## Abstract

In this paper, I examine the decision-theoretic status of risk attitudes. I start by providing evidence showing that the risk attitude concepts do not play a major role in the axiomatic analysis of the classic models of decision-making under risk. This can be interpreted as reflecting the neutrality of these models between the possible risk attitudes. My central claim, however, is that such neutrality needs to be qualified and the axiomatic relevance of risk attitudes needs to be re-evaluated accordingly. Specifically, I highlight the importance of the *conditional variation* and the *strengthening* of risk attitudes, and I explain why they establish the axiomatic significance of the risk attitude concepts. I also present several questions for future research regarding the strengthening of risk attitudes.

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## Notes

For the authoritative exposition of the representational theory of measurement, see the

*Foundations of Measurement*trilogy (starting with Krantz et al. 1971).For simplicity, in this paper,

*C*will always be taken in \({\mathbb {R^+}}\). This is to set aside a specific set of issues, namely, the possible asymmetries between how decision-makers consider gains, and how they consider losses (making the conventional assumption that the former should be mapped onto \({\mathbb {R^+}}\), and the latter onto \({\mathbb {R^-}}\)). Such asymmetries have been discussed at length with respect to the risk attitudes (see in particular the so-called fourfold pattern of risk attitudes emphasized in, e.g.,Tversky and Kahneman 1992, p. 306).For any lottery

*P*, let \(F_P : {\mathbb {R}} \rightarrow [0,1]\) be its cumulative distribution function. I will say that*P*dominates*Q*in the sense of first-order stochastic dominance if \(F_P(x) \le F_Q(x)\) for all \(x \in {\mathbb {R}}\), and this inequality is strict for some*x*. The fact that \(\succcurlyeq \) respects first-order stochastic dominance thus defined (in the sense that \(P \succ Q\) whenever*P*dominates*Q*in the sense of first-order stochastic dominance) entails that \(\succcurlyeq \) is strictly increasing in*C*(i.e., for all \(c,c' \in C\), if \(c > c'\), then \(\delta _c \succ \delta _{c'}\)). This is a natural assumption to make if*c*refers to money or, more generally, any continuous good.For a standard proof, see, e.g., Cerreia-Vioglio et al. 2015, Appendix B, Step 1. This result is essentially an application of Debreu’s theorem (Debreu 1964) regarding the existence of a continuous utility function. As I will highlight later, a certainty equivalent function is nothing but a particular utility function representing \(\succcurlyeq \).

A standard exposition of these definitions can be found in Cohen 1995, Sect. 1.

An alternative definition would require that \(\delta _{E(P)} \succ P\), not for some

*P*, but for all*P*.One can also say that

*P*dominates*Q*in the sense of second-order stochastic dominance, applied to lotteries having the same mathematical expectation. In economics, the concept of mean-preserving spread dates back to Rothschild and Stiglitz 1970. The definition in (1), which corresponds to the simplest case possible, compares most directly to the one presented in Rothschild and Stiglitz 1970, I.1 and III.4. While the two definitions are equivalent, the one given here is simpler, and also less restrictive in that it is compatible with*C*being taken in \({\mathbb {R^+}}\), i.e., having no element in \({\mathbb {R^-}}\) (more on this in footnote 2).The so-called principle of reduction of compound lotteries is assumed in (1) and, more generally, throughout my paper. In particular, I will not investigate whether some risk attitude patterns can be related to patterns of violations of the principle of reduction of compound lotteries. Such violations have been systematically explored in the literature (see, e.g., Segal 1990, and more recently Dillenberger and Segal 2015). However, to my knowledge, they have never been explored with reference to the risk attitude concepts introduced in this section. This would be a particularly interesting (and challenging) research topic.

The defining condition on interquantile intervals is as follows. For any

*P*, let \(F^{-1}_P : (0,1) \rightarrow {\mathbb {R}}\) be its (generalized) inverse distribution function. Then, \(P RR_M Q\) if \(P RR_A Q\) and, for all \(p,q \in (0,1)\) such that \(p< q\), \(F^{-1}_P(q) - F^{-1}_P(p) \le F^{-1}_Q(q) - F^{-1}_Q(p)\). One can also say that*Q*is more dispersed than*P*in the sense of Bickel and Lehman, applied to lotteries having the same mathematical expectation. This kind of risk reduction is called “monotonic”, because, making explicit an underlying state space,*P*and*Q*can be related to “co-monotonic” Savagian acts (for a definition, see Schmeidler 1989, p. 575). For more details on this kind of risk reduction, see, e.g., Chateauneuf et al. 1997, p. 29.Unlike “monotonic”, “moderate” fits the prevailing terminology regarding the “weak” and the “strong” attitudes, and it makes transparent the logical links given in (2) above.

