Purely subjective variational preferences

Abstract

Variational preferences (Maccheroni et al. in Econometrica 74:1447–1498, 2006) are an important class of ambiguity averse preferences, compatible with Ellsberg-type phenomena. In this paper, a new foundation for variational preferences is derived in a framework of two stages of purely subjective uncertainty. A similar foundation is obtained for purely subjective maxmin expected utility (Gilboa and Schmeidler in J Math Econ 18:141–153, 1989). By establishing their axiomatic foundations without the use of extraneous probabilities, the conceptual appeal and applicability of these ambiguity models is enhanced.

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Notes

  1. 1.

    All results can be extended to simple acts with \({\mathcal {S}}\) and \({\mathcal {X}}\) arbitrary.

  2. 2.

    There it is called Weak \({\mathcal {F}}_a\) Independence.

  3. 3.

    Translating the notation of that paper: \(\Sigma _a\) amounts to our second-stage events \({\mathcal {E}}_{{\mathcal {T}}}\) and \(\Sigma _b\) to our first-stage events \({\mathcal {E}}_{{\mathcal {S}}}\), \(\Sigma _a\)-measurability is here called first-stage constant, and \(\Sigma _b\)-measurability is here called second-stage constant.

  4. 4.

    Certainly, in Savage’s expected utility, \({\mathcal {S}}\) must be infinite. It is not necessarily uncountable.

  5. 5.

    Anscombe and Aumann (1962) actually assumed three stages of uncertainty and reduced it to two stages. The presentation follows the modern interpretation. Also, the AA framework has also been used to describe the assumption simply that \({\mathcal {X}}\) is convex, without necessarily committing to the lottery interpretation. In this paper, we maintain the interpretation of a set of lotteries.

  6. 6.

    A similar idea forms the basis of conjoint measurement (Krantz et al. 1971).

  7. 7.

    See, for example, Machina and Schmeidler (1992: 774-775).

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Acknowledgments

I am grateful for the comments of an anonymous reviewer. The usual disclaimer applies.

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Correspondence to Craig S. Webb.

Appendix

Appendix

Proof of Theorem 2

The necessity of the axioms follows from substituting the preference functional and elementary manipulations; hence, the details are omitted here. We show the sufficiency of the axioms for the representation. The standard axioms for subjective expected utility hold for induced preferences over second-stage acts. Hence, subjective expected utility holds over second-stage acts, for some utility u and convex-valued probability measure q over \({\mathcal {E}}_{{\mathcal {T}}}\). Let \(U(f_s)=\int _{\mathcal {S_2}}u(f_s(t))dq(t)\) denote the subjective expected utility of a second-stage act. Let \({\mathcal {S}}=\{s_1,\ldots ,s_n\}\). Let \(U(f):=(U(f_{s_1}), \ldots , U(f_{s_n}))\) and \(U({\mathcal {A}}) := \{ U(f)\in {\mathbb {R}}^n{:} f\in {\mathcal {A}}\}\). For \(\alpha \in [0,1]\) and acts \(f,g\in {\mathcal {A}}\), define \(\alpha U(f) + (1-\alpha )U(g)\) pointwise, so that:

$$\begin{aligned} \alpha U(f) + (1-\alpha )U(g) = (\alpha U(f_{s_1}) + (1-\alpha )U(g_{s_1}) , \ldots , \alpha U(f_{s_n}) + (1-\alpha )U(g_{s_n})). \end{aligned}$$

Note that, because q is convex valued, we are assured that \(U({\mathcal {A}})\) is a convex subset \({\mathbb {R}}^n\). By finiteness of \({\mathcal {X}}\), there is a best outcome \({\overline{x}}\) and worst outcome \(\underline{x}\). By eventwise monotonicity, it can be shown that, for all \(f\in {\mathcal {A}}\), \({\overline{x}} \succcurlyeq f \succcurlyeq \underline{x}\). Normalise U so that \(U(\overline{x})=1\) and \(U(\underline{x})=0\). Consider an act \(f\in {\mathcal {A}}\). By second-stage solvability, there is a \(A\in {\mathcal {E}}_{{\mathcal {T}}}\) such that \(f\sim \overline{x}_A \underline{x}\). If \(h= \overline{x}_A \underline{x}\), notice that that \(U(h_s)=q(A)\) for all \(s\in {\mathcal {S}}\). Define utility function for acts \(\phi {:}{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) as follows: \(\phi (U(f)):= \{ U(h_s) : f\sim \overline{x}_A \underline{x}=h \}\) for all \(f\in {\mathcal {A}}\). Preferences \(\succcurlyeq \) over acts \({\mathcal {A}}\) are represented by the map \(f \mapsto \phi (U(f))\).

