An enormous amount of theoretical and empirical research
effort has been devoted to decision under risk and decision
under uncertainty during the past few decades.
I comment here on two topics that illustrate very different facets of
this work.
Both have assumed that 
on X is a weak order.
The first is a theory of subjective expected utility that relaxes
continuity or an Archimedean axiom to obtain vector-valued
utilities ordered lexicographically along with subjective
probabilities characterized by real matrices rather than real numbers.
The second departs more radically from traditional theories and considers
the role of a binary operation 
of joint receipt.
The principal investigators are Irving LaValle in the
lexicographic domain and Duncan Luce for joint receipt.
The lexicographic story beings with Hausner's (1954) lexicographic
linear utility theory for a weak order
on a mixture space X that can be viewed as a set of
lotteries or it's generalization for (7) that is
closed under a mixture operation.
Hausner assumed that and 
satisfy independence 
;
and proved that
is represented by a linear mapping into a real vector space
ordered lexicographically.
When the vector space has finite dimension, say 
, this gives
that satisfies (7) along with
where 
if the two vectors are not equal and
for the smallest i at which they differ.
When 
, (7) is
with each 
a linear functional on X.
We say that u is parsimonious of dimension m if the
representation can not be satisfied by any linear utility function of
smaller dimension.
Given that u is parsimonious of dimension m, it is unique up to an
affine transformation
where v,u and 
are m-dimensional column vectors, is
fixed, and G is an m-by-m lower triangular matrix (0's above the
main diagonal) with all diagonal entries positive.
Failures of continuity that force 
in (7) and (11)
are analyzed in detail in Fishburn (1982).
The typical failure occurs when 
and 
for all 
, in
which case there is a unique 
such that
An example based on marginal probabilities is 
in which 
is the probability of dying and 
is the probability
that you or your heirs will receive $10.
If no increase in can be compensated for by increasing to 1,
then can not be represented by a linear unidimensional
utility function,
but 
can represent lexicographically.
The extension of linear lexicographic utility to decision under
uncertainty in LaValle and Fishburn (1991, 1992, 1996a) and Fishburn and LaValle (1993)
formulates X as the set of all finite-support probability distributions,
called mixed acts,
on a set A of acts in 
with 
.
In the main state-independent version of our theory, we assume
that every consequence is relevant for every state in S,
that the constant-act set 
is included in A,
and that A has some additional structure.
We denote by 
the set of all lotteries on C and let
denote the marginal distribution in state i of 
.
The preference relation applies to X, and for p and q
in , 
means that
when 
and 
for every consequence 
.
We assume the following axioms:
is a weak order, and satisfy independence,
and a relaxed form of Archimedean axiom which implies that the
lexicographic hierarchy has only finitely many levels.
The axioms imply the existence of linear 
and 
that preserve lexicographically on X and on ,
respectively, with parsimonious dimension J of 
and K of u.
Because on is tantamount to
the restriction of on X to mixed constant acts, we have
, and K < J if preferences between other acts force levels into
the hierarchy not accounted for by u on .
Uniqueness follows the format described after (11).
Subjective matrix probabilities 
for 
rectify 
with 
in the expression
where and 
are J-dimensional column vectors and
is a J-by-K real matrix that premultiplies the K-dimensional
column vector 
.
Matrix begins with 
nonzero columns followed by
zero columns such that the first nonzero entry in
column k for 
is a positive number in row
for some 
.
In addition (LaValle and Fishburn 1996a), 
for some i,
for some i, and the J rows of the J-by-nK matrix
are linearly independent.
The resulting representation is
LaValle and Fishburn (1996b, 1996c) show how to assess the vector utilities and matrix probabilities in (12).
The latter paper also describes admissible transformations of the
matrix-probability distribution 
for any given u that put
in a standard normalized form.
With 
,
we say that is a standard matrix distribution if
the K columns of 
are unit vectors
with the 1's in row positions
left to right.
Thus, if K = J, then is the K-by-K identity matrix.
If K < J then standard has J-K rows of zeros
interspersed among the rows below row 1
of the K-by-K identity matrix.
Our second topic for decision under uncertainty is motivated by situations with holistic alternatives that consist of similar but clearly identifiable pieces received jointly, such as two checks and a bill in today's mail or the good news and bad news parts of a medical diagnosis. A fundamental behavioral question asks how people evaluate such alternatives for preference comparison or choice. Do they tend to combine similar pieces and then evaluate wholes, or do they evaluate pieces and then combine these evaluations to arrive at holistic assessments? And, in either case, what rules or operations govern the combining process?
To consider these questions, let 
denote a nonempty set of basic objects,
such as amounts of money or lotteries,
and let denote a binary operation of joint receipt that applies first
to and then to 
defined recursively by
for 
, so that 
with limit 
.
We assume that 
.
Then X includes at least one joint-receipt level.
We assume also that on X is a weak order.
An elementary case of joint receipt that does not involve decision under
risk or uncertainty takes 
with 
.
We interpret 
as an amount of money and refer to 
as a gain and to 
as a loss.
An early empirical and partly theoretical study of joint receipt
for this case is Thaler (1985), followed by Thaler and
Johnson (1990) and Linville and Fischer (1991).
