
here it is the agent's evaluations, rather than their
consequences that are implicated by the Dutch Book. Still, the
DBA depends upon the theory of preference and utility, and as Kaplan
notes (Kaplan 1996), this does not emerge straightforwardly from
standard presentations of the argument of the sort considered in
section one, nor in the depragmatized versions, as observed in the
previous section. 



The assumptions of the DBA are significant. It is the agent's
system of credences together with the way they are involved in
evaluating options that is implicated by the Dutch Book. To conclude
that simply having incoherent credences is irrational, a further
argument is needed showing that rationality demands that credences be
linked to evaluations of bets as in the DBA, but there is little to
suggest that graded degrees of belief must be so connected. Indeed,
the DBA assumes not only that there is some connection between an
agent's credences and the evaluation of options, but that one
should evaluate a bet as fair if it has an expected value of zero,
using one's credences with the standard expectation rule. This
builds in that an agent's credences are related such that for
each \(H\),

 

\[ \cr(\text{not }H) = 1 - \cr(H). \]


Hedden calls attention to this assumption, which he terms Negation
Coherence, noting that it follows from the claim that credences
satisfy finite additivity and normalization (Hedden 2013). The premise
of the DBA that connects credences with fair betting quotients thus
presupposes a significant part of the probabilist's claim.
 



Even assuming the appropriate kind of connection between credences and
evaluations of options, it may be objected still that the sort of
inconsistency revealed through the Dutch Book vulnerability involved
in violating the probability axioms is not necessarily
irrational, particularly in cases where this is due to the failure to
grasp some subtle or complex logical or other
necessary truths. It is hard to see that rationality requires
agents to attempt to remove such inconsistencies in their belief
systems; indeed for most it would be both hopeless and
counterproductive. Nor does it seem correct to count as more rational
the person who avoids inconsistency, by refusing to have an opinion in
the face of evidence, than the person with less than full confidence
in a logical truth, who takes the incomplete evidence into account.
One option is to insist that coherence is a requirement on ideal
agents. It might be suggested instead that coherence is one of a
number of ideals, whose satisfaction may sometimes conflict, and that
an incoherent agent may still be rational in the sense that they have
optimally managed their opinions in light of the totality of their
goals and constraints. Whatever the conclusion about the irrationality
of being incoherent though, with idealizing assumptions about the
connection between degrees of belief and the evaluation of options,
the DBA succeeds in establishing incoherence as an evaluative defect.
 



There
remain some disputes about how to understand the DBA for probabilism,
and whether it provides an appropriate defense for the claim. As
discussed, some take the DBA to show that the probability axioms are a
condition of consistency for action guiding degrees of belief, where
having incoherent credences involves faulty evaluations of
options. It is sometimes added that incoherence thus indicates a
form of epistemic irrationality. Although incoherence may be
held to depart from a certain ideal of rationality in this way, it
need not mean that it is, all things considered, irrational to violate
the probability axioms, as noted by Huttegger (2013). On this
understanding, the claim that violating the probability axioms is a
form of irrationality proceeds from the claim of inconsistency. 




Others
who accept the idea that the problem with Dutch Book vulnerability is
not that it leads to bad consequences, but that it signals an internal
defect in the agent's credences, simply say that DBAs that meet
certain conditions, including being constructible by a bookie without
requiring more than knowledge of the agent's credences,
establish that the credences leading to a Dutch Book under these
conditions display a form of irrationality, without suggesting that
Dutch Book vulnerability is form of inconsistency. Both ways of
understanding DBAs have been carried over to arguments in support of
additional probabilistic norms.




Taking
the view that Dutch Book vulnerability indicates a form of
inconsistency has the advantage that it makes clear the sort of
irrationality involved in cases of Dutch Book vulnerability, and
avoids the implausible claim that rationality always requires avoiding
Dutch Book vulnerability. However, the view faces a difficulty
in that the Dutch Book vulnerability for those who violate other
proposed norms of probability seems less clearly a matter of
inconsistency, as will be suggested in the following sections. 
Those who understand DBAs as proceeding directly from Dutch Book
vulnerability to the claim of irrationality, without the intermediate
conclusion that Dutch Books indicate inconsistency, must qualify what
is to count as Dutch Book vulnerability and the sort of irrationality
that is at stake. Pettigrew (2020) suggests that the relevant
sort of Dutch Book vulnerability is a sure loss in every world that is
epistemically possible for the agent, and that the conclusion to draw
from this is that someone who is so vulnerable is epistemically
irrational. Insofar as it is epistemic rationality that is of concern,
this formulation has the virtue of not counting the person who could
be made to lose money in every metaphysically possible world, on the
basis of having less than full confidence in some metaphysical truths,
as subject to a proper Dutch Book and so irrational. But, this
still leaves questions about what is to count as epistemically
rational, and how in general this lines up with Dutch Book
vulnerability. Questions remain too about whether the pragmatic
basis of the Dutch Book argument offers appropriate grounds for
defending probabilistic norms of epistemic rationality, and this has
spurred interest in alternative nonpragmatic justifications of such
norms. See Pettigrew (2019).