We now show that \(\phi \) satisfies three properties: monotonicity, mixture continuity and vertical invariance. Call \(\phi \) monotone if \(U(f_i)\geqslant U(g_i)\) for all \(i=1,\ldots ,n\) implies \(\phi (U(f)) \geqslant \phi (U(g))\). Monotonicity of \(\phi \) follows immediately from monotonicity, axiom 2. If, for all acts \(f,g,h\in {\mathcal {A}}\), the sets \(\{ \alpha \in [0,1]: \phi (\alpha U(f)+(1-\alpha ) U(g)) \geqslant \phi (U(h)) \}\) and \(\{ \alpha \in [0,1]{:} \phi (\alpha U(f)+(1-\alpha ) U(g)) \leqslant \phi (U(h)) \}\) are closed in [0, 1], then \(\phi \) is mixture continuous. The derived U is known to be mixture continuousFootnote 7 in that, for all \(f,g,h\in {\mathcal {A}}\), the sets \(\{ \alpha \in [0,1]{:} \alpha U(f)+(1-\alpha ) U(g) \geqslant U(h) \}\) and \(\{ \alpha \in [0,1]{:} \alpha U(f)+(1-\alpha ) U(g) \leqslant U(h) \}\) are closed in [0, 1]. Then, by monotonicity, \(\phi \) inherits mixture continuity.

If, for all acts \(f,g\in {\mathcal {A}}\), first-stage constant act \(h\in {\mathcal {A}}\) and \(\alpha \in [0,1]\), we have \(\phi (\alpha U(f)+(1-\alpha ) U(h)) = \phi (\alpha U(f)) + (1-\alpha )\phi (U(h))\), then \(\phi \) is vertically invariant. An act \(h\in {\mathcal {A}}\) is first-stage constant if, for all \(s,\tilde{s}\in {\mathcal {S}}\), the second-stage acts \(h_s\) and \(h_{\tilde{s}}\) coincide. Consider any act f and first-stage constant act h, \(f,h\in {\mathcal {A}}\) and event A such that \(q(A)=\alpha \), where q is the subjective probability measure obtained above. Using solvability, it can be shown that there exists a first-stage constant act g such that \(f_A h \sim g_A h\). This holds if and only if:

$$\begin{aligned} \phi (U(f_A h))= & {} \phi (\alpha U(f)+(1-\alpha )U(h)) = \phi (U(g_A h)) \\= & {} \phi (\alpha U(g)+(1-\alpha )U(h)). \end{aligned}$$

Because g and h are first-stage constant, \(g_A h\) is first-stage constant and, letting \(\tilde{f}=g_A h\),

$$\begin{aligned} \phi (U(g_A h)) = U(\tilde{f}_s) = \alpha U(g_s)+(1-\alpha )U(h_s) = \alpha \phi (U(g))+(1-\alpha )\phi (U(h)). \end{aligned}$$

By the second-stage sure-thing principle, axiom 6, \(f_A h \sim g_A h\) only if \(f_A \underline{x} \sim g_A \underline{x} \), which holds if and only if (recall \(U(\underline{x})=0\)):

$$\begin{aligned} \phi (U(f_A \underline{x})) = \phi (\alpha U(f)) = \phi (U(g_A \underline{x})) = \phi (\alpha U(g))= \alpha \phi (U(g)). \end{aligned}$$

