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Multiattribute Regret:
Theory and Experimental Study∗
Yoichiro Fujii†, Hajime Murakami‡, Yutaka Nakamura§, and Kazuhisa Takemura¶
Very preliminary version: October 1, 2021
Abstract. The aim of this paper is two-fold. The first part is to generalize the simple
regret model by Bell (1982) and Loomes and Sugden (1982) to cope with the situation in
which decision outcomes are multi-attributed. We propose a model which combines the
simple regret model for ex ante preferences and the additive difference representation for
ex post preferences. Then we show a necessary and sufficient axiomatization of our model
under Savage’s framework. The proposed model is composed of three types of functions.
One is a value function for each attribute. The others are attribute-dependent and holistic
regret-rejoicing functions which respectively capture trade-offs among value-differences of
chosen and forgone outcomes for each attribute and among all attributes. We also discuss
constructive procedures and their consistency. The second part is to conduct an experiment
to measure two attributes case. Participants were asked to join in a work to carry over heavy
bags at some varying distances. The data supports the framing effect, that is, participants
tend to have convex value functions in each attribute and indicate regret averse for weights
and regret seeking for distances in gain frame. On the other hand, participants exhibit
concave value functions for both attributes and regret seeking for weights and regret averse
(almost regret neutral) for distances in loss frame.
Keywords: multiattribute regret theory; decision making under uncertainty; utility
measurement; holistic regret aversion; attribute-dependent regret aversion
JEL Classification Numbers: D81, D91, G41
∗This study is partly supported by Japan Society for the Promotion of Science (JSPS) Kakenhi Grant
Number 17K03637.
†School of Commerce, Meiji University, Japan. e-mail: fujii@meiji.ac.jp
‡Center for Decision Research, Waseda University, Japan. e-mail: h.murakami6@kurenai.waseda.jp
§Professor Emeritus, Graduate School of Systems and Information Engineering, University of Tsukuba,
Japan, e-mail: nakamurayutakads@gmail.com; nakamura@sk.tsukuba.ac.jp
¶Department of Psychology, Waseda University, Japan. e-mail: kazupsy@waseda.jp
1
1 Introduction
Expected utility (EU) theory has been one of the leading standard models in decision mod-
eling under risk and uncertainty. However, we observe a large amount of experimental
evidences in economic and psychological research which presented decision outcomes against
the predictions by EU modeling such as Allais paradox. To overcome those phenomena,
there have been proposed many theoretical and behavioral generalizations of EU modeling.
One of the influential generalizations is prospect theory, which asserts that the reference
point of decision makers changes depending on the situation and that they make risk averse
decisions in gain situations and risk seeking decisions in loss situations (e.g., see Tversky and
Kahneman, 1992; Wakker, 2010). Although prospect theory has explained many decision
making phenomena, it has not fully explained the effect of predicting regret before making
a decision. For example, the anticipation that one will regret adopting this option may
influence behavioral choices.
Bell (1982) and Loomes and Sugden (1982) are the first who introduced regret in decision
making under uncertainty. Regret theory assumes that decision maker’s preference will be
influenced by consequences of foregone alternatives that she does not choose. If the foregone
consequence is preferable to the actual one, she may suffer from regret because she could have
a better choice. In the opposite case, she may feel rejoicing by avoiding worse consequences.
To model regret, Bell, Loomes, and Sugden (hereafter BLS) proposed a simple regret model,
which is constructed by two functions: a value function to evaluate consequences (or their
strength of preferences) and a regret-rejoicing function to capture trade-offs among value-
differences of chosen and forgone outcomes at distinct states. For a recent review of BLS-
related regret theory, see Bleichdodt and Wakker (2015).
Lerner et al. (2015) argued that there is room to introduce emotions into decision anal-
ysis, with a range of emotions including guilt, regret, pride, and happiness. Zeelenberg
and Pieters (2007) focused on the feeling of regret and clarified it in various situations.
Incorporating multidimensional aspects of disclosure into decision models may explain var-
ious decision situations such as medical decisions. In addition, Wallenius et al. (2008) has
shown that multiattribute decisions are made in many situations, and in light of this, regret
modeling in multidimensional situations may deserve to be of research interest.
Regret modeling in decision analysis has recently received much research interest in ap-
plications in finance, insurance, health risks and so forth. For examples, Fujii et al. (2016)
showed that regret theory justifies demand for mixed insurance even expected utility the-
ory cannot explain it. Li and Huang (2017) proposed a regret theoretical travelers’ route
choice model. Diecidue and Somasundaram (2017) provided a new behavioral foundation
using trade-off consistency. Fujii and Osaki (2018) examined the effect of regret feeling in
threshold probability which is the criterion of medical examinations. Fujii and Nakamura
(2021) demonstrated that the equity premium puzzle in financial market may be resolved
when the representative agent is sensitive to regret.
Although the traditional regret theory assumes that the underlying outcomes are mea-
sured by a single attribute such as wealth or income level, we postulate that different at-
tributes may cause different attitudes toward regret feeling. Multiattribute utility theory
2
(hereafter MAUT) has been theoretically developed to investigate various decompositional
forms of von Neumann-Morgenstern utility functions and applied to many real world deci-
sion problems (e.g., see a classical book by Keeney and Raiffa, 1976). Dyer et al. (1992)
and Wallenius et al. (2008) provided comprehensive review and pointed out the importance
of cooperation with behavioral considerations for future research in MAUT. To the best of
our knowledge, however, we observe no theoretical research to investigate decompositional
implications of trade-off conditions among multiple attributes in subjective decision frame-
work.
This paper extends BLS’s simple regret model to cope with multiattribute case, and
presents necessary and sufficient axioms under Savage’s framework. The proposed model is
composed of three types of functions. The first one is a value function for each attribute
and the other two are attribute-dependent and holistic regret-rejoicing functions, which
respectively capture trade-offs among value-differences of chosen and forgone outcomes with
the common attribute-levels except one at distinct states and trade-offs among monotonically
holistic value-differences of all attributes. Our regret model assumes that intransitivity of ex
ante preferences may come from attribute-dependent and/or holistic regret-rejoicing trade-
offs or intransitive ex post preferences captured by additive difference model. We also discuss
quantitative methods to measure those three ingredients in our model.
Finally, we conducted an experimental study of our regret model. We studied a decision
situation in which participants were asked to join in a work to carry over heavy bags at
some varying distances and weights. They were offered a sequence of binary choices between
alternatives depending on two attributes; one is the weight of bags and the other is the
distances to carry. We estimated value functions for each attribute, two attribute-dependent
regret-rejoicing functions and a holistic regret-rejoicing function. We analyzed those func-
tions by the individual level. Although they are found to be essentially nonlinear, we could
not conclude at the present that those functions conclusively exhibit concavity or convexity
property for each participant. The data supports the framing effect, that is, participants
tend to have convex value function in each attribute and indicate regret averse for weights
and regret seeking for distances in gain frame. On the other hand, participants exhibit
concave value functions for two attributes and regret seeking for weights and regret averse
(almost regret neutral) for distances in loss frame.
The paper is organized as follows. In section 2 and 3, we propose multiattribute regret
model and its axioms. We introduce procedure of measurement in section 4. We conduct
an experiment in two attribute case in section 5. We discuss results in section6. Section 7
contains conclusion.
2 Multiattribute Regret
2.1 Model Specification
Let Sbe a nonempty set of states. An act is a mapping from Sinto a nonempty set Xof
outcomes. Let Adenote the set of all acts. The classical simple regret model is described
3
as follows: there exist a real valued function von X, a strictly increasing odd function φon
R, and a probability measure pdefined on a Boolean algebra of subsets of Ssuch that act
fis preferred to act gif and only if
S
φ(v(f(s)) −v(g(s))) dp (s)>0,(1)
where nonlinearity of φ, whose curvature may be interpreted in terms of regret-rejoicing
tradeoffs, causes intransitive ex ante preference patterns (see e.g., Loomes and Sugden, 1991;
Bleichrodt et al., 2010; Humphrey, 2001).
In what follows, let Nn={1, . . . , n}with n≥2 and the outcome set Xwill be replaced
by a set Xof multiattributed outcomes, i.e., X=X1× · · · × Xn, where Xifor i∈Nnis the
set of the i-th attribute levels. Every n-attributed outcome will be denoted by a bold-faced
letter xwith xithe corresponding i-th attribute’s level, so x= (x1, . . . , xn). It has been
argued in the literature that multiattributedness, a characteristic of decision outcomes, may
also cause intransitive ex post preference patterns (see e.g., May 1954; Tversky, 1969). One
model to cope with the intransitivity is known as an additive difference representation in
psychology, which is stated as follows: there exist real valued functions vion Xiand strictly
increasing odd functions τion Rfor i∈Nnsuch that outcome xfor sure is preferred to
outcome yfor sure if and only if
n
i=1
τi(vi(xi)−vi(yi)) >0,(2)
where nonlinearity of τi, whose curvature may be interpreted as tradeoffs among attributes,
causes intransitivity of the ex post preferences. Some axiomatic derivations of model (2) are
known in the literature (see Tversky, 1969; Suppes et. al., 1989; Fishburn, 1992).
Now we combine (1) and (2). We need some notations. Since acts are mappings from
Sinto X, they will be denoted by bold-faced letters, f,g,h, and so on. Given an act f
and a state s∈S,f(s)∈X, so the i-th attribute level may be written by f(s)i. Then
f(s) = (f(s)1, . . . , f (s)n). A constant act is an act ffor which f(s) = xfor all s∈S.
Thus every x∈Xwill be identified with a constant act.
For i∈Nnand f∈ A, the i-th marginal of f, denoted fi, is a mapping from Sinto Xi
defined by
fi(s) = f(s)ifor all s∈S.