## 3. A Dutch Book Argument for Countable Additivity



Given a countable infinity of mutually exclusive and exhaustive ways
\(W\_i\) in which the proposition \(A\) can be true, the
Principle of Countable Additivity requires that

 

\[ \pr(A) = \sum\_{i=1}^{\infty} \pr(W\_i). \]


A Dutch Book argument can be constructed for the principle by
extending the Dutch Book for finite additivity (Adams 1962; Jeffrey
2004; Williamson 1999) by appealing to an infinite set of bets that
each pay $1 if \(W\_i\) is true for the price \(\pr(W\_i)\). In the case
of the Dutch Book argument for countable additivity, there is an
additional reason beyond those for the basic argument to favor the
interpretation of it, as Jeffrey does, as establishing the principle
as a consistency constraint, since the loss is purely theoretical.
There is no threat of an actual loss to an agent since this would
require an infinite series of bets, which cannot all be made and
settled. However, the principle itself is controversial, making the
Dutch Book argument for countable additivity highly provocative. Both
de Finetti (1972) and Savage (1954) argued that the principle should
not be invoked as a constraint on rational degrees of belief. A
consequence of the principle is that it prohibits positive credences
that are uniformly distributed over a countably infinite partition,
and yet it seems at least acceptable, in the absence of a reason to
favor some possibilities over others, to adopt such a distribution.
Intuitively, a uniform distribution ought to count as consistent,
which puts pressure on the idea, at least when it comes to countable
additivity, that Dutch Books reveal a defect that is substantially
analogous to inconsistency. In the background, as with the basic
argument, the Dutch Book argument for countable additivity makes
significant assumptions about how degrees of belief are linked with
preference and assessments of value. For details concerning how these
relate to the demand that an agent's degrees of belief satisfy
countable additivity, see (Seidenfeld and Schervish 1983). For a
recent critical discussion of the argument, see Pettigrew (2020).


## 4. Diachronic Dutch Book Arguments



The basic idea behind the Dutch Book argument for probabilism has been
used in defense of a variety of principles that purport to govern how
beliefs should evolve over time. In each case, it is argued that the
agent who violates the principle is subject to a Dutch Book. Here, the
bets involved are placed at different times, but there is an algorithm
for placing the bets that is available at the outset guaranteeing a
profit to one side. Such arguments are generally referred to as
"Dutch Strategy" arguments.


### 4.1 Conditionalization



The Rule of Conditionalization is commonly endorsed for updating
probabilities in response to new information. It states
that an agent with the probability function \(\pr\_1\) at time \(t\_1\)
who learns \(E\) and nothing stronger at time \(t\_2\)
should obtain her new probability function \(\pr\_2\) from her old
probability function by conditionalizing on \(E\), i.e.
for each proposition \(A\), \(\pr\_2(A) = \pr\_1(A\mid
E)\), where the
conditional probability
\(\pr(A\mid E)\) is given in terms of the ratio formula: \(\pr(A\mid E) = \pr(A
\& E)/\pr(E)\). There
is a Dutch Book (strategy) argument purporting to show that an agent
ought to change beliefs by conditionalization, which was constructed
by Lewis (1999), but was first reported by Teller (1973). If an
agent violates this rule by adjusting her probabilities such that
\(\pr\_2(A) \lt \pr\_1(A\mid E)\), a bookie can inflict a sure loss by
first selling the agent the following three bets:




|  |  |
| --- | --- |
| (Bet 1) | for \(\pr\_1(A\wedge E)\): |
|  | \($1\) | if \(A \wedge E\) |
|  | \($0\) | otherwise |




|  |  |
| --- | --- |
| (Bet 2) | for \(\pr\_1(A\mid E)\pr\_1(\neg E)\): |
|  | \($\pr\_1(A\mid E)\) | if \(\neg E\) |
|  | \($0\) | otherwise |




|  |  |
| --- | --- |
| (Bet 3) | for \([\pr\_1(A\mid E) - \pr\_2(A)]\cdot
\pr\_1(E)\): |
|  | \($\pr\_1(A\mid E) - \pr\_2(A)\) | if \(E\) |
|  | \($0\) | otherwise |



If \(E\) is false, the agent has a net loss of
[\(\pr\_{1}(A\mid E) -\) \(\pr\_{2}(A)\)] \(\pr\_{1}(E)\). If \(E\)
is true, then the bookie buys back a fourth bet:




|  |  |
| --- | --- |
| (Bet 4) | for \(\pr\_2(A)\): |
|  | \($1\) | if \(A\) |
|  | \($0\) | otherwise |



Bets 1 and 2 together amount to a conditional bet on \(A\),
which is called off if \(E\) is
false. In the event that \(E\) is true, this bet is
bought back at the lower price of \(\pr\_{2}(A)\). Bet 3 spreads the
expected winnings of that exchange in order to assure a profit in the
case that \(E\) is false. Thus, if \(E\)
is true the agent suffers a net loss of [ \(\pr\_{1}(A\mid E) -
\pr\_{2}(A)\)] \(\pr\_{1}(E)\). If \(\pr\_{2}(A) \gt \pr\_{1}(A\mid E)\),
then the strategy involves reversing the direction of the above bets.
Where dollars are taken as measures of utility, each bet is fair given
the agent's beliefs at the time offered. There is also a
converse Dutch Book theorem that shows that adhering to
conditionalization avoids Dutch Book vulnerability Skyrms
(1987b).



