10  Utility Functions

A utility function on a set \(X\) is a function \(u: X \rightarrow \mathbb{R}\) that assigns a real number to each outcome in \(X\). For each \(x \in X\), the utility assigned to \(x\) is denoted by \(u(x)\). This raises an important question: How should we interpret these numbers assigned to the elements of \(X\)? What do they reveal about the decision maker’s attitudes toward the outcomes in \(X\)? In this chapter, we will explore two key interpretations.

An important point is that for any utility function \(u: X \rightarrow \mathbb{R}\) on a set \(X\), we can define two relations, \(P_u \subseteq X \times X\) and \(I_u \subseteq X \times X\), as follows: for all \(x, y \in X\),

  1. \(x \mathrel{P_u} y\) if and only if \(u(x) > u(y)\); and
  2. \(x \mathrel{I_u} y\) if and only if \(u(x) = u(y)\).

It is straightforward to see that any utility function \(u: X \rightarrow \mathbb{R}\) on a set \(X\) generates a rational preference \((P_u, I_u)\) on \(X\).

10.1 Ordinal Utility Functions

An important observation is that different utility functions on a set \(X\) can generate the same rational preference over \(X\). For instance, suppose \(X = \{a, b, c\}\) and consider the following three utility functions, \(u_1\), \(u_2\), and \(u_3\), as shown in the table below:

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

Despite their differences in numerical values, all three utility functions produce the same rational preference over \(X\): \(a\) is strictly preferred to \(b\), \(b\) is strictly preferred to \(c\), and \(a\) is strictly preferred to \(c\). In fact, any utility function \(u: X \rightarrow \mathbb{R}\) that satisfies \(u(a) > u(b) > u(c)\) will represent this same preference.

This brings us to the concept of an ordinal utility function. A utility function is considered ordinal when the only information it conveys about the decision maker’s preferences is the relative ordering of the outcomes. The specific numerical values are irrelevant, as long as the ranking of the outcomes remains the same.

A consequence of this is that there is not a single ordinal utility function that represents a decision maker’s preferences on a set \(X\). Instead, if \(u\) is an ordinal utility function that represents the decision maker’s preferences on \(X\), then any utility function that generates the same ordering of the elements of \(X\) also represents the decision maker’s preferences.

10.1.1 Representing Rational Preferences

A notable fact is that ordinal utility functions and rational preferences are two equivalent ways to represent a decision maker’s preferences on a set \(X\). This equivalence can be made more precise with the following definition:

Definition 10.1 Suppose that \(X\) is a set \(P\subseteq X\times X\) and \(I\subseteq X\times X\) are two relations. We say that \((P, I)\) is representable when there is a function \(u_{P,I}:X\rightarrow \mathbb{R}\) such that, for all \(x, y\in X\):

  1. if \(x\mathrel{P} y\), then \(u_{P,I}(x) > u_{P,I}(y)\); and
  2. if \(x\mathrel{I} y\), then \(u_{P,I}(x) = u_{P,I}(y)\).

The central theorem that establishes the fundamental relationship between ordinal utility functions and rational preferences is:

Theorem 10.1 (Basic Representation Theorem) Suppose that \(X\) is a finite set and \(P\subseteq X\times X\) and \(I\subseteq X\times X\). Then, \((P, I)\) is a rational preference on \(X\) if, and only if, \((P, I)\) is representable by a utility function.

Warning

This proof can be skipped on a first reading.

Proof. We leave it to the reader to show that if \((P, I)\) is representable by a utility function, then \((P, I)\) is a rational preference on \(X\). That is, for all utility functions \(u:X\rightarrow \mathbb{R}\), \((P_u, I_u)\) is a rational preference.

We prove the following: For all \(n\in\mathbb{N}\), any rational preference \((P, I)\) on a set of size \(n\) is representable by a utility function \(u_{P,I}:X\rightarrow\mathbb{R}\). The proof is by induction on the size of the set of objects \(X\). The base case is when \(|X|=1\). In this case, \(X=\{a\}\) for some object \(a\). If \((P, I)\) is a rational preference on \(X\), then \(P=\varnothing\) and \(I=\{(a,a)\}\). Then, \(u_{P,I}(a)=0\) (any real number would work here) clearly represents \((P, I)\). The induction hypothesis is: if \(|X|=n\), then any rational preference \((P, I)\) on \(X\) is representable. Suppose that \(|X|=n+1\) and \((P, I)\) is a rational preference on \(X\). Then, \(X=Y\cup \{a\}\) for some object \(a\), where \(|Y|=n\). Note that the restriction of \((P, I)\) to \(Y\), denoted \((P_{Y}, I_{Y})\) where \(P_Y=P\cap (Y\times Y)\) and \(I_Y=I\cap (Y\times Y)\), is a rational preference on \(Y\). By the induction hypothesis, there is a utility function \(u_{P_Y, I_Y}:Y\rightarrow \mathbb{R}\) that represents \((P_Y, I_Y)\). We will show how to extend \(u_{P_Y, I_Y}\) to a utility function \(u_{P, I}:X\rightarrow \mathbb{R}\) that represents \((P, I)\). For all \(b\in Y\), let \(u_{P,I}(b)=u_{P_Y, I_Y}(b)\). For the object \(a\) (the unique object in \(X\) but not in \(Y\)), there are four cases:

