If the decision environment allows you to know the outcomes of unchosen alternatives, it has been argued in the literature of regret, independently initiated by Bell (1982) and Loomes and Sugden (1982), that those outcomes may affect your ex ante preferences in such a way that they may not be predicted by the traditional expected utility paradigm. Although, in the framework of subjective expected utility (SEU) model, every possible outcome of any decision alternative is known to the decision maker (DM), the model and many of its generalizations ignore the possibility that outcome-utility of the chosen alternative may depend on outcomes of unchosen ones.

One exception, whose axiomatic derivations in the various set-ups are known to date (e.g., see Fishburn, 1984, Fishburn, 1989, Fishburn and LaValle, 1987, Nakamura, 1998, Nakamura, 2009, Sugden, 1993) is the SSA (skew-symmetric additive) representation. It says that act f, which maps state space S into a set X of outcomes, is preferred to act g if and only if ∫Sϕfs,gsdπs>0,where π is a subjective probability measure on an algebra BS of subsets of S and ϕ is a skew-symmetric real valued function on X×X. In view of modeling regret, negative (respectively, positive) value of ϕx,y may be interpreted as the subjective evaluation of chosen outcome x against forgone outcome y which is reduced by regret feeling for loosing more preferable y (respectively, enhanced by rejoicing feeling for avoiding less preferable y). The overall evaluation of act f against act g is given by positivity of expected ϕ value, so that we cannot distinguish intrinsic utility-difference of acts from the effects of regret-rejoicing feelings on ex ante preferences. Therefore, the SSA representation is too general to explicitly represent relationship between SEU values and magnitudes of regret-rejoicing feeling.

The aim of the paper is to study and axiomatize a separation of utility-differences of acts and the effects of regret-rejoicing on preferences. Our model is a generalization of SEU with signed threshold , which is interpreted as measuring tradeoffs between regret and rejoicing feelings, i.e., there exist a scale v on X, a utility function u on X, a subjective probability measure π on BS, and a signed threshold function η on Δv=vx−vy:x,y∈X such that act f is preferred to act g if and only if ∫Sufsdπs+∫Sηvfs−vgsdπs>∫Sugsdπs, where η is increasing and odd on Δv and u and v are “ordinally” equivalent. The second term of the above inequality measures a signed expected threshold value between acts f and g. Thus the left hand side totally measures the attractiveness of the chosen act f for which a positive expected threshold value of η enhances SEU value of f because of domination of decision rejoicing against regret, while a negative expected threshold value of η lowers it because of domination of decision regret against rejoicing.

We study axioms for models (1), (2) in Savage’s (1954) framework. An axiom system for (1) was first presented by Fishburn (1989). Sugden (1993) later proposed an extension of Savage’s P4, dubbed weak comparative probability axiom by Machina and Schmeidler (1992), to drop Fishburn’s restricted transitivity axiom. Their axiom systems do not cover non-simple acts. Then Nakamura (1998) axiomatized model (1) to cope with all acts. In this paper, we develop a refinement of Nakamura’s system to drop Fishburn’s transitivity requirement and Savage’s P3, whose redundancy in Savage’s system when all acts are covered was recently discovered by Hartmann (2020). We also add a simple continuity axiom to derive continuity of ϕ in model (1).

Fishburn (1987) argued that the model (1) violates probabilistically sophisticated preferences, dubbed by Machina and Schmeidler (1992), that is, showed that probabilistic sophistication additively decomposes ϕ, i.e., there exists a real valued function u on X such that, for all x,y∈X, ϕx,y=ux−uy. Thus probabilistic sophistication and an SSA axiom system give a characterization of Savage’s SEU. We show that, in addition to our SSA axioms, restriction of probabilistic sophistication to pairs of acts which are regret-free is necessary and sufficient for the existence of a signed-threshold decomposition of a continuous ϕ, i.e., there exist a continuous function v on X, a continuous increasing function u on Rv=vx:x∈X, and a continuous increasing and odd function η on Δv such that, for all x,y∈X, ϕx,y=uvx−uvy+ηvx−vy.The concept of regret-freeness for a pair of acts may be defined to be DM’s judgement for balancing equally probable positive and negative differences of outcomes of the acts, whose magnitudes are measured by a scale v in (3). We present and discuss two alternative axioms to construct v. The first construction is to simply assume that X is a set of monetary values and balancing monetary differences of outcomes invokes regret-freeness, so that v is a linear function on X. The second introduces ex post strength-of-preferences (SoP), which is assumed to be represented by the magnitudes of v-differences. Thus balancing equally probable positive and negative SoP is assumed to define regret-freeness.

