Abstract
I define subjective probabilities for an ambiguity averse agent who is given an exogenous information set \(\varDelta \) containing the true probability law on the state space S. The agent in my model evaluates every uncertain prospect via a mixture of the least favorable scenario in \(\varDelta \) and her unique subjective belief p. This parametric utility structure is characterized by four standard axioms—order, continuity, monotonicity, and Independence—where the last two conditions are modified via the information set \(\varDelta \). Any pair of preferences \(\succeq _1\) and \(\succeq _2\) in my model can be compared in terms of confidence—the weights attached to the subjective beliefs \(p_1\) and \(p_2\), respectively. The corresponding behavioral condition extends Epstein’s comparative ambiguity aversion that leaves some rankings \(\succeq _1\) and \(\succeq _2\) unrelated. Moreover, I relax the wellknown dynamic consistency principle and characterize the Bayesian updating rule for the belief p conditional on any nonnull event in S.
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Notes
If S is finite, then this topology is Euclidean. In general, a sequence \(\{q_n\}\) in \(\mathcal {P}\) converges to \(q\in \mathcal {P}\) in the weak\(^*\) topology if for any event \(A\in \varSigma \), the sequence \(\{q_n(A)\}\) converges to q(A).
Equivalently, p can be mixed with the least favorable belief for the act f in \(\mathrm {cl}(\mathrm {co}\, \varDelta )\) because
$$\begin{aligned} \min _{q\in \mathrm {cl} (\varDelta )} u(f(q)) = \min _{q\in \mathrm {cl}(\mathrm {co}\, \varDelta )}u(f(q)). \end{aligned}$$Note that \(\varDelta E\)Monotonicity implies consequentialism \(f \sim _E f E h\) for all acts \(f, h\in \mathcal {H}\) because \(q(E) = 1\) and \(f(q) = (f E h)(q)\) for all \(q\in \varDelta E\). Thus, \(\succeq _E\) can be viewed as a preference over contingent acts that map E into \(\mathcal {L}\).
For example, take \(S = \{1, 2, 3, 4\}\) and \(\varDelta =\{q, q'\}\) such that
$$\begin{aligned} q = \left( \alpha ^2, (1\alpha )\alpha , 1\alpha , 0 \right) \quad \text {and}\quad q' = ((1\alpha )\alpha , \alpha ^2, 1\alpha , 0 ) \end{aligned}$$for some small \(\alpha >0\). Then, \(\varDelta \{1, 4\}\) consists of just one point (1, 0, 0, 0), but \(\varDelta  \{1, 2\}\) consists of two points \((\alpha , 1\alpha , 0, 0)\) and \((1\alpha , \alpha , 0, 0)\). Thus, conditioning on the event \(E = \{1, 4\}\) reduces the geometric size of \(\varDelta \) to zero, but conditioning on \(E' = \{1, 2\}\) expands the size of \(\varDelta \) from an arbitrarily small number \(\sqrt{2}(\alpha  2\alpha ^2)\) to \(\sqrt{2} (12\alpha )\) that is close to \(\sqrt{2}\) for small \(\alpha \).
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Acknowledgments
I thank Mark Machina, Larry Epstein, Peter Wakker, Marciano Siniscalchi, RUD participants, and anonymous referees for their comments. Early drafts of the paper were circulated under the title “Subjective Probability and Confidence”.
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Appendix: Proofs
Appendix: Proofs
1.1 Proof of Theorem 1
Fix a nonempty set \(\varDelta \subset \mathcal {P}\) and a utility index \(u\in \mathcal {U}\). For any \(f\in \mathcal {H}\), let
Suppose that \(\succeq \) is represented by
for some \(\varepsilon \in [0,1]\) and \(p\in \mathrm {cl}(\mathrm {co}\, \varDelta )\), \(u\in \mathcal {U}\). Order and Continuity are obvious. Note that \(l\in L(f, \varDelta )\) implies \(u(l) \le M(f) \le u(f(p))\), and hence, \(u(l) \le U(f)\). Thus \(\varDelta \)Monotonicity holds. To verify \(\varDelta \)Independence, take \(\alpha \in [0,1]\), acts \(f,g,h\in \mathcal {H}\), and a lottery \(l\in \mathcal {L}\) such that \(f \succeq g\), \( \alpha {f}+(1\alpha ) h\) is more secure than \( \alpha {f}+(1\alpha ) l\), but \( \alpha {g}+(1\alpha ) h\) is less secure than \( \alpha {g}+(1\alpha ) l\). Then
and hence, \( \alpha {f}+(1\alpha ) h\succeq \alpha {g}+(1\alpha ) h\).
