Conditional non-expected utility preferences induced by mixture of lotteries: a note on the normative invalidity of expected utility theory

Geiger, Gebhard Article — Published Version Conditional non-expected utility preferences induced by mixture of lotteries: a note on the normative invalidity of expected utility theory Annals of Operations Research Provided in Cooperation with: Springer Nature Suggested Citation: Geiger, Gebhard (2020) : Conditional non-expected utility preferences induced by mixture of lotteries: a note on the normative invalidity of expected utility theory, Annals of Operations Research, ISSN 1572-9338, Springer US, New York, NY, Vol. 289, Iss. 2, pp. 431-448, https://doi.org/10.1007/s10479-020-03528-5 This Version is available at: https://hdl.handle.net/10419/288401 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Annals of Operations Research (2020) 289:431–448 https://doi.org/10.1007/s10479-020-03528-5 1 3 SHORT NOTE Conditional non‑expected utility preferences induced bymixture oflotteries: anote onthenormative invalidity ofexpected utility theory GebhardGeiger1 Published online: 19 February 2020 © The Author(s) 2020 Abstract This research note is concerned with static choices between alternative mixtures of lotteries with one common mixture component and identical mixture weights. It is shown that the common component induces a conditional preference relation on the underlying lottery space with given (unconditional) preference structure. Induced preferences of this type arise in the comparisons with which the independence axiom of expected utility theory is specifically concerned. Given a few obvious properties of the induced preferences, two basic results are obtained: first, the conditionalisation operation is an order-preserving isomorphism, and, secondly, if the conditional preferences satisfy stochastic dominance preference, they necessarily violate the independence axiom. Together, the two results preclude any possibility of postulating independence consistently for static decision making under risk. The independence axiom is thus generally invalid as a normative principle of rational risky choice. Keywords Rational choice· Risky choice· Normative theory· Expected utility· Independence axiom JEL Classification D81 1 Introduction Despite its notorious lack of empirical validity, expected utility (EU) theory continues to prevail as the standard normative model of economic rationality in theoretical and applied risk and decision analyses. The significance of the model largely rests on the independence axiom of EU theory, which ensures dynamically consistent choices in multiple-stage decision making under risk. Dynamic consistency of risky choice, in turn, is a basic criterion of economic rationality in the sense that it requires an individual´s choices at later stages of a decision process to conform with this individual´s preferred course of action planned at * Gebhard Geiger g.geig[email protected] 1 School ofManagement, Institute ofFinancial Management andCapital Markets, Technical University ofMunich, Arcisstrasse 21, 80333Munich, Germany 432 Annals of Operations Research (2020) 289:431–448 1 3 earlier stages (Karni and Schmeidler 1991; for a related concept of dynamically consistent behaviour, see Hammond 1998; Hammond and Zank 2014). In recent decision research, dynamic choice theory has indeed been extended beyond EU maximisation from various perspectives, especially those of risk, uncertainty and ambiguity (Machina 1989; Sarin and Wakker 1998; Siniscalchi 2011; Nebout 2014). But the extensions have remained essentially restricted to descriptive models of decision making in the sense that they violate, or “sacrifice”, other normative principles of rational choice where necessary to preserve dynamic consistency. The present analysis goes one critical step beyond the descriptive intent of non-EU models. It refutes the normative interpretation of EU theory. To serve as a universally valid principle of rational decision making, EU maximisation would have to take implications of static