Problem Interpretation:

We are dealing with a decision maker (DM) choosing from a finite set $X$ using two numerical utility functions, $u$ and $v$. For any subset $A \subseteq X$ (a choice problem), the DM's choices are determined by these functions in specific ways.

(a) Fix a number $v^*$:

• Procedure:

  1. From $A \subseteq X$, the DM first identifies $a^*$, the option that maximizes $u$ over $A$.
  2. The DM also identifies $b^*$, the option that maximizes $v$ over $A$.
  3. The DM chooses $a^*$ (the maximizer of $u$) if $v(a^*) \geq v^*$.
  4. If $v(a^*) < v^*$, the DM chooses $b^*$ (the maximizer of $v$).

• Does this procedure satisfy the Independence of Irrelevant Alternatives (IIA)?

  • IIA Principle: If the DM chooses an option from a set $A$, and if a subset $B \subset A$ contains that choice, then the DM should still choose the same option from $B$.

  In this case, the procedure violates IIA. Here's why:

  • Suppose the DM chooses $a^*$ from the original set $A$ because $v(a^*) \geq v^*$.
  • Now, consider a subset $B \subset A$ where $a^*$ is excluded. If $v(a^*) < v^*$, the DM would have chosen $b^*$, but since $a^*$ is no longer available in $B$, the DM's choice in $B$ could differ from their choice in $A$. This violates IIA since the removal of irrelevant alternatives (those not chosen initially) influences the choice.

(b) Fix $u^*$:

• Procedure:

  1. From $A \subseteq X$, the DM first identifies $a^*$, the option that maximizes $u$ over $A$.
  2. The DM chooses $a^*$ if $u(a^*) \geq u^*$.
  3. If $u(a^*) < u^*$, the DM chooses $b^*$, the maximizer of $v$ over $A$.

• Does this procedure satisfy the Independence of Irrelevant Alternatives (IIA)?

  • This procedure also violates IIA. Here's why:
    • Suppose the DM chooses $b^*$ because $u(a^*) < u^*$. If the set $B \subset A$ removes $b^*$ but retains $a^*$, the DM might now choose $a^*$, even though this wasn't their original choice in $A$. The removal of irrelevant alternatives changes the DM’s choice, violating IIA.

Conclusion:

Neither (a) nor (b) satisfies the Independence of Irrelevant Alternatives (IIA). Both procedures are dependent on the composition of the choice set, meaning that removing certain options can alter the final decision, even though those removed options weren't originally chosen.

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Would you like further clarification on any specific part of the explanation?

Here are some related questions that might help expand on this topic:

1. What are the implications of the Independence of Irrelevant Alternatives in decision theory?
2. How can utility functions be used in real-world decision-making scenarios?
3. What are examples of choice procedures that satisfy IIA?
4. What are other common axioms in choice theory, and how do they compare to IIA?
5. Can you explain how violations of IIA might manifest in practical decision-making situations?

Tip: Always pay close attention to whether a decision rule's outcome depends on the set of available choices, as this often signals a violation of IIA.