11 From Ordinal to Cardinal Utility

It is not hard to see that if $(P, I)$ is representable by a utility function (see Definition 10.1), then any utility function resulting the same ordering of the objects also represents $(P, I)$. For instance, suppose that $X=\{a, b, c\}$ and $(P, I)$ are rational preferences with $a\mathrel{P} b \mathrel{P} c$ (and $I=\{(a, a), (b, b), (c, c)\}$). Then, the following table gives three utility functions that represent $(P, I)$

	$u_1$	$u_2$	$u_3$
$a$	$3$	$1000$	1.0
$b$	$2$	$900$	0.8
$c$	$1$	$-100$	0.1

Indeed, any function $u:X \rightarrow \mathbb{R}$ such that $u(a) > u(b) > u(c)$ represents $(P, I)$. This means that the only information about the decision maker’s attitude towards the objects that these utility functions provide is the ordering of the objects. In particular, one cannot conclude the following about the decision maker’s preferences about $a$, $b$ and $c$:

The decision maker ranks $b$ closer to $a$ than to $c$ (i.e., the difference in utility of $a$ and $b$ is smaller than the difference in utility of $b$ and $c$).
The utility of $b$ is 8-times the utility assigned to $c$.

Even though statement 1 is true of the utility functions $u_2$ and $u_3$, it is not true of $u_1$. Even though statement 3 is true of the utility function $u_3$, it is not true of the utility functions $u_1$ and $u_2$.

Utility functions that only represent the decision maker’s ordering of the objects are called ordinal utility functions. In many choice situations, utility functions are intended to represent more than simply the ordering of the items. Such utility functions are called cardinal utility functions. There are different types of cardinal utility functions characterized by the what types of comparisons are meaningful:

Interval scale: Quantitative comparisons of objects accurately reflects differences between objects. For instance, temperature is an interval scale: the difference between $75^\circ$F and $70^\circ$F is the same as the difference between $30^\circ$F and $25^\circ$F. However, $70^\circ$F ($=21.11^\circ$C) is not twice as hot as $35^\circ$F ($=1.67^\circ$C).
Ratio scale: Quantitative comparisons of objects accurately reflects ratios between objects. For instance, weight is a ratio scale: 10lb ($=4.53592 kg$) is twice as much as 5lb ($=2.26796 kg$).

11.1 Linear Transformations

Utilities that are related by linear transformations will play an important role in this course.

Definition 11.1 Suppose that $u:X\rightarrow\mathbb{R}$ and $u':X\rightarrow\mathbb{R}$ are two utility functions on a set $X$. We say that $u'$ is a linear transformation of $u$ if there are $\alpha>0$ and $\beta\in \mathbb{R}$ such that for all $x\in X$, \[u'(x) = \alpha u(x) + \beta.\]

To illustrate the importance of Definition 11.1 for Rational Choice Theory, suppose that Ann’s strict preference on the set $\{a, b, c\}$ is: \[a\mathrel{P} b\mathrel{P} c.\] An important question in Rational Choice Theory is how to infer Ann’s preference over lotteries with prizes from the set $X=\{a, b, c\}$ given the strict preference over the set $X$. Clearly, she prefers the lottery $[a:0.1, b:0.9]$ to the lottery $[c:1.0]$ (since $a$ and $b$ are both strictly preferred to $c$, she would prefer a chance to get either $a$ or $b$ to receiving $c$ for sure). However, given only the information about Ann’s strict preference over $X$ we cannot infer how she would rank the lotteries $[a:0.5, c:0.5]$ and $[b:1]$. To infer Ann’s preference between the lotteries $[a:0.5, c:0.5]$ and $[b:1]$, we need to know whether Ann ranks $b$ closer to $a$ than to $c$ or ranks $b$ closer to $c$ than to $a$. That is, we need to know how Ann compares the difference between $a$ and $b$ with the difference between $b$ and $c$. If $u$ is a utility function representing Ann’s preferences, we are interested in how she compares differences in the utilities assigned to $a$, $b$ and $c$: \[\mathrlap{\overbrace{\phantom{a\qquad b}}^{\text{$u(a) - u(b)$}}}a\qquad \mathrlap{\underbrace{\phantom{b\qquad c}}_{\text{$u(b) - u(c)$}}} b \qquad c.\]