  1. \(a \mathrel{P} b\) for all \(b\in Y\). Let \(u_{P,I}(a)=\max\{u_{P_Y, I_Y}(b)\ |\ b\in X'\}+1\).
  2. \(b \mathrel{P} a\) for all \(b\in Y\). Let \(u_{P,I}(a)=\min\{u_{P_Y, I_Y}(b)\ |\ b\in X'\}-1\).
  3. \(a \mathrel{I} b\) for some \(b\in Y\). Let \(u_{P,I}(a)=u_{P_Y, I_Y}(b)\).
  4. There are \(b_1, b_2\in Y\) such that \(b_1 \mathrel{P} a \mathrel{P} b_2\). Let \(u_{P,I}(a)=\frac{u_{P_Y, I_Y}(b_1)+ u_{P_Y, I_Y}(b_2)}{2}\).

Then, it is straightforward to show that \(u_{P, I}:X\rightarrow\mathbb{R}\) represents \((P, I)\) (the details are left to the reader).

The above proof can be extended to relations on infinite sets \(X\). However, this requires additional technical assumptions that are beyond the scope of this course.

10.2 Cardinal Utility Functions

The previous section introduced ordinal utility functions that represent a decision maker’s ranking of a set of objects. For example, each of the utility functions \(u_1\), \(u_2\), and \(u_3\) introduced earlier represents the same ranking of \(\{a, b, c\}\). However, in many choice situations, utility functions are intended to capture more than just the ordering of items.

With ordinal utility functions, we cannot conclude the following about a decision maker’s preferences among \(a\), \(b\), and \(c\):

  1. The decision maker ranks \(b\) closer to \(a\) than to \(c\) (i.e., the difference in utility between \(a\) and \(b\) is smaller than the difference between \(b\) and \(c\)).
  2. The utility of \(b\) is eight times the utility assigned to \(c\).

While statement 1 is true for the utility functions \(u_2\) and \(u_3\), it is not true for \(u_1\). Similarly, while statement 2 is true for the utility function \(u_3\), it is not true for \(u_1\) and \(u_2\).

When a utility function conveys more information about preferences than just the ranking of objects, we refer to it as a cardinal utility function. We distinguish between two types of cardinal utility functions based on the types of comparisons that are meaningful:

  • Interval Scale: This scale allows for meaningful comparisons of differences between objects. For example, temperature is measured on 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 is not twice as hot as \(35^\circ\)F. The key observation is that even though \(70\) is twice \(35\), temperature can be measured in either Fahrenheit or Celsius, where \(70^\circ\)F equals \(21.11^\circ\)C and \(35^\circ\)F equals \(1.67^\circ\)C. However, \(21.11\) is not twice \(1.67\).

  • Ratio Scale: This scale allows for meaningful comparisons of ratios between objects. For instance, weight is measured on a ratio scale: 10 lb (\(=4.53592\) kg) is twice as heavy as 5 lb (\(=2.26796\) kg).

In this course, we focus primarily on cardinal utility functions that measure alternatives on an interval scale.

10.2.1 Linear Transformations

When comparisons between differences in utilities are meaningful, it is no longer true that any utility function representing the same ranking of alternatives accurately represents a decision maker’s preferences. In this case, we focus on utility functions that are related by linear transformations.

Definition 10.2 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 \(x\in X\), \[u'(x) = \alpha u(x) + \beta.\]

A key observation is that if all utility functions representing a decision maker’s preferences are related by linear transformations, then comparisons of differences between utilities are meaningful. For instance, consider two utility functions on \(X=\{a, b, c\}\): \(u: X \rightarrow \mathbb{R}\) with \(u(a) = 2\), \(u(b) = 1\), \(u(c) = 0\) and \(u': X \rightarrow \mathbb{R}\) with \(u'(a) = 5\), \(u'(b) = 4\), \(u'(c) = 1\).

Suppose both \(u\) and \(u'\) represent a decision maker’s preferences on \(X\), where \(a\) is ranked above \(b\), \(b\) is ranked above \(c\), and \(a\) is ranked above \(c\). According to \(u\), \(b\) is ranked evenly between \(a\) and \(c\) because \[u(a) - u(b) = u(b) - u(c).\] However, according to \(u'\), \(b\) is ranked closer to \(a\) than to \(c\) since \[u'(a) - u'(b) < u'(b) - u'(c).\] Consequently, if both \(u\) and \(u'\) represent the decision maker’s preferences, it is not meaningful to compare differences in utilities between the objects.

This issue does not arise if \(u'\) is a linear transformation of \(u\):

Proposition 10.1 Suppose that \(u:X\rightarrow\mathbb{R}\) is a utility function on \(X\) and \(u':X\rightarrow\mathbb{R}\) is a linear transformation of \(u\). Then, for all \(a, b, c, d\in X\),

  1. if \(u(a) - u(b) < u(c) - u(d)\), then \(u'(a) - u'(b)< u'(c) - u'(d)\);
  2. if \(u(a) - u(b) > u(c) - u(d)\), then \(u'(a) - u'(b) > u'(c) - u'(d)\); and
  3. if \(u(a) - u(b) = u(c) - u(d)\), then \(u'(a) - u'(b) = u'(c) - u'(d)\).

