# An example of preferences over a countable set that cannot be represented by a utility function

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

I know a utility function exist that represents preference over a countable set. I know a few types of preferences such as lexicographic, monotonic, quasi linear. I have always dealt with utility function assuming preferences are represented by that utility function. So how to solve this question?

• What countable means? It can be finite countable? Or only infinite countable? Apr 13 '19 at 11:23
• Both cases u can take Apr 13 '19 at 11:25
• I'll give you a tip first, and if you still can't solve the problem, comment later on. Apr 13 '19 at 11:26
• Ok.thnx fr the tips in advance. Apr 13 '19 at 11:27

There is a proposition that says:

If there is a function $$u: X \rightarrow \mathbb{R}$$ that represents the preference relation $$\succsim$$, then $$\succsim$$ is rational.

So, the contrapositive form of this statement is:

If $$\succsim$$ isn't rational, then all functions $$u$$ don't represent $$\succsim$$.

So, it's a good approach to create a preference relation that isn't rational, i.e., the preference must violate transitivity or completeness.

• Whatever your example is, it's probably wrong. The question assumes the relation is transitive and complete. Feb 24 '20 at 0:46