Skip to main content
Log in

Subjective probability, confidence, and Bayesian updating

  • Research Article
  • Published:
Economic Theory Aims and scope Submit manuscript

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 well-known dynamic consistency principle and characterize the Bayesian updating rule for the belief p conditional on any non-null event in S.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Notes

  1. 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).

  2. This definition is analogous to the notions of security premium in Kopylov (2009) and risk premium for monetary gambles in Kreps (1988), p. 74.

  3. Equivalently, p can be mixed with the least favorable belief for the act f in \(\mathrm {cl}(\mathrm {co}\, \varDelta )\) because

    $$\begin{aligned} \min _{q\in \mathrm {cl} (\varDelta )} u(f(q)) = \min _{q\in \mathrm {cl}(\mathrm {co}\, \varDelta )}u(f(q)). \end{aligned}$$
  4. 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 \(\mathcal {L}\).

  5. 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 \(\sqrt{2} (1-2\alpha )\) that is close to \(\sqrt{2}\) for small \(\alpha \).

References

  • Anscombe, F., Aumann, R.: A definition of subjective probability. Ann. Math. Stat. 34, 199–205 (1963)

    Article  Google Scholar 

  • Aryal, G., Stauber, R.: Trembles in extensive games with ambiguity averse players. Econ. Theory 57, 1–40 (2014)

    Article  Google Scholar 

  • Becker, S., Brownson, O.: What price ambiguity? Or the role of ambiguity in decision making. J. Polit. Econ. 72, 62–73 (1964)

    Article  Google Scholar 

  • Berger, J.: An overview of Robust Bayesian analysis. Test 3, 5–124 (1994)

    Article  Google Scholar 

  • Berger, J., Berliner, M.: Robust Bayes and empirical analysis with epsilon contaminated priors. Ann. Stat. 14, 461–486 (1986)

    Article  Google Scholar 

  • Carlier, G., Dana, R., Shahdi, N.: Efficient insurance contracts under epsilon contaminated utilities. Geneva Pap. Risk Insur. Theory 28, 59–71 (2003)

    Article  Google Scholar 

  • Camerer, C., Weber, M.: Recent developments in modeling preferences. J. Risk Uncertain. 5, 325–370 (1992)

    Article  Google Scholar 

  • Eichberger, J., Kelsey, D.: E-capacities and the Ellsberg paradox. Theory Decis. 46, 107–140 (1999)

    Article  Google Scholar 

  • Ellsberg, D.: Risk, ambiguity, and the Savage axioms. Q. J. Econ. 75, 643–669 (1961)

    Article  Google Scholar 

  • Epstein, L.: A definition of uncertainty aversion. Rev. Econ. Stud. 66, 579–608 (1999)

    Article  Google Scholar 

  • Epstein, L., Schneider, M.: Recursive multiple-priors. J. Econ. Theory 61, 1–31 (2003)

    Article  Google Scholar 

  • Epstein, L., Wang, T.: Intertemporal asset pricing and Knightian uncertainty. Econometrica 62, 283–382 (1994)

    Article  Google Scholar 

  • Epstein, L., Zhang, J.: Subjective probabilities on subjectively unambiguous events. Econometrica 69, 265–306 (2001)

    Article  Google Scholar 

  • Gajdos, T., Hayashi, T., Tallon, J.-M., Vergnaud, J.-C.: Attitude towards Imprecise information. J. Econ. Theory 140, 27–65 (2008)

    Article  Google Scholar 

  • Ghirardato, P.: Revisiting Savage in a conditional world. Econ. Theory 20, 83–92 (2002)

    Article  Google Scholar 

  • Ghirardato, P., Maccheroni, F., Marinacci, M.: Differentiating ambiguity and ambiguity attitude. J. Econ. Theory 118, 133–173 (2004)

    Article  Google Scholar 

  • Ghirardato, P., Marinacci, M.: Ambiguity mase precise: a comparative foundation. J. Econ. Theory 102, 251–289 (2002)

    Article  Google Scholar 

  • Gilboa, I., Maccheroni, F., Marinacci, M., Schmeidler, D.: Objective and subjective rationality in a multiple priors model. Econometrica 78, 755–770 (2010)

    Article  Google Scholar 

  • Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18, 141–153 (1989)

    Article  Google Scholar 

  • Halevy, Y., Feltkamp, V.: A Bayesian approach to uncertainty aversion. Rev. Econ. Stud. 72, 449–466 (2005)