Turn to sufficiency. Suppose that \(\succeq \) satisfies Order, Continuity, \(\varDelta \)Monotonicity, and \(\varDelta \)Independence. By Herstein–Milnor’s Theorem, the ranking of lotteries has an expected utility representation \(u\in \mathcal {U}\) that is unique up to a positive linear transformation. If u is constant, then (3.2) is trivial. Assume that u is not constant.
Assume wlog that \(\varDelta \) is convex and closed. Otherwise, replace \(\varDelta \) by its closed convex hull. The security levels M(f) and all the proofs below are unaffected.
Recall that f is more secure than g if and only if \(M(f) \ge M(g)\).
Lemma 1
For all \(\alpha \in [0,1]\) and \(f, g, h\in \mathcal {H}\), \(SP(\alpha , f, h) \ge SP(\alpha , g, h)\) if and only if there is \(l\in \mathcal {L}\) such that \( \alpha {f}+(1\alpha ) h\) is more secure than \( \alpha {f}+(1\alpha ) l\) and \( \alpha {g}+(1\alpha ) h\) is less secure than \( \alpha {g}+(1\alpha ) l\).
Proof
For all \(\alpha \in [0,1]\), \(f, g, h\in \mathcal {H}\), and \(l\in \mathcal {L}\),
By definition of security premia,
Therefore, if \( \alpha {f}+(1\alpha ) h\) is more secure than \( \alpha {f}+(1\alpha ) l\) and \( \alpha {g}+(1\alpha ) h\) is less secure than \( \alpha {g}+(1\alpha ) l\), then \(SP(\alpha , f, h) \ge SP(\alpha , g, h) \). Conversely, suppose that \(SP(\alpha , f, h) \ge SP(\alpha , g, h)\). Take \(l\in \mathcal {L}\) such that \(M\left( \alpha {f}+(1\alpha ) h\right) = M\left( \alpha {f}+(1\alpha ) l\right) \). Then \( \alpha {f}+(1\alpha ) h\) is more secure than \( \alpha {f}+(1\alpha ) l\) and
that is, \( \alpha {g}+(1\alpha ) h\) is less secure than \( \alpha {g}+(1\alpha ) l\).
The preference \(\succeq \) satisfies all conditions in Theorem 1 in Gilboa and Schmeidler Gilboa and Schmeidler (1989). In particular, for all \(\alpha \in [0,1]\) and \(f, g\in \mathcal {H}\),
By Lemma 1 and \(\varDelta \)Independence, \(\succeq \) satisfies Uncertainty Aversion. Thus, there is a unique set convex and closed set \(\varPi \subset \mathcal {P}\) such that \(\succeq \) is represented by
Assume that S and X are finite, and \(\varSigma = 2^S\) (the general case is treated separately). For any \(a\in \mathbb {R}^S\), let
If \(W = V\), then \(\succeq \) is extremely cautious, \(\varPi = \varDelta \), and the utility representation (3.2) holds for \(\varepsilon = 1\) and any \(p\in \varDelta \).
Thus, assume that \(W \ne V\). Lemmas 2 and 3 in Kopylov (2009) establish that

there are unique \(0\le \varepsilon < 1\) and \(p\in \varDelta \) such that
$$\begin{aligned} W(a) = \varepsilon V(a) + (1\varepsilon ) p\cdot a \end{aligned}$$(5.4)for all \(a\in \mathbb {R}^S\), and

p is the only probability measure in \(\mathcal {P}\) such that for all \(f, g\in \mathcal {H}\), if f is more secure than g and \(f(p) \succ g(p)\), then \( f \succ g\).