While
the argument has often been cited
in support of the Rule of Conditionalization as characterized above,
there are significant assumptions and limitations to Lewis'
Dutch Strategy argument in defense of the claim that rational updating
requires conditionalization on the evidence in accordance with the
rule. As
van Fraassen points out, the Dutch Strategy given above does
not yield an argument for the diachronic Principle of
Conditionalization as stated (van Fraassen 1989). There is no strategy
for the bookie that guarantees him a profit from an agent for whom
\(\pr\_{2}(A) \ne \pr\_{1}(A\mid E)\) unless the agent makes a
commitment to violate the rule in a particular way in advance. The
idea that the above bets constitute a betting strategy yielding a
Dutch Book tacitly assumes that the agent's alternative rule \(D\)
is fixed at time \(t\_{1}\), before it is known
whether or not \(E\) is true, and that the agent is
sure to follow the rule if \(E\) is learned.



However, the bookie can guarantee himself a profit in some cases even
if it is not assumed that the agent will follow though by placing a
side bet that he wins in the event that the agent does not change
beliefs as planned. The Dutch Strategy depends upon the fact that the
agent initially has some definite probability that he will shift
probabilities in the event that \(E\), in some
particular way that is not the result of conditionalization on \(E\).
The Dutch Strategy argument in no way establishes
that one must have such a probability, and so at most shows that one
should not have a plan involving a fixed probability to update upon
learning \(E\) in a way that differs from
conditionalizing on \(E\).



Even where it is assumed that an agent is fully committed to following
through on some alternative to the Rule of Condtionalization, there
are questions about what the Dutch Strategy shows. As with the basic
DBA for probabilism, it is possible to devise scenarios in which it
would be rational to leave oneself open to a Dutch Strategy by
announcing a deviant updating rule, and so the Dutch Strategy argument
should not be taken to show that it is strictly irrational to have an
updating rule other than Conditionalization. Rather, the existence of
the strategy points to a tension in adopting such a rule. Shifting
from \(\pr\_{1}\) to \(\pr\_{2}\) by conditionalizing on \(E\)
is equivalent to invariance, i.e. for each
proposition \(A\), \(\pr\_{1}(A\mid E) =\)
\(\pr\_{2}(A\mid E)\). Adopting a rule for updating other than
Conditionalization is thus tantamount to holding at time \(t\_{1}\)
that for some \(A\), \(\pr\_{1}(A\mid E) \ne\)
\(\pr\_{2}(A\mid E)\). Note that the bookie simply takes advantage of
the difference between the agent's probability for A given \(E\)
at \(t\_{1}\) and at \(t\_{2}\). Where the agent is
certain at \(t\_{1}\) that he would update via some deviant rule \(D\),
the rule induces a conditional probability at that
time for \(A\) given \(E\) that differs
from \(\pr\_{1}(A\mid E)\). In such a case, \(\pr\_{2}(A\mid E)\) is
fixed at \(t\_{1}\) by the agent's credal state, and so given
that \(\pr\_{1}(A\mid E) \ne\) \(\pr\_{2}(A\mid E)\), the agent's
credences may be said to exhibit a kind of inconsistency. While this
is problematic in a way that the sort of inconsistency involved
in simply having different levels of confidence for a proposition at
different times is not, it appears less like inconsistency for full
belief than failing to obey the probability axioms, where the agent
has evaluations according to which each in a set of bets is fair at a
single time. While \(\pr\_{2}(A\mid E)\) may be fixed at \(t\_{1}\),
this is not the agent's conditional probability at \(t\_{1}\),
but rather the probability for \(A\) given \(E\)
that she will come to have at \(t\_{2}\) upon learning
\(E\). Such updating plans are part of the
agent's credal system though, for which Dutch Book vulnerability
may plausibly be held to signal a defect.



The Dutch Strategy against the non-conditionalizer, as described
above, also depends upon the assumption that either E will be learned
or the altenative not E will be learned. Otherwise, there is no
guarantee that a loss can be inflicted, even if, on the condition that
E is learned, the agent plans to adjust her credences such that
\(\pr\_2(A) [?]\pr\_1(A\mid E)\). More generally, the existence of a
Dutch Strategy assumes a learning scenario in which the agent's
possible total evidence is one of a set of propositions \(\{E\_1, E\_2,
\ldots, E\_n\}\) that characterize what can be learned. 



Gallow (2019) calls attention to the fact that the Rule of
Conditionalization as stated in the form above does not depend upon
the possible evidential propositions that may be learned, but only the
prior credences and what is actually learned. Constructing a Dutch
Strategy against a non-conditionalizer requires having a rule that
specifies how they will update their credences given the alternative
possible evidence that they might have learned. So, at most the Dutch
Strategy Argument defends the Rule of Conditionalization in an
appropriate learning scenario that specifies the possible evidence
that may be obtained. Gallow also emphasizes that the propositions in
the learning scenario must form a partition, as some presentations
explicitly assume. If they do not, that is some of the \(E\_i\)
overlap, he shows that Dutch Strategies are possible against
conditionalizers. For instance, where \(E\_1\) and \(E\_2\)
are the evidential propositions that may be learned and they overlap,
learning either will raise the conditionalizer's credence in their
conjunction. A Dutch Strategy can be constructed that takes advantage
of the difference between the prior credence for the conjunction and
the foreseeable rise in the conjunction upon either learning
\(E\_1\) or learning \(E\_2\). Gallow suggests that learning
scenarios may exist in which the possible evidence fails to form a
partition, and that evidence can be acquired outside of any learning
scenario at all.