    Article  Google Scholar 

  • Hanany, E., Klibanoff, P.: Updating preferences with multiple priors. Theor. Econ. 2, 261–298 (2007)

    Google Scholar 

  • Hodges, J.L., Lehmann, E.L.: The use of previous experience in reaching statistical decisions. Ann. Math. Stat. 23, 396–407 (1952)

    Article  Google Scholar 

  • Klibanoff, P., Marinacci, M., Mukerji, M.: A smooth model of decision making under ambiguity. Econometrica 73, 1849–1892 (2005)

    Article  Google Scholar 

  • Kopylov, I.: Choice deferral and ambiguity aversion. Theor. Econ. 4, 199–225 (2009)

    Google Scholar 

  • Kreps, D.: Notes on the Theory of Choice. Westview Press, London (1988)

    Google Scholar 

  • Machina, M., Schmeidler, D.: A more robust definition of subjective probability. Econometrica 60, 745–780 (1992)

    Article  Google Scholar 

  • Moreno, E., Cano, J.: Robust Bayesian analysis with epsilon contaminations partially known. J. R. Stat. Soc.: Part B 53, 143–155 (1991)

    Google Scholar 

  • Nishimura, K., Ozaki, H.: Search and Knightian uncertainty. J. Econ. Theory 119, 299–333 (2004)

    Article  Google Scholar 

  • Nishimura, K., Ozaki, H.: An axiomatic approach to \(\varepsilon \)-contamination. Econ. Theory 27, 333–340 (2006)

    Article  Google Scholar 

  • Oechssler, J., Roomets, A.: A test of mechanical ambiguity. J. Econ. Behav. Org. 119, 153–162 (2015)

    Article  Google Scholar 

  • Ramsey, F.: Truth and probability. In: R.B. Braithwaite (ed.) The Foundations of Mathematics and Other Logical Essays. Harcourt, Brace and Co., New York (1931)

  • Savage, L.: The Foundations of Statistics. Wiley, New York (1954)

    Google Scholar 

  • Siniscalchi, M.: Vector expected utility and attitudes toward variation. Econometrica 77, 801–855 (2009)

    Article  Google Scholar 

Download references

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”.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Igor Kopylov.

Appendix: Proofs

Appendix: Proofs

1.1 Proof of Theorem 1

Fix a non-empty set \(\varDelta \subset \mathcal {P}\) and a utility index \(u\in \mathcal {U}\). For any \(f\in \mathcal {H}\), let

$$\begin{aligned} M(f) = \inf _{q\in \varDelta } u(f(q)) = \min _{q\in \mathrm {cl}(\mathrm {co}\, D)} u(f(q)). \end{aligned}$$

Suppose that \(\succeq \) is represented by

$$\begin{aligned} U(f) = (1-\varepsilon )u(f(p)) + \varepsilon M(f) \end{aligned}$$

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

$$\begin{aligned}&a_1 = U\left( \alpha {f}+(1-\alpha ) l\right) - U\left( \alpha {g}+(1-\alpha ) l\right) \ge 0 \\&a_2 = M\left( \alpha {f}+(1-\alpha ) h\right) - M\left( \alpha {f}+(1-\alpha ) l\right) \ge 0 \\&a_3 = M\left( \alpha {g}+(1-\alpha ) l\right) - M\left( \alpha {g}+(1-\alpha ) h\right) \ge 0 \\&U\left( \alpha {f}+(1-\alpha ) h\right) - U\left( \alpha {g}+(1-\alpha ) h\right) = a_1 + \varepsilon (a_2+ a_3) \ge 0 \end{aligned}$$

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 \(l\in \mathcal {L}\),

$$\begin{aligned}&M\left( \alpha {f}+(1-\alpha ) l\right) = \alpha {M(f)}+(1-\alpha ) u(l)\\&M\left( \alpha {g}+(1-\alpha ) l\right) = \alpha {M(g)}+(1-\alpha ) u(l). \end{aligned}$$

By definition of security premia,

$$\begin{aligned} SP(\alpha , f, h) - SP(\alpha , g, h)&= [ M\left( \alpha {f}+(1-\alpha ) h\right) - M\left( \alpha {f}+(1-\alpha ) l\right) ] \\&\quad + [ M\left( \alpha {g}+(1-\alpha ) l\right) - M\left( \alpha {g}+(1-\alpha ) h\right) ]. \end{aligned}$$

