EURO J Decis Process (2013) 1:285–297 DOI 10.1007/s40070-013-0018-1 ORIGINAL ARTICLE

Stochastic dominance, regret dominance and regrettheoretic 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 characterizes 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 preferences 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 set

Regret  Stochastic dominance  Preferences  Risk  Efficient

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

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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

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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 dominated acts, denoted by CSD, CRD and CRTD, respectively. Since regret-theoretic dominance implies both stochastic dominance and regret dominance, it is clear that C RD  C RTD and C SD  C RTD : 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, regrettheoretic 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 ðsÞ ðsÞ c 2 C under state s 2 S is denoted by xc 2 R: Let xc 2 RjCj denote a vector of ðsÞ consequences, where the first element of xc 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., x(s) i for all i 2 Cnfcg):   ðsÞ ðsÞ ðsÞ ðsÞ ðsÞ ; x ; x ; . . .; x ; x ; . . .; x xðsÞ c ¼ xðsÞ c 1 2 c1 cþ1 jCj : The modified utility m is a real-valued function that denotes the utility of receiving x(s) c and foregoing the consequence of unselected acts:     ðsÞ ðsÞ ðsÞ ðsÞ ðsÞ m xðsÞ c ¼ m xðsÞ c ; x1 ; x2 ; . . .; xc1 ; xcþ1 ; . . .; xjCj : (s) for all i 2 Since satisfaction is non-decreasing in x(s) c and non-increasing in xi Cnfcg; 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

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Fig. 1 Efficient sets

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:   X    c1  c2 , ps m xðsÞ  m xðsÞ  0; ð1Þ c1 c2 s

where c1  c2 denotes that c1 is at least as preferred as c2. We say that c1 regrettheoretic 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 multivariate 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 , E½uðXc1 Þ  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 ; . . .; Xc1 ; Xcþ1 ; . . .; XjCj Þ:

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

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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 ; . . .; xjCj Þ 2 L ) ðy1 ; y2 ; . . .; yjCj Þ 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 : X1 s X2 , 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: X1 s X2 , PðX1  xÞ  PðX2  xÞ for all x: It is well known that X1 s X2 if and only if E½uðX1 Þ  E½uðX2 Þ for all u (Shaked and Shanthikumar 2007). Therefore, stochastic dominance of multivariate random variables Xc is necessary and sufficient for regret-theoretic dominance. Theorem 1 Proof

Xc1 s Xc2 , c1  c2 for all u:

The theorem follows from the fact that Xc1 s Xc2 , E½uðXc1 Þ  uðXc2 Þ  0 for all u

and: E½uðXc1 Þ  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 Mu¨ller (2001) that c1 is at least as preferred as c2 by all reasonable decision makers (i.e., non-decreasing utility) described by expected

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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 Xc1 and 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 Xc1 and Xc2 ; respectively. Theorem 5 of Mu¨ller (2001) states that Xc1 s 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 meanregret 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:

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Yc ¼ uðXc Þ:

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 Yc2 s 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 qðYc Þ; where q is some non-decreasing function. Since the satisfaction of a decision is nonincreasing 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 ua ðXc Þ ¼ max Xi  Xc : i2C

Theorem 3

Suppose Yc1 ¼ ua ðXc1 Þ and Yc2 ¼ ua ðXc2 Þ :

Yc2 s Yc1 , E½Xc1   cqðYc1 Þ  E½Xc2   cqðYc2 Þ for all q and c  0: Proof First, we show that E½Xc1   E½Xc2  Yc2 s Yc1 ; E½Yc1   E½Yc2 : This implies that:

if

Yc2 s Yc1 :

When

E½Yc1   E½Yc2      E max Xi  Xc1  E max Xi  Xc2 i2C i2C     E max Xi  E½Xc1   E max Xi  E½Xc2  i2C

i2C

E½Xc1    E½Xc2  E½Xc1   E½Xc2 :

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It follows from standard results in stochastic dominance that q(Yc1) B q(Yc2) for all q when Yc2 s Yc1 : Therefore, Yc2 s Yc1 ) E½Xc1   cqðYc1 Þ  E½Xc2   cqðYc2 Þ for all q and c C 0. Next, we provide the proof for the reverse direction: E½Xc1   cqðYc1 Þ  E½Xc2   cqðYc2 Þ for all q and c  0 ) qðYc1 Þ  qðYc2 Þ for all q ) Yc2 s Yc1 : h

This completes the proof.

