A belief-based definition of ambiguity aversion


This paper proposes a notion of ambiguity aversion and characterizes it in the context of biseparable preferences, which include many popular ambiguity models in the literature. The defined properties suggest that ambiguity aversion is characterized by the properties of its capacity. This formalizes a sharp distinction between ambiguity and risk aversion, where risk aversion is characterized by the properties of its utility index and its probability weighting function.

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  1. 1.

    See Ghirardato and Marinacci (2001) for technique details.

  2. 2.

    Another possible choice of ambiguity neutrality in the context of biseparable preferences are SEU as in Ghirardato and Marinacci (2002).

  3. 3.

    To formally demonstrate that CEU and MEU are special cases of biseparable preference model, we need some additional structural assumption. See Ghirardato and Marinacci (2001) for technique details.

  4. 4.

    It is straightforward that CEU is monotonic. Equation (1) holds with \(\rho =\nu \) pointwise.

  5. 5.

    Actually how to select a ambiguity neutrality is a topic widely debated. Epstein (1999) adopts probabilistically sophisticated preferences as ambiguity neutrality. Ghirardato and Marinacci (2002) adopt SEU as ambiguity neutrality. We refer to their paper for detailed discussion.

  6. 6.

    The behavioral foundation for our definitions is not fully developed as those aforementioned studies. This remains an open question for further study.

  7. 7.

    The readers can consult Abdellaoui et al. (2005) for more details and for implementation in the laboratories.


  1. Abdellaoui, M., Vossmann, F., & Weber, M. (2005). Choice-based elicitation and decomposition of decision weights for gains and losses under uncertainty. Management Science, 51(9), 1384–1399.

    Article  Google Scholar 

  2. Abdellaoui, M., Baillon, A., Placido, L., & Wakker, P. (2011). The rich domain of uncertainty: Source functions and their experimental implementation. American Economic Review, 101(2), 695–723.

    Article  Google Scholar 

  3. Anscombe, F., & Aumann, R. (1963). A definition of subjective probability. Annals of Mathematical Statistics, 34, 199–205.

    Article  Google Scholar 

  4. Baillon, A., Driesen, B., & Wakker, P. (2012). Relative concave utility for risk and ambiguity. Games and Economic Behavior, 75(2), 481–489.

    Article  Google Scholar 

  5. Chateauneuf, A., Cohen, M., & Meilijson, I. (2005). More pessimism than greediness: A characterization of monotone risk aversion in the rank-dependent expected utility model. Economic Theory, 25(3), 649–667.

    Article  Google Scholar 

  6. Chew, S. H., & Sagi, J. (2008). Small worlds: Modeling attitudes toward sources of uncertainty. Journal of Economic Theory, 139(1), 1–24.

    Article  Google Scholar 

  7. Chew, S. H., Karni, E., & Safra, Z. (1987). Risk aversion in the theory of expected utility with rank dependent probabilities. Journal of Economic Theory, 42(2), 370–381.

    Article  Google Scholar 

  8. Choquet, G. (1953). Theory of capacities. Annales de l’institut Fourier, 5, 131–295.

    Article  Google Scholar 

  9. Ellsberg, D. (1961). Risk, ambiguity, and the savage axioms. The Quarterly Journal of Economics, 75(4), 643–669.

    Article  Google Scholar 

  10. Epstein, L. (1999). A definition of uncertainty aversion. The Review of Economic Studies, 66(3), 579–608.

    Article  Google Scholar 

  11. Ergin, H., & Gul, F. (2009). A theory of subjective compound lotteries. Journal of Economic Theory, 144(3), 899–929.

    Article  Google Scholar 

  12. Ghirardato, P., & Marinacci, M. (2001). Risk, ambiguity, and the separation of utility and beliefs. Mathematics of Operations Research, 26(4), 864–890.

    Article  Google Scholar 

  13. Ghirardato, P., & Marinacci, M. (2002). Ambiguity made precise: A comparative foundation. Journal of Economic Theory, 102(2), 251–289.

    Article  Google Scholar 

  14. Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with non-unique prior. Journal of Mathematical Economics, 18(2), 141–153.