Thus we may also write f= (f1, . . . , fn) and f(s) = (f1(s), . . . , fn(s)).
Let ≻be the binary ex ante “is preferred to” relation on A. Our model of multiattributed
regret is stated as follows: there exist real valued functions vion Xifor i∈Nn, strictly
increasing odd functions φand τion Rfor i∈Nn, and a probability measure pdefined on
a Boolean algebra of subsets of Ssuch that, for all f,g∈ A,
f≻g⇐⇒ S
φn
i=1
τi(vi(fi(s)) −vi(gi(s)))dp (s)>0.(3)
4
where nonlinearity of φand the composite functions φ(τi(·)) for i∈Nnmay be respectively
interpreted as “holistic” and “attribute-dependent” regret attitudes. Note also that ex post
preferences are captured by an additive difference model (2), and that when n= 1, (3) is
BLS’s simple regret model.
Two specializations of (3) are given as follows:
(i) when τifor all i∈Nnare linear, we have: for all f,g∈ A,
f≻g⇐⇒ S
φn
i=1
vi(fi(s)) −
n
i=1
vi(gi(s))dp (s)>0,(4)
where the curvature of φmay indistinguishably capture both of holistic and attribute-
dependent regret attitudes;
(ii) when φis linear, we have: for all f,g∈ A,
f≻g⇐⇒
n
i=1 S
τi(vi(fi(s)) −vi(gi(s))) dp (s)>0,(5)
where preferences for regret-rejoicing tradeoffs, which are assumed to be represented by the
curvatures of τ1, . . . , τn, are attribute-dependent.
2.2 Axiomatization
To understand an axiomatic structure of our multiattribute regret model (3) , we divide it
in three steps in the axiomatizing process, in which we shall adopt Savage’s framework (see
Savage, 1954; Fishburn, 1970, Chapter 14) with Xa rectangular subset of Rn. Thus we
shall impose continuity and monotonicity properties on our axiomatization.
Step 1. We note that model (3) is a special case of a so-called skew-symmetric additive
(SSA) representation. Since axiomatizations of SSA representation under Savage’s frame-
work are known (see Fishburn 1989, Sugden 1993, and Nakamura 1998), we shall assume
throughout that (A,≻) admits a Savagean SSA representation, denoted (Φ, p), i.e., there
exist a skew-symmetric function Φ on X×X(i.e., Φ (x,y) + Φ (y,x) = 0 for all x,y∈X)
and a probability measure pon 2Ssuch that, for all f,g∈ A,
f≻g⇐⇒ S
Φ (f(s),g(s)) dp (s)>0,
where Φ is unique up to a positive multiplicative transformation and a unique subjective
probability pis defined on the set of all subsets of Sand convex-ranged, i.e., if p(A)>0
and 0 < λ < 1, then p(B) = λp (A) for some B⊂A.
5
Step 2. Construct a “continuous” nontransitive additive decomposition of Φ (see Propo-
sition 1 below), denoted (φ; Φ1, . . . , Φn), i.e., there exist ncontinuous skew-symmetric func-
tions Φion Xi×Xifor i∈Nnand a strictly increasing odd function φon Rsuch that, for
all x,y∈X,
Φ (x,y) = φn
i=1
Φi(xi, yi).
We note that Model (4) directly obtains by requiring that, for i∈Nn, there exists a strictly
increasing continuous function vion Xisuch that Φi(xi, yi) = vi(xi)−vi(yi) for all xi, yi∈Xi
(see Proposition 7 in Section 3.2).
Step 3. For model (3), we construct an additive difference representation of n
i=1 Φi(xi, yi)
(see Proposition 2 below), i.e., for attribute i, there exist a “strictly increasing” and “con-
tinuous” function vion Xiand a “strictly increasing” and “continuous” odd function τion
Rsuch that, for all xi, yi∈Xi,
Φi(xi, yi) = τi(vi(xi)−vi(yi)) .
Furthermore, Proposition 4 in Section 3.1 characterizes linearity of φ, so that model (5)
obtains.
Our task in this section is to find necessary and sufficient axioms for Steps 2 and 3. For
Step 2, we need several notations and definitions to devise axioms for an additive decompo-
sition of Φ. Multiattributed outcomes in Xare sometimes depicted as follows: for i∈Nn
and x,y∈X,xiy−irepresents an outcome zin Xfor which zi=xiand zj=yjfor all
j̸=i.
An m-partition of Sis a set of mevents, {A1, . . . , Am}, such that Ai∩Aj=∅for all i̸=j
and ∪m
i=1Ai=S. For A⊆S,f∈ A, and x∈X,f(A) = xmeans that f(s) = xfor all
s∈A. Given an m-partition ω={A1, . . . , Am}, an act fwill be denoted by ⟨x1, . . . , xm⟩ω
if f(Ai) = xifor i∈Nm. An m-partition ω={A1, . . . , Am}is said to be uniform if
p(Ai) = 1
mfor i∈Nm, i.e., ⟨x,y⟩ωi∼ ⟨x,y⟩ωjfor all i̸=j, where ωi={Ai, Ac
i}for i∈Nm.
We say that a set of 2-dimensional real vectors, {(α1, β1), . . . , (αK, βK)}, is balanced if
|{i: (αi, βi) = (γ, δ)}| =|{i: (αi, βi) = (δ, γ)}| for all (γ, δ)∈R2. Then a set of pairs of
n-dimensional real vectors, (x1,y1), . . . , xK,yK, is said to be balanced if, for i∈Nn,
(x1
i, y1
i), . . . , xK
i, yK
iis balanced.
Since we require continuity of nontransitive additive decomposition of Φ, we shall impose
the following continuity condition as an axiom, which is understood as applying to all a,b∈
Xand all uniform 2-partitions ω,
Axiom A1.(x,y)∈X2:⟨x,a⟩ω≿⟨y,b⟩ωand (x,y)∈X2:⟨a,y⟩ω≿⟨b,x⟩ω
are closed in X2.
For nontransitive additivity, we need the following axiom, sometimes dubbed a cancellation
axiom or independence axiom, which depends on positive integers Kand applies to all
x1, . . . , xK,y1, . . . , yK,z1, . . . , zK,w1, . . . , wK∈Xand all uniform 2-partitions ω,
6
Axiom A2(K). If (x1,y1), . . . , xK,yK,(w1,z1), . . . , wK,zK is balanced, and
⟨xi,wi⟩ω≿⟨yi,zi⟩ωfor all i∈NK−1,then yK,zKω≿xK,yKω.
Clearly, Axiom A2(K+1) implies Axiom A2(K) for all integers K > 0. This axiom is similar
to the independence axiom devised by Fishburn (1990) to axiomatize nontransitive additive
conjoint measurement.
The following proposition shows that Axioms A1 and A2(4) are necessary and sufficient
for nontransitive additive decomposition of Φ. Note that the under the continuity axiom,
K= 4 in Axiom A2(K) suffices for the decomposition. While Fishburn’s proof for non-
transitive additivity requires quite a lot of intermediate lemmas, our proof, which will be
deferred to Appendix A, is fairly short because of assuming the existence of Φ.
Proposition 1.Suppose that (A,≻)admits a Savagean SSA representation (Φ, p). Then
Axioms A1 and A2 (4) hold if and only if there exist ncontinuous skew-symmetric functions
Φion Xi×Xifor i∈Nnand a strictly increasing odd function φon Rsuch that, for all
x,y∈X,
Φ (x,y) = φn
i=1
Φi(xi, yi).
In what follows, for x,y∈X, we shall write x≥ywhen xi≥yifor i= 1, . . . , n, and
say that xis greater than yif, in addition, x̸=y. Further, we may say that xis a vector
by increasing yby a positive amount x−y≥0, where 0is the zero vector in Rn.
For Step 3 to derive additive difference representations, we need the following two axioms,
understood as applying to all i∈Nn, all x,y,z,x′,y′,z′,a,b∈X, and all uniform
2-partitions ω.
Axiom A3. (monotonicity). If x≥yand x̸=y,then ⟨x,z⟩ω≻ ⟨z,y⟩ω.
Axiom A4.If ⟨xia−i, y′
ib−i⟩ω≿⟨yia−i, x′
ib−i⟩ωand ⟨yia−i, z′
ib−i⟩ω≿⟨zia−i, y′
ib−i⟩ω,
then ⟨xia−i, z′ib−i⟩ω≿⟨zia−i, x′
ib−i⟩ω.
A monotonicity axiom A3 simply says that Φ is strictly increasing in the first argument
when the vector is increased by an positive amount, i.e., for all x,y,a∈X, Φ (x+a,y)>
Φ (x,y) whenever a≥0and a̸=0. Axiom A4 is an adaptation in the present context of the
condition, sometimes called a bi-cancellation, which is a key in the difference measurement.
The implication of Axioms A3 and A4 for a continuous additive decomposition is stated
in the following proposition.
Proposition 2.Suppose that (A,≻)admits a Savagean SSA representation (Φ, p)and
Φhas a continuous additive decomposition (φ; Φ1, . . . , Φn). Then Axioms A3 and A4 hold
if and only if, for attribute i∈Nn,there exist a strictly increasing and continuous function
vion Xiand a strictly increasing and continuous odd function τion Rsuch that, for all
xi, yi∈Xi,
7
Φi(xi, yi) = τi(vi(xi)−vi(yi)) .
The proof of the proposition will be deferred to Appendix A. Hence, we completed an
axiomatic characterization of model (3) with the additional properties of continuity and
monotonicity. In the rest of the paper, we shall study the model (3) whose existence is
guaranteed by Proposition 2.
3 Regret Aversion
In this section, we shall discuss and axiomatize convexity properties of φand τ1, . . . , τnin
model (3).
