IMA Journal of Management Mathematics, Volume Advance Article (3) – Oct 6, 2016

/lp/ou_press/on-bespoke-decision-aid-under-risk-the-engineering-behind-preference-UpxgNYE1YY

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- Oxford University Press
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- © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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- 1471-678X
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- 1471-6798
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- 10.1093/imaman/dpw018
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Abstract Decision-aid as developed from the late Fifties to the mid-Seventies has introduced a substantial change in the previous practice of OR. The spirit of this new deal of a bespoke decision-aid has indeed been that the decision maker’s own preferences, including risk tolerance and attribute selection, as opposed to the standardized goals or discretionarily assigned objectives of the analyst, should be paramount in selecting the most satisfactory strategy. But this way of implementing decision-aid has not received in practice as wide an application as it should have, due to some unduly persistent although erroneous objections. Eliciting the preferences of someone else has thus been regarded as some logically impossible task. The present article argues that such suspicions are today obsolete and that most of the evoked biases or objections can be overcome by using methods of elicitation ignored by textbooks. In particular, the article presents a complete set of non-parametric methods that avoid the above biases and objections. It suggests that a number of fields of application, in wealth management and finance as well as in human resources management and managerial choices, urgently call for such genuinely bespoke decision-aid. 1. Introduction Introductory background: a quick primer on utility theory development After World War II, analytical methods designed to solve military, logistics and engineering problems blossomed, building on the experience accumulated during the war. These methods came to be applied to business decisions, though reservations soon questioned some of the human behaviour aspects, found to be too seldom or too poorly accommodated in the new approach to the field. Not every individual should deal with uncertainty by choosing the highest expected gain and select the best decision accordingly. Not every business organization necessarily abides by maximising expected profit,—provided it could, in the first place, make practical sense of it, in terms of steps to be taken. Indeed, algorithms designed to select decisions should be adjusted (a) from one decision-maker (DM further below) to the other and (b) from one decision context to the other. The von Neumann & Morgenstern (1944),expected utility (EU) approach, soon generalized by Savage (1954) as subjective expected utility, solved at least the first part of the issue. The idea was that attitude toward risk was not the same for all DMs, the utility function being a handy tool to take this fact into account (Schlaifer, 1959). Even this new view was soon regarded as too narrow, but early alternatives challenging EU were insufficiently grounded and too grossly formalized in the Fifties to be immediately convincing. Although the debate was decisively revived in the Seventies, the possibility of extending von Neumann-Morgenstern’s EU to multi-objective decision issues (Keeney & Raiffa, 1976) attracted by then the bulk of the attention and it was not until the Eighties that non-EU approaches paying tribute to both aspects summarized as (a) and (b) above could receive wide scientific recognition. However, from scientific recognition to practical implementation, part of the way still remains to be travelled. This article focuses on this specific and important issue. The article is organized as follows: The rest of Section 1 is devoted to recalling a couple of focal stages in the recent evolution of modern decision theory; Section 2 deals with utility elicitation; section 3 with the weighting function elicitation, while Section 4 develops the extension to Multi-attribute utility encoding. Section 5 concludes and suggests several domains of applications. 1.1. The von Neumann-Morgenstern vision Good sense commands choosing the action entailing the highest outcome. And when it comes to risky situations, it seems at first sight good sense as well to extend the rule of choice to the largest expected outcome. This rule may, as a matter of fact, be sufficient in engineering modelling. But, from a business perspective, it means that a prospect L$$^{1}$$ endowed with 40% chance of making $10000 and 0 otherwise, denoted by the vector (0, 0.6; 10000, 0.4)1 should be regarded as equivalent to a prospect L$$^{2}$$ leading to 50% of making $8000 and 0 otherwise and, in the limit, equivalent as well to a prospect L$$^{3}$$ of picking up a straightforwardly available $4000 (4000; 1). Von Neumann-Morgenstern’s utility theory claims that this is not generally true2 Let us consider for a while only prospect L$$^{2}$$: (0, 0.5; 8000, 0.5). Figure 1 shows in abscissa how ‘certainty equivalents’ (outcomes available with certainty and considered equivalent to the prospect by the individual) to prospect L$$^{2}$$ can differ between three distinct individuals. An individual who is averse to risk will give a lower certainty equivalent X-R$$_{A}$$ to L$$^{2}$$, by subtracting his personal valuation of risk R$$_{A}$$ from the expected outcome X than the one an individual prone to risk will give, the personal valuation of risk R$$_{P}$$ of the latter individual being negative for the same prospect L$$^{2}$$. A ‘risk-neutral’ individual—assumed to be only sensitive to outcome, whether certain or expected, and in this sense indifferent to risk—will value L$$^{2}$$ at exactly 4000, denoted by X-R$$_{N}$$ on Fig.1, his personal valuation of risk R$$_{N}$$ being null. We will therefore have X-R$$_{A}$$$$<$$ X-R$$_{N}$$=4000 $$<$$X-R$$_{P}$$. Fig. 1. View largeDownload slide Three certaintyequivalents of the same prospect L2 for three different individuals: a risk averse one (X-RA), a risk neutral one (X-RN), and a risk-prone one (X-RP). Fig. 1. View largeDownload slide Three certaintyequivalents of the same prospect L2 for three different individuals: a risk averse one (X-RA), a risk neutral one (X-RN), and a risk-prone one (X-RP). If each DM may value differently a same prospect, it does indeed matter to know what the valuation scheme of any DM we might come to advise turns out to be. Von Neumann Morgenstern utility theory claims that points (X-R$$_{\rm A}$$, 0.5), (X-R$$_{B\rm }$$, 0.5) and (X-R$$_{\rm P}$$, 0.5) are each determined by a given function, respectively pertaining to each DM. The concave, convex and linear functions sketched on Fig.1 are possible candidates. But what can be said in more general contexts, and how to determine sufficiently many points of these functions’ graphs? Let us now consider any DM facing three other prospects P$$^{1}$$, P$$^{2}$$,P$$^{3}$$, the outcomes of which are assumed to be all within the interval (0,10000). We do not know whether this particular DM is risk averse or risk prone or risk neutral. We therefore want to discover his or her utility function u(x), where x stands for possible outcomes. Any utility function can be normalized3 by setting u(Min.amount) $$= 0$$ and u(Max. amount) $$= 1$$. We therefore set here u(0) $$=0$$, u(10000) $$=1$$. Traditionally—and still often today—the decision maker was asked about his certainty equivalents $$X_{1}, X_{2}, X_{3}$$ when p(10000) eventually varies from 0 to 1 (this is typically what we call here a probability-entailing