To find an example of preferences over a countable set that cannot be represented by a utility function that returns only integers as values, consider the following scenario:

Setup

Let the countable set be $Z = \{z_1, z_2, z_3, \dots\}$. Define a preference relation $\succ$ on $Z$ such that:

• $z_1 \succ z_2 \succ z_3 \succ \dots$ (i.e., $z_n \succ z_{n+1}$ for all $n \in \mathbb{N}$).

In addition, impose that for any $n \in \mathbb{N}$, the infinite sequence $z_n, z_{n+1}, z_{n+2}, \dots$ is strictly preferred over the finite set $\{z_1, z_2, \dots, z_{n-1}\}$.

Analysis

1. Countable Set and Preference Structure:

   • The set $Z$ is countable since it corresponds to the natural numbers $\mathbb{N}$.
   • The preference relation $\succ$ is transitive and complete, meaning that for any two elements in $Z$, one is strictly preferred over the other.

2. Utility Representation Issue:

   • Suppose there exists a utility function $u: Z \to \mathbb{Z}$ that assigns integer values to each element in $Z$ while preserving the preference order $\succ$. This would mean $u(z_1) > u(z_2) > u(z_3) > \dots$.
   • Because $\mathbb{Z}$ is discrete, there are no integer values strictly between any two consecutive integers, meaning $u(z_n) = u(z_{n+1}) + 1$ for each $n$.

3. Contradiction with Infinite Preference:

   • Consider the infinite sequence $z_n, z_{n+1}, z_{n+2}, \dots$. If this sequence is strictly preferred over any finite set $\{z_1, z_2, \dots, z_{n-1}\}$, then the utility function should reflect that preference.
   • However, since the integers are discrete and bounded below (say, by the smallest utility assigned to some $z_k$), the utility function would fail to represent this preference for the infinite sequence, as adding more elements to a finite set would eventually decrease the utility to a point lower than the value of any finite set.
   • This creates a contradiction, as no integer-valued utility function can simultaneously preserve the strict preference ordering $z_n \succ z_{n+1}$ and the preference for infinite sequences over finite ones.

Conclusion

This example demonstrates that a utility function that returns only integers cannot represent the described preferences over a countable set $Z$. The requirement that an infinite sequence be strictly preferred over any finite subset forces us to consider non-integer (potentially real-valued or ordinal) utility functions, as integers are too coarse to capture the strict preference between finite and infinite sequences while maintaining the transitive order of preferences over individual elements.