A belief-based definition of ambiguity aversion

Abstract

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.

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

Therefore,

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

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

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

and

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. $\square$