choice into account that arise in comparisons between compound lotteries, but are usually ignored in normative interpretations of EU theory. Preferences for compound lotteries will be shown not only to violate the independence axiom under well-defined conditions, but also to exclude any possibility of postulating independence consistently in models of static decision making under risk. Theoretically, this inconsistency arises from conditional preferences induced by alternative mixtures of lotteries with one common mixture component and identical mixture weights. The independence axiom is specifically concerned with convex combinations of lotteries of this type. In practice, the inconsistency occurs in assessments of one risk in the presence of others, commonly called “background risks”. An example will be given in terms of conditional preferences for compound lotteries in which the component lotteries (or “sublotteries”) are resolved at uncertain times. In more technical terms, the scope of the present research note can be delineated as follows. We consider preference comparisons between two mixtures of lotteries p and r, and q and r, respectively. The mixture weights of p and q are the same and, hence, so are those of the common component r. Contrary to what the independence axiom of EU theory proposes, we admit choices between the two compound lotteries to depend not only on the component lotteries, but also on the particular nature and numerical values of the mixture weights. A typical example of the impact of the mixture weights on preference arises where the mixture weight assigned to p (or q) is the probability with which p (or q) is resolved first (i.e., prior to the resolution of r) or else persists and is rejected with the complementary probability in the alternative case of premature resolution of r. In this example, the compound lotteries are, by construction, one-stage and, hence, give rise to static choices, but involve uncertainty of the timing of risk resolution. Realistic examples are familiar from applied statistics (multivariate survival-time analysis) and static investment decision making with uncertain time horizon. Assume now that the decision maker first ranks p and q in preference and then makes his choice between the two compound lotteries. Then, the presence of the common component r in either mixture induces a reference risk so that the decision maker is not, in effect, concerned with the advantage of p over q when considering the impact of p and q on his choice. Rather, he must assess the advantage of p in the presence of r over q in the presence of r, for independently of his choice, he confronts r in case p (or q) fails to be resolved first. The conditionalisation of risk thus entails conditionalisation of preference of which we will show that it is an order-preserving isomorphism: the conditional preferences satisfy the independence axiom excatly if the unconditional preferences do, from which they are derived. Yet, it turns out that, given a few obvious properties of the induced preferences, the latter necessarily violate the independence axiom if they satisfy stochastic dominance preference. Stochastic dominance preference, on its part, is widely viewed as an indispensable normative requirement of rational risky choice weaker than independence. 433 Annals of Operations Research (2020) 289:431–448 1 3 To demarcate the present analysis from previous approaches to conditional risk preferences available in the literature, a few remarks are in order. Although we will represent lotteries as probability distributions (or, equivalently, random variables associated with them), our concept of conditional risk—unlike that used by Pratt (1988), for instance—is different from conditional probability, nor are the conditional preferences we consider of the kind which arise in multi-attribute preference analyses (Keeney and Raiffa 1976; Geiger 2012). Other than in many previous approaches to the subject, background risk is not treated as an exogenous, non-tradable risk (e.g., Franke etal. 2006; Malevergne