Four decades have passed since the seminal works by Bell, Loomes, and Sugden. They proposed a simple model, hereafter dubbed the classical regret model, which requires that u and v in (2) be “cardinally” equivalent. The model and its modifications to cope with multiple choice situations (e.g., see Braun and Muemann, 2004, Quiggin, 1994, Sugden, 1993; and others) are predominantly employed in many empirical investigations of regret effect and many economic applications (e.g., see a recent review found in Diecidue & Somasundaram, 2017). At present several axiom systems for the classical regret model are proposed. They include Fishburn (1992), Diecidue and Somasundaram (2017), Fujii, Murakami, Nakamura, and Takemura (2023), and Nakamura (2023).

On the other hand, the decomposition (3) is rather new in the literature. Nakamura (2016) discusses it by axiomatizing acceptability of statistically dependent risks. Later, Fujii and Nakamura (2021) apply it to economic modeling to investigate the equity premium puzzle which yields the empirically higher risk premium than the one modeling with SEU maximizer as a representative agent predicts. Assuming that the representative agent in a static Lucas economy exhibits preferences represented by (2), they showed that the equilibrium asset price is strictly smaller than the one predicted by modeling with SEU maximizer to explain the empirically high risk premium. They also noted that the equilibrium asset price predicted by the classical regret model is identical to the one under SEU modeling. Thus cardinal equivalence of u and v does not give more predictive power than SEU modeling in an economic application.

The classical regret model assumes that any pair of acts, which are regret-free in our terminology, must be indifferent, while the decomposition (3) requires in addition to regret-freeness that those two acts be probability-equivalent. If we look at the classical regret model as a specialization of SSA preferences, then our question is whether or not there exists a v-scale with which regret-free pairs of acts are indifferent. An advantage of the classical regret model is its simplicity, so that a v-scale in the model can be directly elicited by DM’s ex ante preferences for lotteries (see Bleichrodt, Cillo, & Diecidue, 2010). However, the elicitation method is not devised from the general set-up but assumes that preferences were generated by the classical regret model. Therefore, it is unsettled whether or not the elicited v scale invokes regret-freeness.

Since the decomposition (3) consists of three functions v, u, and η, experimental identifications of those functional forms seem to be hard. We present and discuss how to quantitatively measure those functions from the viewpoint of experimental studies. Assuming SSA preferences, it is shown that, given a v-scale, a collection of odds of events with which DM judges indifference between two binary lotteries is enough to construct functions u and η, where v-scale must be independently elicited. We shall discuss three possible candidates for a v-scale to measure regret-freeness.

The paper is organized as follows. In Section 2, we first examine axioms for an SSA representation under Savage’s framework. Then a continuous and monotonic version of the SSA representation is discussed. Section 3 studies the signed-threshold decomposition (3). First we study the case that X is a nondegenerate, bounded, and closed interval, interpreted as a set of monetary values. We prove that restriction of probabilistic sophistication to pairs of acts which are regret-free separates subjective expected utility and regret-rejoicing tradeoffs. Then we introduce an SoP function to obtain model (3). In Section 4, we present and discuss quantitative measurement of three functions v, u, and η in (3). Section 5 concludes the paper. The appendix includes a lengthy proof of Theorem 4 in Section 3.1, which is the main result of the paper.