question). Following the intuition given above, we now set up a formal definition of a certainty equivalent X to a Prospect P as being defined by: $$u(X)=U(P)=\Sigma_ip_i\ast u(x_i)$$. If, for example, one successively sets p=0.7, p=0.4, p=0.1, we can write: x1∈(0,10000):u(X1)=U(P1)=0.3u(0)+0.7u(10000)=0.7x2∈(0,10000):u(X2)=U(P2)=0.6u(0)+0.4u(10000)=0.4x3∈(0,10000):u(X3)=U(P1)=0.9u(0)+0.1u(10000)=0.1 We have to request the decision maker to provide a specific answer4X* to each one of these questions. Assume his answers are X$$^{1\ast}$$,X$$^{2\ast}$$, X$$^{3\ast}$$. The points of coordinates (X$$^{1\ast}$$, 0.7), (X$$^{2\ast}$$, 0.4), (X$$^{3\ast}$$, 0.1), yield the graph on Fig. 2. This utility function is revealed to the analyst from the DM’s answers as being concave. From this utility function, we can compute any EU of any prospect the DM could have to face, the outcomes of which would fall in the interval (0,10000). Fig. 2. View largeDownload slide Encoding the utility function of a DM using the traditional methodology of certainty equivalents. Fig. 2. View largeDownload slide Encoding the utility function of a DM using the traditional methodology of certainty equivalents. By subtracting the X$$^{i\ast}$$(i=1,2,3) from the expected outcome of each prospect, we can infer the ‘price of the risk’ $$\boldsymbol\pi$$ borne by the DM in the situations respectively corresponding to P$$^{1}$$,P$$^{2}$$, P$$^{3}$$. Figure 2 shows the price of risk for the given individual (called ‘risk premium’ by economists) in the P$$^{2}$$ situation: π2=E(P2)−X2 (1) Building on de Finetti (1952), Arrow & Pratt (1964) have shown that $$\boldsymbol\pi$$ can be recovered from any utility function analytical expression u(x)—through a Taylor expansion of order 2—as: ∀i,πi≃−u′′[x=EU(pi)]u′[x=EU(pi)]σ2(x)2 (2) where the first ratio on the right hand-side is called the Arrow-Pratt coefficient (although de Finetti deserves some of the credit) of Absolute Risk Aversion. From (2), one can see that risk aversion ($$\pi >0$$) translates into a concave utility function and conversely. To sum up, the DM we have to advise will rank the three prospects above in the preference order of their certainty equivalents: $$X^{3*}\prec X^{2*}\prec X^{1*}$$ or, equivalently, in the order of their respective EU (here: $$0.1 < 0.4 < 0.7$$). Preference on prospects is thus formalized as a pre-order $$\preceq$$—read as ‘non preferred or indifferent to’. U($$\bullet$$) is a simple homeomorphism yielding a numerical equivalent on the real line to the preference pre-ordering among prospects of a given DM5. 1.2. The intellectual effervescence of the fifties and sixties Many psychologists and even some economists and statisticians came to challenge EU theory. Some scientists were suggesting a widely different way to model risky situations, like G.L.S. Shackle and his notions of ‘stimulus’ and of ‘surprise’. Notwithstanding Shackle’s efforts (Shackle, 1961), a rigorous formal version of this idea of revision has been lacking, and it was practically abandoned. One other author soon called for alternatives to the mathematical expectation operator in selecting decisions. Allais (1953) made his claim on the ground of what has been termed the ‘Allais Paradox’. Allais himself did not support that denomination, as he pointed out that it was not a paradox in the usual sense of the word, but a simple—though powerful—counterexample. We give here a quick account or it. Subjects were first submitted to one choice question, namely asking to choose between two prospects, A$$^{1}$$ and A$$^{2}$$, denominated in GBP, for example Keeping the standard notation, as used already above and setting M for 10$$^{6}$$: A$$^{1}$$: (10M, 1)$$\qquad \qquad \qquad$$ A$$^{2}$$: (0, 0.01; 10M, 0.89; 50M, 0.10) Having made their choice (irrevocably), subjects were then asked to choose between two other prospects, B$$^{1}$$ and B$$^{2}$$: B$$^{1}$$: (0, 0.89; 10M, 0.11)$$\qquad \qquad \qquad$$ B$$^{2}$$: (0, 0.90; 50M, 0.10) Among A$$^{1}$$ and A$$^{2}$$, a large proportion (above 65% in many cases, sometimes more) of the subjects choose A$$^{1}$$ in the first question, while, in the second question, a heavy proportion of subjects (often close to 90%), choose B$$^{2}$$, among which a number of those having chosen A$$^{1}$$ in the first question, representing often between 50% and 65% of all subjects. This share of the population shows that EU is a rule by which only a minority of individuals follow when choosing among prospects (a bit less than 40%, according to Hey & Orme, 1994). Indeed, choosing A$$^{1}$$ and then B$$^{2}$$ is incompatible with EU behaviour. To see why, a graphical account is illuminating: the Marschak-Machina triangle. Let us indeed remark that only three amounts of outcome are at hand in Allais’ questions: 0, 10M and 50M. Yet, if we consider all possible prospects with this three-tuple support, generically denoted by ($$x^{1}, p_{1}; x^{2}, p_{2}; x^{3}, p_{3})$$, with $$x^{1}<x^{2}<x^{3}$$ being fixed outcomes, while $$p_{1}, p_{2}$$ and $$p_{3}$$ are their (variable) respective probabilities, we may represent all of them by plotting $$p_{1}$$ on the abscissa and $$p_{3}$$ on the ordinate, $$p_{2}$$ being easily recovered by the length of the horizontal segment ending at the hypotenuse of the triangle. P$$^{\circ}$$ stands thus for some specific prospect of this type (Fig. 3). The curve passing through P$$^{\circ}$$ stands for the locus of prospects indifferent to P$$^{\circ}$$ for some DM (a priori, we don’t know the shape of such a locus, we draw thus here some arbitrary curve). ‘Degenerate lotteries’ ($$x^{i}$$, 1) sit on the summits of the triangle. It is straightforward to see that the sense of preferences is toward the NW in this triangle, as the arrow along the hypotenuse indicates. Fig. 3. View largeDownload slide A Marschak-Machina triangle. Fig. 3. View largeDownload slide A Marschak-Machina triangle. This figure has many appealing features, one of which is of particular interest here: If one admits EU theory, all indifference loci are parallel straight lines, the slope of which are a discrete equivalent to the Arrow-Pratt coefficient that is their slope is an index of absolute risk aversion of the DM whose preferences are represented in this triangle. Indeed, under the EU hypothesis, the slope of these indifference loci can be written (substituting 1- $$p_{3}$$–$$p_{1}$$ for $$p_{2}$$, the proof is straightforward): dp3dp1=−[u(x3)−u(x2)]−[u(x2)−u(x1)]u(x3)−u(x2)+1 (3) This expression is a constant, as $$x^{1}, x^{2}$$ and $$x^{3}$$ are fixed. If we ignore the 1 at the end of this expression, we indeed have a ratio between a second order difference in utility to a first order one, what is akin to the Arrow-Pratt coefficient above. A special case is given by assuming that the utility function is linear, that is that the DM is risk neutral in the sense already defined. Then: dp3dp1=−(x3−x2)−(x2−x1)(x3−x2)+1 (4) So, when (3)$$>$$(4), the DM is risk averse, he or she is risk inclined in the opposite case. Now, let us see where our four Allais’ prospects lie in the triangle where $$x^{1}=0, x^{2}=10M, x^{3}=50M$$ (Fig. 4). Looking at the probabilities in the questions above, it is straightforward to see that the quadrangle A$$^{1}$$A$$^{2}$$B$$^{2}$$B$$^{1}$$ is a parallelogram that is that segments A$$^{1}$$A$$^{2}$$ and B$$^{1}$$B$$^{2}$$ are parallel. So, if one admits that indifference loci are parallel straight lines in the triangle, then: Fig. 4. View largeDownload slide (a): Allais’ prospects under EU. (b): Stylized empirical indifference loci in the Marschak-Machina triangle fitting the experimental results. Arrows point toward preferred prospects along the hypotenuse. Fig. 4. View largeDownload