Give an example of preferences over a countable set in which the prefer ences cannot be represented by a utility function that returns only integers as values.

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.

Explore an example of preferences over countable sets that cannot be represented by utility functions returning only integer values. This example involves preference relations and utility theory, demonstrating why integer-valued utility functions fail to capture certain preferences. Suitable for advanced undergraduate students and researchers in mathematical economics and decision theory.

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