stochastic dominance, regret dominance and regret-theoretic dominance

ORI GIN AL ARTICLE

Stochastic dominance, regret dominance and regret-theoretic dominance

Chin Hon Tan • Joseph C. Hartman

Received: 12 November 2012 / Accepted: 9 September 2013 / Published online: 26 September 2013

� Springer-Verlag Berlin Heidelberg and EURO - The Association of European Operational Research

Societies 2013

Abstract It is well known that stochastic dominance alone is insufficient for

ensuring preferences when individuals experience regret. In this paper, we study

two additional notions of dominance: (a) regret-theoretic dominance, which char-

acterizes preferences in regret theory and (b) regret dominance, which characterizes

preferences in mean-risk models with regret-based risk measures. We (a) extend our

understanding of preferences in regret theory to problems with multiple choices

under an infinite number of scenarios, (b) highlight that some notions of regret in the

normative literature, specifically relative regret, can lead to unreasonable prefer-

ences within a mean-risk framework and (c) illustrate how regret dominance can

help reduce the size of conventional efficient sets. Conditions where stochastic

dominance, regret dominance and regret-theoretic dominance are equivalent are

also presented.

Keywords Regret � Stochastic dominance � Preferences � Risk � Efficient

set

Mathematics Subject Classification (2000) 91B06 � 91B08 � 90B50

The authors are grateful to two anonymous referees whose remarks greatly improved this paper.

C. H. Tan (&)

Department of Industrial and Systems Engineering, National University of Singapore, 1 Engineering

Drive 2, Singapore 117576, Singapore

e-mail: [email protected]

J. C. Hartman

Department of Industrial and Systems Engineering, University of Florida, Gainesville, USA

123

EURO J Decis Process (2013) 1:285–297

DOI 10.1007/s40070-013-0018-1

Introduction

The link between stochastic dominance and preferences in expected utility theory is

well known. In particular, first-order stochastic dominance is necessary and

sufficient for an act to be preferred over another by all decision makers described by

utility functions that are non-decreasing in the consequence of the selected act

(Levy 1992). When the true utility function of a decision maker is unknown,

stochastic dominance can be used to identify and eliminate suboptimal choices. The

concept of stochastic dominance dates back more than 80 years (Karamata 1932)

and has been applied to problems in a variety of settings, including finance,

economics, insurance, agriculture and medicine (Levy 2006).

Preferences that are inconsistent with expected utility theory have been observed

experimentally (see, for example Allais 1953). Bell (1982), Loomes and Sugden

(1982) proposed a modified utility function that depends on both consequences and

regret in modeling the satisfaction associated with a decision. Their model,

commonly referred to as regret theory, was initially developed for pairwise

decisions. Subsequently, it was generalized for multiple feasible alternatives in

Loomes and Sugden (1987) and an axiomatic foundation for the theory was

presented in Sugden (1993). More recently, Stoye (2011a) provided a unified

axiomatic framework for preference ordering under minimax regret with no priors

(Stoye 2011b), endogenous priors (Hayashi 2008) and exogenous priors. The

interested reader is referred to Stoye (2012) for a discussion on axiomatic decision

theory and decision making under ambiguity. Bleichrodt et al. (2010) noted a

growing interest in the use of regret in explaining behavior and illustrated how

regret theory can be measured.

In this paper, we present a set of conditions that are necessary and sufficient for

an act to be preferred over another by all decision makers whose satisfaction are

non-decreasing and non-increasing in the consequence of the selected and

unselected acts, respectively. We term this regret-theoretic dominance. The concept

of regret-theoretic dominance is not new. Loomes and Sugden (1987) highlighted

that stochastic dominance implies regret-theoretic dominance when there are only

two independent acts to choose from. Quiggin (1990) showed that stochastic

dominance and regret-theoretic dominance are equivalent when there exists an

invertible bijection that results in state-wise dominance for pairwise problems with

finite states. Subsequently, he presented a set of conditions that is sufficient for

ensuring unanimous preferences in regret theory in problems with multiple feasible

alternatives (Quiggin 1994). This paper presents a set of conditions that is both

necessary and sufficient for unanimous preferences in regret theory in problems

with multiple feasible alternatives.

