A measure of ambiguity (Knightian uncertainty)

Abstract

Uncertain or ambiguous events cannot be objectively measured by probabilities, i.e. different decision-makers may disagree about their likelihood of occurrence. This paper proposes a new decision-theoretical approach on how to measure ambiguity (Knightian uncertainty) that is analogous to axiomatic risk measurement in finance. A decision-theoretical measure of ambiguity is a function from choice alternatives (acts) to non-negative real numbers. Our proposed measure of ambiguity is derived from a novel assumption that ambiguity of any choice alternative can be decomposed into a left-tail ambiguity (uncertainty in the realization of relatively undesirable outcomes) and a right-tail ambiguity (uncertainty in the realization of relatively desirable outcomes). This decomposability assumption is combined with two standard assumptions: ambiguity sources (events) are independent (separable) from outcomes (consequences) and any elementary increase in uncertainty (increasing a more desirable outcome in a binary act) necessarily increases ambiguity.

Appendix

1.1 Proof of proposition 1

Let us first demonstrate that uncertainty measure (1) satisfies assumptions 1–3.

Necessity of assumption 1 If outcome $x \in X$ lies outside the range of outcomes of act $f \in {\mathcal{F}}$ then

$$ \beta \left( {f \wedge x{ }} \right) + \beta \left( {f \vee x} \right) = \beta \left( {f{ }} \right) + \beta \left( {\varvec{x}} \right) = \beta \left( f \right) $$

(3)

with the second equality due to the fact that $\beta \left( {\varvec{x}} \right) = 0$ for measure (1). If outcome $x \in X$ lies inside the range of outcomes of act $f \in {\mathcal{F}}$ then let $x_{i} \in X$ denote the smallest outcome of act $f \in {\mathcal{F}}$ that is greater than $x \in X$. Act $f \wedge x$ is then $\left\{ {x_{1} ,E_{1} ; \ldots ;x_{i - 1} ,E_{i - 1} ;x,{ }\mathop \cup \limits_{j = i}^{n} E_{j} } \right\}$ and act $f \vee x$ is then $\left\{ {x,{ }\mathop \cup \limits_{j = 1}^{i - 1} E_{j} ;x_{i} ,E_{i} ; \ldots ;x_{n} ,E_{n} } \right\}$. Using formula (1) we can write the uncertainty measure of act $f \wedge x$ as

$$ \beta \left( {f \wedge x{ }} \right) = \mathop \sum \limits_{j = 2}^{i - 1} \left[ {v\left( {x_{j} } \right) - v\left( {x_{j - 1} } \right)} \right]\varphi \left( {\mathop \cup \limits_{k = j}^{n} E_{k} } \right) + \left[ {v\left( x \right) - v\left( {x_{i - 1} } \right)} \right]\varphi \left( {\mathop \cup \limits_{k = i}^{n} E_{k} } \right) $$

(4)

and the uncertainty measure of act $f \vee x$ as

$$ \beta \left( {f \vee x} \right) = \mathop \sum \limits_{j = i + 1}^{n} \left[ {v\left( {x_{j} } \right) - v\left( {x_{j - 1} } \right)} \right]\varphi \left( {\mathop \cup \limits_{k = j}^{n} E_{k} } \right) + \left[ {v\left( {x_{i} } \right) - v\left( x \right)} \right]\varphi \left( {\mathop \cup \limits_{k = i}^{n} E_{k} } \right) $$

(5)

Summing Eqs. (4) and (5) together yields

$$ \beta \left( {f \wedge x{ }} \right) + \beta \left( {f \vee x} \right) = \mathop \sum \limits_{j = 2}^{n} \left[ {v\left( {x_{j} } \right) - v\left( {x_{j - 1} } \right)} \right]\varphi \left( {\mathop \cup \limits_{k = j}^{n} E_{k} } \right) = \beta \left( f \right) $$

(6)

Therefore, uncertainty measure (1) satisfies assumption 1.