and Rey 2010) but as one endogenously induced by convex combination of the lotteries to be compared. Likewise, we may ignore preferences induced in choices involving “temporal risk” and “temporal preference”, that is, sequential risk and decision problems in which preference depends on when uncertainty resolves (for review and references, see Machina 1984). There is no interference with the normative view of the independence axiom in the temporal risk case since dynamic consistency, as a criterion of EU preferences, deals with “atemporal” risk preferences (Kreps and Porteus 1979; Machina 1984; Karni and Schmeidler 1991). This non-interference is most obvious in representations of EU preferences as special, atemporal boundary cases of temporal risk preference (Kreps and Porteus 1979; Machina 1984) arising under conditions which also ensure the dynamic consistency of choice (Kreps and Porteus 1978). Indeed, the present analysis, too, applies to preferences depending on when uncertainty resolves, but choices are static, while risks are resolved at uncertain times. Like in our approach, convex mixtures of lotteries have previously been modelled as probabilityweighted averages of bivariate lotteries (uncertainty of resolution time and monetary outcome) and applied in static-decision analyses (for examples and review of the literature, see Martellini and Urošević 2006). But as it seems, their normative implications for EU theory have not been examined so far. After sketching out the conceptual framework of EU theory and some of its most basic normative implications in the next section, we develop formal representations of background risk and conditional preference in Sects.3 and 4. Section5 presents an application of the formalism. In Sect.6, the invalidity of the independence axiom as a normative principle of static, rational risky choice is proved. Section7 concludes, with a few remarks concerning the meaning and significance of the results obtained and their possible extensions to more broadly circumscribed domains of rational risky choice. 2 Normative implications ofEU theory Let P be the convex set of simple probability distributions, or lotteries, defined on a compact real interval I of lottery outcomes. A degenerate lottery which gives x, x ∈ I, with certainty is denoted by x , x ∈ P. Lottery outcomes are evaluated as gains (x > x0) or losses (x < x0) with reference to some neutral point, or “aspiration level” x0, x0 ∈ I (Kahneman and Tversky 1979; Diecidue and van de Ven 2008), which is normalised to zero without loss of generality (x0 = 0, 0 ∈ I). A preference relation ≿ exists which satisfies the familiar axioms of weak order and continuity. If, in addition, the independence axiom is postulated, the more restricted version of independence (replacement of ≿ by strict preference ≻), the indifference version of (1) (replacement of ≿ by ~) and, eventually, EU theory follow (e.g., Hammond 1998). EU theory implies the principle of stochastic dominance preference. (1) p≿q ⇒ 𝛼p+(1−𝛼)r≿𝛼q+(1−𝛼)r,0<𝛼≤1, p∈P,q∈P,r∈P 434 Annals of Operations Research (2020) 289:431–448 1 3 This principle states that p ≻ q if p has first-order stochastic dominance over q, that is, if p ≠ q and Fp(x) ≤ Fq(x), x ∈ I, where Fp and Fq denote cumulative distribution functions. As a postulate weaker than the independence axiom, stochastic dominance preference is a fundamental normative requirement of rational choice, but as such is compatible with violation of the independence axiom (Tversky and Kahneman 1986, p. S253; Machina 1989, pp. 1634–1635). The convex combination αp + (1 − α)r entering (1) is often viewed as the single-stage representation of a two-stage lottery which, in the first stage, gives an α : (1 − α) chance of receiving a ticket for the second-stage lottery p or r. When viewed as a single-stage lottery in P, however, αp + (1 − α)r gives the outcome x, x ∈ I, with probability αp(x)+ (1 − α)r(x). Choices between multiple-stage lotteries can be dynamic or static depending on whether or not they involve contingent decisions to be taken between