1. SUBJECTIVE PROBABILITIES NEED NOT BE
SHARP
Jake Chandler∗
Championed by the likes of Isaac Levi (1980), Peter Walley (1991) and
many others, so-called ‘imprecise’ probabilistic models of rational decison-
making have become increasingly influential in the past few decades, per-
meating disciplines ranging from philosophy to statistics, through engineer-
ing, artificial intelligence and economics.1 These models relax the rational-
ity requirements of standard Bayesian decision theory by representing the
credal states of rational agents, not by a unique probabilistically coherent
credence function, but by a–typically convex–set of such functions–a so-
called ‘credal set’–and opting for some generalisation or other of the stan-
dard Bayesian decision rule. It is argued by proponents of such models that
these notably better capture the rationality constraints that operate on choice
behaviour.
In a recent paper, however, Adam Elga (2010) argues that this liberalisa-
tion of Bayesian orthodoxy has gone a step too far. More specifically, he
claims that the imprecise view falters in its handling of sequential choice
problems, licensing behaviours that are intuitively rationally reprehensible.
The purpose of this note is to point out a number of inadequacies in his ar-
gument, whilst noting that the sequential choice problems that he considers
still pose problems for certain imprecise models that lie at the very liberal
end of the decision-theoretic spectrum.
Elga’s objection, in a nutshell, proceeds as follows. He first asks us to
consider the following pair of decision problems:
The agent faces a choice between A, accepting a bet that pays out
−10$ in case H is true and 15$ in case H is not and ¬A, the status
quo, that pays out 0$ irrespective of whether H is the case.
The agent faces a choice between between B, accepting a bet that pays
∗
Address for correspondence: Center for Logic and Analytic Philosophy, HIW, KU Leuven,
Kardinaal Mercierplein 2, 3000 Leuven, Belgium. Email: jacob.chandler[at]hiw.kuleuven.be
1
For a survey of the imprecise approach, see for instance Miranda (2008) or Troffaes (2007).
1

2. 15$ in case H is true and −10$ in case H is not, and ¬B, the status
quo, that pays out 0$ irrespective of whether H is the case.
As Elga notes, choosing both A and B secures a sure payoff of 5$, a prospect
that is clearly more desirable than the one resulting from choosing both ¬A
and ¬B instead, i.e. making the decision to decline both gambles and run
with the status quo. He then suggests the following principle which, at least
at first glance, could look plausible:
(P) ‘Any perfectly rational agent who is sequentially offered bets A
and B in the above circumstances (full disclosure in advance about
the whole setup, no change of belief in H during the whole process,
utilities linear in dollars) will accept at least one of the bets.’ (ibid,
p. 4)
The problem however, according to Elga, is that whilst (P) is satisfied by
the standard Bayesian model, it is not borne out by any of the most popular
imprecise models of rational choice. Whatmore, he argues, a number of
alternatives that do bear it out, are implausible on independent grounds. If
all this is correct, needless to say, the upshot would seem to be that the
prospects for the imprecise model are rather poor.
Fortunately for the friend of imprecise credences, however, Elga’s argu-
ment is unpersuasive. For one thing, whilst it is indeed true, as Elga claims,
that a number of rules, including E-admissibility and Γ-maximin2 do vio-
late (P), at least one well-known rule, which he omits to discuss, turns out
to satisfy it. This is the somewhat ‘optimistic’:
Γ-Maximax: an act A is rationally permissible if and only if there is no
alternative act B such that the highest expected utility of B according
to any credence function in the credal set is strictly greater than the
highest expected utility of A according to any credence function in
the credal set.
It is easy to see that, according to this principle, if the agent declines the
first bet, then she will be compelled to accept the second. Indeed, after
having declined the first bet and having updated her initial credal set ac-
cordingly, whereas (a) according to all credence functions in the agent’s
resulting credal set, declining the bet will have an expected utility of 0, (b)
for some credence function in that set, accepting the second bet will have a
strictly positive expected utility.
To see why (b) holds, assume for reductio that, for all credence functions
in the credal set immediately prior to reaching the second decision problem,
2
See below for an informal presentation of these
2