Necessity of assumption 2 Let us consider any three events $A,B,C \in$  ∑ such that $A,B,C \notin {\mathbb{C}}$ and any four outcomes $x,y,z,w \in X$ such that x > w, y > w and z > w. If uncertainty measure takes form (1) then equality $\beta \left( {x,A,w} \right)$$= \beta \left( {y,B,w} \right)$ can be written as

$$ \left[ {v\left( x \right) - v\left( w \right)} \right]\varphi \left( A \right) = \left[ {v\left( y \right) - v\left( w \right)} \right]\varphi \left( B \right) $$

(7)

and equality $\beta \left( {y,C,w} \right) = \beta \left( {z,A,w} \right)$ can be written as

$$ \left[ {v\left( y \right) - v\left( w \right)} \right]\varphi \left( C \right) = \left[ {v\left( z \right) - v\left( w \right)} \right]\varphi \left( A \right) $$

(8)

Multiplying Eqs. (7) by (8) and cancelling common terms⁸ yields Eq. (9).

$$ \left[ {v\left( x \right) - v\left( w \right)} \right]\varphi \left( C \right) = \left[ {v\left( z \right) - v\left( w \right)} \right]\varphi \left( B \right) $$

(9)

According to formula (1), the right-hand side of Eq. (9) is $\beta \left( {z,B,w} \right)$ and the left-hand side of Eq. (9) is $\beta \left( {x,C,w} \right)$. Therefore, uncertainty measure (1) satisfies assumption 2.

Necessity of assumption 3 Let us consider any three outcomes $x,y,z \in {\text{X}}$ such that $x > y > z$ and any event $E \in$∑. If uncertainty measure takes form (1) then inequality $\beta \left( {x,E,z} \right) \ge \beta \left( {y,E,z} \right)$ holds whenever

$$ \left[ {v\left( x \right) - v\left( z \right)} \right]\varphi \left( E \right) \ge \left[ {v\left( y \right) - v\left( z \right)} \right]\varphi \left( E \right) $$

(10)

If $x > y$ and desirability function is increasing then $v\left( x \right) > v\left( y \right)$. Thus, inequality (10) holds with a strict inequality if event $E \notin {\mathbb{C}}$. If $E \in {\mathbb{C}}$ then $\varphi \left( E \right) = 0$ and both sides of inequality (10) are equal to zero. Therefore, uncertainty measure (1) satisfies assumption 3.

Let us now prove the sufficiency of assumptions 1–3.

Setting outcome $x = x_{2}$ in assumption 1 yields

$$ \beta \left( f \right) = \beta \left( {x_{1} ,E_{2} ,x_{2} } \right) + \beta \left( {x_{2} ,E_{1} \cup E_{2} ; \ldots ;x_{n} ,E_{n} } \right) $$

(11)

Setting outcome $x = x_{3}$ in assumption 1 yields

$$ \beta \left( {x_{2} ,E_{1} \cup E_{2} ; \ldots ;x_{n} ,E_{n} } \right) = \beta \left( {x_{2} ,E_{1} \cup E_{2} ,x_{3} } \right) + \beta \left( {x_{3} ,E_{1} \cup E_{2} \cup E_{3} ; \ldots ;x_{n} ,E_{n} } \right) $$

(12)

Plugging Eqs. (12) into (11) yields

$$ \beta \left( f \right) = \beta \left( {x_{1} ,E_{2} ,x_{2} } \right) + \beta \left( {x_{2} ,E_{1} \cup E_{2} ,x_{3} } \right) + \beta \left( {x_{3} ,E_{1} \cup E_{2} \cup E_{3} ; \ldots ;x_{n} ,E_{n} } \right) $$

(13)

Iterating the above procedure for outcomes $x_{4} , \ldots ,x_{n - 1}$ we get that uncertainty of any act $f \in {\mathcal{F}}$ can be decomposed into uncertainties of a chain of binary acts:

$$ \beta \left( f \right) = \mathop \sum \limits_{j = 1}^{n - 1} \beta \left( {x_{j} ,\mathop \cup \limits_{i = 1}^{j} E_{i} ,x_{j + 1} } \right) $$

(14)

For any binary act $\left\{ {y,E,z} \right\}$, z > y, and outcome $x \in \left( {y,z} \right)$ assumption 1 implies

$$ \beta \left( {y,E,z} \right) = \beta \left( {y,E,x} \right) + \beta \left( {x,E,z} \right) $$