In the normative literature, it is often assumed that the regret function is known.

For example, the regret associated with a decision is often defined as the difference

between the consequence of the optimal and selected act under the realized state

(see, Kouvelis and Yu 1997 or Aissi et al. 2009). In this paper, we introduce the

notion of regret dominance, which is applicable when the regret function is

available. We note that regret dominance, which assumes a specific expression of

regret, is more restrictive than regret-theoretic dominance. We show that regret

286 C. H. Tan, J. C. Hartman

123

dominance is necessary and sufficient for an act to be preferred by all decision

makers for a wide range of regret-based risk measures within a mean-risk

framework. Furthermore, we highlight that some notions of regret in the normative

literature, in particular relative regret (see, Kouvelis and Yu 1997 or Aissi et al.

2009), can lead to unreasonable preferences and may not be suitable measures of

risk within the mean-risk framework.

Third, we study the relationship between stochastic dominance, regret dominance

and regret-theoretic dominance. Figure 1 illustrates the efficient sets obtained by

eliminating stochastically dominated, regret dominated and regret-theoretic dom-

inated acts, denoted by CSD, CRD and CRTD, respectively. Since regret-theoretic

dominance implies both stochastic dominance and regret dominance, it is clear that

CRD � CRTD and CSD � CRTD: In this paper, we show that CRTD = CSD when

prospects are independent, but stochastic dominance does not guarantee regret

dominance, even under statistical independence. The former generalizes the

observation of Loomes and Sugden (1987) that stochastic dominance implies

regret-theoretic dominance under independence to problems with multiple feasible

alternatives and the latter highlights that regret dominance can potentially be used in

a wide range of problems to identify inferior acts in the absence of stochastic

dominance.

This paper proceeds as follows: we present conditions that define regret-theoretic

dominance and regret dominance in ‘‘Regret-theoretic dominance’’ and ‘‘Regret

dominance’’, respectively. The conditions where stochastic dominance, regret-

theoretic dominance and regret dominance are equivalent are discussed in

‘‘Invariance of stochastic dominance’’. We conclude with a summary of results

and discussion.

Regret-theoretic dominance

Let S and C denote the state space and the set of possible acts, respectively. The

decision maker is to select exactly one act from the set C. The consequence of act

c 2 C under state s 2 S is denoted by xðsÞc 2 R: Let x

ðsÞc 2 R

jCj denote a vector of

consequences, where the first element of xðsÞc corresponds to the consequence of

c under s and the remaining elements correspond to the consequences of the

unselected acts under the same state (i.e., xi(s) for all i 2 Cnfcg):

xðsÞc ¼ xðsÞc ; xðsÞ1 ; x

ðsÞ2 ; . . .; x

ðsÞc�1; x

ðsÞcþ1; . . .; x

ðsÞjCj

� �:

The modified utility m is a real-valued function that denotes the utility of receiving

xc(s) and foregoing the consequence of unselected acts:

m xðsÞc

� �¼ m xðsÞc ; x

ðsÞ1 ; x

ðsÞ2 ; . . .; x

ðsÞc�1; x

ðsÞcþ1; . . .; x

ðsÞjCj

� �:

Since satisfaction is non-decreasing in xc(s) and non-increasing in xi

(s) for all i 2Cnfcg; m is non-decreasing and non-increasing in its first argument and remaining

arguments, respectively. Let ps denote the probability of state s occurring. Decision

Stochastic dominance 287

123

makers are assumed to be rational individuals that maximize expected modified

utility. Stated formally, preferences in regret theory (i.e., Equation (16) in Loomes

and Sugden 1987), or regret-theoretic preferences, are as follows:

c1 � c2 ,X

s

ps m xðsÞc1

� �� m xðsÞc2

� �� 0; ð1Þ

where c1 � c2 denotes that c1 is at least as preferred as c2. We say that c1 regret-

theoretic dominates c2 if c1 � c2 for all m.