Let Ibe a real interval. Consider a skew-symmetric function ϕon I×I. Assume that ϕ
is decomposed as ϕ(α, β) = τ(α−β). Then, in the literature, it is argued that a concept
of regret aversion, which is defined by the inequality, ϕ(α, β ) + ϕ(β, γ) + ϕ(γ , α)<0 for all
α, β, γ ∈Iwhenever α > β > γ, is related to strict convexity of τon the positive domain.
However, the following example suggests that a stronger condition is needed for this claim.
Example 1. Define τon I= [0,1] as follows: for some 1
2< ϵ < 1,
τ(α) = 1
ϵα2if 0 ≤α≤ϵ;
αif ϵ < α ≤1.
Then it is easy to verify that, for all α, β, γ ∈Iwith α > β > γ,
ϕ(α, β) + ϕ(β, γ) + ϕ(γ, α) = τ(α−β) + τ(β−γ) + τ(γ−α)<0.
However, τon Iis not strictly convex. □
It follows from the definition of strict convexity of τon the positive domain that, for all
α, β, γ , δ in {α′−β′:α′≥β′, α′, β′∈I},
τ(α) + τ(β)< τ (γ) + τ(δ)
whenever 0 ≤δ < {α, β}< γ and α+β=γ+δ. Let α′, β′, γ′, δ′∈Isatisfy α=α′−β′,
β=γ′−δ′,γ=α′−δ′, and δ=γ′−β′. Then α′> γ′≥β′> δ′and
τ(α′−β′) + τ(γ′−δ′)< τ (α′−δ′) + τ(γ′−β′),
which is
ϕ(α′, β′) + ϕ(β′, γ ′) + ϕ(γ′, δ′) + ϕ(δ′, α′)<0.(6)
This inequality is necessary and sufficient for strict convexity of τon the positive domain
whenever ϕ(α, β) = τ(α−β). When the strict inequality in (6) is replaced by equality, τ
must be linear.
8
Also, we note a necessary and sufficient condition for τto be a linear function is given
as follows: for all α, β, γ ∈I,
ϕ(α, β) + ϕ(β, γ) + ϕ(γ, α) = 0.(7)
3.1 Attribute-dependent regret
We apply the above necessary and sufficient condition (6) to our model (3). Let ψi(·) =
φ(τi(·)) for i∈Nn. In what follows, let ω={E1, E2, E3, E4}be a 4-partition of S. Then
we note that, for xi, yi, zi, wi∈Xiand some a∈X,
⟨yia−i, zia−i, wia−i, xia−i⟩ω≻ ⟨xia−i, yia−i, zia−i, wia−i⟩ω
⇐⇒ p(E1)φ(τi(vi(yi)−vi(xi))) + p(E2)φ(τi(vi(zi)−vi(yi)))
+p(E3)φ(τi(vi(wi)−vi(zi))) + p(E4)φ(τi(vi(xi)−vi(wi))) >0
⇐⇒ p(E1)ψi(vi(yi)−vi(xi)) + p(E2)ψi(vi(zi)−vi(yi))
+p(E3)ψi(vi(wi)−vi(zi)) + p(E4)ψi(vi(xi)−vi(wi)) >0.
If xi> zi≥yi> wi, then vi(xi)> vi(zi)≥vi(yi)> vi(wi). Therefore, the following axiom,
understood as applying to all xi, yi, zi, wi∈Xi, is necessary and sufficient for the property
that ψiis convex on the positive domain, i.e., we may say that the decision maker is regret
averse for the i-th attribute.
Axiom B1(i).If xi> zi≥yi> wi,then, for a uniform 4-partitions ωand some
a∈X,
⟨yia−i, zia−i, wia−i, xia−i⟩ω≻ ⟨xia−i, yia−i, zia−i, wia−i⟩ω.
The implication of Axiom B1(i) is stated in the following proposition, whose proof im-
mediately follows from the argument in the preceding paragraph.
Proposition 3.Suppose that (A,≻)admits model (3). Then Axiom B1 (i)holds for
i∈Nnif and only if ψion the positive domain is convex.
When φis linear, i.e., model (5) holds, Axiom B(i) characterizes convexity of τion its
positive domain. To derive linearity of φ, we need some notations. For x,y∈X, we shall
write x>yif x≥yand x̸=y. For x∈Xand I={i1, . . . , ik} ⊆ Nnwith i1<· · · < ik,
xImeans an outcome in XI=Xi1× · ·· × Xik, so xI= (xi1, . . . , xik). Then we shall write
a partial summation as
ΦI(xI,yI) :=
i∈I
Φi(xi, yi).
9
Furthermore, for I⊆Nn,J=Nn\I, and x,y∈X,xIyJmeans an outcome in a∈X
for which ai=xifor all i∈Iand ai=yiotherwise. Linearity of φin model (3) follows
from the following axiom, understood as applying to all proper I⊂Nnwith J=Nn\I, all
xI,yI,zI∈XI, and all dJ∈XJ.
Axiom B2.⟨xIdJ,yIdJ,zIdJ⟩ω∼ ⟨yIdJ,zIdJ,xIdJ⟩ωwhenever there are a,b,c∈X
such that
⟨yIaJ,zIbJ⟩ω∼ {⟨xIbJ,yIcJ⟩ω,⟨yIbJ,xIcJ⟩ω}.
An implication of Axiom B2 is given as follows.
Proposition 4.Suppose that (A,≻)admits and SSA representation (Φ, p)and Φhas
a continuous additive decomposition (φ; Φ1, . . . , Φn). Then Axiom B2 implies that φis a
linear function.
Proof. By the hypotheses of Axiom B2, we have
Φ (yIaJ,xIbJ) + Φ (zIbJ,yIcJ)=0,
Φ (yIaJ,yIbJ) + Φ (zIbJ,xIcJ)=0.
By Proposition 1 and strict increasingness of φ, those two are rearranged to give
ΦI(yI,xI)+ΦJ(aJ,bJ)=ΦI(yI,zI)+ΦJ(cJ,bJ),
ΦI(yI,yI)+ΦJ(aJ,bJ)=ΦI(xI,zI)+ΦJ(cJ,bJ),
which are additively combined to get
ΦI(xI,yI) = ΦI(xI,zI)+ΦI(zI,yI).
By the conclusion of Axiom B2, we have
⟨xIdJ,yIdJ,zIdJ⟩ω∼ ⟨yIdJ,zIdJ,xIdJ⟩ω
⇐⇒ Φ (xIdJ,yIdJ) + Φ (yIdJ,zIdJ) + Φ(zIdJ,xIdJ) = 0
⇐⇒ φ(ΦI(xI,yI)) + φ(ΦI(yI,zI)) + φ(ΦI(zI,xI)) = 0
⇐⇒ φ(ΦI(xI,zI)+ΦI(zI,yI)) = φ(ΦI(xI,zI)) + φ(ΦI(zI,yI)) ,
where we substituted the last expression of ΦI(xI,yI) in the preceding paragraph for the
second equation. Since ΦI(xI,zI) and ΦI(zI,yI) can be arbitrary, the last equation implies
that φis a linear function. □
10
3.2 Holistic regret
The holistic regret aversion may be captured by the convexity of φ. Thus we need to
compare the magnitudes of the arguments of φ, i.e., we need to know when ΦN(x,w) is
larger than ΦN(y,z). To this end, we impose the following monotonicity axiom on the
nontransitive continuous additive decomposition of Φ, which is understood as applying to
all x,y,z,w∈X.
Axiom A3*. (monotonicity). If x≥y≥z≥wand either x̸=yor z̸=w,then
⟨x,z⟩ω≻ ⟨w,y⟩ωfor a uniform 2-partition ω.
The implication of the axiom is given as follows: for all x,y,z,w∈X,
ΦN(x,w)>ΦN(y,z)
whenever x≥y≥z≥wand either x̸=yor z̸=w. Clearly, this requirement is more
demanding than the one in Axiom A3.
Assuming Axiom A3*, it follows from (6) that φon the positive domain is convex if and
only if
φ(ΦN(x,w)) + φ(ΦN(z,y)) > φ (ΦN(x,y)) + φ(ΦN(z,w))
for all x,y,z,w∈Xsuch that x>z≥y>wand
ΦN(x,y)+ΦN(z,w) = ΦN(x,w)+ΦN(z,y).
The first inequality can be realized by the preference
⟨y,z,w,x⟩ω≻ ⟨x,y,z,w⟩ω,
and the second equality can (not necessarily) be attained by splitting the equality into two
equalities: for any nonempty proper subset I⊂Nnwith J=Nn\I,
ΦI(xI,yI)+ΦI(zI,wI)=ΦI(xI,wI)+ΦI(zI,yI),
ΦJ(xJ,yJ)+ΦJ(zJ,wJ)=ΦJ(xJ,wJ)+ΦJ(zJ,yJ),
since ΦN(a,b) = ΦI(aI,bI) + ΦJ(aJ,bJ) for all a,b∈X. Those two equalities are realized
by preferences as follows. We focus on a preference-realization for the first equality, since
the second is similarly obtained by putting Jin place of I. Since ΦIand ΦJare continuous,
we may find a,b,c,d∈Xwhich satisfy
ΦI(xI,yI)+ΦI(zI,wI) = ΦI(xI,wI)+ΦI(zI,yI) = ΦJ(cJ,aJ)+ΦJ(dJ,bJ).
Then, for the first and the third terms, ΦI(xI,yI)+ΦI(zI,wI) = ΦJ(cJ,aJ)+ΦJ(dJ,bJ)
if and only if
11
φ(ΦI(xI,yI)+ΦJ(aJ,cJ)) + φ(ΦI(zI,wI)+ΦJ(bJ,dJ)) = 0,
which is ⟨xIaJ,zIbJ⟩ω∼ ⟨yIcJ,wIdJ⟩ωfor a uniform 2-partition ω. For the first and the
second terms, we similarly obtain that ΦI(xI,wI) + ΦI(zI,yI) = ΦJ(cJ,aJ) + ΦJ(dJ,bJ)
if and only if ⟨xIaJ,zIbJ⟩ω∼ ⟨wIcJ,yIdJ⟩ω.
