10 Representing Preferences

The standard interpretation of a utility function in Rational Choice Theory is that it is an indicator of preference. This means that there is an important relationship between utility functions and rational preferences.

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

\(x\mathrel{P_u} y\) when \(u(x) > u(y)\); and
\(x\mathrel{I_u}y\) when \(u(x) = u(y)\).

It is not hard to see that for any utility function \(u:X\rightarrow\mathbb{R}\) on a set \(X\), \((P_u, I_u)\) is a rational preference on \(X\).

The second observation is that every rational preference \((P, I)\) on a set \(X\) can be represented by a utility function.

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 \(x, y\in X\):

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

Putting these two observations together, we have that a pair of relations \((P, I)\) are a rational preference exactly when the relations are representable.

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.

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:

\(a \mathrel{P} b\) for all \(b\in Y\). Let \(u_{P,I}(a)=\max\{u_{P_Y, I_Y}(b)\ |\ b\in X'\}+1\).
\(b \mathrel{P} a\) for all \(b\in Y\). Let \(u_{P,I}(a)=\min\{u_{P_Y, I_Y}(b)\ |\ b\in X'\}-1\).
\(a \mathrel{I} b\) for some \(b\in Y\). Let \(u_{P,I}(a)=u_{P_Y, I_Y}(b)\).
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, additional technical assumptions are needed, which are beyond the scope of this course.

10.1 Exercises

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 \[I=\{(a, a), (b, b), (c, c), (a, d), (d, a)\}.\]
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 \((P, I)\).
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 \(u'(a) - u'(b) < u'(c) - u'(d)\).
Explain what is wrong with the following statement: Ann prefers \(a\) to \(b\) because she assigns higher utility to \(a\) than to \(b\).

Solutions

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 \(u(c)=0\)
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 \(f\circ u(a)= 4 = f\circ b(b)=4\).
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, \[u'(a)-u'(b)=7-4 = 3 > u'(c)=u'(d) = 2-0 =2.\]
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 \(b\).