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!).
I've tried this with a group. It's good for generating endless conversation on principles of economic fairness. It completely fails to be a time-saving mechanism.
There is the completely rational mind and then there are humans - this represents a separation point :)
Ha! Touche! Yes, this has never actually saved time. Well, among me and Bethany Soule (coauthor of the Android app described in the article) and Sharad Goel (co-blogger at Messy Matters) I think it's now routine enough that it saves time.
But it's so the Right Way to do it! I guess it's like trying to get Americans to use the metric system. It would be worth it if it actually worked...
Since one person ends up paying the whole thing, it's only fair if the same group of people goes out together frequently, such that there is a reasonable expectation that everyone will end up paying their fair share eventually.
Another downside is the liquidity requirement. To adequately participate, you need to be able to pay for a fairly large number of meals.
"A common complaint about stochastic schemes is that they’re “only fair if you do it repeatedly with the same group of people”. That’s true if you insist on ex post fairness. We’re usually happy with ex ante fairness. Consider selling me a (perfectly fairly priced) lottery ticket for a dollar. That’s guaranteed to be unfair, ex post. Either you sold me a worthless piece of paper for a dollar, or I got a million dollars and only paid a dollar for it. But the fact that none of us knew which would happen made the one dollar price fair. Same story with venture capital investment, for example. You may need a gambling mentality to be down with it, but it’s quite fair even if only done once. The fact that it averages out in the long term to be perfectly fair ex post is icing on the cake."
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!).