slide (a): Allais’ prospects under EU. (b): Stylized empirical indifference loci in the Marschak-Machina triangle fitting the experimental results. Arrows point toward preferred prospects along the hypotenuse. Either one has (Fig. 4(a), triangle to the left) $$A^2\precsim A^1\Rightarrow B^2\precsim B^1$$ or $$A^1\precsim A^2\Rightarrow B^1\precsim B^2$$, while we have for a large number of subjects $$A^2\precsim A^1$$ and $$B^1\precsim B^2$$ (illustrated on Fig. 4(b), triangle to the right), as experimental research has abundantly documented. A simple EU computation also allows showing the claim. The experiment provides thus a counterexample to EU behaviour revealed by a large part of people. In fact, the experiment shows that in difference loci do not have same slope everywhere in the triangle, a finding of importance for business situations. In particular, when we are in the neighbourhood of an outcome endowed with certainty like A1, risk aversion increases in such a way as not to be representable by any given linear relation. This ‘certainty effect’ experiment has been replicated thousands of times by psychologists, risk engineers (notably though not exclusively in nuclear plants), economists, mathematicians and others, under different protocols, with invariably similar results. Note however that knowing the actual map of indifference loci has far more general implications: A prospect of investment might have a different location in the triangle, depending on the situation of markets, while an environmental issue will often be located in the SW corner, etc. We see that decision contexts do effectively matter to adjust the preference scheme of a given individual. 1.3. The emergence of the Allais-Quiggin-Wakker revision The discussion around the Allais’ Paradox was of poor quality in the Fifties. Some invoked the nature of the utility considered, some others the enormity of the amounts at stake, etc., all these arguments being irrelevant, while nobody perceived the simple meaning that risk preferences just and simply cannot be all along represented by any ‘linear in the probabilities’ functional. Allais’ results were often considered at best as a curiosity of Human behaviour. However, in the 1970’s, psychologists reconsidered the issue, performed many new experiments of their own, notably A. Tversky, P. Slovic and S. Lichtenstein$$\ldots$$ with always the same or very similar results. Three publications gave a new impetus to the line of research: (a) a book by Allais & Hagen (1979), (b) an article by Quiggin (1982), based on an empirical research among Australian farmers and explicitly building on another Allais’ idea of 1952 (in line with the Allais’ Paradox, but not developed further by its author by then), and finally an article by Machina (1982). These three contributions triggered the search for alternatives to the EU model. The model that has as of now emerged as receiving a substantial and increasing consensus (though no majority among economists yet) has been the Rank Dependent Utility (RDU) model. A minimal account of it can be given under the following propositions: (a) DMs do indeed transform outcomes measurement through some utility function (as had been seen since D. Bernoulli and revived by von Neumann and Morgenstern, then Samuelson and Savage) (b) However, DMs do also transform probability distributions under their cumulative (or decumulative) form. The precision is of decisive importance. Psychologists like W. Edwards in the Fifties, followed by several authors (e.g. Bernard, 1964) had indeed modelled the idea that DMs non-linearly transform probabilities of events (as discrete extensions of probability densities). But these ‘separable models’ were flawed in that they necessarily violate first order stochastic dominance (e.g. Munier, 1989)6. The Rank Dependent Model (RDU) was designed to avoid that major flaw. A simple account of the evolution of thought since Pascal, Fermat and Huygens in the 17th Century can be stylized on Figure 5. Let us indeed remark that expected gain can be captured under a simple integral, setting G(x)=1-F(x), where $$F(x)$$ is the usual repartition function: ∫G(x)dx or, under a discrete form (See Fig. 5(a) for a graphical representation in the continuous case): x11 +(x2−x1)(p2+p3+⋯+pn) +(x3−x2)(p3+⋯+pn) +⋯ +(xn−xn−1)pn The contribution of D. Bernoulli, generalized by von Neumann and Morgenstern later consisted in transforming the measuring rod of outcomes on the abscissa in a non-linear way and to write: ∫G[u(x)dx] or, under a discrete form (See Fig. 5(b) for a graphical account in the continuous case): u(x1)1 +[u(x2)−u(x1)](p2+p3+⋯+pn) +[u(x3)−u(x2)](p3+⋯+pn) +⋯ +[u(xn)−u(xn−1)]pn Fig. 5. View largeDownload slide (a) Expected gain as the integral of the decumulative probability distribution of gains, (b) Expected utility as the integral of the decumulative distribution of utilities, (c) RDU as the integral of the transformed decumulative probability distribution of utilities. Arrows indicate the direction of changes introduced by the consideration of utility (horizontal changes, along the abscissa) and then by the concept of the probability transformation function (vertical arrows, along the ordinate). Fig. 5. View largeDownload slide (a) Expected gain as the integral of the decumulative probability distribution of gains, (b) Expected utility as the integral of the decumulative distribution of utilities, (c) RDU as the integral of the transformed decumulative probability distribution of utilities. Arrows indicate the direction of changes introduced by the consideration of utility (horizontal changes, along the abscissa) and then by the concept of the probability transformation function (vertical arrows, along the ordinate). The Quiggin-Allais’s contribution has consisted in transforming also the ordinate measure—which was the real innovation—in a non-linear way and to write: ∫θ{G[u(x)]}dx or, under a discrete form (Fig. 5(c) for the continuous case just above): u(x1)θ1 +[u(x2)−u(x1)]θ(p2+p3+⋯+pn) +[u(x3)−u(x2)]θ(p3+⋯+pn) +⋯ +[u(xn)−u(xn−1)]θ(pn) with $$\theta (0)=0,\theta (1)=1,\theta ^\prime(p)>0$$. The integral of expected gain (a) is transformed, first in the abscissa direction (b) to account for the EU hypothesis, then in the ordinate direction as well (c) to account for the rank dependent utility hypothesis. A new difficulty appears however, regarding our goal of a bespoke decision-aid: To have an account of the preferences of a DM, we now have to elicit not only one, but two functions: the utility function and the probability transformation function7, the difficulty being to disentangle one from the other, for observation—whether statistical or direct—doesn’t allow observing them separately (a parallel to Heisenberg’s inequality could perhaps be attempted here). We shall show in Section 2 below that, thanks to research in the last 20 years, this difficulty has been overcome. 