One of the models for not-necessarily transitive preferences in a multiattribute decision framework is the additive difference model, which is stated as follows: for all n-tuples x=x1,…,xn and y=y1,…,yn in n-Cartesian product set X=X1×⋯×Xn, x is preferred to y if and only if ∑i=1nfiuixi−uiyi>0,where for each i∈Nn=1,…,n, ui and fi are real valued functions respectively on Xi and Δui=uia−uib:a,b∈Xi, and fi is strictly increasing and odd, i.e., fiα+fi−α=0 for all α∈Δui.

Fishburn (1992) presented two axiomatizations of (1) for n≥3, one under a topological assumption and the other for an algebraic structure, based solely on the behavior of a  binary “is-preferred-to” relation on X by utilizing the results of a series of his papers on nontransitive additive conjoint measurement (see Fishburn, 1990a, Fishburn, 1990b, and 1991). He also explored specializations of (1) for the homogeneous case of X1=⋯=Xn in which the sum of (1) reduces to ∑i=1nfiuxi−uyi, or to ∑i=1nπifuxi−uyi with πi>0 for each i∈Nn.

On the other hand, a more demanding axiomatization of (1) for n≥2 was presented by Suppes et al. (1989, Theorem 17.2.3) in which a binary choice probability (BCP) function p on X×X, which is a mapping from X×X into the closed unit interval 0,1 with px,y+py,x=1 for all x,y∈X, is given as a primitive instead of a binary “is-preferred-to” relation. They axiomatized the properties of p and obtained (1) through the representation of p, dubbed the stochastic additive difference representation, as follows: for all x,y∈X, px,y=F∑i=1nfiuixi−uiyi,where F is “strictly increasing” on the domain of F. When x≠y, p-values are interpreted as follows: px,y is the probability that a given decision maker (DM) will choose x rather than y under the forced-choice situation to choose exactly one from x,y. While px,y=1 means that DM always chooses x against y, 12<px,y<1 means that DM is disposed to prefer to take x against y although choosing x does not surely happen.

Since F is assumed to be strictly increasing on the entire domain, (2) is equivalent to asserting that, for all x,y,z,w∈X, px,y≥pz,w⟺∑i=1nfiuixi−uiyi≥∑i=1nfiuizi−uiwi. Then it follows that F is simply given by a strictly increasing transformation of the additive representation in the right-hand side of (3). By properties of a binary relation on X×X (or quaternary relation on X), Suppes et al. (1989) devised their axioms for (3), which consist of a combination of algebraic-difference and additive conjoint structures in Krantz et al. (1971). A drawback of their axiomatization is to fail to cope with extremes of p-values (i.e., we may have ∑i=1nfiuiai−uibi>∑i=1nfiuici−uidi but pa,b=pc,d=1 for some a,b,c,d∈X if p is allowed to take such extreme values). Also their strong solvability axiom assumes that the range of a BCP function p be the open unit interval.

The aim of the paper is to axiomatize properties of p to yield (2) by restricting the domain on which F is strictly increasing. Our approach is to first axiomatize the model (2) with odd functions fi being replaced by skew-symmetric functions ϕixi,yi on Xi×Xi for i∈Nn, dubbed a stochastic skew-symmetric additive (SSA) representation, i.e., for all x,y∈X, px,y=F∑i=1nϕixi,yi,where skew-symmetry of ϕi for i∈Nn means that ϕixi,yi+ϕiyi,xi=0 for all xi,yi∈Xi, and F is strictly increasing on Δ0ϕ1,…,ϕn=∑i=1nϕixi,yi:0<px,y<1 and x,y∈X,which excludes the extreme p-values if they exist. While the relation (3), which replaces the sums of fi-values by the sum of ϕi-values for i∈Nn, does not generally hold true, strict increasingness of F on Δ0ϕ1,…,ϕn allows us to show that (4) is equivalent to the following nested signed-threshold representation of Fishburn’s SSA utility, hereafter denoted ϕ1,…,ϕn;η: there are skew-symmetric functions ϕi on Xi×Xi for i∈Nn and a strictly increasing function η on Rp0, the set of values of p different from 0 and 1, such that, for all x,y∈X and all λ∈Rp0, px,y>λ⟺∑i=1nϕixi,yi>ηλ,where ηλ+η1−λ=0 for all λ∈Rp0. We may say that the function η, which may take positive and negative values, is a signed threshold for SSA utility evaluations and is nested because it depends on referenced p-values, that is, λ in (5).

Instead of attacking direct axiomatization of (4), we devise an axiom system for (5) under an algebraic approach employed by Fishburn’s (1991) axiomatization of SSA representation. He assumed n≥3. We regard the set Rp0 of nonextreme p-values as the n+1th attribute, so that we can reformulate (5) by an SSA representation on n+1 attributes. Thus n≥2 suffices. Once (5) is established, a stochastic additive difference representation (2) with F strictly increasing on Δ0ϕ1,…,ϕn is fairly straightforward by applying the analysis of additive differences by Fishburn (1992).

Last we study homogeneous product sets in the context of finite-state decision making under uncertainty, so that (2) is modified to a stochastic version of the so-called simple regret model developed by Bell (1982) and Loomes and Sugden (1982) by replacing each fiuixi−uiyi by πifuxi−uyi, where f and u are common for all i and πi>0 is interpreted as a subjective probability number of state i. We apply a topological approach employed by Fishburn (1990b). We also extend the domain of ϕi in (5) to multiattributed outcomes and develop an axiom system for a stochastic multiattribute regret representation.