These
observations put significant pressure on the Dutch Strategy argument,
especially when the argument is read, as Gallow does, as drawing the
conclusion that violating the Rule of Conditionalization is
epistemically irrational on the grounds that doing so involves
vulnerability to a Dutch Strategy. The Rule of Conditionalization, as
stated above, says to conditionalize on the evidence E obtained,
irrespective of the evidence that might have been obtained instead, so
in particular it counsels conditionalizing both in learning scenarios
that form a partition and those that do not. But, as Gallow shows,
conditionalizing in the latter case can involve vulnerability to a
diachronic Dutch Book, so Dutch Strategy vulnerability fails to
discriminate between cases in which the rule, as characterized above,
is adopted and cases in which it is violated.
 



One
option for preserving the force to the Dutch Strategy argument for the
Rule of Conditionalization as given is to insist that ideally
rational agents will always obtain evidence in a learning scenario
that forms a partition, which is a view that Gallow plausibly
attributes to Lewis (1999). Even if this is right for highly idealized
epistemic agents, it leaves the status of the Principle of
Conditionalization unclear in more ordinary circumstances and so fails
to establish the principle as a general norm. Moreover, such a defense
of conditionalization would depend upon more than just the claim
that vulnerability to a Dutch Strategy involves irrationality, but
also upon the claim that there are rationality constraints on possible
learning scenarios. 



An alternative is simply to amend the Rule of Conditionalization to
build in that it only applies in learning scenarios that form a
partition. This would allow for a Dutch Strategy argument for the
narrower claim that epistemic rationality requires conditionalizing in
learning scenarios that form a partition, provided that the notion of
epistemic rationality here also makes sense of the Dutch Strategy
vulnerability of conditionalizers in other learning scenarios. 



One more limitation to observe about Lewis' Dutch Strategy
argument for changing credences by Conditionalization stems from his
formulation of the principle itself in terms of the ratio formula,
which requires that the prior probability of the evidence
conditionalized on is non-zero. Rescorla (2019) addresses this by
giving a form of conditionalization, which he terms Kolmogorov
Conditionalization, that permits conditionalizing on evidence with
null priors. He then proves a Dutch Book theorem and converse for it,
where the theorems involve generalized learning scenarios. 


### 4.2 Jeffrey Conditionalization



Jeffrey proposes a further Generalized Conditionalization Rule (also
termed *Probability Kinematics* or *Jeffrey
Conditionalization*) to cover cases in which experience does not
manifest itself in some evidential proposition \(E\)
shifting to one, but rather stems from a shift in probabilities over
some partition \(\{E\_{i}\}\) (Jeffrey 1983). Jeffrey's Rule
states that

 

\[ \pr\_2(A) = \sum\_{i=1}^{n} \pr\_1(A \mid E\_i) \pr\_2(E\_i), \]


which is equivalent to the invariance condition that \(pr\_1(A\mid E\_i)
= pr\_2(A\mid E\_i)\), for each element \(E\_i\) of the partition. To
maintain that updating goes by Jeffrey Conditionalization is to
mandate invariance for each proposition \(A\). This
might seem to be required, as Armendt shows that a Dutch Strategy can
be constructed against an agent who has a rule for updating that
conflicts with Jeffrey's Rule (Armendt 1980). As with the Rule
of Conditionalization though, the Dutch Strategy does not show that
there is any problem with having probabilities over a partition, and
then abandoning one's prior conditional probabilities over that
partition, that is violating invariance; the problem is with adopting
a rule of this sort in advance. As with the simple Rule of
Conditionalization, Skyrms (1987b) shows that there is also a converse
Dutch Strategy argument showing that following Jeffrey's Rule
avoids a Dutch Book. It is built in here that Jeffrey's Rule
applies in cases in which credences shift over a partition, but as
before if the agent's learning scenario does not form a partition,
Gallow (2019) shows that, as with the Simple Conditionalization,
Jeffrey Conditionalizers may be subject to a Dutch Strategy as
well. 



While the Rule of Conditionalization is generally presented as
demanding that agents satisfy invariance for each proposition, and
various authors have rightly complained that this demands an
unreasonable rigidity of belief (Bacchus, Kyburg, and Thalos 1990;
Levi 2002), Jeffrey does not treat the condition as a strict
requirement on rational belief change (Jeffrey 2004). He leaves open
the possibility that when probabilities are shifted over a partition,
the conditional probabilities could change, although this leaves the
unanswered question as to when invariance is reasonable to assume. In
any case, as with the Simple Rule of Conditionalization, the Dutch
Strategy argument here tells in the same way against planned
violations of invariance. 