particular lottery stages. Dynamic consistency of preferences implies that, in a choice between the two-stage lotteries corresponding to αp + (1 − α)r and αq + (1 − α)r, the preferred two-stage lottery gives the preferred second-stage lottery p or q, respectively, when the initial α-lottery gets resolved, with r being rejected. As a normative principle of rational choice, the independence axiom rests on the result that, under a few reasonable restrictions, independence is equivalent to dynamic consistency (Karni and Schmeidler 1991; Hammond 1998). One such reasonable restriction is reduction of compound lotteries, which will be addressed in the concluding section. It means that decision makers are indifferent between multiplestage lotteries and their single-stage, or “reduced”, representations in P (Segal 1990, 1992). 3 Background risk incompound lotteries The independence axiom also raises a static-choice problem, which has been largely ignored in normative interpretations of EU theory, however. To show that this neglect leaves normative interpretations of the independence axiom inconsistent, we first introduce the concept of background risk into the analysis of (1). A lottery with random outcome variable X2 is usually called the background risk of another lottery with random outcome variable X1 if the decision maker faces the risk of X2 while assessing the risk of X1. Typical examples are additive and multiplicative risk-background interactions involving outcome variables X and functions χ of the form X = χ(X1, X2) = X1 + X2 and χ(X1, X2) = X1X2, respectively (Pratt 1988; Tsetlin and Winkler 2005; Franke etal. 2006; Malevergne and Rey 2010). Here, we admit more generally defined functions χ(X1, X2) as well as functions of more than two random variables. Comparison of X and X’, where X’ = χ(X1′, X2) for some X1′, induces a preference ranking between X1 and X1′ conditional on the background risk X2 which can be expressed symbolically as X1 ≿ 𝜒,X2 X1′, and similarly for the more restrictive relations ≻ 𝜒,X 2 and ∼ 𝜒,X 2 . We refer to the random variable X = χ(X1, X2) as X1 in the presence of X2, or, briefly, X1 given X2. In the literature, ≿ 𝜒,X2 has almost invariably been treated as an EU preference relation, excluding violation of independence from the outset (for rare exceptions, see Quiggin 2003; Geiger 2008). However, the equivalence (2) suggests that whether ≿ 𝜒,X2 has, or has not, the EU property depends, besides on ≿, on χ and X2: if χ and X2 can be shown to exist so that ≿ 𝜒,X2 violates (2) X1 ≿ 𝜒,X2 X ′ 1 ⇔𝜒 ( X 1 ,X 2) ≿𝜒 ( X ′ 1 ,X 2) 435 Annals of Operations Research (2020) 289:431–448 1 3 the independence axiom, (2) excludes ≿ as an EU preference relation. This is what our results obtained below demonstrate (see remark following Proposition 4 below). Background risks do not only exist where decision makers face one risk while assessing others. They also arise in weaker cases where the decision maker, while assessing some risks, confronts another risk, but only with some finite probability. Examples of this broader notion of background risk can be found where the timing of the resolution of risk is uncertain, for instance, in the management of risky investments with uncertain time horizon or reliability engineering (Sect.5). We begin to analyse this situation by developing a suitable conceptual framework for risk assessment and choice under constraints that are themselves uncertain. 4 Formal representation ofbackground risk Where not otherwise stated, we denote the probability distributions of X1, X1′ and X2 by p, q and r, respectively, throughout this analysis. The ranges of X1, X1′ and X2 are thus subsets of I. Without further explicit reference, we anticipate that, by construction, all multivariate, composite random functions χ, ζ, … considered below, which will typically be of the form ζ(χ(X1, X2), …), have ranges in I and, therefore, probability distributions in P. We are especially concerned with functions of X1, X1′ and X2 and other variables giving rise to joint probability distributions of the form αp + (1 − α)r. We aim to establish how the comparison αp + (1 − α)r ≿ αq + (1 − α)r translates into the preference between X1 and X1′ given X2. This task amounts to determining the probability distribution of χ(X1, X2), which is derived, but different, from αp + (1 − α)r. More precisely, it is derived from the joint probability distribution of X1, X2 and another real random variable, or, more generally, a random vector T which is supposed to range over a finite subset S of an m-dimensional real interval IT, m ≥ 1. The joint probability distribution of T has finite support S, S ⊂ IT. For every partition of S into subsets S1 and S2 and given T, there clearly exists some α, 0 ≤ α ≤ 1, so that the overall joint probability of t, T = t and t ∈ S1, is α, and analogously for T, S2 and 1 − α. T being constructed in this way shows that the α-lottery can be associated with any suitably distributed T. The latter can be any quantitative constraint on risk measurement, assessment and comparison specifying the impact of a given background risk on risk preference. An example of a vector T of random time variables with m = 2 will be presented in the next section; it is clearly distinct from “temporal” risk in dynamic risky choice problems, however. Let f(x1, x2, t) be the joint probability distribution of X1, X2 and T with the marginal distributions f Xi (xi), fT(t), etc. in the usual notation, and assume a partition of S into subsets S1 and S2 as mentioned above. Consider the random variable Y = ψ(X1, X2, T) with probability distribution g(y), where “ ∑x1,x2, t […]𝜓(x1,x2, t )=y ” means summation over all outcome values of X1, X2 and T satisfying ψ(x1, x2, t) = y for given y. In particular, given ψ(x1, x2, t) = y, it follows g(y) = f Xi (xi) from Eq.(3) for t ∈ Si, i = 1, 2, so that Eq.(4) simplifies to g(y) = f(x1, x2, t). Putting p = f X1 , r = f X2 , and α = ∑ t ∈ S 1 fT(t), one trivially has, (3) 𝜓 (X1,X2,T)= { X1if T=t,t∈S 1 X 2 if T=t,t∈S 2 (4) g (y)= ∑ x 1 ,x 2 ,t [f(x1,x2,t)]𝜓(x1,x2,t)= y 436 Annals of Operations Research (2020) 289:431–448 1 3 Proposition 1 For i = 1, 2, assume stochastic independence of Xi and T. Then, Since equal random variables have the same probability distribution, one has Y = X1 and g(y) = p(y) with probability α, and Y = X2 and g(y) = r(y) with probability 1 − α. Thus, Eq.(3) and Proposition 1 imply g = αp + (1 − α)r. Note that this construction of Y as a random variable with a compound probability distribution makes no reference to the notion of sequential lottery. The probability distribution of X1 in the presence of X2 can now be computed on the basis of Eq.(5). Similarly to the definition of Y, the dependence of X on the α-lottery can generally be expressed as a random function ζ(X, T) determined differently for T-values in S1 and S2, respectively. To define ζ(X, T) in a fashion similar to Eq.(3) means to represent the probability distribution of X1 given X2 as a convex mixture of lotteries. As such, it is in P if the mixture components are. This is a requirement necessary for ζ and χ to have finite ranges in I, and for their probability distributions to be in P, since the preference relations concerned are defined on P. Now, let ζ1 and ζ2 be real functions so that1 Let z(ζ1(x), ζ2(x), t) be the probability distribution of ζ(X, T). The marginal distribution zζ1ζ2(ζ1(x), ζ2(x)) is the probability function of X = χ(X1, X2) to be determined. Since Xi and T are independent, so are ζi(χ(X1, X2)) and T (as for this stochastic-independence property of composite definitions of functions of random vectors, see, e.g., Pfeiffer (1990), esp. p. 251 and Theorem11.3.3). Equations(3) to (6) imply zT(t) = fT(t) so that Equation (7) follows immediately from Eq.(6) in a fashion similar to the derivation of Eq.(5) from Eq.(3), with the replacement of the stochastic independence condition for Xi and T by that of ζi(χ(X1, X2)) and T. The dependence of z 𝜁1𝜁2 on p and r needs to be determined next. Put ω = 1 − α in Eq.