The contrast is also with the more specialized concepts of

*relative*risk attitudes (in which case, “relative” means “relative to a given underlying wealth of the decision-maker”).Comparative risk aversion can be given a much more general definition (see Yaari 1969 and, building on this contribution, Bommier et al. 2012, Sect. 3). This more general definition, which is the truly fundamental one, can be introduced even when one cannot introduce certainty equivalent functions or, indeed, the absolute risk attitudes themselves.

Notice, however, that together with the other defining properties of classic preferences, the respect of VNM independence entails that of first-order stochastic dominance. For a proof of the von Neumann–Morgenstern theorem, see, e.g., Fishburn 1970, Chapter 8.

Betweenness is satisfied if for all \(P,Q \in \Delta (C)\), and any \(\alpha \in (0,1]\), \(P \succcurlyeq Q\) if and only if \(P \succcurlyeq \alpha P + (1-\alpha ) Q \succcurlyeq Q\). On the betweenness branch of non-expected utility theory, see Dekel 1986 and Chew 1989. On the disappointment aversion model in particular, see Gul 1991. Next, some notation is needed to introduce the co-monotonic branch of non-expected utility theory. Given a lottery

*P*, denote by \(\{c^{\text {{p}}}_1,\ldots ,c^{\text {{p}}}_n\}\) its support, ordering it (without loss of generality) so that \(\delta _{c^{\text {{p}}}_1}>\cdots ~>\delta _{c^{\text {{p}}}_n}\), and accordingly, let*P*denote \((c^{\text {{p}}}_1,p_1;\ldots ;c^{\text {{p}}}_n,p_n)\). Take \(Q=(c^{\text {{q}}}_1,p_1;\ldots ;c_i,p_i;\ldots ;c^{\text {{q}}}_n,p_n)\), \(R=(c^{\text {{r}}}_1,p_1;\ldots ;c_i,p_i;\ldots ;c^{\text {{r}}}_n,p_n)\), \(Q'=(c^{\text {{q}}}_1,p_1;\ldots ;c'_i,p_i;\ldots ;c^{\text {{q}}}_n,p_n)\), \(R'=(c^{\text {{r}}}_1,p_1;\ldots ;c'_i,p_i;\ldots ; c^{\text {{r}}}_n,p_n)\) (i.e., the common result \(c_i\) is replaced by the common result \(c'_i\) at the same \(i\text {-th}\) preferential rank). The co-monotonic weakening of VNM independence requires that, for all such \(Q,R,Q',R'\), \(Q \succcurlyeq R\) if and only if \(Q' \succcurlyeq R'\). On this property and the rank-dependent utility model, see, e.g., Chateauneuf 1999. For a classic decision model contained in neither the betweenness, nor the co-monotonic branch of non-expected utility theory, see, e.g., the recent model in Cerreia-Vioglio et al. 2015 (which I will discuss in Sect. 4.2).Given how risk neutrality, the certainty equivalent function, and mathematical expectations are defined, one can check that, for all \(P,Q,R \in \Delta (C)\) and any \(\alpha \in (0,1]\), \(P \succcurlyeq Q\) if and only if \(E(P) \ge E(Q)\) if and only if \(E[\alpha P + (1-\alpha ) R] \ge E[\alpha Q + (1-\alpha ) R]\) if and only if \(\alpha P + (1-\alpha ) R \succcurlyeq \alpha Q + (1-\alpha ) R\). In the above formulation of the contrapositive form of this implication, one should interpret “non-expected utility” in the exclusive sense.

For a review of such results, see, for instance, Chateauneuf et al. 1997, Sect. 3.2.