Hence, a sufficient condition for the convexity of φis given by the following axiom, which
is understood as applying to all proper I⊂Nnwith J=Nn\Iand all x,y,z,w∈X.
Axiom B3.If x>z≥y>w,then ⟨y,z,w,x⟩ω≻ ⟨x,y,z,w⟩ωwhenever there are
a,b,c,d∈Xsuch that c≥a,d≥b,and
⟨xIaJ,zIbJ⟩ω∼ {⟨yIcJ,wIdJ⟩ω,⟨wIcJ,yIdJ⟩ω},
⟨aIxJ,bIzJ⟩ω∼ {⟨cIyJ,dIwJ⟩ω,⟨cIwJ,dIyJ⟩ω}.
An implication of Axiom B3 is given by the following proposition.
Proposition 5. Suppose that (A,≻)admits and SSA representation (Φ, p)and Φhas
a continuous nontransitive additive decomposition (φ; Φ1, . . . , Φn). If Axiom B3 holds, then
φon the positive domain is convex.
Proof. This follows from the argument in the preceding paragraph. □
Less demanding condition can be obtained by the following axiom, understood as apply-
ing to all proper I⊂Nnwith J=Nn\Iand all x,y,z,w∈X.
Axiom B3*(I).If xI>zI≥yI>wI,then
⟨yIeJ,zIeJ,wIeJ,xIeJ⟩ω≻ ⟨xIeJ,yIeJ,zIeJ,wIeJ⟩ω
for a uniform 4-partition ωand some eJ∈XJwhenever
⟨xIaJ,zIbJ⟩ω∼ {⟨yIcJ,wIdJ⟩ω,⟨wIcJ,yIdJ⟩ω}.
Convexity of φobtains when Axioms B3*(I) holds for a particular proper subset I⊂Nn
as shown in the following proposition.
Proposition 6. Suppose that (A,≻)admits and SSA representation (Φ, p)and Φhas
a continuous nontransitive additive decomposition (φ; Φ1, . . . , Φn). Then Axiom B3 *(I)for
some nonempty proper I⊂Nnholds if and only if φon the positive domain is convex.
Proof. Similar to the proof of Proposition 5. □
In the last, we show a necessary and sufficient condition for which holistic regret and
attribute-dependence of regret aversion for the i-th attribute coincides, i.e., linearity of τi
12
holds. This property holds if Φisatisfies (7) in place of ϕ. Thus we need the following axiom,
understood as applying to all xi, yi, zi∈Xiand all a,b,c∈X.
Axiom B4(i). For all xi, yi, zi∈Xiand a uniform 2-partition ω,if ⟨xib−i, yic−i⟩ω∼
⟨yia−i, zib−i⟩ωfor some a,b,c∈X,then ⟨xib−i, yic−i⟩ω∼ ⟨zia−i, yib−i⟩ω.
An implication of Axiom B4(i) is given by
Proposition 7.Suppose that (A,≻)admits an SSA representation (Φ, p)and Φhas a
nontransitive continuous additive decomposition (φ; Φ1, . . . , Φn). Then Axiom B4 (i)holds
for all i∈Nnif and only if, for all i∈Nnand all xi, yi, zi∈Xi,
Φi(xi, yi)+Φi(yi, zi)+Φi(zi, xi) = 0.
Proof. Suppose that ⟨xib−i, yic−i⟩ω∼ ⟨yia−i, zib−i⟩ωfor some a,b,c∈X. Then
Φi(xi, yi)+Φi(yi, zi) = Φ−i(a−i,b−i)+Φ−i(b−i,c−i)
By Axiom B2(i), ⟨xib−i, yic−i⟩ω∼ ⟨zia−i, yib−i⟩ω, so that
Φi(xi, zi) = Φ−i(a−i,b−i)+Φ−i(b−i,c−i).
Hence Φi(xi, yi)+Φi(yi, zi)+Φi(zi, xi) = 0. □
4 Constructive Procedures
We show how to construct vi, τi, and φin model (3) with additional properties such as
continuity and monotonicity. Assume that state space Sis generated by a probability wheel.
Let Mbe a (finite or infinite) set of consecutive integers.
(i) Construction of vi
Let ω={E, E c}be a uniform 2-partition of S. We note that, for all a,b∈X, all
xi, yi, zi, wi∈Xi, and i∈Nn,
⟨xia−i, wib−i⟩ω≿⟨yia−i, zib−i⟩ω(8)
⇐⇒ p(E)φ(Φi(xi, yi)) + p(Ec)φ(Φi(wi, zi)) ≥0
⇐⇒ Φi(xi, yi)+Φi(wi, zi)≥0
⇐⇒ τi(vi(xi)−vi(yi)) ≥τi(vi(zi)−vi(wi))
⇐⇒ vi(xi)−vi(yi)≥vi(zi)−vi(wi).
Then a subset Xω
i=xk
i:k∈M⊂Xiis said to be an equally vi-spaced sequence generated
by ωif xk+1
i> xk
iand xk−1
ia−i, xk+1
ia−iω∼xk
ia−ifor all k∈M, where yi=ziand a=b
13
Figure 1: Construction of vion Xi: an equally vi-spaced sequence xℓ
i:ℓ∈Min Xiwith
vixk
i= 0 for some k∈Mand vixk+ℓ
i=ℓ.
in (8) and xk
imay be called a conditional certainty equivalent to a lottery yielding xk−1
ior
xk+1
iwith equal probability. Thus, for all k∈M,
vixk−1
i−vixk
i=vixk
i−vixk+1
i.
Hence, with no loss of generality (w.n.l.g), we may define
vixk+1
i−vixk
i= 1
for all xk
i, xk+1
i∈Xω
i(see Figure 1).
Consistency of Xω
i: for any equally vi-spaced sequence Xω
iconstructed in (i), the consis-
tency requires that, for all xk
i, xℓ
i∈Xω
i, all integers m > 0, and all a,b∈X,
xk+m
ia−i, xℓ
ib−iω∼xk
ia−i, xℓ+m
ib−iω.
(ii) Construction of φ(τi(·))
Let ψi(·) = φ(τi(·)). Given Xω
iin (i), we assume that vi(x0
i) = 0 and vixk+1
i−
vixk
i= 1 for all k∈M. Then w.n.l.g. (see Figure 2), we can scale ψias follows: for all
x−1
i, x0
i, x1
i∈Xω
i,
14
Figure 2: Construction of ψion vixk
i−vi(x0
i) : k∈M, where xk
i:k∈Mis an
equally vi-spaced sequence, and it is assumed that vi(x0
i) = 0 and vixk
i−vi(x0
i) = k
for k=−3,−2,−1,0,1,2,3.
ψivix1
i−vix0
i =ψi(1) = 1,
ψi(0) = 0,
ψivix−1
i−vix0
i =ψi(−1) = −1.
For k≥2, our task is to find an event Aksuch that, for a 2-partition σk={Ak, Ac
k},
x−1
ia−i, xk
ia−iσk∼x0
ia−i,
which gives
p(Ak)ψivix−1
i−vix0
i+p(Ac
k)ψivixk
i−vix0
i= 0,
so that p(Ak)ψi(−1) + p(Ac
k)ψi(k) = 0. Hence,
ψi(k) = p(Ak)
1−p(Ak).
Consistency of ψi(k) for k > 1: for k > 1, find an event A−ksuch that, for a 2-partition
σ−k=A−k, Ac
−k,
x−k
ia−i, x1
ia−iσk∼x0
ia−i,
15
which gives
p(A−k)ψivix−k
i−vix0
i+pAc
−kψivix1
i−vix0
i= 0,
so that p(A−k)ψi(−k) + pAc
−kψi(1) = 0. Hence,
ψi(−k) = −1−p(A−k)
p(A−k).
Since ψimust be odd, by the consistency, we must have
p(Ak)
1−p(Ak)=1−p(A−k)
p(A−k).
(iii) Construction of τi.
Let ω={E, E c}be a uniform 2-partition of S. We note that, for all x,y,z∈X,
⟨x,z⟩ω≿⟨z,y⟩ω
⇐⇒ p(E)φn
i=1
τi(vi(xi)−vi(zi))+p(Ec)φn
i=1
τi(vi(zi)−vi(yi))≥0
⇐⇒ φn
i=1
τi(vi(xi)−vi(zi))≥φn
i=1
τi(vi(yi)−vi(zi))
⇐⇒
n
i=1
τi(vi(xi)−vi(zi)) ≥
n
i=1
τi(vi(yi)−vi(zi)) .
Then for a given z∈X, let Vi(z) = {vi(xi)−v(zi) : xi∈Xi}for i∈Nn. We define a
binary relation ≿zon V(z) = V1(z)× · ·· × Vn(z) as follows: for all α,β∈V(z),
α≿zβ⇐⇒ ⟨x,z⟩ω≿⟨z,y⟩ω,
where αi=vi(xi)−τi(zi) and βi=vi(yi)−τi(zi) for i∈Nn. Thus (V(z),≿z) admits an
additive representation: for all α,β∈V(z),
α≿zβ⇐⇒
n
i=1
τi(αi)≥
n
i=1
τi(βi).
To construct τifor i∈Nn, we may apply the conjoint scaling method (see Keeney
and Raiffa, 1976). We need to specify tradeoffs between two attributes. Thus for distinct
i.j ∈Nn, we let xiyia−ij represent an outcome z∈Xfor which zi=xi,zj=xj, and zk=ak
for all k̸=i, j.