1.4. The new issue raised by multi-objective decisions since the mid-Seventies Meanwhile, a decisive progress had been accomplished in the Seventies, when R. Keeney and H. Raiffa provided the clever and subtle development of a ‘Multi-attribute Utility Theory’ (MAUT). They basically showed that, under some not innocent but relatively general assumptions, multi-objective decision problems could be solved without loosing the benefit of EU theory. They studied preferences defined along different axes, each axis bearing a particular ‘attribute’. Each attribute refers to one aspect of the preferences (e.g. cars can be selected according to their look, speed, robustness and price). We denote these attributes as x$$_{1}$$, x$$_{2}$$, x$$_{3}$$, each attribute being statistically distributed according to some distribution in a given interval [x$$_{i}^{0}$$, x$$_{i}^{\ast}$$]. They showed that, under the specific hypothesis of ‘utility independence’8, decision problems of this type and the corresponding decision-aid could be dealt with in four steps: (a) Eliciting a univariate (or ‘partial’, in the context of multiple attributes) utility function on each axis: $$u_{1} (x_{1}), u_{2} (x_{2}), u_{3} (x_{3})$$ from the DM (b) Eliciting ‘scaling constants’, that is fixed coefficients used to specify the ‘multi-attribute utility’ function under one of three forms: additive, multiplicative, multi-linear: - Additive form: $$U (x_{1}, x_{2},x_{3})=k^{1}u_{1 }(x_{1})+ k^{2}u_{2 }(x_{2})+ k^{3}u_{3 }(x_{3})$$ - Multiplicative form: $$U (x_{1}, x_{2},x_{3})=\prod_{i=1}^{i=3}[1+Kk^i u_i(x_i)]\frac1K-\frac1K$$ - Multi-linear form: U(x1,x2,x3) =k1u1(x1)+k2u2(x2)+k3u3(x3)+k12u1(x1).u2(x2) +k13u1(x1).u3(x3)+k23u2(x2).u3(x3)+k123u1(x1).u2(x2).u3(x3) where all $$k^{i}$$s and $$K$$ are constant coefficients to be elicited from the DM, as parts of his (her) preferences. (3) The ‘multi-attribute utility theory’ (MAUT) shows then that von Neumann-Morgenstern rationality axioms lead to maximize the expected multi-attribute utility function in the same way they led (above) to maximize the expected (one-dimensional) utility. (4) Though not necessary from a theoretical point of view, in practice; however, the latter maximization, to be feasible without too many computing difficulties as soon as we go further than two attributes, requires statistical independence between the probability distributions of the different attributes. This remark is important (see below). The essential objections raised today against any bespoke decision-aid based on MAUT could therefore be summarized in five propositions, according to which: (a) It would be impossible to elicit any ‘partial’ utility function of some DM free of very significant biases (as in the one-dimensional case); (b) Even assuming that the former were possible, it would be impossible to elicit both the DM’s utility function and probability transformation function without even more important biases; (c) Even assuming that (a) and (b) were possible, it would be impossible to elicit from the DM the scaling constants—necessary to any multi-attribute utility function—without significant biases again, as their determination seems to require probability entailing questions (beyond the two-attribute case); (d) Even assuming (a)–(c) possible, one could not—in practice—rank available strategies according to either one of two rationality rules, that is EU or RDU of the multi-attribute utility functional, unless all statistical distributions on attributes were statistically independent. And finally, (e) Even assuming that all three elicitations (a)–(c) were possible, it would not be possible—as an alternative to (d)—to directly elicit any multi-attribute utility function to bypass objection (d). We proceed now to show that these objections are not well grounded, except perhaps (d), to which fortunately (e) provides a solution. For practical purposes, these objections do not constitute anymore serious obstacles to bespoke decision-aid. As probability-entailing questions are most often considered to be the source of biases in elicitation, we shall insist on the possibility to do the job without having recourse to any of them—except in a restricted sense at one given stage. We take up these issues in the order above, and hence start with utility elicitation. 2. Eliciting utility functions without any probability In Section 1.1, the reader will have noticed that we made—for the sake of exposition—‘hidden assumptions’ as to how to obtain certainty equivalents to some prospects (Fig. 1) and that we didn’t discuss the method used to elicit a utility function (Fig. 2). Indeed, the method used there makes use of questions based on changes in probabilities discretionarily set by the analyst. Two main objections can be raised against this type of protocols: (a) It is far from certain that every DM follows an EU rule, as was implicitly assumed in computing certainty equivalents reported on Fig. 2; and (b) In some cases, subjects do not even understand what ‘probability’ means, even when they are business managers (March & Shapira, 1987). Replacing probability by ‘chances in hundred cases’ (Gigerenzer & Hoffrage, 1995) helps, but can be far from sufficient in all cases. Fortunately, we need not resort to such concepts or expressions to encode univariate utility functions (the latter being ‘partial utility’ functions in the case of multi-objectives decision making). What we need instead is to discover some event (and his complementary event), which can be described in terms meaningful to the DM, in view of his personal experience (based on his or her profession or former profession or sport practiced, club membership, intellectual interests, etc.). The event should be as much as possible—the ideal is: entirely—out of the hands of the DM (to avoid biases due to voluntary risk-taking). For example, it may be ‘the proportion of dropouts in general education—or in a given curriculum—after a (given) number of years’ if somebody has long been interested in education or it may refer to ‘the average chance of being detected by radar when speeding on the roads of country X’ if somebody is familiar (in one way or the other) with speed on roads reports, etc. Whether the individual evaluates ‘correctly’ in terms of frequencies the event finally selected is secondary, as long as the event is (i) given and (ii) not considered as either almost impossible or almost certain (to avoid several biases). Call $$E$$ the event selected, and E its complementary event. We subject then the DM to the comparison of two prospects, each one yielding to him something he or she is interesting in having more of. Money is the simplest and the safest choice in this respect, except for special cases. The analyst can then choose two discretionary numbers, say R and r, with r$$<$$R, used as illustrated on Fig. 6. We are giving here a basic exposition of the Wakker & Deneffe (1996) seminal contribution. The upper branch on the left as well as on the right side stands for ‘Event $$E$$ obtains’, the lower branch for ‘Event E obtains’9, x$$^{0}$$ is the lower bound (here: amount of money) of the argument of the utility function u(x) we seek to elicit, R and r are fixed quantities as mentioned. The aim is to determine the amount x$$^{1\ast}$$, which will make the left bet equivalent to the right one. As r$$<$$R by construction, x$$^{1\ast }>$$x$$^{0}$$, whatever the probability of $$E$$ implicitly evaluated by the DM. We do not determine x$$^{1\ast}$$ as an answer to a direct question, which would entail evaluation biases, as already mentioned. We rather proceed by successive comparisons of the two bets: we start with an arbitrary number x$$^{1'}$$, if the left bet is preferred, we increase $$x^{1'}$$ and conversely, etc. until we find two values between which the preference is reversed. If these two values are sufficiently close to each other as to satisfy the degree of precision sought, we arbitrarily take the middle of their interval as being a satisfactory proxy for x$$^{1\ast}$$. Once we have x$$^{1\ast}$$, we substitute it for the initial value x$$^{0}$$ and substitute an x$$^{2}$$ unknown value for x$$^{1}$$ on a figure similar to Fig. 6. We determine x$$^{2\ast}$$ in the same way as done for x$$^{1\ast}$$, etc. We finally obtain a sequence x$$^{0}$$, x$$^{1\ast}$$, x$$^{2\ast}$$, $$\ldots$$, x$$^{n\ast}$$$$\ldots$$, called by convention a normalized sequence. We stop when x$$^{n\ast}$$ is equal to (or above) the end value we want for the argument (amount of money) of the utility function, say x$$^{\ast \ast}$$. More