Start with a strict preference relation ≻ that is asymmetric and transitive, (but not negatively
transitive). Define weak preference ⪰ by
x ⪰ y ⇐⇒ ¬(y ≻ x),
and indifference ∼ by
x ∼ y ⇐⇒ ¬(x ≻ y) and ¬(y ≻ x).
Give an example of such a strict preference where the derived ⪰ is not a rational preference
relation.

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.

Start with a strict preference relation ≻ that is asymmetric and transitive, (but not negatively transitive). Define weak preference ⪰ by x ⪰ y ⇐⇒ ¬(y ≻ x), and indifference ∼ by x ∼ y ⇐⇒ ¬(x ≻ y) and ¬(y ≻ x). Give an example of such a strict preference where the derived ⪰ is not a rational preference relation.

Explore an example of a strict preference relation that is asymmetric and transitive but not negatively transitive, and how it leads to an incomplete weak preference relation. This topic is relevant for understanding preference theory in economics and decision theory.

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