To provide an example of preferences over a countable set that cannot be represented by a utility function returning only integer values, let's consider a countable set $S = \mathbb{Q}$, the set of all rational numbers.

Example of Preferences

Let's define a preference relation $\succ$ over $S = \mathbb{Q}$ as follows:

1. For any $q_1, q_2 \in \mathbb{Q}$, we say $q_1 \succ q_2$ if and only if $q_1 > q_2$.

This preference relation represents the usual "greater than" ordering on the rational numbers.

Why No Integer-Valued Utility Function Exists

To represent these preferences with a utility function, we would need a function $u: \mathbb{Q} \to \mathbb{Z}$ such that:

• $q_1 \succ q_2 \iff u(q_1) > u(q_2)$.

However, this is impossible with a function that takes only integer values. Here’s why:

1. Density of Rational Numbers: The set $\mathbb{Q}$ is dense in itself, meaning that between any two rational numbers, there exists another rational number. More formally, for any $q_1, q_2 \in \mathbb{Q}$ with $q_1 < q_2$, there exists a $q_3 \in \mathbb{Q}$ such that $q_1 < q_3 < q_2$.

2. Utility Function's Discreteness: If $u: \mathbb{Q} \to \mathbb{Z}$ is a utility function, it assigns an integer to each rational number. But integers are not dense; there is no integer strictly between two consecutive integers. Therefore, there cannot be a strict utility ordering $u(q_1) > u(q_2)$ corresponding to every possible pair $q_1 > q_2$ because we would run out of available integers to assign utility values uniquely in a way that reflects the density of $\mathbb{Q}$.

3. Contradiction: Because $\mathbb{Q}$ has infinitely many values between any two given points and $\mathbb{Z}$ does not, there is no way to assign integer values to $\mathbb{Q}$ while preserving the strict order given by $\succ$. Hence, there is no integer-valued utility function that can represent the preferences $\succ$.

Conclusion

Thus, the preference relation $\succ$ over the set of rational numbers $\mathbb{Q}$, where $q_1 \succ q_2 \iff q_1 > q_2$, cannot be represented by any utility function that returns only integer values.

Would you like more details on this, or have any questions?

5 Related Questions:

1. What is a utility function, and why is it used to represent preferences?
2. How can a utility function with real values represent preferences over a dense set like $\mathbb{Q}$?
3. Can there be a utility function that represents preferences over $\mathbb{Q}$ using only a subset of rational numbers?
4. What is the difference between countable and uncountable sets in the context of preferences?
5. How do we construct utility functions for more complex sets, such as function spaces?

Tip: When dealing with preference relations, consider whether the density of the set and the discreteness of the utility function's range are compatible.