All the notation and the proofs apply as is with the axiom of Cautious Independence renamed to \(\varDelta \)Independence.
Representation (5.4) can be rewritten in the required form (3.2). Moreover, if \(\succeq \) is not extremely cautious, then this representation is unique up to a positive linear transformation of u, and p is the only probability measure that satisfies the condition (3.3).
Extend the construction of the utility representation (3.2) for an arbitrary state space S with an algebra of events \(\varSigma \). For any \(A\in \varSigma \), let
Consider two cases.
Case 1. \(\pi (A) = \pi _* (A)\) for all events \(A\in \varSigma \). Let \(\varepsilon = 1\) and take any \(p\in \varDelta \).
Case 2. \(\pi (A) > \pi _* (A)\) for some \(A\in \varSigma \). Define \(\varepsilon <1\) by
For all events \(B\in \varSigma \), take
In both cases, for any finite subalgebra \(\varSigma ' \subset \varSigma \) such that \(A\in \varSigma '\), the finite case of Theorem 1 implies that p is finitely additive on \(\varSigma '\), and the preference \(\succeq \) restricted to \(\varSigma '\) measurable acts is represented by (3.2) with the triple \((u, \varepsilon , p)\). Thus, p is finitely additive on all of \(\varSigma \), and \(\succeq \) is represented by (3.2) with the triple \((u, \varepsilon , p)\) on all of \(\mathcal {H}\). The inclusion \(p\in \mathrm {cl}(\mathrm {co}\, \varDelta )\) follows from the fact that p is the limit of a net \(p_{\varSigma '}\in \mathrm {cl}(\mathrm {co}\, \varDelta )\) in the weak* closed set \(\mathcal {P}\).
1.2 Proofs of Theorems 2 and 3
Let preferences \(\succeq _1\) and \(\succeq _2\) have utility representations (3.2) with tuples \((\varepsilon _1, p_1, u_1)\) and \((\varepsilon _2, p_2, u_2)\). Each of the conditions (i), (ii), or (iii) of Theorem 2 implies that \(\succeq _1\) equals \(\succeq _2\) on the domain \(\mathcal {L}\) of all lotteries, and hence, that the risk utility indices \(u_1\) and \(u_2\) can be taken equal \(u_1 = u_2 = u\).
Let \(\varPi _1 = \varepsilon _1 \varDelta + (1\varepsilon _1)\{p_1\}\) and \(\varPi _2 = \varepsilon _2 \varDelta + (1\varepsilon _2)\{p_2\}\). GMM show that for all \(f, g\in \mathcal {H}\),
Each of the conditions of Theorem 2 is equivalent to the inclusion \(\varPi _1 \subset \varPi _2\). For (i), it is established by GMM, for (ii) by Ghirardato and Marinacci (2002), and for (iii), it is shown directly via
Turn to Theorem 3. Suppose that \(\varDelta \) is convex and closed and is not an interval (if the convex closed hull of \(\varDelta \) is an interval, then \(\varDelta \) is also an interval).
Let the preferences \(\succeq _1\) and \(\succeq _2\) have representations (3.2) with tuples \((u_1, p_1, \varepsilon _1)\) and \((u_2, p_3, \varepsilon _2)\), respectively. Take x, y such that \(x \succ _1 y\) and \(x \succ _2 y\). Let \(u = u_1 = u_2\) on \(\mathcal {L}(x, y)\).
Suppose that \(\varepsilon _2 \le \varepsilon _1\). Take any \(f, g\in \mathcal {H}\) such that \( \tfrac{f + g}{2} \in \mathcal {L}(x, y)\). Then
Thus, either \(u(f(p_2))) \ge u(f(p_1))\) or \(u(g(p_2)) \ge u(g(p_1))\). Wlog, assume that \(u(f(p_2)) \ge u(f(p_1))\). Let
Then,
Thus, \(U_1(f) \ge u(l)\) implies that \(U_2(f) \ge u(l)\) as well. It follows that \(\succeq _2\) is more confident than \(\succeq _1\).
