Suppose you go out to eat lunch with a group of friends every day. Say for simplicity’s sake you have N sets of lunchmates, and you eat with each group every N days on a regular cycle.
On a day i of this periodic schedule, the lunch is shared by ni of your friends, plus you. You have a total of F lunchmate friends, and some of those friends may occur in more than one lunch group. Everybody who is a member of each group always shows up for lunch.
The rule for paying for each lunch is that every member of a group pays in turn. So if you and Bob and Sally go to lunch on Mondays, every third Monday one of you will buy everybody’s lunch. This is true in every group, regardless of its size or composition.
For every lunch group with ni friends, you will end up paying for everyone’s lunch every ni+1 days.
So: Is there any arrangement of your F friends into N subsets, such that you end up paying less (over a long time, on average) than any of your friends? Does it matter if some friends are members of more than one lunch group? You are, after all, member of all the lunch groups.
There will be a follow-on question, when or if somebody deigns to respond 
I’ve meaning to go away and think about this for weeks, but never really have. I guess I’m not clear about the assumptions, though. Certainly if I order more expensive food (on average) than all my friends, then I will end up scamming them. So are we assuming everyone’s bills are roughly equal? If so, then it seems that on average over time you’ll come out a wash.
Am I missing something?
Why not assume to begin with that the variation in meal costs is very small over time?
I see (having revisited this recently) there might be some clearing up in order. Here’s an example:
[and repeat]
Suppose in each set there is a strict cyclic order of who pays in a given meal, not necessarily alphabetical.