### 4.3 The Principle of Reflection



A Dutch Strategy argument has also been given for the controversial
Principle of Reflection, which requires that for any proposition \(A\)
and any future time \(t\), an
agent's current probability for \(A\) conditional
on later assigning it probability \(r\) is itself \(r\),
that is pr \((A\mid \pr\_{t}(A) = r) = r\). In
violating Reflection, the agent fails to endorse certain of her
possible future judgments, which as van Fraassen showed (van Fraassen
1984), opens the door to a Dutch Strategy. If the agent violates
Reflection in that \(\pr(A\mid \pr\_{t}(A) = r) \gt r\), the bookie
sells the following bets:




|  |  |
| --- | --- |
| (Bet 1) | for \(\pr(A \wedge \pr\_t(A) = r)\): |
|  | \($1\) | if \((A \wedge \pr\_t(A) = r)\) |
|  | \($0\) | otherwise |




|  |  |
| --- | --- |
| (Bet 2) | for \(x\,\pr(\pr\_t(A)\ne r)\): |
|  | \($x\) | if \(\pr\_t(A)\ne r\) |
|  | \($0\) | otherwise |




|  |  |
| --- | --- |
| (Bet 3) | for \((x-r)\,\pr(\pr\_{t}(A) = r)\): |
|  | \($x-r\) | if \(\pr\_t(A) = r\) |
|  | \($0\) | otherwise |



where \(x = \pr(A \mid \pr\_{t}(A) = r)\).



In the event that \(\pr\_{t}(A) = r\), the bookie then buys a fourth
bet from the agent on \(A\) with a prize of 1, with a
price of the agent's new probability.



The idea that the existence of such a strategy shows that agents
should satisfy Reflection has been widely disputed, even by many who
take the Dutch Book arguments for probabilism and conditionalization
as having force. Much of the discussion stems from a variety of
counterexamples to Reflection in which it appears irrational not to
violate the principle. One example, due to Talbott, turns on the fact
that real agents lack perfect recall (Talbott 1991). He considers a
person who had spaghetti \((S)\) the previous night, say November 16,
and contemplates the probability that she will give to her having
eaten spaghetti on that day in year's time. Since she eats
spaghetti on average one out of every ten days, she thinks it likely
that \(\pr\_{t}(S) = 0\).1. But surely she should not assign a
probability of 0.1 to her having had spaghetti the previous night
conditional on the claim that \(\pr\_{t}(S) = 0\).1, since she is
nearly certain (say \(\pr\_{t}(S) = 0\).99) of what she had for dinner
the previous night.



The forgetting that makes it reasonable to violate Reflection in
Talbott's spaghetti dinner example involves one type of limited
future impairment. Christensen gives a different case in which an
agent has just consumed a mind-altering drug (LSQ) that he expects in
hour's time will make him believe that he can fly, without
conferring upon him any such power (Christensen 1991). However, this
is the only effect of the drug, so that the agent's faculties
will remain otherwise acute. Before the drug takes effect, the agent
should not take his probability that he can fly conditional on his
coming to believe it in an hour to be high, as required by Reflection,
because what he will think when under the influence is irrelevant to
his ability to fly. A similar example involving Ulysses and the Sirens
is discussed by van Fraassen (1995).



It is important in both of these examples that it is not only
reasonable to violate Reflection, but that to do otherwise would
involve failing to properly account for the evidence about the
unreliability of one's future beliefs. Christensen's own
response involves arguing that because the bets in diachronic Dutch
Book arguments are made over time, unlike in the synchronic argument
for the probability axioms, they do not reveal a form of inconsistency
that is generally irrational. One of several troubles with this is
that it would require rejecting the claim that the Dutch Strategy
against agents who plan to violate Conditionalization reveals any sort
of problematic inconsistency. Other problems with Christensen's
response are discussed in (Vineberg 1997).



Others have tried to respond to the counterexamples by distinguishing
features of the Dutch Strategy argument for Reflection that do not
involve dismissing all diachronic Dutch Book arguments. For instance,
Hitchcock emphasizes that it is important to consider what the bookie
knows in assessing the relevance of Dutch Books on an agent's
rationality (Hitchcock 2004). He observes that unlike the case in
which an agent violates the probability axioms, where the bookie can
determine and place the bets with only the knowledge of the
agent's beliefs, in Talbott's example, the bookie may need
to exploit information that is unavailable to the agent in order to
make the Dutch Book.



Taking into account what a bookie needs to know in order to guarantee
himself a profit fails to distinguish adequately between the Dutch
Books that appear to be norms of rationality and those that do not.
First, as Hitchcock acknowledges, this won't work to resolve the
tension between the Dutch Strategy argument for Reflection and many of
the other counterexamples. In Christensen's example, taking LSQ
does not have any effect on memory, it merely changes the
agent's evaluation of his ability to fly, that is, he changes
his view of a very limited bit of evidence, so that the Dutch Strategy
can be carried out without the bookie having relevant information that
the agent lacks.