In regret theory, the satisfaction of the decision maker in any given state can be

influenced by the consequences of all acts in that state. Therefore, the possible states

are explicitly represented and individually accounted for in the computation of the

decision maker’s expected utility in Loomes and Sugden’s model. However,

obtaining dominance results for regret theory models of this form is challenging

(see, Quiggin 1990, 1994).

Next, we present an alternate, but equivalent, representation of the model where

the utility of a decision maker is expressed in terms of random variables that

implicitly, rather than explicitly, account for the possible states. The main advantage

of this representation is that it enables us to obtain the set of necessary and sufficient

conditions for unanimous regret-theoretic preferences by applying known multi-

variate stochastic dominance results.

Let Xc denote the prospects of act c where Xc is a random variable describing the

consequences of c according to some known probability distribution function. We note

that no independence assumptions are made regarding the prospects Xi for all i 2 C:Regret-theoretic preferences, as described by Eq. (1), can be re-expressed as follows:

c1 � c2 , E½uðXc1Þ � uðXc2

Þ�� 0; ð2Þ

where u is a real-valued non-decreasing function and Xc is a multivariate random

variable defined as follows:

Xc ¼ ðXc;�X1;�X2; . . .;�Xc�1;�Xcþ1; . . .;�XjCjÞ: ð3Þ

Note that the elements associated with unselected acts are defined as the negative of

their prospects. This definition allows us to define a non-decreasing function

Fig. 1 Efficient sets


123

u, rather than a function that is both non-increasing and non-decreasing in its

arguments, which simplifies the proofs that are presented in this paper.

Consider two multivariate random variables, X1 and X2: We say that X1

stochastically dominates X2 if and only if:

PðX1 2 LÞ�PðX2 2 LÞ for all lower sets L � RjCj; ð4Þ

where L is a lower set such that:

ðx1; x2; . . .; xjCjÞ 2 L) ðy1; y2; . . .; yjCjÞ 2 L when yi� xi for all i:

For univariate random variables, X1 stochastically dominates X2 if and only if:

PðX1� xÞ�PðX2� xÞ for all x:

For notation simplicity, we let s denote stochastic dominance relations between

two random variables such that the following holds for any two multivariate random

variables X1 and X2 :

X1sX2 , PðX1 2 LÞ�PðX2 2 LÞ for all lower sets L � RjCj;

and the following holds for any two univariate random variables X1 and X2:

X1sX2 , PðX1� xÞ�PðX2� xÞ for all x:

It is well known that X1sX2 if and only if E½u X1ð Þ� �E½u X2ð Þ� for all

u (Shaked and Shanthikumar 2007). Therefore, stochastic dominance of multivar-

iate random variables Xc is necessary and sufficient for regret-theoretic dominance.

Theorem 1 Xc1sXc2

, c1 � c2 for all u:

Proof The theorem follows from the fact that

Xc1s Xc2

, E½uðXc1Þ � uðXc2

Þ� � 0 for all u

and:

E½uðXc1Þ � uðXc2

Þ� � 0, c1 � c2:

h

Loomes and Sugden (1987) highlighted that stochastic dominance between two

univariate random variables (i.e., the relation denoted by s) is insufficient for

ensuring preferences in regret theory. Theorem 1 highlights that stochastic

dominance between two appropriately defined multivariate random variables [see,

Eq. (3)] is necessary and sufficient for unanimous regret-theoretic preferences.

Unlike the conditions presented in Quiggin (1990, 1994), the conditions presented

in Theorem 1 are both necessary and sufficient for problems involving multiple

choices. In addition, they are easy to verify and are applicable when Xi is

continuous.

Next, we illustrate how Theorem 1 can be applied to acts with normally

distributed consequences. Suppose X1 and X2 are normally distributed. It follows

from Theorem 4 of Muller (2001) that c1 is at least as preferred as c2 by all

reasonable decision makers (i.e., non-decreasing utility) described by expected


123

utility theory if and only if l1 C l2 and r11 = r22, where li denotes the expected

value of Xi and rij denotes the covariance between Xi and Xj. However, these

conditions are, in general, not sufficient for ensuring preferences between acts with

normally distributed consequences when decision makers are influenced by feelings

of regret.

