Preface: I think this is interesting (and hopefully at least one other mathematician will agree!), but it's entirely possible that y'all will consider this too low-brow for MO. There isn't a completely definite answer, but I think there can probably be a near-consensus. If you're able, feel free to re-title if you think you have something more appropriate. Also, I have no idea what tag(s) to apply. Community-wiki?
I'm trying to set up a winner-takes-all bet with four other friends about the order in which we end up getting married. So that nobody has to simultaneously shell out $100 and face the prospect of living alone for the rest of their life, it seems prudent to end the bet and determine the winner after 3 of us have gotten married. Here was my original scoring scheme:
123 > 132 > 213 > 231 > 312 > 321 > 12* > 13* > 1*2 > 1*3 > 21* > 23* > 2*1 > 2*3 > 31* > 32* > 3*1 > 3*2 > *12 > *13 > *21 > *23 > *31 > *32 > * 1 * > * 2 * > * 3 * > **1 > **2 > **3
[For example, the first ordering 123 > 132 means the following. Suppose I list 1: Aaron, 2: Ben, 3: Carlos. Then, I am better off if they get married in the order Aaron, Ben, Carlos than if they come out in order Aaron, Carlos, Ben.]
[Note that if you want to put asterisks around a number you need to include spaces, otherwise they'll disappear and the number will be italicized. Although I'm sure there's a way to avoid this.]
I made this based off of the naive initial assumption that getting the top three picks should beat anything else. But one friend pointed out that probably it should be that 12*>321 (for example), and so the question becomes:
- What are reasonable parameters for the ordering?
- How should they be prioritized?
- What ordering (if any) does this yield?
With regards to the third question, I wouldn't be at all surprised if some variant of Arrow Impossibility comes into play, but we'll cross that bridge when we get there...