Consider, e.g., the second paradox. Recalling (2), assume by way of contradiction that the decision-maker is weakly risk seeking. Then, by definition of weak risk seeking, \(Q_2 \succcurlyeq \delta _{E(Q_2)}\). From this, the fact that \(E(Q_2)=400\), \(P_2 \succ Q_2\), and transitivity, it follows that \(\delta _{100} \succ \delta _{400}\). This contradicts the fact that \(\succcurlyeq \) is strictly increasing in

*C*.This can be established by algebraic examples. Let

*D*and \(D'\) be two decision-makers to whom the rank-dependent utility model applies. Let them be characterized by the same utility function, \(u(c)=\sqrt{c}\), together with the probability weighting functions \(w_D(p)=1 - \sqrt{1-p}\) and \(w_{D'}(p)=\sqrt{p} / (\sqrt{p} + \sqrt{1-p})^2\), respectively (\(w_D\) comes from Segal 1987, p. 149, while \(w_{D'}\), with a typical inverse-S shaped graph, comes from Tversky and Kahneman 1992, p. 309). The functional form of rank-dependent utility (which I will recall in footnote 34) implies that both*D*and \(D'\) have the paradoxical preferences. However, it can be proved that*D*is strongly risk averse (see Chew et al. 1987, Corollary 2), while \(D'\) is not even weakly so (see Chateauneuf and Cohen 1994, Corollary 1).The above is meant not as a precise historical statement, but as a suggestive rational reconstruction.Seidl (Seidl 2013) reviews the history of the St. Petersburg paradox up to the present.

This is due to the characterization of absolute risk attitude in expected utility (which primarily leads back to Rothschild and Stiglitz 1970). Shortly after von Neumann and Morgenstern’s groundbreaking axiomatization of expected utility (von Neumann and Morgenstern , 1947, Appendix), Friedman and Savage (Friedman and Savage 1948) were among the first to stress the compatibility between expected utility and risk seeking. This compatibility had previously been denied, most prominently by Marshall (see, e.g., Marshall 1890, Mathematical Appendix, Note IX).

To my knowledge, Machina was the first to explicitly introduce conditional certainty equivalents (Machina 1982, p. 288).

In particular, the descriptions of the Allais paradoxes referred to in footnote 21 can be made precise using the concept of conditional certainty equivalent.

See Chew and Epstein 1989 (corrected by Chew et al. 1993). Thus, Machina is justified in calling the independence axiom “the requirement of constant conditional certainty equivalents” (Machina 1982, p. 298). Although it is not fully axiomatized (but see the recent results in Cerreia-Vioglio et al. 2016), the decision model proposed by Machina would also support the present analysis. Its key assumption (Machina 1982, Hypothesis II, p. 300) is about comparative risk attitude.

Kahneman and Tversky (Kahneman and Tversky 1979, p. 267) consider a similar variation, yet without making explicit the link with the absolute risk attitude concepts.

Notice that unlike any of the Allais cases considered hitherto, this particular variation features no certain option (like \(P_1, P_2, or P_3\)).

Recall the algebraic examples in footnote 20.

*D*illustrates that one can fail to respect VNM independence while displaying all the degrees of risk aversion distinguished hitherto.It follows from Jensen’s inequality that, in expected utility, a decision-maker is weakly risk averse if and only if the utility function \(u: C \rightarrow {\mathbb {R}}\), the expectation of which represents her preferences, is concave. Rothschild and Stiglitz (Rothschild and Stiglitz 1970) show the less immediate result that, in expected utility, a decision-maker is strongly risk averse if and only if \(u: C \rightarrow {\mathbb {R}}\) is concave. Then, given (2), all the risk aversion concepts must be equivalent.

Clearly, this rigidity alone cannot characterize expected utility. Consider, e.g., the class of all preference relations respecting the co-monotonic weakening of VNM independence. Building on the results mentioned in footnote 20, one could easily define a subclass of classic preference relations displaying the two following properties. First, all relations respect the axiom in (4). Second, no relation respects VNM independence.

The dual model is due to Yaari (Yaari 1987). Its transformation function concerns not the direct probability values, but the decumulative ones. The dual model is a variant of the more general rank-dependent utility model that generalizes expected utility, as described in Sect. 3. On the absolute risk attitudes in the dual model, see, e.g., Chateauneuf et al. 1997, p. 35. Segal and Spivak (Segal and Spivak 1990) shed light on why the dual model is more flexible than expected utility, despite its being, like expected utility, a one-parameter model.