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 e such that uðXc Þ ¼ X e  Xc : We term this property a univariate random variable X e and Xc. separability. Note that there is no independence restriction between X 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 ¼ Example 1

maxi2C Xi  Xc : Xc

Relative regret in mean-regret models. ðs Þ

ðs Þ

Consider two acts, c1 and c2, and two states, s1 and s2, where xc11 ¼ 1; xc12 ¼ ðs Þ xc21

ðs Þ xc22

5; ¼ 2; ¼ 3; ps1 ¼ 0:5 and ps2 ¼ 0:5: Computing the relative regret of c1 Xi and c2 (i.e., Yc ¼ maxXi2C  1), we get P(Yc1 = 0) = P(Yc1 = 1) = P(Yc2 = 0) = c which implies that Y c 1 s Yc 2 : However, P(Yc2 = 0.67) = 0.5, E½Xc1  ¼ 3 [ E½Xc2  ¼ 2:5: Since E½Xc1  [ E½Xc2 ; a decision maker with a sufficiently 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 nonseparable regret may not be suitable as measures of risk, at least within the meanrisk 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

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when regret is separable. In this section, we identify conditions where stochastic dominance in prospects Xi implies regret dominance and regret-theoretic dominance. First, we prove the following lemma which states that the marginal distributions of Xc1 stochastically dominate the marginal distributions of Xc2 if and only if Xc1 stochastically dominates Xc2. Let Xi ðkÞ denote the kth element of Xi : Lemma 1 Proof

Xc1 s Xc2 , Xc1 ðkÞ s Xc2 ðkÞ for all k:

First, we show that: Xc1 s Xc2 ) Xc1 ðkÞ 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 Xc1 ð1Þ ¼ Xc1 and Xc2 ð1Þ ¼ Xc2 ; Eq. (6) holds for k = 1. Since Xc1 ð2Þ ¼ X2 and Xc2 ð2Þ ¼ X1 and X2 s  X1 ; Eq. (6) holds for k = 2. For k [ 2; Xc1 ðkÞ 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, Xc1 s Xc2 , c1  c2 for all u:

Proof When Xi are independent, stochastic dominance of each marginal distribution is necessary and sufficient for multivariate stochastic dominance. Therefore, it follows from Lemma 1 and Theorem 1 that Xc1 s Xc2 , Xc1 ðkÞ s Xc2 ðkÞ for all k , Xc1 s Xc2 , c1  c2 for all u: 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 ; . . .; XjCj Þ ¼

jCj X

ui ðXi Þ:

i¼1

Theorem 5

Xc1 s Xc2 , E½uðXc1 Þ  uðXc2 Þ  0 for all additive u:

Proof Stochastic dominance of each marginal distribution is necessary and sufficient for E½uðXc1 Þ  uðXc2 Þ  0 for all additive u (Levy and Paroush 1974). Therefore, it follows from Lemma 1 that

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Xc1 s Xc2 , Xc1 ðkÞ s Xc2 ðkÞ for all k , E½uðXc1 Þ  uðXc2 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 distribution 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: Xc1 s Xc2 ) Yc2 s 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:

Xc1 s Xc2 )c1  c2 for all non-decreasing u )E½uðXc1 Þ  uðXc2 Þ  0; for all non-decreasing u )E½uðXc1 Þ  uðXc2 Þ  0; for all non-decreasing u )E½qðuðXc1 ÞÞ  qðuðXc2 ÞÞ  0; for all non-decreasing q )E½qðYc1 Þ  qðYc2 Þ  0; for all non-decreasing q )Yc2 s 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 unidirectional 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: PðXc1 ¼ 1Þ ¼ 0:30 PðXc1 ¼ 2Þ ¼ 0:15 PðXc2 ¼ 1Þ ¼ 0:20 PðXc2 ¼ 2Þ ¼ 0:30

PðXc1 ¼ 3Þ ¼ 0:55 PðXc2 ¼ 3Þ ¼ 0:50:

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

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PðYc1 ¼ 0Þ ¼ 0:685

PðYc1 ¼ 1Þ ¼ 0:165

PðYc1 ¼ 2Þ ¼ 0:15

PðYc2 ¼ 0Þ ¼ 0:695

PðYc2 ¼ 1Þ ¼ 0:195

PðYc2 ¼ 2Þ ¼ 0:11:

Since P(Yc1 B y) B P(Yc2 B y) for all y; Yc1 s 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, C RD  C RTD ¼ C SD 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 T dominated (i.e., CSD C RD 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 characterizes 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.

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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 regretdominated 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-266-000068-133, respectively.

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