The problem asks whether it is possible to find a complete, transitive, convex, and continuous binary relation $\succeq^*$ on $\mathbb{R}^2_+$ (the positive orthant of $\mathbb{R}^2$) that satisfies the given conditions:

1. $\begin{pmatrix} 3 \\ 1 \end{pmatrix} \succeq^* \begin{pmatrix} x \\ x \end{pmatrix}$,
2. $\begin{pmatrix} 1 \\ 3 \end{pmatrix} \succeq^* \begin{pmatrix} x \\ x \end{pmatrix}$,

for all $x \in [0, \infty)$.

Solution Outline:

1. Understand the conditions for the binary relation $\succeq^*$:

• Completeness: For any two points $a, b \in \mathbb{R}^2_+$, either $a \succeq^* b$, $b \succeq^* a$, or both (indifference).
• Transitivity: If $a \succeq^* b$ and $b \succeq^* c$, then $a \succeq^* c$.
• Convexity: If $a \succeq^* b$, then any convex combination of $a$ and $c$ (e.g., $\lambda a + (1-\lambda)c$) also satisfies the relation with $b$.
• Continuity: The relation $\succeq^*$ is preserved under limits; for example, if $a_n \to a$ and $b_n \to b$, and $a_n \succeq^* b_n$, then $a \succeq^* b$.

2. Analyze the specific requirements:

• The inequalities given state that the vectors $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$ are always preferred over $\begin{pmatrix} x \\ x \end{pmatrix}$, for any $x \geq 0$.

• This implies that no matter how large $x$ becomes, $\begin{pmatrix} x \\ x \end{pmatrix}$ cannot "catch up" to either $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$ or $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$ under the binary relation $\succeq^*$.

3. Feasibility of such a relation:

• Consider monotonicity: In $\mathbb{R}^2_+$, it is often assumed that larger values in both dimensions should be "preferred" under reasonable assumptions (e.g., $\begin{pmatrix} x \\ x \end{pmatrix}$ becomes arbitrarily large).

• However, the conditions contradict this:

  • For sufficiently large $x$, $\begin{pmatrix} x \\ x \end{pmatrix}$ dominates $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$ in at least one dimension.
  • A complete and transitive relation cannot rank $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$ above $\begin{pmatrix} x \\ x \end{pmatrix}$ for arbitrarily large $x$ without violating continuity.

4. Conclusion:

It is not possible to construct such a binary relation $\succeq^*$ that satisfies all the conditions. The issue arises from the incompatibility between the requirements for convexity, transitivity, and the stated preference ordering.

Would you like a more detailed explanation or help with related questions?

Follow-up Questions:

1. What does it mean for a binary relation to be convex in $\mathbb{R}^2_+$?
2. How do transitivity and continuity work together in preference relations?
3. Why does $\begin{pmatrix} 3 \\ 1 \end{pmatrix} \succeq^* \begin{pmatrix} x \\ x \end{pmatrix}$ lead to a contradiction when $x \to \infty$?
4. Can we weaken any of the conditions (completeness, convexity, etc.) to make such a relation feasible?
5. How would this problem change if $\mathbb{R}^2_+$ were replaced with $\mathbb{R}^2$?

Tip:

When analyzing binary relations, always check for potential contradictions between completeness and other properties like continuity and convexity. These often cause inconsistencies in preference modeling.