Therefore, collecting the above results, \(\phi (\alpha U(f)+(1-\alpha ) U(h)) = \phi (\alpha U(f)) + (1-\alpha )\phi (U(h))\); hence, \(\phi \) is vertically invariant. This completes the proof of Theorem 1. \(\square \)

Proof of Theorem 5

Axioms 1-6 have been shown, in Theorem 2, to be equivalent to preferences admitting an invariant second-stage expected utility representation. If, for all acts \(f,g\in {\mathcal {A}}\) and \(\alpha \in [0,1]\) we have \(\phi (\alpha U(f)+(1-\alpha ) U(g)) \geqslant \alpha \phi (U(f)) + (1-\alpha )\phi (U(g))\), then \(\phi \) is concave. We show that second-stage uncertainty aversion is equivalent to concavity of \(\phi \).

Take any acts \(f,g\in {\mathcal {A}}\) with \(f\sim g\), so that \(\phi (U(f))=\phi (U(g))\). By second-stage solvability, there exists A(s) such that \(f_s \sim (\overline{x}_{A(s)}\underline{x})_s\) and there exists C(s) such that \(f_s \sim (\overline{x}_{C(s)}\underline{x})_s\), for all \(s\in {\mathcal {S}}\). This holds if and only if \(U(f_s)=q(A(s))\) and \(U(g_s)=q(C(s))\) for all \(s\in {\mathcal {S}}\). Because q is convex valued, we can find, for all \(s\in {\mathcal {S}}\), second-stage events B(s) such that \(q(B(s)) = \frac{1}{2} q(A(s)) + \frac{1}{2}q(C(s))\). Notice that, for all \(s\in {\mathcal {S}}\), B(s) is a second-stage event average of A(s) and C(s). Then, an act h with \(h_s = (\overline{x}_{B(s)}\underline{x})_s\) for all \(s\in {\mathcal {S}}\) is a second-stage act average of f and g with \(U(h) = \frac{1}{2} U(f) + \frac{1}{2} U(g)\). By second-stage uncertainty aversion, axiom 7, \(h\succcurlyeq f\), which holds if and only if \(\phi (U(h))= \phi \big (\frac{1}{2} U(f) + \frac{1}{2}U(g)\big ) \geqslant \phi (U(f))\). Let \({\mathcal {D}}\) be the set of dyadic rationals. Applying the above finitely many times, it can be shown that, for all \(\alpha \in [0,1]\cap {\mathcal {D}}\), \(\phi (U(f))=\phi (U(g))\) implies \(\phi (\alpha U(f)+(1-\alpha ) U(g)) \geqslant \phi (U(f))\). \(\phi \) is mixture continuous; hence, \(\phi \) is quasi-concave. By Lemma 25 of MMR, \(\phi \) is concave (see also Theorem 4 of Cerreia-Vioglio et al. (2014)). By Lemma 26 of Maccheroni et al. (2006: 1476–1477), \(\phi \) has the following representation:

$$\begin{aligned} \phi (U(f))= & {} \min _{p\in {\mathcal {P}}_{{\mathcal {S}}}}\bigg ( \int _{{\mathcal {S}}}U(f)dp(s) + c(p) \bigg )\\&=\min _{p\in {\mathcal {P}}_{{\mathcal {S}}}}\bigg ( \int _{{\mathcal {S}}} \int _{{\mathcal {T}}} u(f(s,t))dp(s)dq(t) + c(p) \bigg ) \end{aligned}$$

where \(u:{\mathcal {X}}\rightarrow {\mathbb {R}}\) is a strictly \(\succcurlyeq \)-increasing utility function and \(c:{\mathcal {P}}_{{\mathcal {S}}}\rightarrow [0,\infty ]\) is a grounded, convex and lower semicontinuous function. Therefore, preferences over \({\mathcal {A}}\) satisfying the axioms of statement 1 of Theorem 5 are variational preferences. For the uniqueness results, cardinality of u and uniqueness of q are well known. That c is a ratio scale follows from MMR’s corollary 5. \(\square \)