Still, perhaps we might take the fact that a book can be made without
having knowledge beyond that of the agent as at least a necessary
condition for a telling book. Various cases, besides that of the
spaghetti dinner, give plausibility to the idea that a Dutch Book only
establishes the unreasonableness of violating a particular principle
when the book is made with no special knowledge not possessed by the
agent. Consider an agent who violates the probability axioms in virtue
of failing to recognize some complicated logical truth. Arguably this
is not unreasonable, and setting up a Dutch Book in this case would
require knowledge that the agent lacks. But, insofar as the
irrationality in violating the axioms stems from the fact that it is
supposed to be a form of inconsistency, as both Hitchcock and
Christensen maintain, then the fact that a bookie might need (logical)
knowledge that the agent lacks in order to set up the book, is
irrelevant to whether the book has force. Whether an agent's
beliefs are *inconsistent*, unlike the question of whether she
is to be regarded as reasonable, is independent of what she or anybody
else knows.



Briggs offers another way of distinguishing those Dutch Books that
mark the violation of a genuine norm from those that do not (Briggs
2009). The contention is that whereas the Dutch Strategy against those
who violate Conditionalization reveals what Briggs terms incoherence,
which is taken to be a form of inconsistency, the Dutch Strategy
against violators of Reflection reveals merely self-doubt, which
involves an agent who suspects herself of incoherence. Briggs assumes
that an agent's credences "condone" accepting
certain bets as fair, and that an agent's credences admit of a
Dutch Book just in case those credences condone a set of bets that
yield a net loss to the agent in every world where the agent has those
credences and accordingly would condone the bets. Violating the
probability axioms, Conditionalization or Reflection makes one
vulnerable to a Dutch Book in this sense. Briggs observes that the
bets in the Dutch Books that may be constructed against someone
violating either the probability axioms or Conditionalization will
yield a net loss in every possible world; but, the bets in the Dutch
Strategy against the violator of Reflection will not result in a loss
in certain worlds in which the agent's beliefs differ from those
that she has in the actual world. Accordingly, Briggs suggests that
incoherence is marked by Dutch Books that involve bets that result in
a loss in every world, and thus that violation of Reflection does not
involve incoherence. 



This way of attempting to save the Dutch Strategy argument for
Conditionalization, while jettisoning that for Reflection, is
criticized by Mahtani (2012), who argues that Briggs' test
mistakenly counts certain cases of mere self-doubt as
incoherence. Mahtani proposes a new way of understanding
Dutch Book arguments as revealing incoherence, according to which an
agent is incoherent if and only if she would accept as fair a set of
bets that would result in a loss under any interpretation of the
claims involved (Mahtani 2015). With this understanding, violation of
the probability axioms reveals incoherence, but violations of
Reflection and cases of self-doubt in which the agent isn't sure
of her own credences do not count as incoherence, as they do not
involve proper Dutch Books that result in a loss under any
interpretation. As she discusses, this way of understanding Dutch
Books leaves it open as to whether there is a proper Dutch Book
argument for Conditionalization.



Both Briggs and Mahtani extract the condition of incoherence from the
features of certain bets. However, it remains unclear as to
how precisely the concept of incoherence, in their sense, is to be
understood, although they seem to take it as a form
of inconsistency. Still, the discussion is important
for there is some intuition that violating Reflection is less similar
to holding inconsistent beliefs than violating the probability axioms.
When it comes to Conditionalization, Briggs takes the Dutch Book
argument as establishing the full principle that the rule will be
followed, which appears too strong, since the consistency required by
rationality does not seem to require having a plan for updating nor to
strictly require following through on such a plan. Indeed, precisely
what sort of inconsistency is involved in cases where one is
susceptible to a Dutch Strategy on account of having an alternative
updating rule to Conditionalization remains unclear, and is perhaps a
point in favor of Mahtani's analysis upon which the status of
Conditionalization remains inconclusive. Against Mahtani, Pust
(2021) claims that her reading of logical form is too narrow, and that
given a more plausible account of logical form, cases of self-doubt
will be subject to Dutch Books that establish incoherence. Moreover,
Pust argues that the point applies to Dutch Books against violators of
Reflection, so that they too will count as incoherent. If this is
right, Mahtani's new understanding of Dutch Book arguments would
fail to distinguish as intended between Dutch Books
to undermine the Dutch Strategy argument for Reflection.
 The proposals of both Briggs and Mahtani must contend with
the subtle connection between credences and bets. Mahtani associates
credences with bets that the agent would accept as fair, but as
previously noted, having credences need not go along with accepting
bets as fair in the way needed to connect the violation of various
probabilistic norms with a Dutch Book. While Briggs acknowledges that
there is a complex relationship between credences and bets, the notion
that credences condone certain bets also appears to be in the spirit
of the depragmatized DBAs, which, as observed, suffer from various
difficulties. 



Pettigrew (2020) offers another way of jettisoning the Dutch Strategy
argument for Reflection, while retaining that for Conditionalization.
 He observes that the Reflection principle is synchronic, whereas
the Dutch Strategy argument for it is diachronic. As a result,
he claims that the Dutch Strategy argument here fails to target
Reflection. Instead, he says that since violators of Reflection
can't follow conditionalization, if they have an updating rule, it is
this departure from Conditionalization that is implicated by the Dutch
Stategy. This approach to distinguishing the Dutch Strategy argument
for Reflection from that for Conditionalization and the probability
axioms does not involve some sort of consistency, but rather it is in
line with taking Dutch Strategy arguments to move directly from Dutch
Book vulnerability to irrationality. The trouble is that
violators of Reflection are subject to a Dutch Strategy no matter what
future credences they come to have, regardless of whether they
adopt a particular updating rule, so given the understanding of
Dutch Strategy vulnerability as indicating some form of irrationality,
it is hard to see here why the Dutch Strategy argument for Reflection
doesn't defend the principle as a norm.