(7) and define2 Consistently with the interpretation of zζ1ζ2(ζ1(x), ζ2(x)) as the probability distribution of X1 given X2, p∣ωr means p given r (“p in the presence of r”), with r being the background risk of p. The condition r ≠  0 (i.e., r(0) < 1) excludes the trivial case in which r gives zero with certainty, that is, the case of no background risk at all. Let the conditionalisation operation (5) fX1X2 ( x1,x2 ) = ∑ t∈S f ( x1,x2,t ) =𝛼p(x1)+(1−𝛼)r(x2),0≤𝛼≤ 1 (6) 𝜁 (X,T)= { 𝜁1(X)if T=t,t∈S 2 𝜁 2 (X)if T=t,t∈S 1 (7) z𝜁1𝜁2( 𝜁 1 (x),𝜁 2 (x) ) =(1−𝛼)z 𝜁1 (𝜁 1 (x)) + az 𝜁2 (𝜁 2 (x )) (8) ( p∣𝜔r)(x)=z𝜁 1 𝜁 2( 𝜁 1 (x),𝜁 2 (x) ) ,r≠  0, x∈ I 1 Observe that the subsequent notation for the T-dependence of ζ is inverted as compared to that of ψ as specified on the right-hand side of Eq.(3) (X1 → ζ2(X), X2 → ζ1(X)). The change is adopted for technical reasons and is consistent with Eq.(3); it admits a more natural and more coherent notation in the following paragraphs, especially in Eqs.(9) and (15) below, but otherwise has no theoretical significance. . 2 Reference to ω rather than α in the following definition is for reasons of convenience. It admits a more compact and consistent formalism, as will be obvious from Eqs.(8) to (15) below and from the realistic interpretation of ω in the application example of Sect.5. 437 Annals of Operations Research (2020) 289:431–448 1 3 ∣ωr: P → Pω, r be a function operating to the left on the function variable p in “p∣ωr”, where P∣ωr is the range of ∣ωr, Pω, r ⊂ P. From Eqs.(7) and (8) follows It remains to determine p∣1r and p∣0r. To this goal, a few more properties of p∣ωr have to be specified. First, consider the boundary case ω = 1. In this special case, one has α = 0 so that the comparison of αp + (1 − α)r and αq + (1 − α)r reduces to the trivial indifference r ~ r for all p, q and r. Hence, p and q in the presence of r are unconstrained by r, meaning Secondly, recall that the aspiration level x0 has been assumed to be an exogenously fixed parameter. As such, it is not related to probability and does not vary with the background risk (Diecidue and van de Ven 2008). In the normalisation x0 = 0 adopted above, this implies  0 ∣ωr =  0 , and conversely, if p∣ωr =  0 for some p and every ω, 0 ≤ ω ≤ 1, then p∣1r = p =  0 as a special case for ω = 1 so that altogether Thirdly, if p = r, this means that p obtains with certainty. This is equivalent to ω = 0 so that Considering Eqs.(3) and (9), one has Proposition 2 For given r, r(0) < 1, and 0 ≤ ω ≤ 1, ∣ωr: P → Pω, r is a linear function. The proof of Proposition 2 is outlined in the Appendix together with the proofs of Propositions 3–5 stated below. From Proposition 2, it follows immediately that, if {x1, …, xn} is the support of p and pi = p(xi), then for p as a finite convex combination of the degenerate distributions x 1, …, x n.3 Given Eqs.(9) and (10), our next result completes the formal representation of p∣ωr, Proposition 3 For every r, r(0) < 1, |0r has the idempotent property (p∣0r)∣0r = p∣0r, and Because of this idempotence, which is a familiar characteristic of projection operations (e.g., Yanai etal. 2011), the transformation ∣0r: P → P0, r can be understood as a parallel projection which maps P onto P0, r along hypersurfaces p(0) = constant in convex subsets Δ ⊂ P with r ∈ Δ and  0 ∈ Δ (Fig.1). (9) p∣𝜔r=𝜔p∣1r+(1−𝜔)p∣0r,r ≠  0, 0 ≤ 𝜔 ≤ 1 (10) p∣1r=p,p∈P (11) p∣𝜔r= 0 ⇔ p= 0 (12) p∣0p=p, (13) p∣𝜔r= (∑ i≤n pixi ) ∣𝜔r= ∑ i≤n pi(xi∣𝜔r),r≠ 0, (14) ( p ∣ 0r )( 0 )= p ( 0 ) ( p∣0r)(x)=r(x)1−p(0) 1−r(0) ,x≠ 0 3 To keep the notation simple, we do not consistently distinguish between the particular values x1, x2, … of the real variable X and the real values x1, x2 the random variables X1 and X2 may respectively take on. The special meaning attached to the xi’s should always be clear from the context. 438 Annals of Operations Research (2020) 289:431–448 1 3 Inserting the results (10) and (14) into Eq.