Excerpt that describes the actual mechanism:

Start with any item on the bill. The person who ordered that item pays the bill with probability equal to the cost of that item divided by the subtotal. Flip the appropriately biased coin; if that person is it, then you're done. If not, then subtract that item from the subtotal and repeat, recursively, with another arbitrary item. If you start with expensive items then you'll probably find the person who's paying after a handful of items, but it doesn't matter for fairness what order you pick things in. You won't have to figure out all the confusing drinks and appetizers (yet the outcome is as fair as if you had!).