Theorem 2 Suppose Xc1and Xc2

are multivariate normal random variables. c1 �c2 for all u if and only if:

A1: l1 C l2,

A2: r11 = r22 and

A3: r1i = r2i for i = 3, 4,…, |C|.

Proof Let R1 and R2 denote the covariance matrix of Xc1and Xc2

; respectively.

Theorem 5 of Muller (2001) states that Xc1s Xc2

if and only if l1 C l2 and

R1 ¼ R2: It is not hard to see that R1 ¼ R2 if and only if A2 and A3 are true.

Therefore, it follows from Theorem 1 that A1–A3 are necessary and sufficient for

c1 � c2 for all u. h

When comparing between two acts with normally distributed consequences, it is

sufficient to consider their respective mean and variance to ensure preference by all

reasonable decision makers described by expected utility theory. However, if

decision makers experience feelings of regret, Theorem 2 highlights that c1 and c2

must also have the same correlation with all acts in C in order to ensure preference

when Xi for all i 2 C are jointly normally distributed.

Regret dominance

In the previous section, we highlight that multivariate stochastic dominance is

necessary and sufficient for unanimous regret-theoretic preferences. An act that is

preferred over another by any regret-theoretic decision maker (i.e., decision maker

who is described by regret theory) is also preferred by any decision maker that is

described by expected utility theory. However, a regret-theoretic decision maker

may prefer an act that is rejected by all decision makers described by expected

utility theory. Hence, the efficient set (i.e., set of non-dominated acts) obtained by

regret-theoretic dominance rules can, in some cases, be too large for practical

purposes. In this section, we introduce a more restrictive form of dominance, which

we term regret dominance, and highlight its relationship with preferences in mean-

regret models (i.e., mean-risk models with a regret-based risk measure). In

particular, we show that regret dominance is necessary and sufficient for unanimous

preferences in mean-regret models under absolute regret and other similar

expressions of regret, but is insufficient for ensuring preferences in mean-regret

models under relative regret.

Let Yc denote the regret prospects associated with act c, where Yc is a function of

Xc: Let u denote the regret function:


123

Yc ¼ uð�XcÞ: ð5Þ

Since regret is non-increasing in Xc and non-decreasing in Xi for all i 2 Cnfcg; u is

non-decreasing in �Xc: We say that c1 regret dominates c2 when Yc2s Yc1

:Let rc denote the risk associated with act c. The mean-risk framework seeks an

act c that maximizes the following objective:

gðcÞ ¼ E½Xc� � crc;

where c is some constant. Conventional risk measures are based on the prospects of

selected acts and the value of c is defined as zero, positive and negative for a risk

neutral, risk averse and risk seeking decision maker, respectively (Krokhmal et al.

2011). Clearly, such risk measures are insufficient for modeling feelings of regret,

which can be viewed as a form of risk that a decision maker would seek to reduce,

even at a premium (Bell 1983). We define a regret-based risk measure as follows:

rc ¼ qðYcÞ;

where q is some non-decreasing function. Since the satisfaction of a decision is non-

increasing in regret, the value of c is non-negative in mean-regret models. We note

that mean-regret models have been used to analyze decisions in a variety of settings

(see, for example, Muermann et al. 2006; Irons and Hepburn 2007; Michenaud and

Solnik 2008; Syam et al. 2008; Nasiry and Popescu 2012), even though feelings of

regret may not be explicitly linked to risk, but seen merely as an emotion that results

in dissatisfaction, in some of these papers.

