I'm stuck trying to figure out what it means, economically (and what we're trying to demonstrate with it), for the employer to have three possible wage levels to choose from, rather than two…

For the two-wage-level case, we have:

		Worker
		Work	Home
Employer	$w^M=3$	$\boxed{{\displaystyle 9}},\boxed{{\displaystyle 3}}$	0, 1
	$w^L=2$	8, 2	0, 1

Which gives

$$\begin{array}{rl}{\mathcal{A}}_{E\to W}& =\left|\frac{2-3}{3}\right|=\frac{1}{3}\\ {\mathcal{I}}_{E\to W}& =\left|\frac{8-9}{9}\right|=\frac{1}{9}\\ {\mathcal{C}}_{E\to W}& =\frac{1/3}{1/9}=3\end{array}$$

and

$$\begin{array}{rl}{\mathcal{A}}_{W\to E}& =\left|\frac{0-9}{9}\right|=1\\ {\mathcal{I}}_{W\to E}& =\left|\frac{1-3}{3}\right|=\frac{2}{3}\\ {\mathcal{C}}_{W\to E}& =\frac{1}{2/3}=\frac{3}{2}\end{array}$$

But when we add a third level to the possible wages that can be posted, I don't understand the point. We can add a higher wage that's not a Nash, for example, by using:

		Worker
		Work	Home
Employer	$w^H=4$	8, 4	0, 1
	$w^M=3$	$\boxed{{\displaystyle 9}},\boxed{{\displaystyle 3}}$	0, 1
	$w^L=2$	8, 2	0, 1

But the higher wage is irrelevant here, in that it doesn't change the employer's capacity, since 1 unit of positive change (from 3 to 4) is the same as 1 unit of negative change (from 3 to 2), once we take the absolute value in computing

$\mathcal{A}$ and

$\mathcal{I}$.

But if we instead add an additional lower wage, we get

		Worker $W$
		Work	Home
Employer $E$	$w^M=3$	$\boxed{{\displaystyle 9}},\boxed{{\displaystyle 3}}$	0, 1
	$w^L=2$	8, 2	0, 1
	$w^B=1$	7, 1	0, 1

Then the

$w^L$ becomes irrelevant, and the new

$w^B$ just plays the same role that

$w^L$ played in the two-wage-level game:

$$\begin{array}{rl}{\mathcal{A}}_{E\to W}& =\left|\frac{1-3}{3}\right|=\frac{2}{3}\\ {\mathcal{I}}_{E\to W}& =\left|\frac{9-7}{9}\right|=\frac{2}{9}\\ {\mathcal{C}}_{E\to W}& =\frac{\frac{2}{3}}{\frac{2}{9}}=\frac{2}{3}\cdot \frac{9}{2}=3\end{array}$$

and

$$\begin{array}{rl}{\mathcal{A}}_{W\to E}& =\left|\frac{0-9}{9}\right|=1\\ {\mathcal{I}}_{W\to E}& =\left|\frac{1-3}{3}\right|=\frac{2}{3}\\ {\mathcal{C}}_{W\to E}& =\frac{1}{2/3}=\frac{3}{2}\end{array}$$

So, is it that I'm missing something crucial–-like, did I mess up the payoffs for the working vs. not-working? Or, is it that the monopsony effects we want to highlight don't "emerge" until you have a competitive market with

$N$ workers, for example?