Fix a∈Xand distinct i, j in Nn. Assume vk(ak) = 0 for all k∈Nn. Take any positive
αi
1∈Vi(a) with vi(x1
i) = α1
ifor x1
i∈Xi. A sequence {αi
k:k∈NKi}in Vi(a) for some
16
Figure 3: Construction of τiand τjfor i̸=j(conjoint sclaing method): We assume that
vi(ai) = vj(aj) = 0, vi(α1
i) = 1 with αk
i=vi(x1
i) for some x1
i∈Xi. For each k, we shall
find xk
iand xk
jfor which αk
i=vixk
i−vi(ai) and αk
j=vjxk
j−vj(aj).
integer Ki>0 (Kimay be infinite) is said to be an increasing standard sequence if there
exists an α1
j∈Vj(a) such that vjx1
j=α1
jand
x1
ia−i,aω∼a, x1
ja−jω,
(i.e., x1
ia−i≿ax1
ja−j, so that τi(vi(x1
i)) = τjvjx1
j), and then αk
ifor k > 1 are recursively
defined as follows: given αk
iwith vixk
i=αk
i,αk+1
isatisfies
xk+1
ia−i,aω∼a, xk
ix1
ja−ij .
Similarly, a increasing standard sequence αj
k:k∈NKjin Vj(a) is defined as follows: given
αk
jwith vjxk
j=αk
j,αk+1
jsatisfies
xk+1
ja−j,aω∼a, xk
jx1
1a−ij
Therefore, τiand τjare scaled as follows (see Figure 3): for all k > 0,
τiαk
i=τivixk
i−vi(ai)=k,
τjαk
j=τjvjxk
j−vj(aj)=k.
Consistency: Given increasing standard sequences, {αi
k:k= 1,2, . . . , Ki}in Vi(a) and
αj
k:k∈NKjin Vj(a), we may similarly construct decreasing standard sequences:
17
αi
k:k=−1,−2, . . . , −K′
i,
αj
k:k=−1,−2, . . . , −K′
j.
Then consistency requires that αi
−k=αi
kand αj
−k=αj
k.
Furthermore, on the grids in {αi
k:k= 0,±1,±2, . . .} × αj
k:k= 0,±1,±2, . . ., the fol-
lowing must holds true:
τiαk+1
i+τjαℓ
j=τiαk
i+τjαℓ+1
j
⇐⇒ φτiαk+1
i+τjαℓ
j=φτiαk
i+τjαℓ+1
j
⇐⇒ p(E)φτiαk+1
i+τjαℓ
j+p(Ec)φτi−αk
i+τj−αℓ+1
j= 0
⇐⇒ p(E)φτivixk+1
i−vi(ai)+τjvjxℓ
j−vj(aj)
+p(Ec)φτivi(ai)−vixk
i+τjvj(aj)−vjxℓ+1
j= 0
⇐⇒ xk+1
ixℓ
ja−ij , x0
ix0
ja−ij ω∼x0
ix0
ja−ij , xk
ixℓ+1
ja−ij ω.
5 Experimental Study
5.1 Participant
Participants were 15 students (12 female, mean age of participants was 21.93 years) from
department of psychology of the Waseda University, Tokyo, Japan. Each participant had a
pilot session before the actual session.
5.2 Incentives
Participants were paid a flat fee of 2,500 Japanese yen (about 24 US dollars) book cards
for their participation. In addition, participants were told to play one of their choices for
real at the end of all sessions. All outcomes were loss. Before participants carry a weight,
experimenter explained again to participants that they can quit experiment any time, if they
didn’t want to carry a weight.
5.3 Procedure
The study was composed of choice session using computer and interview session lasting two
and a half hours on average. Each session started with an explanation of the task, read aloud
by the experimenter. Participants could see this explanation on their computer screen. The
experimenter explained that at the end of all sessions, they would play one of their choices
for real. Then two practice questions followed. After these practice questions, participants
18
were asked to walk 10 m with a 2kg weight. Participants were not asked directly for their
indifference values. Instead, indifference values were determined through a series of binary
choices. A computer program, using Psychopy (Peirce et al., 2019), following the procedure
outlined in Gonzalez and Wu (1999) was used in this study. Each binary choice corresponded
to an iteration in a process for eliciting the indifference values, described in Appendix B.
Figure 4 gives five examples of the way the choice questions were presented. Participants
at first faced two options, neutrally labeled A and B (Figure 4a), and were asked to choose
between these options by clicking on their preferred option. Indifference was not allowed. If
they chose option A, new options were appeared at a next row (Figure 4b). Participants were
asked to choose between these new options. New rows were appeared until participants chose
option A six times (Figure 4c) or chose option B (Figure 4d). If the option B was chosen,
a second screen was presented using a narrower range (Figure 4e). Then they moved on to
the next question. If they chose option B at the first row (Figure 4a), an error message was
shown, because option B in the first row was always inferior to option A. The experimenter
explained why the error message was shown.
The experiment consisted of six construction tasks for vi, τiand ψi, i = 1,2, in gain
and loss frame, consistency tasks for viand choice tasks to test axiom B1(i). Table 1
shows the stimuli used in construction tasks. Two attributes were measured for each func-
tion. To measure value function in gain (loss) frame, we elicited a standard sequence
{x0
i, x1
i, ..., xn
i}, i = 1,2. As explained in section 4, the measurement method amounted
to finding values xk−1
iso that < xk−1
ia−i, xk+1
ia−i>ωand xk
ia−i, i = 1,2, were equiva-
lent. To measure τi, we elicited a sequence of money amount z1
i, ..., z4
i, i = 1,2. The
measurement method amounted to finding values zk+1
ithat led to indifference between
< zk+1
ia0
−i, z0
ia0
−i>ωand < z0
ia0
−i, zk
1a1
−i>ω. To measure ψi, we elicited a sequence of
probabilities p(A2), ..., p(An), i = 1,2, that led to indifference between < x−1
ia−i, xk
ia−i>σk
and x0
ia−i.
19
(a)
(b)
(c)
(d)
(e)
Figure 4: Five examples of a screen and mouse faced in the first part (attribute 1, distances)
20
Table 1: Measurement method
21
5.4 Data analysis
This subsection describes an analysis of the experimental data, mentioned in previous sub-
section. We will show aggregate and individual data for vi, τi, ψiand φ. To investigate
curvature of these functions at the individual level, the data were analyzed under specific
parametric assumption, the power family. The power family is defined by xrfor r > 0.
We scaled these functions such that vi(x0
i) = 0 and vi(max(xk
i)) = 1, in the gain frame.
It is well known that r < 1 corresponds to concave function, r > 1 to convex function,
and r= 1 to linear function. Estimations were computed by nonlinear least square method
for each participant separately. The parametric estimates were used to obtain parametric
classification of the individual functions. Using the coefficients, a participant was classified
as concave(convex) if his power coefficient was smaller (larger) than or equal to 0.99 (1.01)
and linear if 0.99 < r < 1.01.
5.4.1 Analysis of attribute-dependent value function vi
To analyze the shape of viat the individual level, we fitted the individual vi=xαfunctions
parametrically through a power function and classified participants based on their estimated
power coefficients α.
5.4.2 Analysis of attribute-dependent regret-rejoicing function τi
To compute τi, the vi(zk
i), k = 1, ..., 4, i = 1,2, had to be determined. This was done by
interpolation through the estimated power coefficients of vi. Then we fitted the individual
τi=xβfunctions parametrically through a power function and classified participants based
on their estimated power coefficientsβ.
5.4.3 Analysis of holistic and attribute-dependent regret ψi
To analyze the shape of ψiat the individual level, we fitted the individual ψi=xγfunc-
tions parametrically through a power function and classified participants based on their
estimated power coefficients γ. Some participants reached the maximum p(Ak∗) = .99 (e.g.:
p(Ak) = .60, .70, .80, .99, .99, k = 2, ..., 5). When we fitted those sequences, sequences of
{p(A2), ..., p(Ak∗)}were used (e.g.: p(Ak) = .60, .70, .80, .99, k = 2, ..., 4).
5.4.4 Analysis of holistic regret φ
As explained in section 4, ψi(k) = φ(τi(k)) = (τi(k))δand ψi(k) = p(Ak)/[1 −p(Ak)] in
gain frame ( ψi(−k) = [1 −p(A−k)]/p(A−k) in loss frame). This implies that p(Ak)/[1 −
p(Ak)] = φ(τi(k)) in gain frame [1−p(A−k)]/p(A−k) = φ(τi(−k)) in loss frame). To compute
φ, τi(k), i = 1,2, had to be determined. This was done by interpolation through the estimated
power coefficients of τi. The holistic regret φis not attribute-dependent. Theoretically,
p(Ak)/[1 −p(Ak)] = φ(τ1(k)) and p(Ak)/[1 −p(Ak)] = φ(τ2(k)) should have the same shape.
Both values were used to fit φfunction parametrically through a power function and classified
participants based on their estimated power coefficients. When we fitted φfunction, almost
22
(a) Loss frame (b) Gain frame
Figure 5: Estimated vifunctions based on mean coefficients
same range of τi(k) were used. If τ1(k) = {1,2, ..., 3}and τ2(k) = {1,2, ..., 2.5,4,5}, then
τ1(k) = 1,2, ..., 3 and τ2(k) = {1,2, ..., 2.5,4}were used.
6 Results
6.1 Aggregate Findings
Figure 5, 6, 7 and 8 shows vi, τi, ϕiand φfunctions, respectively, based on the mean and
median estimated power coefficients. The broken line is drawn for comparison and represents
the case of linear value function. The solid lines and dotted lines represent mean and median
coefficients, respectively. The red (blue) lines represent the case of attribute 1, distances
(attribute 2, weights).