If all we know is that $u$ represents Ann’s preference over $X$, then it is not meaningful to compare the utility differences $u(a)-u(b)$ with $u(b)-u(c)$. The problem is that there are different utility functions that both represent Ann’s preferences, but differ in the ranking of the differences in utilities. For instance, both of the utility functions $u:X\rightarrow\mathbb{R}$ with $u(a)=2, u(b)=1, u(c)=0$ and $u':X\rightarrow\mathbb{R}$ with $u'(a)=5, u'(b)=4, u'(c)=1$ represent Ann’s preference on $X$. However, according to $u$, $b$ is ranked evenly between $a$ and $c$ since $u(a)-u(b)=u(b)-u(c)$, but according to $u'$, $b$ is ranked closer to $a$ than to $c$ since $u(a)-u(b) < u(b)-u(c)$.

The crucial observation is that if all the utility functions that represent a decision maker’s preferences are related by linear transformations, then comparisons of differences between utilities are meaningful.

Proposition 11.1 Suppose that $u:X\rightarrow\mathbb{R}$ and $u':X\rightarrow\mathbb{R}$ is a linear transformation of $u$. Then, for all $a, b, c, d\in X$,

if $u(a)-u(b) < u(c) -u(d)$, then $u'(a)-u'(b)< u'(c)-u'(d)$;
if $u(a)-u(b) > u(c) -u(d)$, then $u'(a)-u'(b) > u'(c)-u'(d)$; and
if $u(a)-u(b) = u(c) -u(d)$, then $u'(a)-u'(b) = u'(c)-u'(d)$.

Proof. We only prove item . since the proofs of 2 and 3 are similar. Suppose that $u':X\rightarrow\mathbb{R}$ is a linear transformation of $u:X\rightarrow\mathbb{R}$ and $a, b, c, d\in X$. Then there are real numbers $\alpha>0$ and $\beta$ such that for all $x\in X$, $u'(x)=\alpha u(x) + \beta$. Suppose that $u(a) - u(b) < u(c) - u(d)$. Then, since $\alpha>0$, we have that $\alpha(u(a) - u(b)) > \alpha(u(c) - u(d))$. We show that $u'(a) - u'(b) < u'(c) - u'(d)$ as follows:

\[\begin{align*} u'(a) - u'(b) &= (\alpha u(a) +\beta) - (\alpha u(b) + \beta)\\ &= \alpha (u(a) - u(b)) \\ &< \alpha(u(c) - u(d))\\ &= (\alpha u(c) + \beta) -(\alpha u(d) +\beta)\\ &= u'(c) - u'(d) \end{align*}\]

11.2 Exercises

Suppose that $X=\{a, b, c\}$ and that $u:X\rightarrow \mathbb{R}$ with $u(a)= 3$, $u(b)=2$ and $u(c)=0$. Which of the following utilities are linear transformations of $u$?

$a$ $b$ $c$

$u_1$ $32$ $22$ $2$

$u_2$ $0.75$ $0.5$ $0$

$u_3$ $9$ $4$ $0$

$u_4$ $-1$ $0$ $2$

	\(u_1\)	\(u_2\)	\(u_3\)
\(a\)	\(3\)	\(1000\)	1.0
\(b\)	\(2\)	\(900\)	0.8
\(c\)	\(1\)	\(-100\)	0.1

	\(a\)	\(b\)	\(c\)
\(u_1\)	\(32\)	\(22\)	\(2\)
\(u_2\)	\(0.75\)	\(0.5\)	\(0\)
\(u_3\)	\(9\)	\(4\)	\(0\)
\(u_4\)	\(-1\)	\(0\)	\(2\)