# Preference*

Let $\succeq$ be a relation defined on $\mathbb{R}_+^n$.

### Completeness

$\succeq$ is complete if for any $x$ and $y$ in the $\mathbb{R}_+^n$, we have $x \succeq y$, $y \succeq x$, or both.

### Transitivity

$\succeq$ is transitive if for any $x$, $y$, and $z$ in the $\mathbb{R}_+^n$ and $x \succeq y, y \succeq z$, then we have $x \succeq z$.

**Note:** Some scholars define rational preference if a preference is complete and transitive.

### Continuity

$\succeq$ is continuous if for any $x$ in the $\mathbb{R}_+^n$, both the upper contour set $=\{y \in \mathbb{R}_+^n| y \succeq x\}$ and lower contour set $=\{z \in \mathbb{R}_+^n| x \succeq z\}$ are closed.

### Continuous but not Rational Preference


For any $(x_1,x_2)$ and $(y_1,y_2)$ in the $\mathbb{R}_+^2$, we define $\succeq$ as

$$(x_1,x_2) \succeq (y_1,y_2) \text{ iff } x_1 \ge x_2 \text{ and } 1\le x_2 \le 2, 1\le y_2 \le 2$$

Hence, if $x_2 \notin [1,2]$ or $y_2 \notin [1,2]$, $(x_1,x_2)$ and $(y_1,y_2)$ are not comparable.

Clearly, $\succeq$ is not complete. We need to show $\succeq$ is continuous.

For any $(z_1,z_2)$ in the $\mathbb{R}_+^2$, if $z_2 \notin [1,2]$, then its upper contour set and lower contour set are both empty and closed.

If $z_2 \in [1,2]$, then its upper contour set is $\{(x_1,x_2)| x_1 \ge z_1, 1\le x_2 \le 2 \}$ and its lower contour set is $\{(x_1,x_2)| x_1 \le z_1, 1\le x_2 \le 2 \}$. Both sets are closed.