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).