(9) gives the desired representation of p in the presence of r,4 Recall that q is the probability distribution of X1′. The comparison X1 ≿ 𝜒,X2 X1′ implicitly defines a preference ranking of p and q conditional on r. It can be written as p ≿ω, r q, while (2) goes over into This definition means that in a choice between αp + (1 − α)r and αq + (1 − α)r, the mixture component r induces a preference ranking ≿ω, r between p and q conditional on the background risk r. For independently of his choice, the decision maker always faces r while assessing p and q, except in case ω = 1. He is not concerned with the advantage of p over q, but the relative advantage of p given r over q given r, or, formally, p∣ωr ≿ q∣ωr. 5 Example: uncertain risk resolution time In practice, often decision makers cannot be sure when, exactly, the risks they are facing will be resolved. This situation is critical for the management of risky investments with uncertain time horizon, for example (Martellini and Urošević 2006; Blanchet-Scalliet etal. 2008). It also induces time duration of risk as a random variable in many static decision tasks. In fact, “static” in the sense of “non-dynamic” means non-sequential, but (15) p∣ 𝜔 r=𝜔p+(1−𝜔)p∣ 0 rr≠  0, 0 ≤𝜔≤ 1 (16) p≿ 𝜔,r q⇔p∣ 𝜔 r≿q∣ 𝜔 r,r≠  0, 0 ≤𝜔≤ 1 Fig. 1 Convex set Δ of lotteries with possible outcomes x1 < x2 = 0 < x3, given background risk r. The dashed line is P0, r 4 An equation similar to (15) is used in Geiger (2008) as a plausible model of p given r, but not rigorously derived. Yet the interpretation of the parameter ω differs from that in Geiger (2008), where it is defined as the complementary probability obtained by substituting ω → 1 − ω. Our formal results are clearly compatible with our previous approach despite this change in notation. 445 Annals of Operations Research (2020) 289:431–448 1 3 Considering (11), which entails ( x k∣0r)i = 0 for i ≠ k, one now compares the expressions for p∣0r and (p∣0r)∣0r in a component-by-component fashion to obtain after an obvious rearrangement of terms. If the matrix of the coefficients ( x j∣0r)i − δ ji in (A.4) has maximum rank n − 1, one has (p∣0r)j = 0 for all j ≠ k. Then, (p∣0r)k = 1 and p∣0r = p =  0 from (11). If the rank of the matrix in (A.4) is less than n − 1, every solution of (A.4) must satisfy (p∣0r)i = λri for some real λ, λ > 0, since r is a solution of Eq. (A.4). One immediately finds λ = (1 − pk)/(1 − rk) with the use of the above result (p∣0r)k = pk. Proof of Proposition 4 (i) By definition of Pω, r, ∣ωr maps P onto Pω, r. Considering Eqs.(14) and (15) and the condition0 < ω, the reverse of ∣ωr exists. Hence, ∣ωr: P → Pω, r is an isomorphism for 0 < ω. Since it is also linear, one has Since P is a convex set, βp′ + (1 − β)p″ is in P, while p′∣ωr, p″∣ωr and (βp′ + (1 − β)p″)∣ωr are in Pω, r so that the right-hand side of (A.5) is in Pω, r as well. Hence, Pω, r is a convex set. (ii) (P, ≿ω, r) and (Pω, r, ≿) are trivially isomorphic under ∣ωr, by the definition (16) and the convexity of P and Pω, r. (iii) Because of this isomorphism, ≿ trivially satisfies the axioms of EU preference if and only if ≿ω, r does. In particular, if U: Pω, r → ℝ is a linear functional and Uω, r (p) = U(p∣ω r), then U represents ≿ on Pω, r if and only if Uω, r represents ≿ω, r on P. (iv) Since Pω, r ⊂ P, ≿ trivially satisfies stochastic dominance preference on Pω, r if it satisfies stochastic dominance preference on P. Proof of Proposition 5 The proof proceeds in two steps. In either step, stochastic dominance preference is supposed to hold for ≿. We first construct a simple counterexample to the EU property of ≿ω, r for a special class of neutral background risks r ~  0 , r ≠  0 . Thereby, we conveniently restrict the analysis to a convex set Δ of lotteries, Δ ⊂ P, with three possible outcomes x1 < x2 = 0 < x3 and r ∈ Δ, r(0) = r2, p(0) = p2 in Eq.