Turn to sufficiency. Wlog assume that S is finite and \(\varSigma = 2^S\). (Otherwise, take a finite algebra \(\varSigma ' \subset \varSigma \) such that \(\varDelta \) restricted to \(\varSigma '\) is not an interval. Identify \(S'\) with the elements of the finest partition generated by \(\varSigma '\). Replace \((S, \varSigma )\) with \((S', 2^{S'})\).)
Suppose that \(\succeq _2\) is more confident than \(\succeq _1\), but \(\varepsilon _2 > \varepsilon _1\). Consider two cases.
Case 1. \(p_1 = p_2 = p\). Take \(q, r\in \varDelta \) such that \(p, q, r\in \mathbb {R}^S\) are linearly independent. Then, there exist \(a, b \in \mathbb {R}^S\) and \(\alpha , \beta \in \mathbb {R}\) such that
It follows that \((r q) \cdot a > 0\) and \((rq ) \cdot b >0\). Thus \(r \cdot a = p\cdot a > q\cdot a \) and \(r \cdot b > p \cdot b = q\cdot b \), and
where \(c = a+b\). Take \(r', q' \in \varDelta \) such that
for all \(p' \in \varDelta \). Take a constant vector \(\gamma \in \mathbb {R}^S\) such that
Wlog let u be such that the vectors c and \(\gamma c\) belong to \([u(y), u(x)]^S\). Then, there exist acts \(f, g: S \rightarrow \mathcal {L}(x,y)\) such that \(u\circ f = c \) and \(u\circ g=\gamma c\). By construction, \( \tfrac{f + g}{2}\in \mathcal {L}(x,y)\) and \(U_1(f) = U_1 (g) = u(l)\) for some \(l \in \mathcal {L}(x,y)\). Note that \(p \cdot c > q'\cdot c\) and \(p \cdot (\gamma c) > r' \cdot (\gamma c)\). As \(\varepsilon _1 < \varepsilon _2\), then
Hence, \(f \succeq _1 l \succ _2 f\) and \(g\succeq _1 l\succ _2 g\). Thus \(\succeq _2\) is not more confident than \(\succeq _1\).
Case 2. \(p_1 \ne p_2\). Take \(r\in \varDelta \) such that \(p_1, p_2, r\in \mathbb {R}^S\) are linearly independent. Then there is \(c\in \mathbb {R}^S\) and \(\alpha \in \mathbb {R}\) such that
Thus, \(r \cdot c > p_1 \cdot c = p_2 \cdot c\). If there is \(q\in \varDelta \) such that \(p_1 \cdot c > q \cdot c\), then one can construct f and g analogously to Case 1 and show that \(\succeq _2\) is not more confident than \(\succeq _1\).
Assume that \(p' \cdot c \ge p_1\cdot c\) for all \(p'\in \varDelta \). Take \(r' \in \varDelta \) such that \(r' \cdot c \ge p'\cdot c\) for all \(p'\in \varDelta \) as well. Let \(d = c + \delta (p_1  p_2)\) for arbitrary small \(\delta >0\). Then, \((p_1  p_2) \cdot d > 0\), and hence, \(p_1 \cdot d > p_2 \cdot d\). Analogously to Step 1, construct acts \(f, g\in \mathcal {H}\) such that \(u \circ f = d\), \(u \circ g = \gamma  d\), \( \tfrac{f + g}{2} \in \mathcal {L}(x, y)\), and \(U_1(f) = U_1(g) = u(l)\) for some \(l \in \mathcal {L}(x, y)\). As \(\varepsilon _2 > \varepsilon _1\), and \(p_1 \cdot d > p_2 \cdot d\), then \(U_2(f) < U_1(f) = u(l)\). As \(\varepsilon _2 > \varepsilon _1\) and \(r' \cdot (c) < p_1 \cdot (c) = p_2 \cdot (c)\), then \(\delta \) can be taken sufficiently small so that \(U_2(g) < U_2(g) = u(l)\) as well. Thus, \(\succeq _2\) is not more confident than \(\succeq _1\).
