There are a number of instances of sports teams intentionally losing matches in order to secure a more favorable situation in a playoff round. While this doesn't happen terribly often, when it does it's usually pretty disappointing even for fans of teams who both win and benefit from the win -- and certainly for fans of the sport generally.
So it would be useful if there was a system for seeding the playoff round which was not susceptible to "tactical losing." Unfortunately I can't think of any such rule which seems fair, as long as there is more than one team in the playoffs.
So the question, albeit ill-defined, is this: Is there an analogue of Arrow's theorem for sports tournaments/leagues? (Or perhaps more appropriately an analogue of the related Gibbard-Satterthwaite theorem.)
I can make this a little bit more precise, but probably not totally (and there may be other ways of making the question more rigorous). We'll model the results of the regular season as a directed multigraph on the set of teams $S$, with an edge from $u$ to $v$ for each time team $u$ defeated team $v$. Is there a function from such multigraphs to ordered lists of size $n$ (N.B. that the order isn't meant to represent the relative strength of the teams, but is just a proxy for the extra structure of the playoffs) which satisfies the following (roughly defined) conditions:
- Path-independence: If there is a directed edge from $u$ to $v$ and a directed edge from $v$ to $u$, then the function is invariant under swapping the directions of these two edges.
- Universality. At weakest, this condition ought to state that for each underlying multigraph $G$ and each team, there's some orientation $G'$ of the multigraph such that that team makes the playoffs.
- Weak independence of irrelevant alternatives. Suppose $G, G'$ differ only in the orientations of edges between $u$ and $v$. Then, if any team $w \in S$ is in exactly one of $f(G), f(G')$, one of $u$ or $v$ must be in exactly one of $f(G), f(G')$. (Intuitively, this says that the only way that changing the result of an individual game changes who's in the playoffs is if it causes one of the teams playing the game to drop out of or enter the playoffs.)
- No tactical losing. This is hardest to define, and the big reason why this is a soft question. Is there a reasonable way to make this condition rigorous that leads to an Arrow-type theoreM?