They focused in part on the hedonic editing rule
where 
is strictly increasing, preserves , and has its
origin fixed by u(0) = 0.
This indicates pre-evaluation aggregation in u(x+y) as well as post-evaluation
aggregation in u(x) + u(y),
with addition as the combining operation in each case.
Thaler (1985) found that subjects tend to have 
when x
and y are losses, but 
when x and y are gains, and these agree with (13) if u is
convex
in losses and concave in gains.
Fishburn and Luce (1995) provides a complete analysis of (13)
under the assumption that u is convex in gains and either convex or concave in
losses.
The option for (13) is then clear except in the mixed loss and gain
region where x > 0 > y.
Our results for the mixed region, which depend on limiting slopes of u
at the origin and 
, show for most cases that there is a continuous
curve in 
that separates
and 
when (13) holds.
Research that followed Thaler (1985) in Thaler and Johnson (1990), Linville and
Fischer (1991), and -- within a setting of certainty equivalents for
monetary lotteries -- in Luce (1995) and Cho and Luce (1996),
shows that (13) is not viable in many situations.
Part of the difficulty arises in the mixed region,
where individuals' assessments of
joint receipts are not generally well understood.
Another difficulty can be seen in the conjecture of Tversky and
Kahneman (1992) that , which was sustained at least in
the loss and gain regions separately in Cho and Luce (1996).
If everywhere, it would gut (13) by
effectively excluding .
Also, as Tversky and Kahneman (1992) notes, if in fact we assume that
,
as was done in part of Luce and Fishburn (1991), and if
, then u(x) = kx for some k > 0, and
this linear form is supported neither by intuition nor by empirical research.
Empirical and theoretical investigations of joint receipt of lotteries or acts in decision under risk or uncertainty include, in addition to the certainty-equivalence approach of Luce (1995) and Cho and Luce (1996), Slovic and Lichtenstein (1968), Luce (1991), Luce and Fishburn (1991, 1995) and Cho, Luce and von Winterfeldt (1994). We comment briefly on representational aspects of Luce and Fishburn (1991, 1995) for a joint-receipt axiomatization of what they refer to as rank- and sign-dependent linear utility. A similar representation without the joint-receipt operation was proposed independently in Tversky and Kahneman (1992) and axiomatized in Wakker and Tversky (1993) under the rubric of cumulative prospect theory.
A central part of the representation in Luce and Fishburn (1991) is based on a
qualitative structure 
where
is a weak order on X, is a joint receipt operation on X, and e
denotes the status quo consequence.
Several axioms for the qualitative structure imply that there exists
that satisfies (1) along with u(e) = 0 and
where 
and 
are positive constants and 
and 
are constants.
If u is unbounded and is monotonic in the sense that 
and
and 
and 
, then
and 
.
On the other hand, if and if u is bounded and
is monotonic, then (Luce and Fishburn 1995) 
and
.
In both cases, if is commutative and associative,
then 
.
The weighted additive forms in (14) for the mixed cases of 
and 
were
adopted as a compromise between and
more
complex possibilities.
To expand the formulation to the context of decision under uncertainty,
we can take as the set of all
acts in that assign only finite numbers of consequences to the
states, define as above, and assume that
is associative so that can be replaced by the set 
of all finite sequences 
for which
and 
for all i.
Representation (14) applies under this expansion with
.
Given X in this form, Luce and Fishburn (1991) describes conditions
which imply algebraic forms for the utility of acts that
involve subjective probabilities.
We illustrate with 
,
as in a monetary context, with 
increasing in c.
Let cEd denote the act in that yields c if event
obtains and yields d otherwise.
Assuming that E is neither empty nor the universal event and that
e = 0 with u(0) = 0,
one form is
with 
.
This applies separately to gains, where 
and
, and to losses, where 
and 
.
For each , let 
and 
,
and define 
by
A main part of the representation in Luce and Fishburn (1991) separates gains from losses in the decomposition
with 
for 
and ,
and with the 
unique
and u unique up to a proportionality transform that fixes u(e) at 0.
Additional conditions that allow further refinements for 
and 
based on (14) are described in the reference.
Luce and Fishburn (1995) focuses on the monetary context with
e = 0 and considers first the effect of
for monetary amounts with monotonic.
We adopt the first and last lines of (14) for joint gains and
joint losses respectively, so that:
We assume 
and 
, which imply u(x+y) < u(x) + u(y)
for gains and u(x+y ) > u(x) + u(y) for losses, given
.
These inequalities are consistent with studies, including Kahneman and
Tversky (1979) and others surveyed in Fishburn and Kochenberger (1979), that observe
increasing marginal utility for losses and decreasing marginal
utility for gains.
We show in Luce and Fishburn (1995) that these hypotheses imply exponential
expressions for utility of gains and utility of losses:
We also consider lotteries without presuming
but maintaining the parts of (14) just noted.
Let x denote a lottery on gains
with 
and
.
We identify fairly reasonable assumptions which imply the rank-dependent
form (Quiggin 1993)
in which 
and 
is a continuous
and increasing map
from [0,1] onto [0,1].
A similar form with 
in place of applies to lotteries
on losses.
The issue of mixed gains and losses is more problematic as
described in Section 5 in Luce and Fishburn (1995).