sophisticated stopping rules have since then been developed, but this one is sufficient for most practical cases. We then set u(x$$^{0}$$) = 0 and u(x$$^{n\ast}$$) = 1, a normalization which von Neumann Morgenstern utility theory allows choosing (see above). Fig. 6. View largeDownload slide The ‘Tradeoff’ questioning (adapted from Wakker & Deneffe, 1996). Fig. 6. View largeDownload slide The ‘Tradeoff’ questioning (adapted from Wakker & Deneffe, 1996). Assume our DM is of the EU maximizing type. Then, whatever i*, we may write: p(E)u(xi∗)+[1−p(E)]u(R)=p(E)u(x(i+1)∗)+[1−p(E)]u(r)which can be rewritten as:u(x(i+1)∗)−u(xi∗)=1−p(E)p(E)[u(R)−u(r)] (5) Whatever event $$E$$ and whatever its probability $$p(E)$$ as perceived by the DM, the second member of equation (5) is a constant: In any normalized sequence, utility intervals between two successive terms of the sequence are of equal length. This result is all the more important, because it is robust with respect to a large number of rationality rules beyond the EU rule, in particular with respect to a number of non-EU rules, among which the rank dependent utility rule. Indeed, $$p(E)$$ may be ‘transformed’ in a number of ways by the DM, but that will not change the fact that the right side of equation (5) is some constant (as utility is normalized, the absolute values on the ordinate of Fig. 7 are here irrelevant). Therefore, the terms of the normalized sequence can be associated to equally distant points of the utility image between 0 and 1. This method of encoding a utility function does not mention the concept of probability, does not presuppose any mathematical specification of the utility function (it is a non-parametric elicitation procedure) and—above all—does not presuppose any specific rationality rule followed by the DM. It represents a huge progress with respect to what was done in Section 1.1 above10 and allows discarding traditional objections to utility encoding, as recalled above. The graph of the utility function can then be drawn as on Fig. 7 (assuming that $$n* = 5$$, for example’s sake). Fig. 7. View largeDownload slide Graph of the utility function, using the trade-off method. Fig. 7. View largeDownload slide Graph of the utility function, using the trade-off method. We may then look for a mathematical expression of the utility function, if one is needed by our problem, and correspondingly fit some analytical specification on the points obtained. Going the other way around—as seems these days to be often done in pure research—only can add some specification error. 3. Eliciting probability transformation functions with minimal biases It would be foolish, not to make use of such a precise encoding of the utility function. A contribution designed at GRID laboratory (Abdellaoui, 2000) allows deriving a procedure to elicit the probability transformation function of a DM. Referring to the example and with the same notation as on Fig. 7 above, let us compare prospects: (x1∗;1) and (x0,(1−p1);xn∗,p1) (6) This comparison amounts, under rank dependent rationality rule, to comparing $$\theta (1)*u(x^{1})$$ to $$\theta (p^{1})*u(x^{n\ast})$$. By closing in, once again, we look for the value $$p^{1\ast}$$ of $$p^{1}$$ for which the indifference holds in the eyes of the DM between both prospects. Such an indifference implies, given the chosen normalization of utility, $$\theta (p^{1})=u(x^{1})=0.2$$. Hence, in differences of the (6) type allow eliciting the probability transformation function (Fig. 8). We may set $$p^{1\ast}= \theta^{-1}(p^{1})$$. For such last-rank low payments as $$x^{1\ast}$$, a pessimistic DM will tend to assess a larger RDU weight than the corresponding frequency of the poor outcome. However, an optimistic DM might also assess a larger RDU weight than the corresponding probability, but this time in the case of first-rank high payments like $$x^{n\ast}$$. In frequent cases, though not always (Balcombe & Fraser, 2015) the transformation function may look like on Fig. 8, on the abscissa of which first-rank outcomes probability intervals (e.g. as would be p$$^{n\ast}$$) are to the left, last-rank outcomes probability intervals (like the p$$^{1\ast}$$ shown by the arrow on Fig. 8) to the right. Fig. 8. View largeDownload slide Example of a probability transformation function. Fig. 8. View largeDownload slide Example of a probability transformation function. In eliciting this probability transformation function, we are asking about comparisons entailing one probability within each comparison. It may thus appear that this stage in preference elicitation cannot dispense with a series of probability entailing questions, thus restraining the significance of our claim in this article. But two observations seriously counterbalance this remark: - Far from having to explore an entire probability distribution, we ask here only 3–5 questions in most practical cases—indeed their number can be reduced to 2 or 3. - The comparisons required bear on outcomes already evaluated by the DM in the utility elicitation phase, which makes them way more accessible to any DM than comparisons between non-familiar outcomes subject to probabilities taking many different values. Yet, a source of potential error comes from the fact that we use assessments of the utility function (equal to the weights implicitly used by the DM) as bases of comparison. Potential errors in the former assessment might therefore propagate in the latter. However, as we have seen, the trade-off method, when correctly practiced, yields quite robust results and one can see this as assuring little error in the elicitation. 4. Extensions to multi-objective utility functions In a classic article, Miyamoto & Wakker (1996) have shown that the axioms used by Keeney & Raiffa (1976) in an EU framework can be straightforwardly used in most non-EU environments. We therefore feel safe in looking for a way to implement decision-aid with all sorts of DM, some of them potentially using an RDU rationality rule. But little has specifically been said regarding the way to avoid probability biases in the elicitation of DM’s preferences in the multi-objective-RDU case. In raising this question, we have to keep in mind that DM’s preferences, in MAUT, do not only regard utility and probability transformation functions, but also contain as decisive components of these preferences the scaling constants. We skip here the elicitation of partial utility functions, which has been shown in section 2 above. We show here that scaling constants can also be elicited without recourse to any probability entailing questions, though this is not the beaten path used to elicit them. 4.1. Deriving scaling constants without any probability-entailing question While we already know the answer to the question raised in this paragraph when we deal with the limited case of two attributes, the general case is not as simple. In the two-attribute case—ignoring the additive specification form of the MAU function, which is trivial—we have only three scaling constants to elicit. Finding three independent first order linear equations yielding one single solution can solve therefore the question. But one such equation pre-exists in fact as an identity: either yielding K from k$$^{1}$$ and k$$^{2}$$ in the multiplicative case, or yielding k$$^{3}$$ from the (differently valued in the multi-linear form) coefficients k$$^{1}$$ and k$$^{2}$$ in the multi-linear case (for the specification of these respective cases, see 1.4. above). We know, indeed, from Keeney & Raiffa (1976) that, in the multiplicative two-attribute case, we have11 k1+k2+Kk1k2+(K)2k1k2≡1 and in the multi-linear one, the even simpler first order linear equation: k1+k2+k12≡1 For the two other independent equations, we use the