    Article  Google Scholar 

  15. Keynes, J. (1921). A treatise on probability. London: Macmillan.

    Google Scholar 

  16. Klibanoff, P., Marinacci, M., & Mukerji, S. (2005). A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.

    Article  Google Scholar 

  17. Knight, F. (1921). Risk, uncertainty and profit. Boston: Houghton Mifflin.

    Google Scholar 

  18. Maccheroni, F., Marinacci, M., & Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica, 74(6), 1447–1498.

    Article  Google Scholar 

  19. Machina, M., & Schmeidler, D. (1992). A more robust definition of subjective probability. Econometrica, 60(4), 745–780.

    Article  Google Scholar 

  20. Machina, M., & Schmeidler, D. (1995). Bayes without Bernoulli: Simple conditions for probabilistically sophisticated choice. Journal of Economic Theory, 67(1), 106–128.

    Article  Google Scholar 

  21. Marinacci, M. (2002). Probabilistic sophistication and multiple priors. Econometrica, 70(2), 755–764.

    Article  Google Scholar 

  22. Quiggin, J. (1991). Comparative statics for rank-dependent expected utility theory. Journal of Risk and Uncertainty, 4(4), 339–350.

    Article  Google Scholar 

  23. Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57(3), 571–587.

    Article  Google Scholar 

  24. Siniscalchi, M. (2009). Vector expected utility and attitudes toward variation. Econometrica, 77(3), 801–855.

    Article  Google Scholar 

  25. Strzalecki, T. (2011). Axiomatic foundations of multiplier preferences. Econometrica, 79(1), 47–73.

    Article  Google Scholar 

  26. Wakker, P. (2004). On the composition of risk preference and belief. Psychological Review, 111(1), 236–241.

    Article  Google Scholar 

  27. Wakker, P., & Deneffe, D. (1996). Eliciting von Neumann–Morgenstern utilities when probabilities are distorted or unknown. Management Science, 42(8), 1131–1150.

  28. Yaari, M. (1969). Some remarks on measures of risk aversion and on their uses. Journal of Economic Theory, 1(3), 315–329.

    Article  Google Scholar 

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I am deeply indebted to David Schmeidler for his guidance and generous advices. I also wish to thank Ani Guerdjikova, Edi Karni, Dan Levin and Horst Zank for helpful comments and discussion. I am very much thankful for the editor and two referees, who help improve this paper greatly.

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Correspondence to Xiangyu Qu.

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This research is supported by the grant ANR 2011 CHEX 006 01 of the Agence Nationale de Recherche.

Appendix: Proofs

Appendix: Proofs

Proof of Proposition 1

Suppose that \(\succsim \) is a PSBP. There exist outcomes \(z^*\succ z\) in \(X\) by assumption. Since \(\succsim \) is probabilistically sophisticated, by Machina and Schmeidler (1992) theorem, there exists a unique, finitely additive non-atomic probability measure \(p\) on \(\fancyscript{A}\) such that for any \(E,F\in \fancyscript{A}\),

$$\begin{aligned} z^*Ez\succsim z^*Fz\Leftrightarrow p(E)\ge p(F). \end{aligned}$$

Furthermore, \(\succsim \) is also biseparable. Its associated capacity \(\rho \) has the property that for any \(E,F\in \fancyscript{A}\),

$$\begin{aligned} z^*Ez\succsim z^*Fz\Leftrightarrow \rho (E)\ge \rho (F). \end{aligned}$$

Therefore, for any \(E,F\in \fancyscript{A}\),

$$\begin{aligned} p(E)\ge p(F)\Leftrightarrow \rho (E)\ge \rho (F). \end{aligned}$$

Thus, there is a monotone function \(\phi :[0,1]\rightarrow [0,1]\) with \(\phi (0)=0\) and \(\phi (1)=1\) such that \(\rho =\phi \circ p\). This shows that \(\rho \) is a distorted probability. \(\square \)

Proof of Theorem 1

I first prove the part (a). Given a representation \(V\) of a biseparable preference \(\succsim \) with utility index \(u\) and capacity \(\rho \). Suppose that \(\succsim \) is ambiguity averse. Consider a PSBP induced by \(u\) and \(\phi \circ p\) where \(\phi \circ p\ge \rho \) pointwise. For every \(x\in X\) and \(f\in \fancyscript{F}\),

where \(u(x)=V(\bar{x})\) for all \(x\in X\). Hence \(p\in core(\phi ^{-1}(\rho ))\).

Consider set \(M(\succsim )\). Following the above statement, it is straightforward that ambiguity aversion implies the non-emptiness of \(M(\succsim )\). Now we show the reverse part by assuming that \(M(\succsim )\) is non-empty. According to the definition of \(M(\succsim )\), we have for all \(x\in X\) and \(f\in \fancyscript{F}\),

Therefore, \(\succsim \) is ambiguity aversion.

