To construct an example where the derived weak preference relation ⪰ is not rational, we need to focus on the failure of completeness in the derived relation ⪰. A rational preference relation is both transitive and complete. Since we are given that the strict preference ≻ is asymmetric and transitive but not negatively transitive, the derived weak preference ⪰ can fail to be rational due to the lack of completeness.

Key points:

• Asymmetry of ≻ means that if $x \succ y$, then $y \not\succ x$.
• Transitivity of ≻ means that if $x \succ y$ and $y \succ z$, then $x \succ z$.
• Completeness of ⪰ requires that for any two elements $x$ and $y$, either $x \⪰ y$ or $y \⪰ x$.
• Negative transitivity (which we assume does not hold) means that if $x \not\succ y$ and $y \not\succ z$, then $x \not\succ z$. This failure can lead to incompleteness in the weak preference.

Example:

Let the set of alternatives be $A = \{ a, b, c \}$, and define the strict preference relation ≻ as follows:

• $a \succ b$
• $b \succ c$

Thus, the strict preference ≻ is asymmetric and transitive (since $a \succ c$ by transitivity of ≻).

Now let's look at the derived weak preference relation ⪰ defined by: $x \⪰ y \iff \neg(y \succ x)$

Derived Weak Preferences:

• $a \succ b$ implies $a \⪰ b$ and $b \not\⪰ a$.
• $b \succ c$ implies $b \⪰ c$ and $c \not\⪰ b$.
• However, neither $a \succ c$ nor $c \succ a$ holds directly because ≻ is only strict between $a \succ b$ and $b \succ c$. Therefore, we have $a \not\succ c$ and $c \not\succ a$, which means neither $a \⪰ c$ nor $c \⪰ a$.

Failure of Completeness:

In this example, there is no weak preference relationship between $a$ and $c$. We cannot say $a \⪰ c$ or $c \⪰ a$, which means the derived weak preference relation ⪰ is incomplete. Hence, it is not a rational preference relation.

Conclusion:

The derived weak preference relation ⪰ fails to be rational because it is incomplete. The key point is that, since ≻ is not negatively transitive, it leads to situations where two elements cannot be compared using the weak preference ⪰.

Would you like further details or have any questions about this concept?

Here are 5 related questions to explore further:

1. How does the failure of negative transitivity in strict preferences affect rationality?
2. Can you give an example where a weak preference relation is both transitive and complete?
3. What are the differences between strict, weak, and indifference relations in preference theory?
4. Can a weak preference relation ⪰ be complete but fail to be transitive? Why or why not?
5. How does one formally define rationality in terms of preference relations?

Tip: A rational preference relation requires both completeness and transitivity, but a strict preference doesn't guarantee completeness when derived from a weak preference.