1.3 Proofs of Theorem 4, Theorem 5, and Theorem 6
Suppose that E is a nonnull event, preferences \(\succeq \) and \(\succeq _E\) satisfy Axioms 1–4 with information sets \(\varDelta \) and \(\varDelta E\), respectively, and \(\varDelta \)Dynamic Consistency holds.
Turn to Theorem 4. By Theorem 1, \(\succeq \) and \(\succeq _E\) have representations (3.2) with components \((\varepsilon , p, u)\) and \((\lambda , p_E, u)\) where \(p\in \mathrm {cl}(\mathrm {co}\, \varDelta )\) and \(p_E\in \mathrm {cl}(\mathrm {co}\, \varDelta E)\), respectively. If \(\succeq _E\) is extremely cautious, then \(p_E\) is arbitrary and can be taken equal to pE. Suppose that \(\lambda <1\).
Assume that \(p_E \ne pE\). Let
I claim that there are acts \(f, g\in \mathcal {H}\) such that
To construct such f and g, take an event \(A \subset S\) such that \(p_E(A) > (pE)(A)\). Let \(\pi _*(A) = \min _{q\in \varDelta E}q(A)\) and \(\pi _*(A^c) = \min _{q\in \varDelta E}q(A^c)\). Take vectors \(a, b\in \mathbb {R}^S\) such that
By construction, \(p_E\cdot a > (pE)\cdot a\), \((pE)\cdot b > p_E \cdot b\), \(p_E\cdot a \ge V_E(a) = 0\), and \(p_E\cdot b \ge V_E(b) = 0\), If \(p_E\cdot a = p_E \cdot b \), then take \(f, g\in \mathcal {H}\) such that \(u(g) = a\) and \(u(f) = b\) (the range of u is assumed to contain \([1, 1]\).) If \(p_E\cdot a \ne p_E \cdot b \), then take \(f, g\in \mathcal {H}\) such that \(u(g) = (p_E\cdot b) a\) and \(u(f) = (p_E\cdot a) b\).
Take \(l\in \mathcal {L}\) such that \(u(l) = V_E(u(f)) = V_E(u(g))\). Then, for all \(q\in \mathrm {cl}(\mathrm {co}\, \varDelta )\),
As u(f(qE)) can be arbitrarily close to \(V_E(f)\),
Similarly, \(M(gEl) = u(l)\).
It follows from (5.6) and (3.2) that fEl is less secure than gEl, f is more secure than g on E, \(fEl \succeq gEl\) because \((pE)\cdot u(f) > (pE)\cdot u(g)\), but \(g \succ _E f\) because \(p_E \cdot u(f) < p_E\cdot u(g)\) and \(\lambda <1\).
This is a contradiction with \(\varDelta \)Dynamic Consistency. Thus, \(p_E = pE\).
To prove Theorem 6, proceed analogously, but take l such that \(u(l) \le \min _{s\in S} u(f(s))\) and \(u(l) \le \min _{s\in S} u(g(s))\). As E is surprising, then \(\inf _{q\in \varDelta } q(E) = 0\), and hence, \(M(fEl) = M(gEl) = u(l)\). Therefore, \(p_E \ne pE\) contradicts \(\varDelta \)Dynamic Consistency as well.
To prove Theorem 5, observe first that \(f E l \succeq l\) holds in my model if and only if \(\min _{q\in \varPi E} u(f(q)) \ge u(l)\). Let \(\succeq '_E\) be the preference represented by \(\min _{q\in \varPi E} u(f(q))\). Then, property (4.2) is identical to the comparative ambiguity aversion (3.7) applied to rankings \(\succeq _1 = \succeq '_E \) and \(\succeq _2 = \succeq _E \). It follows that (4.2) holds if and only if \( \varPi _E \subset \varPi E \), that is,
Direct computation shows that
The inequality
follows.
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Kopylov, I. Subjective probability, confidence, and Bayesian updating. Econ Theory 62, 635–658 (2016). https://doi.org/10.1007/s0019901509290
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DOI: https://doi.org/10.1007/s0019901509290