The proof of (b) is obvious. \(\square \)

Proof of Proposition 2

The proofs of (ii) and (iii) follow easily from (i). Hence, we only prove (i) here. Since CEU is also biseparable, that ambiguity aversion implies non-emptiness of \(core(\phi ^{-1}(\nu ))\) follows directly from Theorem 1. we now show the reverse part. Suppose that \(core(\phi ^{-1}(\nu ))\ne \emptyset \). Let \(V\) be a representation of \(\succsim \) with utility index \(u\) and probability measure p, where \(p\in core(\phi ^{-1}(\nu ))\). So for all \(x\in X\) and \(f\in \fancyscript{F}\),

This shows that this CEU preference is ambiguity averse. \(\square \)

Proof of Proposition 3

The proofs of (ii) and (iii) follow easily from (i). We only show part (i). Since MEU is also biseparable, it is also straightforward to see that \(K\) is non-empty. To show the reverse, suppose that \(K\ne \emptyset \). Choose a probability measure \(p\in K\). Then for all \(x\in X\) and \(f\in \fancyscript{F}\),

$$\begin{aligned} u(x)\ge \int \limits _S u(f(s))\mathrm {d}p(s)\Rightarrow u(x)\ge \min _{q\in K}\int \limits _S u(f(s))\mathrm {d}q(s). \end{aligned}$$

Hence, the MEU is ambiguity averse. \(\square \)

Proof of Proposition 4

Let \(\succsim _1\) and \(\succsim _2\) be two biseparable preferences with utility index \(u_1\) and \(u_2\) and capacities \(\rho _1\) and \(\rho _2\), respectively. Suppose that \(\succsim _2\) is more ambiguity averse than \(\succsim _1\). We need to show that \(\rho _1(A)\ge \rho _2(A)\) for all \(A\in \fancyscript{A}\). Consider an intermediary biseparable preference \(\succsim ^*\) with corresponding utility index \(u^*=u_2\) and capacity \(\rho ^*=\rho _1\) eventwise So \(\succsim ^*\) and \(\succsim _1\) are ambiguity equivalent, and \(\succsim ^*\) and \(\succsim _1\) are cardinal equivalent. Since \(\succsim _2\) is more ambiguity averse than \(\succsim _1\), we have \(V^*(f)\ge V_2(f)\) for all \(f\in \fancyscript{F}\) by definition of comparative uncertainty.

Let \(x,y\in X\) with \(u_2(x)>u_2(y)\) and let \(A\in \fancyscript{A}\). For the binary act \(xAy\), we have \(V^*(xAy)\ge V_2(xAy)\). That is, \(u^*(x)\rho ^*(A)+u^*(y)(1-\rho ^*(A))\ge u_2(x)\rho _2(A)+u_2(y)(1-\rho _2(A))\). So \(\rho ^*(A)\ge \rho _2(A)\). Therefore, \(\rho _1(A)\ge \rho _2(A)\). Since the above argument is true for any \(A\in \fancyscript{A}\), we have \(\rho _1\ge \rho _2\) eventwise. \(\square \)

Proof of Theorem 2

First, we prove statement (i). Let \(\succsim _1\) and \(\succsim _2\) be CEU orderings with capacities \(\nu _1\) and \(\nu _2\), and utility indexes \(u_1\) and \(u_2\), respectively. The Proposition 4 automatically implies the IF part. We only need to prove the ONLY IF part. Suppose \(\nu _1(A)\ge \nu _2(A)\) for all \(A\in \fancyscript{A}\).

Consider a biseparable preference \(\succsim ^*\) with canonical representation \(V^*\), which is defined as

To prove that \(\succsim _2\) is more ambiguity averse than \(\succsim _1\), it is sufficient to show that for all \(f\in \fancyscript{F}\), \(V^*(f)\ge V_2(f)\). To this end, note that \(V^*\) and \(V_2\) are cardinal equivalent. Without loss of generality, suppose act \(f\) is realized at \(n\) states, \(s_1,\ldots ,s_n\) and \(u(f(s_1))\ge \cdots \ge u(f(s_n))\). Therefore,

$$\begin{aligned}&V^*(f)-V_2(f)\\&\qquad =\sum ^{n-1}_{i=1}(u_2(f(s_i))-u_2(f(s_{i+1}))) [\nu _1(\{s_1,\ldots ,s_i\})-\nu _2(\{s_1,\ldots ,s_i\})]. \end{aligned}$$

Notice that the first and second terms in the parenthesis are both nonnegative. Therefore, \(V^*(f)\ge V_2(f)\). Hence, by definition \(\succsim _2\) is more ambiguity averse than \(\succsim _1\).