Proof. We only prove item 1. 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:

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

10.3 Exercises

  1. Find three utility functions that represent the rational preference relation \((P, I)\) on \(X=\{a, b, c, d\}\), where \[P=\{(a, b), (b, c), (a, c), (d, b), (d, c)\}\] and \[I=\{(a, a), (b, b), (c, c), (a, d), (d, a)\}.\]

  2. Suppose that \((P, I)\) is a rational preference on \(X\) and that \(u:X\rightarrow \mathbb{R}\) represents \((P, I)\).

    1. True or False: Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be the function where for all \(x\in\mathbb{R}\), \(f(x)=2x + 3\). Then, \(f\circ u\) represents \((P, I)\).
    2. True or False: Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be the function where for all \(x\in\mathbb{R}\), \(f(x)=-5x\). Then, \(f\circ u\) represents \((P, I)\).
    3. True or False: Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be the function where for all \(x\in\mathbb{R}\), \(f(x)=x^2\). Then, \(f\circ u\) represents \((P, I)\).

  3. True or False: Suppose that \(X=\{a, b, c, d\}\) and \((P, I)\) is a rational preference with \[P=\{(a,b), (b, c), (c, d), (a, c), (a, d), (b, d)\}\] and \[I=\{(a, a), (b, b), (c, c)\}.\] Further, suppose that \(u\) and \(u'\) both represent \((P, I)\). Then, if \(u(a) - u(b) < u(c) - u(d)\), then \(u'(a) - u'(b) < u'(c) - u'(d)\).

  4. Explain what is wrong with the following statement: Ann prefers \(a\) to \(b\) because she assigns higher utility to \(a\) than to \(b\).

  5. 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 \(u\)?

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

  1. Find three utility functions that represent the rational preference relation \((P, I)\) on \(X=\{a, b, c, d\}\), where \[P=\{(a, b), (b, c), (a, c), (d, b), (d, c)\}\] and \[I=\{(a, a), (b, b), (c, c), (a, d), (d, a)\}.\]

    1. \(u(a) = u(d) = 3\), \(u(b)=2\), and \(u(c)=1\)
    2. \(u(a) = u(d) = 300\), \(u(b)=2\), and \(u(c)=0\)
    3. \(u(a) = u(d) = 1\), \(u(b)=0.5\), and \(u(c)=0\)

  2. Suppose that \((P, I)\) is a rational preference on \(X\) and that \(u:X\rightarrow \mathbb{R}\) represents \((P, I)\).

    1. True or False: Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be the function where for all \(x\in\mathbb{R}\), \(f(x)=2x + 3\). Then, \(f\circ u\) represents \((P, I)\).

      True: if \(u(x) \geq u(y)\) then \(2u(x) + 3 \geq 2u(y) + 3\)

    2. True or False: Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be the function where for all \(x\in\mathbb{R}\), \(f(x)=-5x\). Then, \(f\circ u\) represents \((P, I)\).

      False: Suppose that \(a\mathrel{P} b\) and \(u(a) = 2 > u(b) = 1\). Then \(f\circ u(a)= - 10 < f\circ b(b)=-5\).

    3. True or False: Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be the function where for all \(x\in\mathbb{R}\), \(f(x)=x^2\). Then, \(f\circ u\) represents \((P, I)\).

      False: Suppose that \(a\mathrel{P} b\) and \(u(a) = 2 > u(b) = -2\). Then \(f\circ u(a)= 4 = f\circ b(b)=4\).

  3. True or False: Suppose that \(X=\{a, b, c, d\}\) and \((P, I)\) is a rational preference with \[P=\{(a,b), (b, c), (c, d), (a, c), (a, d), (b, d)\}\] and \[I=\{(a, a), (b, b), (c, c)\}.\] Further, suppose that \(u\) and \(u'\) both represent \((P, I)\). Then, if \(u(a) - u(b) < u(c) - u(d)\), then \(u'(a) - u'(b) < u'(c) - u'(d)\).

    False: Suppose that \(u(a)=4, u(b)=3\), \(u(c)=2\), and \(u(d)=0\) and that \(u'(a)=7, u'(b)=3\), \(u'(c)=2\), and \(u'(d)=0\). Both \(u\) and \(u'\) represent \((P, I)\). We have that \[u(a)-u(b)=4-3 = 1 < u(c)-u(d) = 2-0=2.\] However, \[u'(a)-u'(b)=7-4 = 3 > u'(c)=u'(d) = 2-0 =2.\]

  4. Explain what is wrong with the following statement: Ann prefers \(a\) to \(b\) because she assigns higher utility to \(a\) than to \(b\).

    In standard rational choice models, a utility function \(u\) represents a decision maker’s preference. In this case, assigning a higher utility to an object \(a\) than to \(b\) does not mean anything else except that \(a\) is preferred to \(b\).