Recall the notation of footnote 15. Denote by \(G_P\) the decumulative distribution function of lottery

*P*. The axioms of rank-dependent utility are satisfied if and only if there exist two strictly increasing functions \(u : C \rightarrow {\mathbb {R}}\) and \(w : [0,1] \rightarrow [0,1]\), the latter normalized at 0 and at 1, such that one can analyze*v*in (3) as \(v(P)=\sum _{i=1}^{n}\bigl [w(G_P(c_{i+1})) - w(G_P(c_{i}))\bigr ]u(c_i)\). Expected utility corresponds to when \(w(p)=p\), for all \(p \in [0,1]\). The dual model corresponds to when \(u(c)=c\), for all \(c \in C\). For a review on the absolute risk attitudes in rank-dependent utility, see Chateauneuf et al. 1997, and see Ryan 2006 for further results. Unlike the strong risk attitudes ( Chew et al. 1987) and the moderate risk attitudes (Chateauneuf et al. 2005), the weak risk attitudes have not yet been characterized in the rank-dependent utility model. However, the partial results available (Chateauneuf and Cohen 1994—see also Cohen and Meilijson 2014) suffice to establish that any intermediate risk attitude can be accommodated.The cautious expected utility model is due to Cerreia-Vioglio and co-authors (Cerreia-Vioglio et al. 2015). Its key axiom is the following weakening of VNM independence: for any \(c \in C\), all \(P,R \in \Delta (C)\), and any \(\alpha \in (0,1], P \succcurlyeq \delta _c\) if and only if \(\alpha P + (1-\alpha ) R \succcurlyeq \alpha \delta _c + (1-\alpha ) R\).

Notice this implication of the axiom in footnote 35: for all \(P,R \in \Delta (C)\), and any \(\alpha \in (0,1], P \succcurlyeq \delta _{E(P)}\) if and only if \(\alpha P + (1-\alpha ) R \succcurlyeq \alpha \delta _{E(P)} + (1-\alpha ) R\). In the case of classic preferences, the preference on the right-hand side proves equivalent to strong risk seeking (see Chew and Mao 1995, p. 413). Thus, in the cautious expected utility model, weak risk seeking and strong risk seeking are equivalent. However, for a decision-maker to be weakly risk averse in this model, it suffices that one of her utility functions is concave, while strong risk aversion requires that all of them are concave (Cerreia-Vioglio et al. 2015, Theorem 3).

Recently, Dean and Ortoleva (Dean and Ortoleva 2017, p. 386–389) have introduced a new model that is to some extent dual to the cautious expected utility model. In this new model, decision-makers are characterized by one utility function and a set of probability weighting functions. They choose cautiously in the sense above, but with reference to rank-dependent utility, instead of expected utility. The properties of this model as regards the absolute risk attitudes are currently unknown. It would be particularly interesting to investigate them.

Such results would be particularly instructive, because the betweenness and the co-monotonic branches of non-expected utility theory are disjoint, in the sense of having in common only the expected utility model itself (see, e.g., Chew and Epstein 1989, p. 208).

See Chew and Mao 1995 (with slightly different continuity requirements than those assumed here). The key property in this characterization is Schur-concavity (see, e.g., Marshall et al. 2010, Chapter 3), displayed by the function

*v*in (3) when restricted to a distinguished subset of lotteries, namely, the set of all equiprobable lotteries. Apart from the characterization results in Chateauneuf and Lakhnati 2007, I am aware of only one other result established at a similar level of generality, which is to be found in Cerreia-Vioglio et al. 2016, Proposition 2 (the equivalence between (ii) and (iv)). There, it is established that the strong risk aversion of a classic preference relation is equivalent to the weak risk aversion of one of its distinguished subrelations, namely, its largest subrelation respecting VNM independence (together with transitivity and continuity, but not necessarily completeness). The authors of this result do not compare it with that of Chew and Mao.

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## Acknowledgements

For helpful comments or discussions, I am thankful to two reviewers, Mohammed Abdellaoui, Richard Bradley, Michèle Cohen, Mikaël Cozic, Eric Danan, David Dillenberger, Louis Eeckhoudt, Raphaël Giraud, Philippe Mongin, Pietro Ortoleva, and Fanyin Zheng. All errors and omissions are mine. Research for this paper was funded by the Ecole Normale Supérieure-Ulm, the Université Cergy-Pontoise, and the Wagemann Foundation.

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Baccelli, J. Risk attitudes in axiomatic decision theory: a conceptual perspective.
*Theory Decis* **84**, 61–82 (2018). https://doi.org/10.1007/s11238-017-9636-6

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DOI: https://doi.org/10.1007/s11238-017-9636-6