I have this egg cooker;

And it has presented me with a mystery.
I wasn’t sure if I should ask in Cafe Society (quisine) or General Questions (physics), but to cook eggs, the more eggs one cooks, the less water that’s required. When the water boils away it shuts off. So it cooks longer with fewer eggs!
Howzat?
said egg cooker.
By the way, it works great for boiled eggs, but I’ve had little luck with poached.

My parents had a similar contraption when I was in high school. That '70s relic required less water for soft, more water for hard, with different levels for soft & hard depending if one was boiling or poaching.

Your link didn’t work for me, I got a Cuisinart site, but it wanted a log in.

Anyway, can’t you just put less water in with fewer eggs?

That’s a search page. I don’t know why it came up instead of the product page.
Anyway, it works fine when used as instructed. I just wonder why you’d use less water (so less time) for more eggs!
Just had an epiphany, gotta think. Ah bee bock.

WAG: The water equalizes the rate of heat transfer, since it takes heat that would otherwise go into the other eggs.

I read the instructions more carefully.
For any number of eggs, boiled, the cooking time is 8-9 minutes for soft, 13-14 for medium, and 16-18 for hard. That makes sense, and would probably satisfy Julia Child.
What confounds me is this:
Number of eggs Tbsp of water
7 hard 6 1/2
5 hard 6 3/4
3 hard 7
1 hard 8
And so on for medium and soft. Always more water but the same time for fewer eggs at each degree of doneness.
As Lute Skywatcher mentions it takes 8 tbsp water for 7 hard cooked and 1 1/2 for soft.
In all cases the thing shuts off when all the water is steamed away.
7e + 6.5w = 13-14 minutes to steam.
1e + 8w = 13-14 minutes to steam.