The significance of distinguishing between different types of Dutch
Books so as to mark off some as pointing to a form of incoherence or
irrationality is unclear. Whereas one or another
proposal may comport better with our intuitions about what is
intuitively incoherent (inconsistent), or irrational, if we retain
Ramsey's idea of credences as guides to action, then any way of
distinguishing between Dutch Books here looks problematic, since any
credences that lead to a sure loss are, at least to some extent,
defective as a guide to action. 



Rather than defend the idea that violations of Reflection can be
rational by attempting to distinguish the Dutch Strategy argument for
the principle from other Dutch Book arguments that are desirable to
retain, an alternative is to take each kind of Dutch Book as pointing
to a defect, and then argue that the most reasonable attitude in
some circumstances is to hold on to less than ideal
credences. The view that Dutch Book vulnerability generally
involves a departure from the ideal of consistency is
endorsed in defense of Reflection by Huttegger (2013). The
idea is that whereas the Dutch Strategy argument does establish
Reflection as a kind of consistency constraint, this doesn't require
that in bad circumstances departure from the ideal cannot
be the most rational option overall. This response does
assume that violations of Reflection can sensibly be held as a form of
inconsistency, and this remains controversial.


## 5. Other Uses of Dutch Book Arguments


### 5.1 Sleeping Beauty



An interesting application of Dutch Strategy arguments has been made
by Hitchcock to the perplexing Sleeping Beauty problem (Hitchcock
2004). In the problem, Beauty is to undergo an experiment in which she
will be put to sleep on Sunday night, after which a fair coin will be
tossed. If it comes up heads, she will be awakened just once for a
short time on Monday, and if tails she will also be awakened again on
Tuesday. In each case, after being awakened, she will be put back to
sleep with drugs that will erase her memory of having been awake. On
Wednesday, she is awakened and the experiment ends. Beauty is aware of
these facts about the experiment, so that when she wakes up during the
experiment she will not know whether it is Monday or Tuesday. The
problem is to determine what her probability should be for heads upon
awakening within the experiment. Elga introduced the problem into the
philosophical literature (Elga 2000), and argued that upon awakening
Beauty should shift her probability from the value of 1/2
that she assigned on Sunday night to &frac13;. Lewis argued
shortly thereafter that Beauty's probability upon awakening
should remain at 1/2 (Lewis 2001), and additional
arguments have followed on both sides. Hitchcock shows that there is a
Dutch Strategy against either answer, but then argues that there is
only force to the one against Beauty if she sticks with
1/2.



The strategies serve as potential counterarguments to the two ways of
answering the problem, rather than showing that it is irrational to
switch to &frac13; or stick with 1/2 per se,
since the individual strategies depend upon the plan to have a
particular probability upon awakening. If Beauty plans to switch to
probability &frac13; for heads, the Dutch Strategy is just an
application of the general strategy against someone who violates
Reflection. Hitchcock argues that this would not establish
Beauty's irrationality, since in this case the bookie must know
during the experiment what day it is in order to implement the
strategy, which is information that Beauty lacks. On the other hand,
if Beauty sticks with 1/2, a strategy can be carried out
without the bookie having to know more than Beauty by having the
bookie undergo the experiment with her and offer a bet against heads
on Sunday and a bet on heads whenever Beauty and the bookie awake
within the experiment. If the coin lands heads, the bookie sells
Beauty just one bet on heads, with her gain totaling less than her
loss on the first bet against heads. If the coin lands tails, the
bookie sells a bet on heads once on Monday and the same bet again on
Tuesday, so as to again produce a loss for Beauty. Hitchcock takes
this book to tell against the answer 1/2, and in favor of
&frac13;.



One trouble with defending the answer of &frac13; in this way
is that what a bookie must know to assure himself a profit does not
characterize what is essential to a proper Dutch Book. There is also a
particular reason to suppose that Beauty's predicament is one in
which reasonable degrees of belief and betting quotients can come
apart, because the third bet needed to make book in the switching case
is only offered when it is sure to be a losing bet (discussed in
Vineberg 2004--see the Other Internet Resources, and Bradley and
Leitgeb 2006). In this case the bookie exploits features of the
experiment, so that the strategy can be carried out without his
possessing knowledge of the day. But the fact that the bets leading to
book are triggered by the circumstances suggests that it is not just
Beauty's beliefs on which the strategy depends, and hence the
loss does not reflect a purely internal defect, despite the fact that
the bookie can implement the strategy without having access during the
experiment to knowledge that Beauty lacks. Draper and Pust also argue
that Hitchcock's use of a Dutch Book argument is subject to an
objection from evidential decision theory (Draper and Pust 2008).