6.1.1 Attribute-dependent value function vi
The figure 5 shows that at the aggregate level value functions for both attributes were convex
(concave) in the gain (loss) frame. The mean (SD) power coefficients in loss frame were 2.2
(1.88) for distances and 1.56 (1.24) for weights. The mean (SD) power coefficients in gain
frame were 1.64 (2.02) for distances and 2.44 (3.69) for weights.
6.1.2 Attribute-dependent regret-rejoicing function τi
The figure 6 shows that at the aggregate level attribute-dependent regret-rejoicing function
for distances was convex (concave or almost linear) in the gain (loss) frame and for weights
23
(a) Loss frame (b) Gain frame
Figure 6: Estimated τifunctions based on mean coefficients
was concave (convex) in the gain (loss) frame. The mean (SD) power coefficients in loss
frame were 1.02 (0.46) for distances and 0.73 (0.27) for weights. The mean (SD) power
coefficients in gain frame were 1.52 (1.10) for distances and 0.72 (0.34) for weights.
6.1.3 Holistic and attribute-dependent regret ψi
The figure 7 shows that at the aggregate level holistic and regret-rejoicing functions for both
attributes were convex (concave) in the gain (loss) frame. The mean (SD) power coefficients
in loss frame were 6.33 (5.29) for distances and 2.17 (2.82) for weights. The mean (SD)
power coefficients in gain frame were 10.12 (7.83) for distances and 4.59 (3.63) for weights.
To investigate axiom B1(i), participants were asked to choose between
< yia−i, zia−i, wia−i, xia−i>ωv.s. < xia−i, yia−i, zia−i, wia−i>ω
where x1= 0m, z1= 166.7m, y1= 333.3m, w1= 1000m, a−1= 3kg, for distances, x2= 0kg,
z2= 1kg, y2= 2kg, w2= 6kg, a−2= 500m, for weights, and ω={white ball, yellow ball,
green ball, blue ball }in both attributes. If the stimuli were perceived in loss (gain) frame, 8
(9) participants were the same shape between the parametric classification and classification
by axiom B1(i) for distances, and 10 (5) participants were the same shape for weights. In
three out of four conditions (for distances in loss and gain frame, and for weights in loss
frame), the majority of choices in axiom B1(i) were consistent.
6.1.4 Holistic regret φ
The figure 8 shows that at the aggregate level holistic and attribute-dependent regret function
was convex (concave) in the gain (loss) frame. The mean (SD) power coefficients were 10.55
24
(a) Loss frame (b) Gain frame
Figure 7: Estimated ψifunctions based on mean coefficients
(16.61) in loss frame and 22.59 (46.06) in gain frame.
6.2 Individual Findings
6.2.1 Attribute-dependent value function vi
For attribute distances (weights), 9 (6) participants were the same parametric classification
in loss and gain frame. Four (6) participants were the opposite parametric classification in
loss and gain frame. The parametric classification showed modest evidence for concavity in
loss frame and convexity in gain frame. For distances, 4 participants were concave in loss
and convex in gain. For weights, 5 participants were concave in loss and convex in gain.
6.2.2 Attribute-dependent regret-rejoicing function τi
For attribute distances (weights), 7 (4) participants were the same parametric classification
in loss and gain frame. Six (10) participants were the opposite parametric classification in
loss and gain frame. The parametric classification showed modest evidence for convexity in
both frame for distances. For weights, the parametric classification showed strong evidence
for convexity in loss frame and concavity in gain frame.
6.2.3 Holistic and attribute-dependent regret ψi
For attribute distances (weights), 1 (5) participants were the same parametric classification
in loss and gain frame. Fourteen (10) participants were the opposite parametric classification
in loss and gain frame. The parametric classification showed strong support for concavity in
25
(a) Loss frame (b) Gain frame
Figure 8: Estimated φfunctions based on mean coefficients
loss frame and convexity in gain frame especially for distances. For distances, 14 participants
were concave in loss and convex in gain. For weights, 7 participants were concave in loss
and convex in gain.
6.2.4 Holistic regret φ
Two participants were the same parametric classification in loss and gain frame. Thirteen
participants were the opposite parametric classification in loss and gain frame. The para-
metric classification showed strong support for concavity in loss frame and convexity in gain
frame.
26
Table 2: The pattern of parametric classifications in loss and gain frame in individual level
27
7 Discussion and Conclusion
This paper aimed to construct a multiattribute regret theory and to measure preferences in
two attributes case. We provided necessary and sufficient axioms for our proposed model.
Our model was composed of three types of functions; value function to evaluate outcomes
in each attribute, attribute-dependent, and holistic regret-rejoicing function. The last two
functions captured trade-offs among value-differences of chosen and forgone outcomes for
each attribute and among all attributes respectively.
We conducted an experiment in two attributes case to join in a work to carry over heavy
bags at some varying distances. We found that participants were affected by the framing
effect in aggregate level. Participants exhibited convex preferences for two attributes and
regret averse for weights and regret seeking for distances in gain domain while participants
were cancavity for two attributes and regret seeking for weights and regret averse (almost
regret neutral) for distances in loss domain. On the other hand, we found that about half the
number of participants were not affected by the framing effect in each attribute level such
as value function and attribute-dependent regret-rejoicing function. Interestingly, the fram-
ing effect was observed in holistic regret-rejoicing function, that is, almost all participants
exhibited opposite attitude toward regret.
Wallenius et al. (2008) showed accomplishments and prospective research topics using
multiattribute utility theory. It is natural that decision makers are affected by the feeling of
rejoicing or regret because decision outcomes in different alternatives may differ in different
state. Our proposed model will be applied to various fields. In our experimental study, the
number of participants was limited. In future study, Larger sample sizes would be favorable
in examining the proposed model. We studied the decision situation where participants were
asked to join in a work to carry weights at some distances. Further research could investigate
other decision situation and it would also be beneficial in testing our model.
Appendix A
This appendix provides proofs of Propositions 1 and 2.
Proof of Proposition 1. Let (Φ, p) be a Savagean SSA representation of (A,≻). To
prove necessity of Axioms A1 and A2(K) for all integers K > 0, we suppose that Φ has a
continuous additive decomposition (φ; Φ1, . . . , Φn). Let ω={E, Ec}be a uniform 2-partition
of S. Then for all x,y,a,b∈X,
⟨x,b⟩ω≿⟨y,a⟩ω⇐⇒ p(E) Φ (x,y) + p(Ec) Φ (b,a)≥0
⇐⇒ Φ (x,y)≥Φ (a,b)
⇐⇒ φn
i=1
Φi(xi, yi)≥φn
i=1
Φi(ai, bi)
⇐⇒
n
i=1
Φi(xi, yi)≥
n
i=1
Φi(ai, bi).
28
Since Φifor i∈Nnare continuous,
(x,y) :
n
i=1
Φi(xi, yi)≥
n
i=1
Φi(ai, bi)={(x,y) : ⟨x,a⟩ω≿⟨y,b⟩ω},
(x,y) :
n
i=1
Φi(ai, bi)≥
n
i=1
Φi(xi, yi)={(x,y) : ⟨a,y⟩ω≿⟨b,x⟩ω}
are closed in X2. Thus Axiom A1 obtains.
To see necessity of Axiom A2(K), assume that (x1,y1), . . . , xK,yK,(w1,z1), . . . , wK,zK
is balanced. Then we have that, for j∈Nn,
K
i=1
Φjxi
j, yi
j+
K
i=1
Φjwi
j, zi
j= 0,
so that
n
j=1 K
i=1
Φjxi
j, yi
j+
K
i=1
Φjwi
j, zi
j
=
K
i=1 n
j=1
Φjxi
j, yi
j+
n
j=1
Φjwi
j, zi
j
= 0.
If n
j=1 Φjxi
j, yi
j+n
j=1 Φjwi
j, zi
j≥0 for i∈NK−1(i.e., ⟨xi,wi⟩ω≿⟨yi,zi⟩ωfor
i∈NK−1), then
n
j=1
ΦjxK
j, yK
j+
n
j=1
ΦjwK
j, zK
j≤0,
so xK,wKω≿yK,zKω. Hence, Axiom A2(K) obtains.
To prove sufficiency of Axioms A1 and A2(4), we suppose that Axioms A1 and A2(4)
hold. Let ω={E, Ec}be a uniform 2-partition of S. We note that, for all x,y,z,w∈X,
⟨x,w⟩ω≿⟨y,z⟩ω⇐⇒ p(E) Φ (x,y) + (1 −p(E)) Φ (w,z)≥0
⇐⇒ Φ (x,y)≥Φ (z,w).
Thus we define a binary relation ≥∗on X2as follows: for all x,y,z,w∈X,
(x,y)≥∗(z,w)⇐⇒ ⟨x,w⟩ω≿⟨y,z⟩ω.
29
Then it clearly follows that, for all x,y,z,w∈X,
(x,y)≥∗(z,w)⇐⇒ Φ (x,y)≥Φ (z,w),
so that ≥∗is a weak order. Furthermore, ≥∗satisfies the following two conditions, understood
as applying to all x,y,z,w,a,b,c,d,a′,b′,c′,d′∈Xand i∈Nn.
C1. (x,y)∈X2: (x,y)≥∗(z,w)and (x,y)∈X2: (z,w)≥∗(x,y)are closed in
X2.
C2. If
xia−i, x′
ia′
−i≥∗yib−i, y′
ib′
−i,
zib−i, z′
ib′
−i≥∗wia−i, w′
ia′
−i,
yic−i, y′
ic′
−i≥∗xid−i, x′
id′
−i,
then zic−i, z′
ic′
−i≥∗wid−i, w′
id′
−i.