Now we prove statement (ii). Again the IF part is directly implied by Proposition 4. We only need to prove the ONLY IF part. Let \(\succsim _1\) and \(\succsim _2\) be MEU ordering with set of probabilities \(K_1\) and \(K_2\). Suppose that \(K_1\subseteq K_2\). Let \(\succsim ^*\) be a biseparable preference that can be represented by a canonical functional \(V^*(f)=\min _{p\in K_1}\int _S u_2(f(s))\mathrm {d}p\). So \(\succsim ^*\) and \(\succsim _1\) are ambiguity equivalent. The biseparable preferences \(\succsim ^*\) and \(\succsim _2\) are cardinal equivalent. To prove that \(\succsim _2\) is more uncertainty averse than \(\succsim _1\), we need to show that \(V^*(f)\ge V_2(f)\) for all \(f\in \fancyscript{F}\). But this is straight forward because the probability that minimizes the expected utility of act \(f\) with respect to \(\succsim ^*\) is always in the set of probabilities \(K_2\). \(\square \)

Proof of Theorem 3

We first prove (i). Let \(\succsim _1\) be a MEU preference represented by set of probabilities \(K\) and utility index \(u_1\). Let \(\succsim _2\) be a CEU preference, represented by capacity \(\nu \) and utility index \(u_2\).

Suppose that \(K\subseteq core(\phi \circ \nu )\), where \(\phi :[0,1]\rightarrow [0,1]\) is bijective and increasing. We want to show that \(\succsim _2\) is more ambiguity averse than \(\succsim _1\). To this end, let \(\succsim ^*\) be an intermediary preference such that it has CEU preference represented by capacity \(\nu \) and utility index \(u^*\), where \(u^*= u_1\).

Let \(x\in X\) and \(f\in \fancyscript{F}\) be such that \(x\succsim _1 f\). Then

$$\begin{aligned} u_1(x)\ge \min _{p\in K}\int u_1(f)\mathrm {d}p. \end{aligned}$$

Since \(K\subseteq core(\phi \circ \nu )\), for each \(p\in K\), we have


$$\begin{aligned} \min _{p\in K}\int u_1(f)\mathrm {d}p\ge \int u^*(f)\mathrm {d}\nu . \end{aligned}$$

Hence, \(\succsim _2\) is more uncertainty averse than \(\succsim _1\) by definition.

Now suppose that \(\succsim _2\) is more ambiguity averse than \(\succsim _1\). We want to show that \(K\subseteq core(\phi \circ \nu )\) for some bijective and increasing function \(\phi :[0,1]\rightarrow [0,1]\). Suppose this is not the case. Then there exists a \(p\) in \(K\) such that \(p\notin core(\nu )\), where \(\phi \) is an identity function. That is, there exists an event \(A\) in \(\fancyscript{A}\) such that \(p(A)<\nu (A)\). Let \(x,y\in X\) be such that \(x\succsim _1 y\). Consider a binary act \(xAy\). It is not hard to see that

$$\begin{aligned} \min _{p\in K}\int u_1(xAy)\mathrm {d}p<u_1(x)p(A)+u_1(y)(1-p(A)). \end{aligned}$$


By continuity, there exists a \(z\in X\) such that \(u_1(z)=u_1(x)p(A)+u_1(y)(1-p(A))\). Obviously \(u_1(z)\ge \min _{p\in K}\int u_1(xAy)\mathrm {d}p\). But . It contradicts that \(\succsim _2\) is more ambiguity averse than \(\succsim _1\). Hence, we have \(K\subseteq core(\phi \circ \nu )\).

The proof of (ii) is similar. \(\square \)

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Qu, X. A belief-based definition of ambiguity aversion. Theory Decis 79, 15–30 (2015). https://doi.org/10.1007/s11238-014-9452-1

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  • Ambiguity aversion
  • Biseparable preference
  • Maxmin expected utility
  • Choquet expected utility

JEL Classification

  • D80
  • D81