### 5.2 The Principal Principle


Recently, Richard Pettigrew (2020) has offered a Dutch Book
argument for Lewis' Principal Principle as an additional
constraint on rational credences. The Principal Principle requires
that an agent's credences about chances be such that her
credence for the proposition, \(A\) *conditional on the chance of* \(A\)
*being* \(r\), is equal to \(r\). Supposing that \(\text{cr}\)
is the agent's credence function and \(\text{ch}\) a chance
function such that \(\text{ch}(A) = r\) says that *the chance
of* \(A\) *is* \(r\), the requirement is that
\(\text{cr}(A/\text{ch}(A) = r) = r\). The principle has a similar
structure to the Principle of Reflection, with the chance of \(A\) at
the current time t in place of the agent's credence in \(A\) at a
future time \(t\). So, it is not so surprising that if
\(\text{cr}(A/\text{ch}(A) = r) \neq r\), there is a collection of
bets in accordance with those credences that produce a kind of Dutch
Book. For details see Pettigrew (2020). However, there are
significant differences between the arguments for the Principal
Principle and that for the Reflection Principle. First, since the
agent's future credences do not figure in the Principal
Principle, the bets are part of a Dutch Book proper, rather than a
Dutch Strategy. While this avoids some concerns about Dutch Strategy
arguments, a problematic difference is that the bets, which are given
in terms of chances do not guarantee a sure loss, but only an expected
one, whatever the chances may be.


For Ramsey and de Finetti, the point of appealing to a Dutch Book
argument was to establish probabilistic constraints using precise
terms that were tied to verifiable conditions, making it crucial that
the bets constituting a book are given definite payoff conditions and
can be settled. However, the book against the violator of the
Principal Principle involves bets on chances, which lack the
definitive settlement conditions that would produce a sure loss. To
the extent that the underlying problem associated with Dutch Book
vulnerability lies with the agent's evaluations rather than
merely with their pragmatic consequences, as discussed previously, the
lack of a sure loss here may not be crucial. Still, the motivation for
taking the Dutch Book vulnerability in cases where the probability
axioms are violated as an indication of irrationality appears to
derive from the idea that it leads to a sure loss, so that the merely
expected loss associated with violating the Principal Principle, is
less clearly irrational. In particular, credences that are associated
with a mere expectation of loss, rather than a sure loss, are less
clearly held to indicate an inconsistency in the agent's action
guiding evaluations. Such credences may still be irrational, but
clarifying the source seems to require appealing to the nature of
chance, which remains unsettled.


### 5.3 Dutch Books and Rational Choice



Dutch Books have also been used to show that there are limitations on
the proper applications of credences within the standard decision
theoretic framework. Given that the available evidence often appears
to dictate only vague or interval-valued credences, it has been argued
that rationality permits having such unsharp credences. Elga uses a
kind of reverse Dutch Book involving a series of bets that are sure to
result in a net gain for the agent to argue that credences must be
sharp (Elga 2010). The idea is that a rational agent must accept at
least one of the bets in the series, but that there does not appear to
be a satisfactory set of decision rules for unsharp credences with
this consequence.



McGee constructs another type of Dutch Book with implications for
decision theory. Suppose that an agent's probability function is
not concentrated at finitely many points. Such a state of opinion, in
which the agent does not rule out all but a finite number of
possibilities, seems both consistent and reasonable, yet assuming that
an agent's utilities are unbounded, McGee shows that there is an
infinite series of bets, each with positive expected value, which
together assure a loss (McGee 1999). Assuming that it is reasonable to
have probabilities that are not concentrated at finitely many points,
McGee calls into question the general adequacy of the decision
theoretic framework, and in particular the principle of utility
maximization that his Dutch Book argument presumes.


## 6. Conclusion



It remains defensible that the Dutch Book argument shows that the
probability axioms are something like consistency constraints on
partial belief, at least given some substantial decision-theoretic
assumptions, unlike the idea that their violation is a pragmatic
liability. But this interpretation is less successful for the various
other norms for which Dutch Book arguments have been constructed. The
Dutch Strategy arguments for the Principle of Conditionalization and
for Jeffrey's Rule do not fully support the norms, but only the
weaker claim that one should not commit in advance to following an
alternative rule under some additional conditions, and the sense in
which it may be said to be inconsistent to do so seems different from
what is involved in violating the axioms. The other claims considered
here for which Dutch Book arguments have been constructed are even
less plausibly regarded as consistency constraints, and thus seem to
be a different sort of requirement on ideal rationality. This suggests
the need to identify 'consistency' with a narrower set of
conditions than mere Dutch Book vulnerability, if we are not to
abandon the idea of the probability axioms as a consistency constraint
that extends the ordinary concept and, perhaps, reject the DBA
altogether. A more fine-grained analysis of the various Dutch Books
might yield a way of marking off those that point to genuine
inconsistency from those exhibiting some other sort of defect, but a
satisfactory way of accomplishing this remains illusive.



Among those who endorse at least certain Dutch Book arguments,
some regard them as establishing rationality requirements without
embracing the idea that Dutch Book vulnerability involves a kind of
inconsistency. This avoids the problem of making sense of the
inconsistency involved in Dutch Book vulnerability, but faces
challenges in specifying the sort of rationality that is at stake.
 Responses to the Dutch Strategy argument for Reflection have
prompted a particularly wide array of attitudes towards
Dutch Book vulnerability. It is striking that nearly a century
after the publication of Ramsey's paper, there remains so much
disagreement and continued work on how to understand Dutch Book
arguments.