(15)

Equation (15) can be rearranged as Eq. (16).

$$ \beta \left( {x,E,z} \right) = \beta \left( {y,E,z} \right) - \beta \left( {y,E,x} \right) $$

(16)

Equation (16) is known as additive Sincov functional equation (e.gAczél 1966; Cинцoв 1903) and it is solved as follows. Since the left-hand side of Eq. (16) is independent of outcome y then the difference of two uncertainty measures on the right-hand side of Eq. (16) must be also independent of y. In other words, we can evaluate the difference of two uncertainty measures on the right-hand side of Eq. (16) for any fixed outcome y. Therefore, uncertainty of a binary act can be written as

$$ \beta \left( {x,E,z} \right) = \rho \left( {E,z} \right) - \rho \left( {E,x} \right) $$

(17)

where $\rho \left( {E,z} \right) \equiv \beta \left( {\overline{y},E,z} \right)$ for some fixed outcome $\overline{y} \in X$. Using Eq. (17) we can rewrite Eq. (14) as follow

$$ \beta \left( f \right) = \mathop \sum \limits_{j = 1}^{n - 1} \left[ {\rho \left( {\mathop \cup \limits_{i = 1}^{j} E_{i} ,x_{j + 1} } \right) - \rho \left( {\mathop \cup \limits_{i = 1}^{j} E_{i} ,x_{j} } \right)} \right] $$

(18)

where $\rho :{ }$ ∑$\times X \to {\mathbb{R}}_{ + }$ is an arbitrary function.

If assumption 2 holds then according to theorem 1 in Debreu (1960, p. 18) uncertainty of any binary act can be written as a multiplicatively⁹ separable function of sources of uncertainty (events) and outcomes. In other words, there exist functions $v_{1} :XxX \to {\mathbb{R}}$ and $\varphi_{1} :$ ∑$xX \to {\mathbb{R}}$ such that

$$ \beta \left( {x,A,w} \right) = v_{1} \left( {x,w} \right)*\varphi_{1} \left( {A,w} \right) $$

(19)

for all x > w and $A \in$. Using Eqs. (17) and (19) we can then write Eq. (20).

$$ \beta \left( {x,E,z} \right) = \rho \left( {E,z} \right) - \rho \left( {E,x} \right) = \beta \left( {z,S\backslash E,x} \right) = v_{1} \left( {z,x} \right)*\varphi_{1} \left( {S\backslash E,x} \right) $$

(20)

Rearranging Eq. (20) yields (21).

$$ \rho \left( {E,z} \right) = \rho \left( {E,x} \right) + v_{1} \left( {z,x} \right)*\varphi_{1} \left( {S\backslash E,x} \right) $$

(21)

The left-hand side of Eq. (21) is independent of outcome x. Hence, the whole right-hand side of Eq. (21) can be evaluated for any fixed value of outcome x so that

$$ \rho \left( {E,z} \right) = \tilde{\rho }\left( E \right) + v\left( z \right)*\varphi \left( {S\backslash E} \right) $$

(22)

where $\tilde{\rho }\left( E \right) \equiv \rho \left( {E,\overline{x}} \right)$, $v\left( z \right) \equiv v_{1} \left( {z,\overline{x}} \right)$ and $\varphi \left( {S\backslash E} \right) \equiv \varphi_{1} \left( {S\backslash E,\overline{x}} \right)$ for some fixed outcome $\overline{x} \in X$. Using Eq. (22) we can rewrite Eq. (18) as

$$ \beta \left( f \right) = \mathop \sum \limits_{j = 1}^{n - 1} \left[ {v\left( {x_{j + 1} } \right) - v\left( {x_{j} } \right)} \right]\varphi \left( {\mathop \cup \limits_{i = j + 1}^{n} E_{i} } \right) $$

(23)

If assumption 3 holds then for any three outcomes $x,y,z \in {\text{X}}$ such that $x > y > z$ and any event $E \in$∑ the following inequality holds

$$ \beta \left( {x,E,z} \right) \ge \beta \left( {y,E,z} \right) $$

(24)