definitions of $$k^{1}$$ and $$k^{2}$$, denoting by U the MAU function: k1=U(x1∗,x2∘) and k2=U(x1∘,x2∗) We start with the ordering question between both expressions. Assume $$k^{1}>k^{2}$$. Then, we ask for the following comparisons (consider the multi-linear case, the multiplicative one is similar): (1) $$U(x_1^+,x_2^\circ) \leq or \geq U(x_1^\circ,x_2^*)$$?, for various values of $$x_{1}^{+}$$. By closing in again, we stop when we find $$x_{1}^{\prime}$$ such that equality prevails. The equality can be rewritten as $$k^{1} u_{1 }(x_{1}^{\prime})+0 = k^{2}$$. As we know $$u_{1}(\bullet)$$ from the elicitation of partial utility functions, we can compute $$u_{1 }(x_{1}^{\prime})$$ and consider it as a given constant. We then have our first equation. (2) $$U(x_1^*,x_2^\circ) \leq or \geq U(x_1^{++},x_2^*)$$? Let us call $$x_{1}^{\prime\prime}$$ the value ($$x_{1}^{\prime\prime}$$) assuring equality. By the same application of the definitions, we get our second independent equation as: k1[1−u1(x1′′)]=k2+k12u1(x1′′) The two-attribute case is thus easy to solve. Yet, we cannot directly extend the method to the general case. Consider, for example, the three-attribute case under the usual multiplicative form. We have four coefficients to estimate. The functional can be written: KU(x1,x2,x3)+1=∏i=1i=3[Kkiui(xi)+1] One could try proceeding in a similar way as in the two-attribute case above and write three independent equations from three in differences derived in the same way as just above for respectively x$$_{1}^{+}$$, x$$_{1}^{++}$$, x$$_{2}^{+}$$,such that: (A) k1u1(x1+)=k2;(B) k1u1(x1++)=k3; and (C) k2u2(x2+)=k3 However, one of these expressions fails to be independent from the two others. Fortunately, to solve the difficulty, we can proceed as follows. We keep ($$A$$) and ($$C$$) as above but replace ($$B$$) by ($$B$$’) below, in the following way. Consider first the following comparison: U(x10,x20,x3a) versus U(x10,x2,x3b) where the fixed values of $$x_{3}^{a}$$ and $$x_{3}^{b}$$ are—as much as possible—focal values to the DM. For example, if $$x_{3}$$ is energy consumption, these values may represent two different thresholds of energy cost (given a discrete tariff scheme) and/or two different thresholds in environmental protection regulation, which the DM knows well and which mean something clear to him/her. Then, we vary $$x_{2}$$and, by ‘closing in’ again, we try to find the value of $$x_{2,}$$ which will insure equality of the two terms compared. Denote that value by $$x_{2}^{1}$$. We can then write: U(x10,x20,x3a) =U(x10,x21,x3b)⇔[Kk3u3(x3a)+1] =[Kk2u2(x21)+1]⋅[Kk3u3(x3b)+1] (7) Set $$u_2(x_2^1)=\psi; u_3(x_3^a)=\chi^a; u_3(x_3^b)=\chi^b$$ all constants, the numerical values of which we know how to compute, having estimated partial utilities above. One can then derive the third independent equation sought for: (B′) K=χa−χbk2ψχb−1k3χb (8) This equation yields a unique solution for K when k$$^{3}$$ is given, and conversely. We classically can then derive K from the second order equation (9) and select then the appropriate value from the two solutions offered (Keeney & Raiffa, 1976), checking the meaning of it with the DM. K2(k1k2k3)+K(k1k2+k2k3+k2k3)+(k1+k2+k3−1)=0 (9) We also can substitute (A), (C) and (B’) in this last equation. We get a second order equation in $$k^{3}$$ and can check with the DM the appropriate solution to retain. There is a powerful argument here to select the latter strategy: it is easier to associate the DM to the choice of the relevant solution for $$k^{3}$$ than for K, the meaning of the latter being more difficult to explain to a DM than the one of $$k^{3}$$. To the best of our knowledge, no one did ever care to develop and examine the value of this way of solving the issue of scaling constants’ elicitation12. The solution developed here for the determination of the scaling constants does not require any probability entailing question and can be applied to the general case, with n attributes. In this general case, however, the multi-linear form is avoided for practical reasons: the number of coefficients to be estimated becomes quickly too large and too complex to elicit: There are cognitive limits of the DM—not to mention the costs of computations—we should be careful with. 4.2. The difficulty of ‘bottom-up’ strategy in applying an RDU-MAUT model At this point, advising the DM as to what investment to choose seems to be an easy task: We maximize either EU or RDU—according to the rationality rule, the partial utility functions, the probability transformation function(s) and finally the scaling constants, all elicited from the DM with almost no probability-entailing question, remember we want a bespoke decision-aid. However, two difficulties appear, which make this optimization rather hazardous. (1) The problem is one in multiple-integral optimization. This can become very complex, even in the special and simpler case of EU, unless there is stochastic independence between the different attributes probability distributions. In the latter case, we can write our optimization problem as: max(x~1,x~2,x~3)EU(x~1,x~2,x~3)=∏i=1i=3[Kk1Eui(x~1)+1] which amounts to maximizing a weighted sum of one-variable integrals, a much more easily tractable problem. But the hypothesis of stochastic independence is not always relevant, even accepting some approximation. This is well-known, although more than often ignored in practical applications. (2) Much less known and dealt with is the fact that, when eliciting partial utility probability transformation functions, one usually gets different probability transformation functions for the different attributes (Beaudouin et al., 1999). This raises a much deeper problem, way more difficult to overcome. Because it is not clear, contrary to what was tentatively conjectured in the last cited article, that this empirical fact could be compatible with the utility independence hypothesis, making MAUT methodology altogether inadequate—too restrictive—to handle the problem. 4.3. Overcoming the difficulty: a ‘top-down’ strategy in applying an RDU-MAUT model One can make use of an alternative way of doing things, avoiding at the same time both difficulties mentioned in Subsection 4.2. The idea is to use the same elicitation method as in Section 2 above, but within the (easy to meet) constraint that we shall use the same number of steps and the same ranking for every partial utility function13. We get therefore ‘standard sequences’ containing the same number of elements, say $$n^*+1$$.We then encode scaling constants as done in Section 4.1 above. And we proceed as follows: We use the same step for every attribute, say step $$s_i$$ (s $$=1, 2, , n^*+1$$). We may encode the probability transformation function bearing on the global MAU function, using the same methodology as in Section 3 above. For example, for step i, we may, using closing in comparisons, elicit $$p^{i'}$$, s.t.: [(x1∘,x2∘,x3∘),(1−pi′);(x1∗,x2∗,x3∗),pi′)]≈[(x1i,x2i,x3i),1]which yields: θ(1)∗U(x1i,x2i,x3i)=θ(pi′)U(x1∗,x2∗,x3∗)+[1−θ(pi′)U(x10,x20,x30)] and finally: θ(pi′)=U(x1i,x2i,x3i) As we know $$u_{1 }(x_{1}), u_{2 }(x_{2}), u_{3 }(x_{3}),$$ as well as the scaling constants (see above), we can compute $$U(x_{1}^{i},x_{2}^{i},x_{3}^{i})$$ and hence $$\theta (p^{i'})$$. As we know $$p^{i'}$$ from the DM’s answers,we have one point of the probability transformation function $$\theta (\bullet)$$. We thus can encode non-parametrically, point by point, the probability transformation function relating to the MAU function directly. This procedure avoids at the same time both difficulties noted in Section 4.2 above. The same remark is in order here as in the end of Section 3 above, with the same important attenuation applying again. It must however be admitted that, unless we disregard the fact that the DM might have different probability transformation functions for the different attributes (a case which we documented as frequent), we do not have, as of now, any feasible alternative solution to the top-down strategy we just described, if we want to maintain the discipline of using the DM’s own preferences to help him (her) introducing consistency into his (her) business choices. 