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

$$\begin{aligned} M\left( \alpha {g}+(1-\alpha ) l\right) - M\left( \alpha {g}+(1-\alpha ) h\right) = SP(\alpha , f, h) - SP(\alpha , g, h) \ge 0, \end{aligned}$$

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 \(f, g\in \mathcal {H}\),

$$\begin{aligned} SP(\alpha , f, g) \ge 0 = SP(\alpha , g, g). \end{aligned}$$

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

$$\begin{aligned} U(f) = \min _{q\in \varPi } u(f(q)). \end{aligned}$$
(5.2)

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

$$\begin{aligned} V\left( a\right) = \min _{q\in \varDelta } q\cdot a \quad \text {and}\quad W\left( a\right) = \min _{q\in \varPi } q\cdot a. \end{aligned}$$
(5.3)

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

$$\begin{aligned} \pi (A) = \min _{q\in \varPi }q(A) \quad \text {and}\quad \pi _*(A) = \min _{q\in \varDelta }q(A). \end{aligned}$$

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

$$\begin{aligned} \varepsilon = \frac{1-\pi (A) -\pi (A^c)}{1- \pi _* (A) - \pi _*(A^c)}. \end{aligned}$$

For all events \(B\in \varSigma \), take

$$\begin{aligned} p(B) = \frac{\pi (B) - \varepsilon \pi _*(B)}{1-\varepsilon }. \end{aligned}$$

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 \(f, g\in \mathcal {H}\),

$$\begin{aligned} \begin{aligned} f \succeq _1^* g \quad \Leftrightarrow \quad f(q) \succeq _1 g(q) \quad \text {for all }q\in \varPi _1\\ f \succeq _2^* g \quad \Leftrightarrow \quad f(q) \succeq _2 g(q) \quad \text {for all }q\in \varPi _2. \end{aligned}\end{aligned}$$
(5.5)

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

$$\begin{aligned}&p_2 \in \tfrac{\varepsilon _1- \varepsilon _2}{1-\varepsilon _2}\mathrm {cl}(\mathrm {co}\, \varDelta ) + \tfrac{1 - \varepsilon _1}{1-\varepsilon _2} \{p_1\} \quad \Leftrightarrow \quad \\&\quad (1-\varepsilon _2)\{p_2\} + \varepsilon _2 \mathrm {cl}(\mathrm {co}\, \varDelta ) \subset (1-\varepsilon _1)\{p_1\} + \varepsilon _1 \mathrm {cl}(\mathrm {co}\, \varDelta ). \end{aligned}$$

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 xy 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

$$\begin{aligned} \tfrac{f(p_1) + g(p_1)}{2} = \tfrac{f + g}{2}(p_1) = \tfrac{f + g}{2}(p_2) = \tfrac{f(p_2) + g(p_2)}{2} . \end{aligned}$$

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

$$\begin{aligned} M(f) = \min _{q\in \varDelta } u(f(q)) \le u(f(p_1)) \le u(f(p_2)). \end{aligned}$$

Then,

$$\begin{aligned} U_2(f) = \varepsilon _2 M(f) + (1-\varepsilon _2) u(f(p_2))\ge & {} \varepsilon _1 M(f) + (1-\varepsilon _1) u(f(p_2))\\\ge & {} \varepsilon _1 M(f) + (1-\varepsilon _1) u(f(p_1)) = U_1(f). \end{aligned}$$

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

$$\begin{aligned} r - q = \alpha (p - r) + a \quad \text {and}\quad (p-r) \cdot a = 0\\ r- q = \beta (p- q) + b \quad \text {and}\quad (p-q) \cdot b = 0. \end{aligned}$$

It follows that \((r- q) \cdot a > 0\) and \((r-q ) \cdot b >0\). Thus \(r \cdot a = p\cdot a > q\cdot a \) and \(r \cdot b > p \cdot b = q\cdot b \), and

$$\begin{aligned} r \cdot c > p \cdot c > q\cdot c \end{aligned}$$

where \(c = a+b\). Take \(r', q' \in \varDelta \) such that

$$\begin{aligned} r' \cdot c \ge p' \cdot c \ge q' \cdot c \end{aligned}$$

for all \(p' \in \varDelta \). Take a constant vector \(\gamma \in \mathbb {R}^S\) such that

$$\begin{aligned} \varepsilon _1 (q'\cdot c) + (1-\varepsilon _1) (p\cdot c) = \varepsilon _1 (r'\cdot (\gamma -c )) + (1-\varepsilon _1) (p \cdot (\gamma -c )). \end{aligned}$$