In the normative literature, regret is commonly expressed as the difference

between the consequence of the optimal and selected act. Following the terminology

of Kouvelis and Yu (1997), we refer to this as absolute regret and let ua denote the

absolute regret function:

Yc ¼ uað�XcÞ ¼ maxi2C

Xi � Xc:

Theorem 3 Suppose Yc1¼ uað�Xc1

Þ and Yc2¼ uað�Xc2

Þ :

Yc2s Yc1

, E½Xc1� � cqðYc1

Þ�E½Xc2� � cqðYc2

Þ for all q and c� 0:

Proof First, we show that E½Xc1� �E½Xc2

� if Yc2s Yc1

: When

Yc2s Yc1

; E½Yc1� �E½Yc2

�: This implies that:

E½Yc1� �E½Yc2

�

E maxi2C

Xi � Xc1

� ��E max

i2CXi � Xc2

� �

E maxi2C

Xi

� �� E½Xc1

� �E maxi2C

Xi

� �� E½Xc2

�

�E½Xc1� � � E½Xc2

�E½Xc1

� �E½Xc2�:


123

It follows from standard results in stochastic dominance that q(Yc1) B q(Yc2) for all

q when Yc2s Yc1

: Therefore, Yc2s Yc1

) E½Xc1� � cqðYc1


Þ for

all q and c C 0. Next, we provide the proof for the reverse direction:

E½Xc1� � cqðYc1


Þ for all q and c� 0

) qðYc1Þ� qðYc2

Þ for all q

) Yc2s Yc1

:

This completes the proof. h

Theorem 3 highlights that regret dominance is necessary and sufficient for

unanimous preferences in mean-regret models under absolute regret. The proof

follows from the fact that, under absolute regret, a regret-dominated act must yield a

lower expected value. It is not difficult to see that regret dominance is also necessary

and sufficient for unanimous preferences under any notion of regret where there exists

a univariate random variable eX such that uð�XcÞ ¼ eX � Xc: We term this property

separability. Note that there is no independence restriction between eX and Xc.

Under non-separable regret, regret dominance does not necessarily imply higher

expected value and a regret-dominated act may be preferred under the mean-regret

framework. One example of a non-separable regret is relative regret, defined as the

ratio of the absolute regret and the consequence of the selected act (Kouvelis and Yu

1997):

Yc ¼maxi2C Xi � Xc

Xc

:

Example 1 Relative regret in mean-regret models.

Consider two acts, c1 and c2, and two states, s1 and s2, where xðs1Þc1 ¼ 1; x

ðs2Þc1 ¼

5; xðs1Þc2 ¼ 2; x

ðs2Þc2 ¼ 3; ps1

¼ 0:5 and ps2¼ 0:5: Computing the relative regret of c1

and c2 (i.e., Yc ¼ maxi2C Xi

Xc� 1), we get P(Yc1 = 0) = P(Yc1 = 1) = P(Yc2 = 0) = -

P(Yc2 = 0.67) = 0.5, which implies that Yc1s Yc2

: However,

E½Xc1� ¼ 3 [E½Xc2

� ¼ 2:5: Since E½Xc1�[E½Xc2

�; a decision maker with a suffi-

ciently small c will prefer c1, even though it is regret dominated by c2.

Example 1 illustrates how a regret-dominated act may be preferred when regret is

non-separable. Stochastic dominance is considered by many as a basic tenet of

rational decision making (see, for example, Charness et al. 2007). We believe that

regret dominance is also a reasonable requirement for choice preference and the

counter-intuitive result in Example 1 suggests that relative regret and other non-

separable regret may not be suitable as measures of risk, at least within the mean-

risk framework.

Invariance of stochastic dominance

In the previous two sections, we showed that regret-theoretic dominance implies

regret-theoretic preferences and regret dominance implies mean-regret preferences


123

when regret is separable. In this section, we identify conditions where stochastic

dominance in prospects Xi implies regret dominance and regret-theoretic domi-

nance. First, we prove the following lemma which states that the marginal

distributions of Xc1stochastically dominate the marginal distributions of Xc2

if and

only if Xc1 stochastically dominates Xc2. Let XiðkÞ denote the kth element of Xi:

Lemma 1 Xc1s Xc2

, Xc1ðkÞ s Xc2

ðkÞ for all k:

Proof First, we show that:

