The probability you use is the value of that item divided by the total, no? So on average you'll get halfway down the menu, and have to have figured out who ordered half the things, in slow bits and pieces.
Thinking about it, I think I have a faster way to do it: 1. generate a random number between 0 and bill amount. 2. "Count" that much money down the bill. 3. Whoever ordered the item you stop on pays.
But to clarify, the method in the article is quite a bit better than you think:
"On average you’ll only traverse half the bill — and that’s half in terms of dollar value, not number of items. So much less than half the items if most of the bill is concentrated in a few expensive entrees."
I actually think it's kind of fun, with the multiple trials for who will pay. But I'm eager to try your idea now!
EDIT: You're totally right, this is a faster fair way! It's not as simple as it sounds though: if you land on, say, a $2 drink of which there are multiple such on the bill then you need a protocol to disambiguate. You could randomize again among the people who got one of those items.
What I like most about the method in the article is that you can avoid the confusing smaller items with high probability. Maybe the way to achieve that with your method is to walk down the items in decreasing order of price. That also forces you to identify sets of things that all cost the same, so you can do the second randomization if necessary. (You could also just pick a canonical order of the people, like alphabetical, or where they're seated, since there's enough randomness in the original random number.)
You're even righter than I thought. This is definitively faster than the method in the article. And with your method there's no point in rearranging the items. You're no more likely to have to deal with a confusing small item than with the method in the article.
The only possibly worse aspect is that, when the item is chosen, you have to make sure to identify everyone who got a copy of that item. With the method in the article you always consider items one at a time, in isolation.