Moreover, this inequality (24) holds with a strict inequality if and only if $E \notin {\mathbb{C}}$. Using Eq. (23) we can rewrite inequality (24) as inequality (10). Three cases are possible. First, if $\varphi \left( E \right) > 0$ then we must have $v\left( x \right) > v\left( y \right)$ for all $x > y$. Second, if $\varphi \left( E \right) < 0$ then we must have $v\left( x \right) < v\left( y \right)$ for all $x > y$. Thus, either desirability function is increasing and $\varphi :$ ∑$\to {\mathbb{R}}_{ + }$ or desirability function is decreasing and $\varphi :$ ∑$\to {\mathbb{R}}_{ - }$. The latter case is easily transformed into the former by defining a new function $\tilde{v}\left( x \right) = - v\left( x \right)$ for all x ∊ X and $\tilde{\varphi }\left( E \right) = - \varphi \left( E \right)$ for all events E ∊ ∑. Third, if $\varphi \left( E \right) = 0$ then inequality (10) holds as equality. According to assumption 3 this is only possible when $E \in {\mathbb{C}}$. Thus, $\varphi \left( E \right) = 0$ for all events $E \in {\mathbb{C}}$. Q.E.D.

1.2 Proof of proposition 2

Let us consider an act f that constitutes an elementary increase in uncertainty over an act $g = \left( {x_{1} ,E_{1} ; \ldots ;x_{n} ,E_{n} } \right)$ in the sense of Grant and Quiggin (2005, p. 121, definition 1). In this case, act f takes the form

$$ f = \left( {x_{1} - b,E_{1} ; \ldots ;x_{i - 1} - b,E_{i - 1} ;x_{i} + a,E_{i} ; \ldots ;x_{n} + a,E_{n} } \right) $$

for some $i \in \left\{ {2, \ldots ,n - 1} \right\}$ and $a,b > 0$. Uncertainty measure (1) of such act f is given by

$$ \begin{aligned} \beta \left( f \right) & = \mathop \sum \limits_{j = 2}^{i - 1} \left[ {v\left( {x_{j} - b} \right) - v\left( {x_{j - 1} - b} \right)} \right]\varphi \left( {\mathop \cup \limits_{k = j}^{n} E_{k} } \right) + \left[ {v\left( {x_{i} + a} \right) - v\left( {x_{i - 1} - b} \right)} \right]\varphi \left( {\mathop \cup \limits_{k = i}^{n} E_{k} } \right) \\ & \;\;\; + \mathop \sum \limits_{j = i + 1}^{n} \left[ {v\left( {x_{j} + a} \right) - v\left( {x_{j - 1} + a} \right)} \right]\varphi \left( {\mathop \cup \limits_{k = j}^{n} E_{k} } \right) \\ \end{aligned} $$

For a linear desirability function, $v\left( x \right) = x$ this uncertainty measure simplifies into

$$ \begin{aligned} \beta \left( f \right) & = \mathop \sum \limits_{j = 2}^{n} \left[ {x_{j} - x_{j - 1} } \right]\varphi \left( {\mathop \cup \limits_{k = j}^{n} E_{k} } \right) + \left[ {a + b} \right]\varphi \left( {\mathop \cup \limits_{k = i}^{n} E_{k} } \right) \\ & = \beta \left( g \right) + \left[ {a + b} \right]\varphi \left( {\mathop \cup \limits_{k = i}^{n} E_{k} } \right) \\ \end{aligned} $$

If $\mathop \cup \limits_{k = i}^{n} E_{k} \in {\mathbb{C}}$ then $\varphi \left( {\mathop \cup \limits_{k = i}^{n} E_{k} } \right) = 0$ and $\beta \left( f \right) = \beta \left( g \right)$. If $\mathop \cup \limits_{k = i}^{n} E_{k} \notin {\mathbb{C}}$ then $\varphi \left( {\mathop \cup \limits_{k = i}^{n} E_{k} } \right) > 0$ and $\beta \left( f \right) > \beta \left( g \right)$ because $a + b > 0$. Thus, $\beta \left( f \right) \ge \beta \left( g \right)$ for a linear desirability function $v\left( x \right) = x$. Q.E.D.