Xc1s Xc2

) Xc1ðkÞ s Xc2

ðkÞ for all k: ð6Þ

We prove this result by showing that Eq. (6) holds for each k. Without loss of

generality, let acts c1 = 1 and c2 = 2. Since Xc1ð1Þ ¼ Xc1

and Xc2ð1Þ ¼ Xc2

; Eq. (6)

holds for k = 1. Since Xc1ð2Þ ¼ �X2 and Xc2

ð2Þ ¼ �X1 and �X2s � X1; Eq. (6)

holds for k = 2. For k [ 2; Xc1ðkÞ ¼ Xc2

ðkÞ ¼ �Xk: Therefore, Eq. (6) holds for

k [ 2 as well. The proof in the reverse direction is straightforward. h

Theorem 4 When Xi are independent, Xc1s Xc2


Proof When Xi are independent, stochastic dominance of each marginal distribu-

tion is necessary and sufficient for multivariate stochastic dominance. Therefore, it

follows from Lemma 1 and Theorem 1 that

Xc1s Xc2

, Xc1ðkÞ s Xc2

ðkÞ for all k, Xc1s Xc2


h

Theorem 4 states that when prospects are independent, stochastic dominance is

necessary and sufficient for regret-theoretic dominance. This implies that, when

prospects are independent, unanimous preferences in expected utility theory is

consistent with unanimous preferences in regret theory. If one act is preferred over

another by all individuals described by expected utility theory, it will also be

preferred by all individuals described by regret theory. If expected utility theory

predicts that there exists an individual that prefers one act over another, regret

theory also predicts the existence of such an individual.

Theorem 5 highlights that unanimous preferences in regret theory is also

consistent unanimous preferences in expected utility theory when we limit ourselves

to additive u, where u can be expressed as the sum of |C| non-decreasing univariate

functions ui as follows:

uðX1;X2; . . .;XjCjÞ ¼XjCj

i¼1

uiðXiÞ:

Theorem 5 Xc1s Xc2


Þ� � 0 for all additive u:

Proof Stochastic dominance of each marginal distribution is necessary and

sufficient for E½uðXc1Þ � uðXc2

Þ�� 0 for all additive u (Levy and Paroush 1974).

Therefore, it follows from Lemma 1 that


123

Xc1s Xc2

, Xc1ðkÞ s Xc2

ðkÞ for all k


Þ� � 0 for all additive u:

h

The proofs of Theorems 4 and 5 are based on known results in multivariate

stochastic dominance. Specifically, stochastic dominance of each marginal distri-

bution is necessary and sufficient for unanimous preferences when prospects are

independent or when u is additive. The following theorem highlights that stochastic

dominance relationships are preserved under consequence-to-regret transformations

when acts are independent.

Theorem 6 When Xi are independent:

Xc1s Xc2

) Yc2s Yc1

;

where Yc denotes the regret prospects of act c as defined in Eq. (5).

Proof It follows from Theorem 4 that when Xi are independent:

Xc1s Xc2

)c1 � c2 for all non-decreasing u

)E½uðXc1Þ � uðXc2

Þ� � 0; for all non-decreasing u

)E½uð�Xc1Þ � uð�Xc2

Þ�� 0; for all non-decreasing u

)E½qðuð�Xc1ÞÞ � qðuð�Xc2

ÞÞ� � 0; for all non-decreasing q

)E½qðYc1Þ � qðYc2

Þ� � 0; for all non-decreasing q

)Yc2s Yc1

:

h

Theorem 6 highlights that stochastic dominance also imply regret dominance

when prospects are independent. We note that proving this result, even for the

special case of relative regret, from first principles is challenging because the regret

function for relative regret involves a max operator and the division of a dependent

random variable. Our result, which is based on a proof that invokes the necessity

and sufficiency of stochastic dominance, holds for all reasonable forms of regret

(i.e., non-decreasing u). It is important to note that the result of Theorem 6 is uni-

directional and the reverse may not be true, even when Xi are independent. This is

illustrated in Example 2.

Example 2 Non-stochastically dominated act that is regret dominated.