C1 follows from Axiom A1. To see that C2 holds, we rewrite it in terms of ≿as follows:
if
xia−i, y′
ib′
−iω≿x′
ia′
−i, yib−iω,
zib−i, w′
ia′
−iω≿z′
ib′
−i, wia−iω,
yic−i, x′
id′
−iω≿y′
ic′
−i, xid−iω,
then zic−i, w′
id′
−iω≿z′
ic′
−i, wid−iω. This claim follows from Axiom A2(4), since the set
xia−i, x′
ia′
−i,zib−i, z′
ib′
−i,yic−i, y′
ic′
−i,z′
ic′
−i, zic−i,y′
ib′
−i, yib−i,
w′
ia′
−i, wia−i,x′
id′
−i, xid−i,wid−i, w′
id′
−i
is balanced.
It is well known (see Wakker, 1989, Theorem III.6.6.) that a weak order ≥∗on X2that
satisfies C1 and C2 is represented by a continuous additive representation, i.e., there exist
ncontinuous functions Φion Xi×Xifor i∈Nnsuch that, for all x,y,z,w∈X,
(x,y)≥∗(z,w)⇐⇒
n
i=1
Φi(xi, yi)≥
n
i=1
Φi(zi, wi).
Furthermore, Φiare unique up to similar positive transformations, i.e., for i∈Nn, Φ′
i(xi, yi) =
αΦi(xi, yi)+βifor some α > 0 and βi. Since Φ (x,x) = 0 for all x∈X, (x,x)≥∗(y,y) and
(y,y)≥∗(x,x) for all x,y∈X, so that n
i=1 Φi(xi, xi) = n
i=1 Φi(yi, yi). Thus we can
assume that by appropriate choices of β1, . . . , βn, Φ (xi, xi) = 0 for all xi∈Xiand i∈Nn.
To assure that Φiare skew-symmetric, we note by skew-symmetry of Φ that
30
(x,y)≥∗(z,w)⇐⇒ Φ (x,y)≥Φ (z,w)
⇐⇒ Φ (w,z)≥Φ (y,x)
⇐⇒ (w,z)≥∗(y,x)
⇐⇒
n
i=1
Φi(wi, zi)≥
n
i=1
Φi(yi, xi)
⇐⇒
n
i=1
Φ′
i(xi, yi)≥
n
i=1
Φ′
i(zi, wi),
where Φ′
i(xi, yi) = −Φi(yi, xi) for i∈Nn. Thus Φ′
i(xi, yi) = αΦi(xi, yi) for some α > 0, so
αΦi(xi, yi) = −Φi(yi, xi),
αΦi(yi, xi) = −Φi(xi, yi).
Therefore, α2Φi(xi, yi) = Φi(xi, yi), so α= 1. Hence, Φi(xi, yi) = −Φi(yi, xi).
Hence, it easily follows that there exists a strictly increasing odd function φon Rsuch
that, for all x,y∈X,
Φ (x,y) = φn
i=1
Φi(xi, yi).
This completes the sufficiency proof. □
Proof of Proposition 2. Suppose that (A,≻) admits a Savagean SSA representation
(Φ, p) and Φ has a continuous nontransitive additive decomposition (φ; Φ1, . . . , Φn).
Necessity of Axiom A3 immediately follows from strict increasingness of φ,vi, and τi
for i∈Nn. For necessity of Axiom A4, assume that ⟨xia−i, y′
ib−i⟩ω≿⟨yia−i, x′
ib−i⟩ωand
⟨yia−i, z′
ib−i⟩ω≿⟨zia−i, y′
ib−i⟩ω. Then
φ(τi(vi(xi)−vi(yi))) + φ(τi(vi(y′
i)−vi(x′
i))) = 0,
φ(τi(vi(yi)−vi(zi))) + φ(τi(vi(z′
i)−vi(y′
i))) = 0,
which are rearranged to give
τi(vi(xi)−vi(yi)) + τi(vi(y′
i)−vi(x′
i)) = 0,
τi(vi(yi)−vi(zi)) + τi(vi(z′
i)−vi(y′
i)) = 0.
Again, those two equations are rearranged to give
31
vi(xi)−vi(yi) + vi(y′
i)−vi(x′
i)=0,
vi(yi)−vi(zi) + vi(z′
i)−vi(y′
i)=0,
which are additively combined to give
vi(xi)−vi(zi) + vi(z′
i)−vi(x′
i) = 0.
Hence, φ(τi(vi(xi)−vi(zi))) + φ(τi(vi(z′
i)−vi(x′
i))) = 0, which means ⟨xia−i, z′ib−i⟩ω≿
⟨zia−i, x′
ib−i⟩ω.
To prove sufficiency of Axioms A3 and A4, we suppose that Axioms A3 and A4 hold.
Monotonicity of Φifollows from Axiom A3 and Proposition 1.
Let ω={E, E c}be a uniform 2-partition. Then it follows from Proposition 1 that
⟨xia−i, wib−i⟩ω≿⟨yia−i, zib−i⟩ω
⇐⇒ p(E)φ(Φi(xi, yi)) + p(Ec)φ(Φi(wi, zi)) ≥0
⇐⇒ φ(Φi(xi, yi)) ≥φ(Φi(zi, wi))
⇐⇒ Φi(xi, yi)≥Φi(zi, wi).
Thus we define a binary relation ≥ion Xi×Xias follows: for all xi, yi, zi, wi∈Xi,
(xi, yi)≥i(zi, wi)⇐⇒ ⟨xia−i, wib−i⟩ω≿⟨yia−i, zib−i⟩ωfor some a,b∈X.
We show that ≥isatisfies the following four conditions, understood as applying to all xi, yi, zi,
x′
i, y′
i, z′
i∈Xi:
D1. ≥iis a weak order, i.e., it is complete and transitive.
D2. If (xi, yi)≥i(x′
i, y′
i), then (y′
i, x′
i)≥i(yi, xi).
D3. If (xi, yi)≥i(x′
i, y′
i) and (yi, zi)≥i(y′
i, z′
i), then (xi, zi)≥i(x′
i, z′
i).
D4. {(x′
i, y′
i) : (x′
i, y′
i)≥i(xi, yi)}and {(x′
i, y′
i) : (xi, yi)≥i(x′
i, y′
i)}are closed in Xi×Xi.
The completeness of ≥ifollows from the completeness of ≿. Transitivity of ≥iand
conditions D2-D3 are respectively rewritten in terms of ≿as follows:
D1*. If ⟨xia−i, wib−i⟩ω≿⟨yia−i, zib−i⟩ωand ⟨zia−i, w′
ib−i⟩ω≿⟨wia−i, z′
ib−i⟩ω, then
⟨xia−i, w′
ib−i⟩ω≿⟨yia−i, z′
ib−i⟩ω.
D2*. If ⟨xia−i, y′
ib−i⟩ω≿⟨yia−i, x′
ib−i⟩ω, then ⟨y′
ia−i, xib−i⟩ω≿⟨x′
ia−i, yib−i⟩ω
D3*. The same as Axiom A4.
D4*. The two sets,
32
{(x′
i, y′
i) : ⟨x′
ia−i, yib−i⟩ω≿⟨y′
ia−i, xib−i⟩ω},
{(x′
i, y′
i) : ⟨xia−i, y′wib−i⟩ω≿⟨yia−i, x′
ib−i⟩ω},
are closed in Xi×Xi.
For D1*, {(xia−i, yia−i),(zia−i, wia−i),(yia−i, xia−i),(wib−i, zib−i),(w′
ib−i, z′
ib−i),
(z′
ib−i, w′
ib−i)}is balanced. By Axiom A2(3), D1* holds true. For D2*, {(xia−i, yia−i),(x′
ia−i, y′
ia−i),
(y′
ib−i, x′
ib−i),(yib−i, xib−i)}is balanced. By Axiom A2(2), D2* holds true. D3* follows
from Axiom A1.
It follows from Krantz et. al. (1971, Chapter 4) that Conditions D1-D4 hold if and
only if there exists a strictly increasing continuous function vion Xisuch that, for all
xi, yi, zi, wi∈Xi,
(xi, yi)≥i(zi, wi)⇐⇒ vi(xi)−vi(yi)≥vi(zi)−vi(wi).
By definition of ≥i,
(xi, yi)≥i(zi, wi)⇐⇒ Φi(xi, yi)≥Φi(zi, wi).
Hence, ϕi(xi, yi) = τi(vi(xi)−vi(yi)) for some strictly increasing odd function τi. This
completes the proof.
Appendix B
Method for Eliciting the indifference values. In the measurement of viwith loss
frame, x−k−1
iwas elicited through choices between A=< x−k−1
ia−i, x−k+1
ia−i>ωand B=
x−k
ia−i. We stopped the measurement of viwith loss frame when −k−1≥4 and x−k−1
1≥
800m (x−k−1
2≥4.8kg), which is 80% of the reference point of gain frame (1000m in attribute
1 and 6kg in attribute 2), or −k−1 reached 51. Figure 9 gives an example of the procedure for
the elicitation of x−1
1through comparisons between A=< x−1
13kg,0m3kg >ωand B=50m
3kg. From response in Figure 9(a), the indifference value should be somewhere between
50m and 70m. Then in Figure 9 (b), a second step was presented using a narrower range
(e.g., 50m to 70m in increments of 4m). From response in Figure 9(b), We recorded as
indifference value the midpoint between 66 and 70, that is, 68. Figure 10 gives another
example of the procedure for the elicitation of x−1
1in first step. From response in Figure 10
(a), the indifference value must be larger than 150m. Then as in Figure 10 (b), a second
screen of first step was presented using a wide range (e.g., 150m to 250m in increments of
20m).