3. the second bet has an expected utility that is not strictly positive. Given the
payoffs, this amounts to the claim that, for any such function Cr, Cr(H) ≤
2/5. Then, given the condition that stipulates that there is ‘no change of
belief in H during the whole process’, it immediately follows that for any
credence function Cr in the credal set immediately prior to reaching the first
decision problem, we also have Cr(H) ≤ 2/5, From this we can then infer
that, according to any such credence function, the first bet had an expected
utility that was strictly positive, and hence, by Γ-Maximax, was mandatory
to accept. But, by hypothesis, the first bet was not accepted and the agent is
assumed to be rational. Contradiction.
But even if we set this issue to one side, problems remain. Indeed, there
seem to be strong grounds to hold that (P) is simply too strong a constraint
in the first place. More specifically, since the bets display a certain inter-
dependence in the payoffs, the rationality of the choice made at the first
decision point may well hinge in part on what the agent thinks that she will
do at the second. In particular, consider the issue from the perspective of
the popular ‘pessimistic’ counterpart of Γ-Maximax, endorsed by Gilboa &
Schmeidler (1989):
Γ-Maximin: an act A is rationally permissible if and only if there is no
alternative act B such that the lowest expected utility of B according
to any credence function in the credal set is strictly greater than the
lowest expected utility of A according to any credence function in the
credal set.3
It is clear that beliefs about future actions make a crucial difference here. To
illustrate the point, let us assume that our agent finds herself in a maximal
state of agnosticism with respect to H, so that the credences in H in the
initial credal set span the interval [0, 1].
Assume for example that, for whatever reason, at the first decision point,
(a) the agent believes that if she declines the first bet, she will accept the
second, and that, conversely, if she accepts the first bet, she will decline
the second. Since, with respect to the initial credal set, the lowest expected
utility associated with declining the first bet and accepting the second is the
same as the one associated with accepting the first bet and declining the
second, namely −10$, Γ-Maximin leaves it open to her as to whether or not
to accept the first bet.
But now assume that, at the first decision point, (b) the agent believes that
she will accept the first bet if and only if she will accept the second. Clearly,
3
Note that Γ-Maximin, whilst popular, is not uncontentious. Seidenfeld (2004), in particular,
criticises it on the basis that it can sometimes license a negative valuation of cost-free information.
3

4. Γ-Maximin recommends accepting the first bet: with respect to the initial
credal set, the lowest expected utility associated with accepting both bets
is 5$, whereas the lowest expected utility associated with declining both
bets is 0$. In this sequential choice problem, according to Γ-Maximin, the
recommendations of rationality are crucially sensitive to the agent’s beliefs
about her future choices.
We have seen that if the agent is state of mind (b), reasoning by backward
induction, she will follow Elga’s recommendation to accept at least one
of the bets. But under which conditions is there any rational compulsion
whatsoever for her to hold these views about her future self? Well, one
could easily imagine her reasoning as follows:
Suppose that I decline the first bet. What will I do when I get to the
second decision point? Well, given that I will have declined the first
bet, and will be none the wiser about whether or not H, I will think to
myself that if I decline the second bet, the expected utility will be 0$
come what may, whereas if I accept the second bet, for all I know, the
expected utility could be −10$. Being a sensible person, I will err on
the side of caution and decline the second bet. So If I decline the first
bet, I will decline the second one as well.
But now suppose that I accept the first bet. What then? Well, again,
given that I will have accepted the first bet, and will be none the wiser
about whether or not H, I will think to myself that if I decline the
second bet, the expected utility could well be −10$, for all I know,
whereas if I accept the second bet, I am guaranteed an expected payoff
of 5$. Being a sensible person, I will err on the side of caution and
accept the second bet: If I accept the first bet, I will accept the second
one as well.
Underlying the agent’s reasoning are a number of assumptions about her
future state of mind, many of which were left implicit. These include: (i)
the belief that her opinions with respect to H would remain unchanged be-
tween the first and the second decision point, (ii) the belief that she would
remember, her first decision at the time of the second and (iii) the belief that
her future self would follow the recommendations of Γ-Maximin.
It is of course far from clear to what extent the agent’s holding these
views is rationally mandatory. But the above considerations suggest a more
convincing, weaker, alternative to (P):
(P− ) Any perfectly rational agent who is sequentially offered bets A
and B in the above circumstances (full disclosure in advance about the
4