Consider two independent prospects Xc1 and Xc2 where:

PðXc1¼ 1Þ ¼ 0:30 PðXc1

¼ 2Þ ¼ 0:15 PðXc1¼ 3Þ ¼ 0:55

PðXc2¼ 1Þ ¼ 0:20 PðXc2

¼ 2Þ ¼ 0:30 PðXc2¼ 3Þ ¼ 0:50:

Suppose that Yc ¼ maxi2C Xi � Xc: The probability distribution of Yc1 and Yc2

are:


123

PðYc1¼ 0Þ ¼ 0:685 PðYc1

¼ 1Þ ¼ 0:165 PðYc1¼ 2Þ ¼ 0:15

PðYc2¼ 0Þ ¼ 0:695 PðYc2

¼ 1Þ ¼ 0:195 PðYc2¼ 2Þ ¼ 0:11:

Since P(Yc1 B y) B P(Yc2 B y) for all y; Yc1s Yc2

: However, Xc2 does not

stochastically dominate Xc1 since P(Xc1 B 2) \ P(Xc2 B 2).

Let CSD, CRD and CRTD denote the efficient sets obtained by eliminating

stochastically dominated, regret-dominated and regret-theoretic dominated acts,

respectively. In Example 2, neither act stochastically dominates the other.

Therefore, CSD = {c1, c2}. However, c2 regret dominates c1, which implies

CRD = {c2}. It has been highlighted that CSD can sometimes be large and may not

be useful in practice (Levy 2006). Example 2 highlights that CSD may contain

regret-dominated acts and thus, regret dominance can potentially be used to further

reduce choices.

In addition, it follows from Theorem 4 that CRTD = CSD when prospects are

independent. Therefore, CRD � CRTD ¼ CSD when prospects are independent.

When prospects are dependent, CSD is a subset of CRTD and it is not difficult to

come up with an example where a stochastically dominated act is not regret

dominated (i.e., CSDT

CRD may be non-empty). The relationships between the

efficient sets obtained by various dominance rules are illustrated in Fig. 1.

Discussion and summary

It is well known that stochastic dominance is necessary and sufficient for unanimous

preferences in expected utility theory but is insufficient for ensuring regret-theoretic

preferences. In this paper, we illustrate how regret-theoretic dominance character-

izes preferences in regret theory. The conditions that are presented in this paper

extend our understanding of preferences predicted by regret theory to problems

involving multiple choices under an infinite number of states (e.g., acts with

normally distributed consequences) and can be used by behavioral researchers to

further examine the validity of regret theory as a descriptive choice model.

In ‘‘Regret dominance’’ we introduce the concept of regret dominance. We prove

that regret dominance is necessary and sufficient for unanimous preferences in the

mean-regret framework when regret is separable. Various researchers have

considered the effects of regret within a mean-risk framework, but generally

restricted to absolute and relative regret. We hope that this work, which highlights

that mean-regret preferences are consistent with regret dominance for all types of

separable regret, will encourage researchers to adopt other definitions of regret in

modeling behavior. An interesting area of future research is to study how different

definitions of regret affect preferences and the differences between behavior of

individuals described by conventional risk measures and regret-based risk measures.

In addition, we highlight that mean-regret preferences may not be consistent with

regret dominance when regret is non-separable, which suggests that relative regret is

not a suitable risk measure within the mean-risk framework. Another interesting

area of research will be to identify other classes of regret, besides separable regret,

where regret dominance is consistent with mean-regret choice preference.


123

Finally, we generalize the observation by Loomes and Sugden (1987) that

stochastic dominance implies regret dominance when prospects are independent to

problems involving multiple choices. In addition, we highlight that an efficient set

that is obtained by conventional stochastic dominance rules may contain regret-

dominated acts, even when prospects are independent. This implies that regret

dominance can potentially be used by decision analysts to reduce the size of

conventional efficient sets. An interesting area of future research is to study the

conditions where regret dominance is effective in trimming conventional efficient

sets.

Acknowledgments The authors also gratefully acknowledge support from the National Science

Foundation and the National University of Singapore under Grant No. CMMI-0813671 and R-266-000-

068-133, respectively.

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