Proof of Theorem 7

If preferences satisfy axioms 1–6, then, following the proof of Theorem 5, preferences are represented by a functional \(f\mapsto \phi (U(f))\). We have established that \(\phi \) is monotonic, mixture continuous and vertically invariant, and \(\phi \) is concave if and only if preferences satisfy second-stage uncertainty aversion. Now assume certainty equivalents exist for all acts. Fix \(U(\overline{x})=1\) and \(U(\underline{x})=0\). Take any \(a=(a_1,\ldots ,a_n)\) and \(b=(b_1,\ldots ,b_n)\), with \(a,b\in [0,1]^n\). For all \(a_i,b_i\), \(i=1,\ldots ,n\), by convex valuedness of q, there exist events \(A_i\) and \(B_i\) with \(q(A_i)=a_i\) and \(q(B_i)=b_i\). Let x(E) denote the certainty equivalent of \(\overline{x}_{E}\underline{x}\), for \(E\in {\mathcal {E}}_{{\mathcal {T}}}\). Then, the act f with \(f_{s_i} = x(A_i)\) for all \(i=1,\ldots ,n\) has utility vector \(U(f)=a\), and the act g with \(g_{s_i} = x(B_i)\) for all \(i=1,\ldots ,n\) has utility vector \(U(g)=b\). Notice that f and g are second-stage constant acts. By convex valuedness of q, there exists a second-stage event C with \(q(C)=\frac{1}{2}\). Then, the act \(f_C g\) generates utility vector \(U(f_C g) = \frac{1}{2}a+\frac{1}{2}b\). Second-order risk aversion holds, hence \(f\sim g\) only if \(f_C g\succcurlyeq f\). Equivalently: \(\phi (U(f_C g)) = \phi \big (\frac{1}{2}a+\frac{1}{2}b\big ) \geqslant \phi (U(f))=\phi (a)\). Because \(\phi \) is mixture continuous, \(\phi \) is quasi-concave. Monotonicity and vertical invariance have been shown. Hence, under axioms 1–6 and existence of certainty equivalents, second-order risk aversion holds if and only if \(\phi \) is concave. \(\square \)

Proof of Theorem 8

The necessity of the axioms involves only substitution of the preference representation. We prove the sufficiency of the axioms for the representation. Axioms 1–7 hold; hence, preferences admit a representation \(f\mapsto \phi (U(f))\), with \(\phi \) monotonic, mixture continuous, vertically invariant and concave. Second-stage constant independence allows us to establish that \(\phi \) is linearly homogeneous: For all \(f\in {\mathcal {A}}\) and \(\alpha \in [0,1]\), we have \(\phi (\alpha U(f))=\alpha \phi (U(f))\).

Let \(f\in {\mathcal {A}}\) be an act and let g be a first-stage constant act such that \(f\sim g\), or equivalently \(\phi (U(f))=\phi (U(g))\). There exist acts \(\tilde{f}\) and \(\tilde{g}\) such that \(\tilde{f}\) is a second-stage act average of f and \(\underline{x}\) and \(\tilde{g}\) is a second-stage act average of g and \(\underline{x}\). The second-stage constant independence axiom implies \(\tilde{f}\sim \tilde{g}\), or equivalently \(\phi (\frac{1}{2}U(f)) = \phi (\frac{1}{2}U(g)) = \frac{1}{2}\phi (U(g))\), where the second equality exploits that g is first-stage constant. Hence, \(\phi (\frac{1}{2}U(f)) = \frac{1}{2}\phi (U(f))\). The same technique can be used iteratively to show that \(\phi (\alpha U(f)) = \alpha \phi (U(f))\) for all \(\alpha \in [0,1]\cap {\mathcal {D}}\). Linear homogeneity follows as \(\phi \) is mixture continuous. It follows from Lemma 3.5 of Gilboa and Schmeidler (1989) that \(\phi \) is a subjective maxmin expected utility representation. \(\square \)

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Webb, C.S. Purely subjective variational preferences. Econ Theory 64, 121–137 (2017). https://doi.org/10.1007/s00199-016-1003-2

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Keywords

  • Variational preferences
  • Ambiguity aversion
  • Subjective uncertainty

JEL Classification

  • D81