5. whole setup (FD), no change of belief in H during the whole process
(NC), utilities linear in dollars, certainty of future rationality (CFR),
introspection (I), certainty of perfect recall (CPR) and certainty of
lack of change of belief in H (CNC)) will accept at least one of the
bets.4
I take this principle to be intuitively compelling. Indeed, more generally,
the following would seem to hold: if you are rational and can count on your
future self making a rational choice that is informed by your decisions, you
will avoid making a sequence of choices that is strictly dispreferrable to
some available alternative. Whatmore, clearly, to the extent that Elga deems
even (P) to be plausible, he will grant that (P− ) is so too.
But now, once the additional conditions are suitably cashed out in formal
terms, it can be easily verified that Γ-Maximin satisfies (P− ). The supple-
mentary conditions are naturally interpreted as follows, where Γ denotes
the credal set of the agent at the time immediately prior to making the first
decision, and Γ∗ denotes the credal set of the agent at the time immediately
prior to making the second:
(CFR) implies that, for any function in Γ, the credence accorded to
the agent’s making, at the time of the second decision problem, a deci-
sion in line with the relevant principle of rational choice (for instance
Γ-Maximax or Γ-Maximin, depending on one’s prefered recommen-
dation) is 1.
(I) implies that, for any function Cr in Γ, the credence accorded to the
agent’s credal set being Γ is 1.
(CNC) entails, in conjunction with the previous condition, that, for
any function Cr in Γ, if there exists a function Cr in Γ such that
Cr (H) = r, then the credence assigned by Cr to their being a function
Cr in Γ∗ such that Cr (H) = r is 1.
(CPR) requires it to be the case that, for any function Cr in Γ and
any of the options available in the first decision problem, Cr assigns
a credence of 1 to (a) its being the case that, for any function Cr
in Γ∗ , Cr assigns a credence of 1 to the option having been chosen,
conditional on (b) its having been chosen.
4
Strictly-speaking, these assumptions can be somewhat weakened but the strong versions suf-
fice to make the present point.
5

6. (FD), which Elga left somewhat unspecified, simply amounts to its
being the case that, for any function in Γ, the credence accorded to the
second decision problem being encountered is 1
With this in hand, we can now provide an informal sketch of the proof:
Let Γ be such that, for every real number r in an arbitrary interval [a,
b], there is a function Cr in Γ such that Cr(H) = r. It will be shown
that, depending on the value of a, one or the other of the two following
disjuncts hold: either chosing ¬A will mandate chosing B, or chosing
A will be mandatory in the first place.
Either (1) a > 2/5 or (2) a ≤ 2/5. Assume (1). Now, by (NC), if
¬A were to be chosen at the first point, according to Γ∗ , a choice of
B at the second point would have, at worst, an expected utility of
15a − 10(1 − a) = 25a − 10$, whereas a choice of ¬B would have, at
worst, an expected utility of 0$. Since (1) is equivalent to 25a−10 > 0,
it follows from Γ-maximin that if ¬A were to be chosen at the first
point, then B would be chosen at the second. This is the first of our
disjuncts.
So now assume (2), i.e. that a ≤ 2/5. This is equivalent to 0 ≥ 25a−10.
With respect to Γ, the left hand term of this last expression is the
lowest expected payoff for chosing ¬A and then ¬B and the right hand
term is the lowest expected payoff for chosing ¬A and then B. It
follows that, whatever credences are assigned by the various functions
in Γ to B’s being chosen, conditional on ¬A’s being chosen, according
to each of these functions, the expected utility of chosing ¬A will be
no higher than 0.
Now by (FD), (CPR), (I) and (CNC), it follows that (3): Every cre-
dence function in Γ accords a credence of 1, conditional on A’s being
chosen at the first decision point, to the proposition that, according to
Γ∗ , a choice of ¬B at the second point would have, at worst, an ex-
pected utility of −10b + 15(1 − b) = 15 − 25b$, whereas a choice of B
would have, at worst, an expected utility of 5$.
By (CFR) and Γ-Maximin, it then follows from (3) that, whatever cre-
dences are assigned by the various functions in Γ to B’s being chosen,
conditional on ¬A’s being chosen, according to each of these func-
tions, the expected utility of chosing ¬A will be no lower than 5. It
then follows by Γ-Maximin that ¬A will not be chosen. We now have
the second disjunct. QED5
5
Note that it is easy to see from this that, as suggested above, if we set a = 0 and b = 1, we
6