(14). In the second step, we show that Proposition 5 holds for all r, r ≠  0 , if it holds for some such r, r ~  0 , and 0 < ω < 1. − Step 1 Assume stochastic dominance preference for ≿ω, r. Recall that we also suppose the axioms of weak order and continuity to hold for ≿. Use ∣ωr and ≿ to represent ≿ω, r according to (16), define Δω, r = Δ ∩ Pω, r so that r ∈ Δω, r, and choose 0 < ω < 1 (Fig.3). We have the result (Becker and Sarin 1987) that the two axioms, together with the principle of stochastic dominance preference, imply the existence of a real-valued functional U: P → ℝ, though not necessarily an EU functional, which uniquely represents ≿ (i.e., satisfies U(p) ≥ U(q) if and only if p ≿ q) on P and, hence, on Δω, r. The uniqueness is up to strictly increasing (i.e., bijective and order-preserving) transforms of U (Weymark 2005). Because of stochastic dominance preference, for p2 = q2 < 1 it holds that p∣ωr ≻ q∣ωr if and only if (p∣ωr)3> (q∣ωr)3, or, equivalently, (p∣ωr)3/(p∣ωr)1 > (q∣ωr)3/(q∣ωr)1, meaning that p∣0r= ∑ i≤n (p∣0r)i xiby (A3) =(p∣0r)∣0r =∑ j≤n (p∣0r)j(xj∣0r)by (A2) = ∑ i,j ≤ n (p∣0r)j(xj∣0r)i xiby (A3 ) (A.4) ∑� j ≤ n (p∣0r)j((xj∣0r)i−𝛿ji)=0, i≠k . (A.5) ( 𝛽p � +(1−𝛽)p ��) ∣ 𝜔 r=𝛽p � ∣ 𝜔 r+(1−𝛽)p �� ∣ 𝜔 r,0≤𝛽≤ 1 446 Annals of Operations Research (2020) 289:431–448 1 3 preference strictly increases with (p∣ω r)3 and with ρω, r(p) = (p∣ωr)3/(p∣ωr)1 along parallel straight line segments p2 = constant < 1 in Δω, r. The ratio ρω, r(p) is invariant under the transformation p∣ω r → (p∣ω r)’, For given p2 < 1, the inverse transformation of (A.6) exists. Hence, under (A.6) parallel straight line segments in Δω, r with constant, but different values of p2 are isomorphic, with respect to preference, to the boundary segment p2′ = 0 and, hence, to each other (Fig.3). Observe that the real-valued functional U which represents ≿ on P and, hence, on Δω, r can be normalised to be unique. Rewrite U(p∣ωr) as U(ρω, r(p), p2), considering that ρω, r(p) and p2 uniquely determine p∣ωr. Then U(ρω, r(p2′), 0) and U(ρω, r(p), c) respectively represent the isomorphic preference orderings on the straight line segments p2′ = 0 and p2 = c, 0 ≤ c < 1, with U(ρω, r(p2′), 0) and U(ρω, r(p), c) strictly increasing in ρω, r. To exhibit this isomorphic property, U(ρω, r(p), p2) must be a strictly increasing transform of U(ρω, r(p), 0) for given p2 and take on the general form where V(p2) > 0 and b is a real constant which can be used to extend Eq. (A.7) to the case p2 = 1 and then normalise U to U(ρω, r(  0 ), 1) = 0. Observe that this extension and normalisation imply V(1) = b = 0 and that U(ρω, r(p), 0) is a well-defined expression even if p2 ≠ 0 since ρω, r(p) is independent of p2. To see that V(p2) strictly decreases with increasing p2, consider the special case p3 = 0 in which preference and, hence, U(ρω, r(p), p2) strictly increase with p2, by stochastic dominance preference. Considering r ~  0 and the normalisation U(ρω, r(  0 ), 1) = 0 with b = 0 in Eq. (A.7), one has Equation (A.8) implies U(ρω, r(r), 0) = 0, because of r2 < 1 and V(r2) > 0 for r2 < 1. Now, one has U(ρω, r(p), 0) < U(ρω, r(r), 0) = 0 for p3 = 0 < r3 so that, according to Eq. (A.7), V(p2) must strictly decrease for U(ρω, r(p), p2) to increase strictly with p2, and, clearly, so must V(p2) for arbitrary p. Since V(p2) and U(ρω, r(p), 0) are both strictly monotonic and, hence, non-constant functions, U(ρω, r(p), p2) is non-linear in the probabilities, by Eq. (A.7). Since U(ρω, r(p), p2) represents ≿ω, r uniquely on Δ, ≿ω, r violates the independence axiom on Δ and, hence, on P. As an immediate consequence of Propositions 4(i) and 4(ii), (Δω, r, ≿) is a non-EU preference structure, too (Fig.2).8 − Step 2 Let ≿ω, r be given as in Step 1, and assume that ≿Ω, s satisfies stochastic dominance preference on P for some arbitrary s except s =  0 , where 0 < Ω < 1. Observe that  0 ∈ PΩ, s (see Eq.(11)) and that there exists some r’, r’ ∈ PΩ, s, so that r’ ~  0 but r’ ≠  0 . Now construct a convex subset Δ’ ⊂ PΩ, s similar to Δ in Step 1, define (ΔΩ, r’)’ = Δ’ ∩ (PΩ, s)Ω, r’ and, following the proof carried out in Step 1, proceed to show that ≿Ω, r’ violates independence on Δ’ and, hence, on PΩ, s. From 4(ii) one immediately concludes that ((PΩ, s)Ω, r’, ≿) is a non-EU preference structure. Recall that (PΩ, s)Ω, r’ is a convex subset of PΩ, s so that (PΩ, s, ≿) and, once more by Proposition 4(ii), (P, ≿Ω, s) are non-EU preference structures. Hence, ≿Ω, s violates the independence axiom on P. 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