In the measurement of τiwith loss frame, the procedure was the largely same as the elic-
itation method for vi.z−1
iwas elicited through choices between A=< z−(k+1)
ia0
−i, z0
ia0
−i>ω
and B=< z0
ia0
−i, z−k
1a−1
−i>ω, k = 0, ..., −3, for attribute 1 (distances) and k=−1, ..., −3, for
attribute 2 (weights). Figure 11 gives an example of the procedure for the elicitation of z−1
1
33
(a) First step using a wide range (e.g., 50m to 150m in in-
crements of 20m)
(b) Second step using a narrower range (e.g., 50m to 70m in
increments of 4m)
Figure 9: Example of the elicitation of x−1
1
through comparisons between A=< z−1
10kg,50m0kg >ωand B=<50m0kg,50m0.5kg >ω.
From response in Figure 11(a), the indifference value should be somewhere between 50m and
70m. Then in Figure 11(b), a second step was presented using a narrower range (e.g., 50m
to 70m in increments of 4m). From response in Figure 11(b), We recorded as indifference
value the midpoint between 66 and 70, that is, 68. Figure 12 gives another example of the
procedure for the elicitation of z−1
1in first step. From response in Figure 12 (a), the indif-
ference value must be larger than 150m. Then as in Figure 12 (b), a second screen of first
step was presented using a wide range (e.g., 150m to 250m in increments of 20m).
In the measurement of ψiwith loss frame, the procedure was the largely similar. We
elicited the value of A−kfor which indifference held between A=< x−k
ia−i, x1
ia−i>σk and
B=x0
ia−i, k = 2, ..., n, where σk={A−k, Ac
−k}and {x−k
i, x1
i, x0
i}were the outcomes of the
standard sequence elicited in the measurement of viwith loss frame. The probability of A−k
was assumed to be A−k/100. Figure 13 gives an example of the procedure for the elicitation
of A−2through comparisons between A=< x−2
13kg,0m 3kg>σkand B=50m 3kg, and A−3
through comparisons between A=< x−3
13kg,0m 3kg>σkand B=50m 3kg. In this example,
the elicited value for x−2
1and x−3
1were 100m and 150m respectively. From response in Figure
13(a), the indifference value should be somewhere between 0 and 10 white balls. Then in
Figure 13(b), a second step was presented using a narrower range (e.g., 0 to 10 white balls
in increments of 2). From response in Figure 13(b), We recorded as indifference value the
midpoint between 6 and 8, that is, 7. As showed in Figure 13(c), a first step for the elicitation
34
(a) First step using a wide range (e.g., 50m to 150m in in-
crements of 20m)
(b) Second screen of first step using a wide range (e.g., 150m
to 250m in increments of 20m)
Figure 10: Another example of the elicitation of x−1
1in first step
of A−3was passed with a message for participants (e.g., ”There is only one response held
consistency. You don’t need to choose here. This screen will be changed in a few seconds.”),
because A−3should be less than A−2. Then in Figure 13(d), the second step was presented
using a narrower range (e.g., 0 to 8 white balls in increments of 2). From response in Figure
13(d), We recorded as indifference value the midpoint between 6 and 8, that is, 7.
Figure 14 gives an example of the procedure for the investigation of axiom B1(i) through
comparisons between A=<333.3m 3kg,166.7m 3kg,1000m 3kg, 0m 3kg>ωand B=<0m 3kg,
333.3m 3kg, 166.7m 3kg, 1000m 3kg>ω. As showed in Figure 14(a), each row represents
ω={white ball, yellow ball, green ball, blue ball}. The order of attributes in each row
was randomized. Each row was presented for 5 seconds. Figure 14(b) gives an example of
presented a first row. After the last row was presented, as in Figure 14(c), participants were
asked to choose between A and B by clicking the left or right box.
35
(a) First step using a wide range (e.g., 50m to 150m in in-
crements of 20m)
(b) Second step using a narrower range (e.g., 50m to 70m in
increments of 4m)
Figure 11: Example of the elicitation of z−1
1in first step
36
(a) First screen of first step using a wide range (e.g., 50m to
150m in increments of 20m)
(b) Second screen of first step using a wide range (e.g., 150m
to 250m in increments of 20m)
Figure 12: Another example of the elicitation of z−1
1in first step
37
(a) First step for the elicitation of A−2using a wide range
(e.g., 0 to 51 white balls in increments of 10 except 40 to 51)
(b) Second step for the elicitation of A−2using a narrower
range (e.g., 0 to 10 white balls in increments of 2)
(c) First step for the elicitation of A−3
(d) Second step for the elicitation of A−3using a narrower
range (e.g., 0 to 8 white balls in increments of 2)
Figure 13: Example of the elicitation of A−2and A−3in attribute 1 when x−2
1=100m and
x−3
1=150m
38
(a)
(b)
(c)
Figure 14: Example of the investigation of axiom B1(i) for distances.
39
References
1. Bell, D.E. (1982) Regret in decision making under uncertainty, Operations Research
30(5), 961-981.
2. Bleichrodt, H., Cillo, A., and Diecidue, E. (2010) A quantitative measurement of regret
theory. Management Science 56(1), 161-175.
3. Bleichrodt, H. and Wakker, P.P. (2015) Regret theory: a bold alternative to the alter-
natives. The Economic Journal 125(583), 493-532.
4. Diecidue, E. and Somasundaram, J. (2017) Regret theory: a new foundation. Journal
of Economic Theory 172, 88-119.
5. Dyer, J.S., Fishburn, P.C., Steuer, R.E., Wallenius, J., and Zionts, S. (1992) Multiple
criteria decision making, Multiattribute utility theory: The next ten years. Manage-
ment Science 38(5), 645-654.
6. Fishburn, P.C. (1970) Utility Theory for Decision Making. Wiley, New York.
7. Fishburn, P.C. (1989) Non-transitive measurable utility for decision under uncertainty,
Journal of Mathematical Economics 18(2), 187-207.
8. Fishburn, P.C. (1990) Continuous nontransitive additive conjoint measurement. Math-
ematical Social Sciences 20(2), 165-193.
9. Fishburn, P.C. (1992) Additive differences and simple preference comparisons. Journal
of Mathematical Psychology 36(1), 21-31.
10. Fujii, Y. and Osaki, Y. (2018) Regret-sensitive Treatment Decisions. Health Economics
Review 8(1), 1-14.
11. Fujii, Y., Okura, M., and Osaki, Y. (2016) Regret, Rejoicing, and Mixed Insurance.
Economic Modelling 58, 126-132.
12. Fujii, Y., and Nakamura, Y. (2021) Regret-sensitive equity premium. International
Review of Economics & Finance 76, 302-307.
13. Gonzalez, R., and Wu, G. (1999) On the shape of the probability weighting function.
Cognitive Psychology 38(1), 129-166.
14. Humphrey, S.J. (2001) Non-transitive choice: event-splitting effects or framing effects?
Economica 68(269), 77-96.
15. Keeney, R.L. and Raiffa, H. (1976) Decisions with Multiple Objective: Preferences and
Value Tradeoffs. Cambridge University Press, Cambridge.
40
16. Krantz, D.H., Luce, R.D., Suppes P., and Tversky A. (1971) Foundations of Mea-
surement, Volume I: Additive and Polynomial Representations. Academic Press, New
York.
17. Lerner, J. S., Li, Y., Valdesolo, P., and Kassam, K. S. (2015) Emotion and decision
making. Annual Review of Psychology, 66(1), 799-823.
18. Li, M. and Huang, H-J. (2017) A regret theory-based route choice model. Transport-
metrica A: Transport Science 13(3), 250-272.
19. Loomes, G., and Sugden, R. (1982) Regret theory: an alternative theory of rational
choice under uncertainty. The Economic Journal 92(368), 805-824.
20. Loomes, G. and Sugden, R. (1991) Observing violations of transitivity by experimental
methods. Econometrica 59(2), 425-439.
21. May, K.O. (1954) Intransitivity, utility, and the aggregation of preference patterns.
Econometrica 22(1), 1-13.
22. Nakamura, Y. (1998) Skew-symmetric additive representations of preferences. Journal
of Mathematical Economics 30(3), 367-387.
23. Peirce, J. W., Gray, J. R., Simpson, S., MacAskill, M., H¨ochenberger, R., Sogo, H.,
Kastman, E., and Lindeløv, J. K. (2019). PsychoPy2: experiments in behavior made
easy. Behavior Research Methods 51(1), 195-203.
24. Savage, L.J. (1954) The Foundations of Statistics. Wiley, New York, 2nd revised ed.,
Dover, New York.
25. Sugden, R. (1993) An axiomatic foundation for regret theory. Journal of Economic
Theory 60(1), 159-180.
26. Suppes, P., Krantz, D.H., Luce, R.D. and Tversky, A. (1989) Foundations of Measure-
ment, Volume II: Geometrical, Threshold, and Probabilistic Representations. Academic
Press, New York. (Chapter 17).
27. Tversky, A. (1969) Intransitivity of preferences. Psychological Review 76(1), 31-48.
28. Tversky, A., and Kahneman, D. (1992) Advances in prospect theory: cumulative rep-
resentation of uncertainty. Journal of Risk and Uncertainty 5(4), 297-323.
29. Wakker, P.P. (1989) Additive Representations of Preferences: A New Foundation of
Decision Analysis. Kluwer Academic Publishers, Dordrecht.
30. Wakker, P.P. (2010) Prospect Theory: For Risk and Ambiguity. Cambridge University
Press, Cambridge.
41
31. Wallenius, J., Dyer, J.S., Fishburn, P.C., Steuer, R.E., Zionts, S., and Deb, K. (2008)
Multiple criteria decision making, multiattribute utility theory: recent accomplish-
ments and what lies ahead. Management Science 54(7), 1336-1349.
32. Zeelenberg, M., and Pieters, R. (2007) A theory of regret regulation 1.0. Journal of
Consumer Psychology 17(1), 3-18.
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