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

$$\begin{aligned} U_2(f)&= \varepsilon _2 (q'\cdot c) + (1-\varepsilon _2) (p\cdot c) < \varepsilon _1 (q'\cdot c) + (1-\varepsilon _1) (p\cdot c) = U_1(f)\\ U_2(g)&= \varepsilon _2 (r'\cdot (\gamma -c)) + (1-\varepsilon _2) (p \cdot (\gamma -c)) < \varepsilon _1 (r'\cdot (\gamma -c))\\&\quad + (1-\varepsilon _1) (p \cdot (\gamma -c))= U_1(g). \end{aligned}$$

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

$$\begin{aligned} r - p_1 = \alpha (p_1 - p_2) + c \quad \text {and}\quad c \cdot (p_1 - p_2) = 0. \end{aligned}$$

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 non-null 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 p|E. Suppose that \(\lambda <1\).

Assume that \(p_E \ne p|E\). Let

$$\begin{aligned} V_E(a) = \min _{q\in \varDelta |E} q \cdot a. \end{aligned}$$

I claim that there are acts \(f, g\in \mathcal {H}\) such that

$$\begin{aligned} p_E \cdot u(f) < p_E\cdot u(g), \quad (p|E)\cdot u(f) > (p|E)\cdot u(g), \text { and }V_E(u(f)) = V_E(u(g)). \end{aligned}$$
(5.6)

To construct such f and g, take an event \(A \subset S\) such that \(p_E(A) > (p|E)(A)\). 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

$$\begin{aligned} a_s = {\left\{ \begin{array}{ll} 1 - \pi _*(A) &{} \quad \text {if }s\in A \\ - \pi _*(A) &{} \quad \text {if }s\not \in A \end{array}\right. } \quad \text {and}\quad b_s = {\left\{ \begin{array}{ll} -\pi _*(A^c) &{} \quad \text {if }s\in A \\ 1-\pi _*(A^c) &{} \quad \text {if }s\not \in A. \end{array}\right. } \end{aligned}$$

By construction, \(p_E\cdot a > (p|E)\cdot a\), \((p|E)\cdot b > p_E \cdot b\), \(p_E\cdot a \ge V_E(a) = 0\), 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 \(q\in \mathrm {cl}(\mathrm {co}\, \varDelta )\),

$$\begin{aligned} u(fEl(q))= & {} q(E) u(f(q|E)) + (1-q(E))u(l)\\\ge & {} q(E) V_E(f) + (1-q(E))u(l) = u(l). \end{aligned}$$

As u(f(q|E)) can be arbitrarily close to \(V_E(f)\),

$$\begin{aligned} M(fEl) = \inf _{q\in \varDelta } u(fEl(q)) = u(l). \end{aligned}$$

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 \((p|E)\cdot u(f) > (p|E)\cdot u(g)\), 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, \(p_E = p|E\).

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, \(p_E \ne p|E\) 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,

$$\begin{aligned} \lambda \mathcal {P}(E) + (1-\lambda ) \{p|E\} \subset (\varepsilon \mathcal {P}+ (1-\varepsilon )\{ p\}) | E. \end{aligned}$$

Direct computation shows that

$$\begin{aligned} (\varepsilon \mathcal {P}+ (1-\varepsilon ) \{p\}) | E = \frac{\varepsilon }{(1-\varepsilon )p(E) + \varepsilon } \mathcal {P}(E) + (1-\varepsilon ) \frac{p(E)}{(1-\varepsilon ) p(E) + \varepsilon } \{p|E\}. \end{aligned}$$

The inequality

$$\begin{aligned} 1- \lambda \ge (1-\varepsilon ) \frac{p(E)}{(1-\varepsilon ) p(E) + \varepsilon } \end{aligned}$$

follows.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kopylov, I. Subjective probability, confidence, and Bayesian updating. Econ Theory 62, 635–658 (2016). https://doi.org/10.1007/s00199-015-0929-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00199-015-0929-0

Keywords

JEL Classification

Navigation