What does $\overline{z}\mathbb{1}$ and $\underline{z}\mathbb{1}$ mean?

Question

I'm working on some paper concerning auction analysis. I have trouble with understanding what is the meaning of symbols:

$\overline{z}\mathbb{1}$ and $\underline{z}\mathbb{1}$

Do you have any idea? Thank you.

Answer 1

This appears to be from Robert Wilson, Strategic Analysis of Auctions. (Surprisingly you can find this by just Googling the phrase "indicate nature's choice".) The passage here is on p. 6 of the linked pdf.

I believe $\underline{z}$ and $\overline{z}$ are just scalars, and $\mathbb{1}$ is the vector of all 1s. Furthermore $\le$ is being used on vectors to mean componentwise inequality: that is, $(x_1, x_2, \ldots, x_n) \le (y_1, y_2, \ldots, y_n)$ is defined to mean $x_i \le y_i$ for $1, 2, \cdots, n$. So the cell $Z = \{ z \: | \: \underline{z} \mathbb{1} \le z \le \overline{z} \mathbb{1} \}$ is just the set of vectors $z$ where every element is between $\underline{z}$ and $\overline{z}$.