5. Concluding remarks and applications Decision makers are often eager to benefit from decision - aid, but they are tired with recommendations produced by what could be termed standard packages, while they are deeply convinced of the distinctive features of their organization and of its situation in their market in the given context. They would like to see their preferences, based on their experience, qualify the recommendations to receive from decision-aid—which does not mean hearing recommended what they already had in mind, of course. But they do not know how to do it, they do not know how to ask for it. Indeed, they often do not even think that it could be done. Rank Dependent-Multi-Attribute Utility (RDU-MAU) is the most general version at hand of decision analysis and it can translate both risk aversion—in the sense where everyone dislikes seeing his outcomes unpredictably fluctuate up and down—as well as the personal appraisal of the DM due to the particular macro - and micro-context of his or her market. In financial investment choice, there is little reason to consider the so-called historical probabilities as intangible, according to the change in macro-economic context, for example. The problem is hence to use RDU-MAU in a way which makes decision-aid a bespoke help to the Decision Maker. This article shows that there are ways to reach that goal. Decision-aid is not necessarily prescriptive, although it is thought of as prescriptive in most cases14. For example, helping wealth managers deliver a ‘suitable’ investment recommendation to investors—as is required by either prudential rules and / or by sincere compliance with regulations like MIFID II15—means eliciting the ‘risk profile’ of the investor and building on it an adapted portfolio. What ‘suitability’ simply means is that the portfolio has to be a bespoke recommendation, resting on the effective risk profile of the investor as revealed by the investor, not as intuitively seen—or interpreted from conversations or vague questionnaires—by the clientele officer or, as is more frequently the case, dictated by investment managers. Appropriate software pieces derived from the methodologies developed here can do way quicker, better—and on firmer scientific grounds16—than the ad hoc at best little informative questionnaires used by still too many wealth managers, whatever the institution they are in. Similarly, in some general managerial problems, decision-aid can also be the main descriptive information provided, as in hiring problems, for example in several industries. For example, nuclear plants may want to avoid recruiting aggressively risk loving employees. Casinos houses may have less stringent but not very different requirements. Lottery organizations may as well want to avoid certain kinds of preferences toward risk or, in a more sophisticated move, may want to know the personal characteristics and motivations of their clients—a direct marketing behaviour. Still, most industrial decision-aid situations require some prescriptive usage of the model: what investment should be selected from the opportunities offered? What strategy should be preferable?17 etc. But CEOs would value that, while the needed data or estimates be provided by experts—from within and / or from the outside of their organization—the processing of these data in view of reaching a decision could integrate as well, in an operational and possibly quantitative way, their own experience, judgments and expectations as to the near future. They know the specific character of their organization—notably its risk policy and attitude toward risk and uncertainty. They do not want to learn how external specialists, just landed a couple of days ago for some short term mission, see them. Everyone should use the best of its competencies, often derived from experience at least as much as from sophisticated knowledge. Embedding at least some of the experience of the DM into decision-aid is indeed a good idea. Alas, this still too rarely happens. There is a deep need for bespoke decision-aiding. References Abdellaoui, M., (2000) Parameter-free elicitation of utility and probability weighting functions. Manag. Sci., 46 , 1497 – 1512 . Google Scholar CrossRef Search ADS Abdellaoui, M. & Munier, B. (1994) The closing in method: an experimental tool to investigate individual choice patterns under risk. Models and Experiments in Risk and Uncertainty ( Munier B. & Machina, M. J. eds). Boston: Dordrecht, Kluwer, pp. 141 – 155 . Google Scholar CrossRef Search ADS Abdellaoui, M. and Munier, B. (1998) The risk-structure dependence effect: experimenting with an Eye to decision-aiding. Ann. Oper. Res., 80 , 237 – 252 . Google Scholar CrossRef Search ADS Allais, M. (1953) Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’école américaine. Econometrica, 21 , 503 – 546 . Google Scholar CrossRef Search ADS Allais, M. & Hagen, O. (1979) The Expected Utility Hypotheses and the So-Called Allais’ Paradox . Baker, R. (2010) Risk aversion in maintenance : a utility-based approach. IMA J. Manag. Math., Dordrecht: Reidel Publishing Company 21 , 319 – 332 . Google Scholar CrossRef Search ADS Balcombe, K. & Fraser, I. (2015) Parametric preference functionals under risk in the gain domain: a Bayesian analysis. J. Risk Uncertain., 50 , 161 – 187 . Google Scholar CrossRef Search ADS Bernoulli, D. (1738) Specimen Theoriae Novae de Mensura Sortis. In: Commentarii Academiae Scientiarum Imperialis Petropolitanae. (Proceedings of the Imperial Academy of the Sciences), vol. 5 , pp. 175 – 192 , Saint Petersburg: Academia Scientiarum. (English translation by L. Sommer.) Exposition of a New Theory of Risk. Econometrica, 22 , pp. 23 – 36 . De Finetti, B. (1952) Sulla Preferibilitá. Giornale Degli Economisti e Annali di Economia, 11 , 685 – 709 . Gigerenzer, G. & Hoffrage, U. (1995) How to improve bayesian reasoning without instructions: frequency formats. Psychol. Rev., 102 , 684 – 704 . Google Scholar CrossRef Search ADS Hey, J. & Orme, C. H. (1994) Investigating generalizations of expected utility theory using experimental data. Econometrica, 62 , 1291 – 1326 . Google Scholar CrossRef Search ADS Kahneman, D. & Tversky, A. (1979) Prospect theory: an analysis of decision under risk. Econometrica, 47 , 263 – 291 . Google Scholar CrossRef Search ADS Keeney, R. & Raiffa, H. (1976) Decisions with Multiple Objectives. New York: Wiley . Machina, M. J. (1982) Expected utility analysis without the independence axiom. Econometrica, 50 , 277 – 323 . Google Scholar CrossRef Search ADS Munier, B. (1989) New models of decision under uncertainty. Eur. J. Oper. Res., 38 , 307 – 317 . Google Scholar CrossRef Search ADS Munier, B. (1996) Comments on Peter Hammond’s ‘consequentialism, rationality and game theory. The Rational Foundations of Economic Behaviour, IEA Proceedings ( Arrow, K. J. Colombatto, E. Perlman M. eds). London: MacMillan, pp. 43 – 47 . Munier, B. & Tapiero, C. H. (2008) Risk attitude. Encyclopedia of Quantitative Risk Assessment and Analysis ( Melnick, E. L. and Everitt, B. S. ed.). vol. 4 . New York: Wiley, pp. 1512 – 1524 . Miyamoto, J.M. & Wakker, P. (1996) Multi-attribute utility theory without expected utility foundations. Oper. Res., 44 , 313 — 326 . Google Scholar CrossRef Search ADS Pratt, J. (1964) Risk aversion in the small