7. It is worth noting, however, that (P− ) is violated by at least three further
rules. These include:
E-Admissibility: an act A is permissible if and only if there is a cre-
dence function Cr in the agent’s credal set such that there is not al-
ternative act B such that the expected utility of B according to Cr is
strictly greater than the expected utility B according to Cr.
Maximality: an act A is permissible if and only if there is no alternative
act B such that for any credence function Cr in the agent’s credal set,
the expected utility of B according to Cr is higher than the expected
utility of A according to Cr.
Interval Dominance: an act A is permissible if and only if there is no
alternative act B such that such that the lowest expected utility of B
according to any credence function in the credal set is strictly greater
than the highest expected utility of A according to any credence func-
tion in the credal set.
Regarding the first rule, consider for instance the case of extreme agnosti-
cism in which a = 0 and b = 1. As we have seen above, since a ≤ 2/5,
chosing ¬A leaves it open whether or not B will subsequently be chosen.
Whatmore, for all credence functions in Γ, there is no constraint on the cre-
dence that B will be chosen at the second point conditional either on A or
on ¬A having been chosen at the first. Consider for instance the scenario
in which ¬A was chosen at the first decision point. There exists a credence
function in Γ∗ according to which choosing B has a higher expected utility
than choosing ¬B (let the credence accorded to H be 0). But there also
exists a credence function in which this relation is inverted (let the cre-
dence accorded to H be 1). Consider now the scenario in which A was
chosen. Again, there exists a credence function in Γ∗ such that choosing B
has a higher expected utility than choosing ¬B (let the credence accorded
to H be 0). But there also exists a credence function in which this relation
is inverted (let the credence accorded to H be 1). From this observation,
it quickly follows that E-Admissibility violates (P− ). But then the rule is
therefore also violated by both Maximality and Interval Dominance, since
it is well known that if an act is permissible according to the first rule, it is
thereby permissible according to the subsequent two.6
recover the result that chosing A, followed by chosing B, is rationally mandated by Γ-Maximin,
given the assumptions in place.
6
See Troffaes (2007). In addition, note that E-admissibility is striclty more permissive than
Γ-Maximax and that Maximality is strictly more permissive than Γ-Maximin. There is no
7

8. To conclude, Elga’s todeserklärung for the imprecise view is somewhat
premature: the case for sharpness has simply not been made. First of all, (P)
happens to be satisfied by Γ-Maximax, a fact that Elga omitted to mention.
Secondly, it seems plausible to argue that (P) was too strong in the first
place and that its more sensible replacement, (P− ), is notably satisfied by
the fairly popular Γ-Maximin. Interestingly, however, it is worth noting that
a number of well-known decision rules do not satisfy even the weakened
principle. This arguably provides prima facie grounds to reject them in
favour of something more stringent.
Acknowledgments
I am grateful to Richard Bradley, Igor Douven and Wlodek Rabinowicz for
helpful feedback on an earlier version of this paper.
References
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Levi, I. (1980). The Enterprise of Knowledge: An Essay on Knowledge,
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Gilboa, I. and D. Schmeidler (1989). Maxmin Expected Utility with Non-
Unique Prior. J. Math. Econ. 18, pp. 141-53.
Miranda, E. (2008). A Survey of the Theory of Coherent Lower Previsions.
International Journal of Approximate Reasoning, 48(2), pp. 628–58.
Seidenfeld, T. (2004). A Contrast between Two Decision Rules for Use with
(Convex) Sets of Probabilities: γ-Maximin versus E-Admissibility. Syn-
these 140(1/2), pp. 69-88.
Troffaes, M. C. M. (2007). Decision Making under Uncertainty Using Im-
precise Probabilities. International Journal of Approximate Reasoning 45,
pp. 17-29.
Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. Lon-
don: Chapman and Hall.
such straightforward relationship, however, between E-Admissibility and Γ-Maximin: some E-
admissible acts are not Γ-Maximinimal and vice versa.
8