First, it requires an
unchanging evaluative function \(u\) defined over the atoms of the
propositional space, viz. Thus for all doxastically
changed preference orderings, the preferences over worlds remain
identical. Second, the model only considers the effects of a belief
change to certainty. But it is plausible that one’s preference
– say, for a vacation in Florida – changes just because
one believes that it is more likely that there will be a hurricane
next week.
- Steedman and
Krause (1986) have shown that there is no rule for deriving total
preferences from a preference vector that satisfies four seemingly
plausible conditions and also yields a transitive and complete total
preference ordering. - The most convenient way to represent problems with multiple preference
aspects is to introduce a vector \(\langle \succcurlyeq_1 ,\ldots
,\succcurlyeq_n \rangle\) whose elements are the preference relations
we have to take into account. - But unlike Buchak,
they suggest that what explains Allais’ preferences is that the
value of wining nothing from a chosen lottery partly depends on what
would have happened had one chosen differently. - It is hard to deny that Ulysses makes a wise choice in being tied to
the mast.
It
employs a series of objects that are so arranged that we cannot
distinguish between two adjacent members of the series, whereas we can
distinguish between members at greater distance (Armstrong 1939,
Armstrong 1948, Luce 1956). Since one cannot taste the
difference between C999 and
C998, they are equally good (of equal value),
C999∼C998. For the same
reason, we have C998∼C997,
etc. all the way up to
C1∼C0, but clearly
C0
≻ C999. This
contradicts transitivity of indifference, and therefore also
transitivity of weak preference. Now, Savage’s theory is neutral about how to interpret the
states in \(\bS\) and the outcomes in \(\bO\). Expected utility theory has been criticised for not allowing for value
interactions between outcomes in different, mutually incompatible
states of the world.
Humeans often took this distinction
between beliefs and desires not only to imply that beliefs alone
cannot motivate action, but also that desires are not open to similar
rational criticism as beliefs. Therefore, Humeans conclude,
preferences can only be criticised if they are
extrinsic—i.e. Are instrumentally derived from other
preferences on the basis of beliefs—or inconsistent. This is a
very minimal position on preference criticism, but it has been
questioned regardless.
Numerical Representation of Preference
Last, even if future preferences do not differ from
present preferences on these two accounts, future preferences may
differ because they are formed from another point of view than present
preferences are. A large number of rationality properties have been proposed for
choice functions. The most famous journal voucher argument in favour of preference transitivity is the
money pump argument. Ramsey
(1928a, 182), who pointed out that if a subject’s behaviour violated
axioms of probability and preference, then “[h]e could have a
book made against him by a cunning better and would then stand to lose
in any event”.
Valuational change models (section
7.4) investigate how a change in an agent’s basic evaluations
leads to a change in her preferences. This is what happens when people involved in
negotiations or discussions approach each other’s views in ways that
make their preference relations less conflicting. The Delphi method is
a systematized procedure that can be used to reduce interindividual
differences in preferences.
The vNM theorem is a very important result for measuring the strength
of a rational agent’s preferences over sure options (the
lotteries effectively facilitate a cardinal measure over sure
options). But this does not get us all the way to making rational
decisions in the real world; we do not yet really have a decision
theory. The theorem is limited to evaluating options that come with a
probability distribution over outcomes—a situation decision
theorists and economists often describe as “choice under
risk” (Knight 1921). But there are at least four arguments to the effect that
people’s preferences really do change over time. External
influence models attempt to establish general links between external
events and agents’ preference formations.
2 On rational desire
Hume distinguished reason
from the passion, and argued “that reason alone can never be a
motive to any action of the will; that it can never oppose passion in
the direction of the will” (Treatise, Book II, Part
III, Section III). Humeans often took this distinction between beliefs
and desires to imply not only that beliefs alone cannot motivate
action, but also that desires are not open to similar rational
criticism as beliefs. Therefore, Humeans conclude, preferences can
only be criticised if they are extrinsic – i.e.
instrumentally derived from other preferences on the basis of beliefs
– or inconsistent. Such criticism of extrinsic preferences would
seem ultimately to be a criticism of false beliefs, and it could be
argued that it is therefore not really criticism of preferences
(Broome 1993).
4 Valuational preference change
In preference-to-choice and
choice-to-choice procedures, the outcome is an element of that
alternative set, or possibly a tie outcome (meaning that no social
choice was made). In preference-to-preference procedures, the outcome
is a preference relation over the set of alternatives. If the choice function is defined over all relevant subsets of the
set of alternatives,
≽R
is always complete.
There are substantial differences
between these approaches and their respective assumptions. Completeness (connectedness) is commonly assumed in many applications,
not least in economics. The Bayesian decision maker is assumed to make her choices in
accordance with a complete preference ordering over the available
options.
Secondly, it is necessary to distinguish between those agents who
indeed have preferences as state of minds—e.g. The former category may choose on the basis of
their preferences, and hence the above-discussed effort can aim at
eliciting the preferences on which their choices are based. The latter
category, despite their lack of states of mind, may nevertheless
exhibit behaviour that can be interpreted as relational choice. In
those cases, one can only speak of preferences reconstructed
from choice, without claiming that these agents actually have
preferences at all.
To arrive at a single overall
evaluation of A, the agent needs to employ a formation rule
F, which in itself has nothing to do with choosing between
alternatives but with evaluating preferences on a meta-level according
to the agent’s specific character. Steedman and Krause (1986) discuss
different types of formation rules, which map https://intuit-payroll.org/ a bundle
(≽1,
≽2,
…,
≽n)
onto a single preference relation. The first formation rule describes a
very cautious character, who considers an alternative at least as good
as another only if she considers it at least as good in every
aspect. The assumption of completeness is useful in many applications, not
least in economics.
Instead of adding specific belief-postulates to Jeffrey’s
theory, as Joyce suggests, one can get the same uniqueness result by
enriching the set of prospects. It should moreover be evident, given the discussion of the Sure Thing
Principle (STP) in
Section 3.1,
that Jeffrey’s theory does not have this axiom. Since states
may be probabilistically dependent on acts, an agent can be
represented as maximising the value of Jeffrey’s desirability
function while violating the STP. Moreover, unlike Savage’s,
Jeffrey’s representation theorem does not depend on anything
like the Rectangular Field Assumption. The agent is not required to
have preferences over artificially constructed acts or propositions
that turn out to be nonsensical, given the interpretation of
particular states and outcomes.
3 Constructing preference from choice
The question
arises as to whether this framework is adequate for handling more
complex scenarios, in particular those involving a series or sequence
of decisions; these are referred to as sequential decision
problems. From the perspective of decision-making, unawareness of unawareness is
not of much interest. After all, if one is not even aware of the
possibility that one is unaware of some state or outcome, then that
unawareness cannot play any role in one’s reasoning about what
to do. However, decision-theoretic models have been proposed for how a
rational person responds to growth in awareness (that is
meant to apply even to people who previously were unaware of their
unawareness). In particular, economists Karni and Vierø (2013,
2015) have recently extended standard Bayesian conditionalisation to
such learning events. Their theory, Reverse Bayesianism,
informally says that awareness growth should not affect the ratios of
probabilities of the states/outcomes that the agent was aware of
before the growth.