and in the large. Econometrica, 32 , 122 – 136 . Google Scholar CrossRef Search ADS Quiggin, J. (1982) A theory of anticipated utility. J. Econ. Behav. Org., 3 , 323 — 343 . Google Scholar CrossRef Search ADS Quiggin, J. (1993) Generalized Expected Utility, The Rank-Dependent Model. Boston: Kluwer Acad. Publ . Google Scholar CrossRef Search ADS Savage, L. J. (1954) The Foundations of Statistics . New York: Wiley . Shackle, G. L. S. (1961) Decision Order and Time in Human Affairs, 2nd edn. Cambridge, UK: Cambridge University Press, 1969 . Schlaifer, R. (1959) Statistics for Business Decisions . New York: Wiley . Von Neumann, J. & Morgenstern, O. (1944) Theory of Games and Economic Behavior, 3rd edn. Princeton: Princeton University Press, 1953 . Wakker, P. & Deneffe, D. (1996) Eliciting von Neumann-Morgenstern utilities when probabilities are distorted or unknown. Manag. Sci., 42 , 113 – 150 . Google Scholar CrossRef Search ADS Zhang, A. (2012) The terminal real wealth optimization problem with index bonds: equivalence of real and nominal portfolio choices for the constant relative risk aversion utility. IMA J. Manag. Math., 23 , 29 – 39 . Google Scholar CrossRef Search ADS Footnotes 1Notations: Prospects are denoted as special vectors encompassing a number $$n$$ of potential outcomes $$x_{i}$$, each one immediately associated to its probability $$p_{i}$$. Note that outcomes are conventionally ranked from the lowest, $$x_{1}$$ to the highest, $$x_{n}$$. In the special case where only one outcome is to be envisioned ($$n=1$$), one has a probability equal to 1 and one speaks then of a “degenerate” lottery or prospect. 2Notations: We use further down exponents to characterize specific values of some variable or some parameter, while we use indices to distinguish between different variables and/or functions. Thus, L$$^{2}$$ is a specific value of the prospect (or lottery) L = (0, 1-p; x, p). U($$\bullet$$) will stand for the utility of a prospect, while u($$\bullet$$) stands for the restriction of U to “degenerate lotteries” (certain outcomes). As is known, von Neumann-Morgenstern’s theorem states that $$U(L)=\Sigma_ip_i\ast u(x_i)$$. 3 Due to the specific set of axioms chosen by von Neumann and Morgenstern (see fn. 5 for a compacted version of these axioms), which allow defining utility functions uniquely up to an affine positive transform. Think, for an example in Physics, of the Fahrenheit or Celsius scales in temperature measurement! Note also that the specific normalization referred to here makes the image of u(x) and of E[u(x)] equal to the probability of receiving the maximal outcome on Figs 1 and 2. 4 Under this direct form, questions would represent evaluation tasks, to which our brain is little adapted, and would be flawed by biases and little robustness. One therefore should ask these questions as comparisons between varying sums, “closing in” on the X$$_{1}^{\ast}$$, X$$_{2}^{\ast}$$, X$$_{3}^{\ast}$$ values. This fact is well documented by many psychologists. Examples of such procedures are numerous. Abdellaoui & Munier (1994) provide an example. 5von Neumann & Morgenstern (1944) unearthed the early work by Bernoulli (1738) but gave to it different foundations by building on a set of axioms. The axioms in the original work of von Neumann & Morgenstern (1944) are complicated. They boil down to 1) Existence of a pre-order on the set of prospects; 2) Lottery (or prospects) composition: Given $$L^1, L^2\in \boldsymbol{L}, \alpha \in [0,{\bf 1}], (\alpha L^1+\alpha L^2)\ \in L$$, where L is the set of prospects already defined; 3) The specific continuity condition known as the Archimedean property: $$\forall L^1\preceq L^2\preceq L^3 \in \boldsymbol{L}, \exists\alpha \in [0,{\bf 1}]: L^{\bf 2}\sim\alpha L^3+(1-\alpha)L^1 $$ and finally the much debated independence axiom:$$L^1 \preceq L^2, L^3\neq L^1, L^2 \Rightarrow \alpha L^3+(1-\alpha)L^1\preceq \alpha L^3+(1-\alpha)L^2$$. This last axiom makes computations easy, but its behavioural foundations can indeed be challenged. 6 It was in particular the case of “Prospect Theory” by Kahneman & Tversky (1979). Later Tversky & Kahneman (1992) had the idea to draft RDU into their own former theory, which, adding on the way a couple of additional hypotheses, some of them previously suggested by Allais (1953), led to the celebrated “Cumulative Prospect Theory”. The seminal papers remain Quiggin (1982) and Allais (1988) himself – a very striking case of Mertonian discovery, witnessed by the author of the present paper in 1986 at a conference he had organized in Aix-en-Provence, France. These two authors should receive credit for the new theory of risk (Fig. 5(c) below). 7 In the particular case where this function is the identity function, we are back to EU. Hence, RDU encompasses EU as a particular case. 8 Keeney and Raiffa’s “Utility independence” hypothesis states that, if the values taken by (n-1) attributes change, the certainty equivalent - to the DM - of the probability distribution of the remaining attribute doesn’t change, i.e. preferences on the prospect represented by the latter attribute are independent of the specific values taken by the (n-1) others. This is in general quite weaker than saying that the one preferred value of the remaining attribute doesn’t change – which is called “preferential independence”. “Utility independence” is yet sufficient to establish Keeney and Raiffa’s generic results. 9 One can as well state the probabilities of event $$E$$ and E, as some fixed p and (1-p)for example. But it is not necessary. 10 Some more recent elicitation methods – seen as improvements by researchers - do not, in our opinion, represent any advantage for business applications, especially when they assume beforehand a given mathematical specification of the functions involved, whatever the case. 11 This identity is a second-order equation and therefore yields two solutions. But one can be simply discarded (see below). 12Section 4.1 builds on an unpublished research paper of ours, leading to equation (8). 13 We build here on part of a joint unpublished mimeo at GRID with N. Makhoul in the mid-2000. 14 In finance, in particular, a huge literature has developed since the Sixties in portfolio theory. Risk aversion of the DM matters a lot for the results, as shown in Zhang (2012) (this Journal, 2010). 15 The Markets in Financial Instruments Directive II has been adopted in 2014, after many years of discussions with professional organisations, to both insure transparency of operations and protect the retail individual investors. This Art. 25 of the Directive makes compulsory for all kinds of wealth managers to determine the risk profile, meaning in particular the risk appetite or aversion of every individual investor they advise. (It will be enforced after January 1st, 2017). 16 A well-designed software can reveal within an average of 10 minutes, adjusting in an interactive way to each particular investor a bespoke questionnaire, not only the appropriate curves, but also a generalized index of subjective risk aversion of the said investor and an index of his objective capacity to bear the risk, together with associated recommendations as to how to conceive of an appropriate distribution of portfolio’s returns for a suitable recommendation. 17Baker (2010) has developed an interesting example in industrial maintenance, although with discretionarily selected utility functions. In real world applications, similar models should take advantage of the elicitation techniques described in the present paper. © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

IMA Journal of Management Mathematics – Oxford University Press

**Published: ** Oct 6, 2016

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