Baseball elimination problem: Is second place possible? The lecture here describes how to tell if a given team is excluded from topping a baseball league using max-flow. For baseball, the team with the most wins tops the league and every game can either be a win or a loss. An obvious extension to this problem is determining if a team can come second. Is it possible to modify the solution to answer this question? Taking this further, one might ask what is the best possible rank a team can end up with?
 A: This is just the obvious common-sense approach, probably not very efficient.
You want to know if it's mathematically possible for your team, Detroit, to finish no worse than a tie for second place. This can be answered by solving four subproblems: Is it mathematically possible for your team to finish ahead of or tied with every team except NY? And the same question with NY replaced successively by Baltimore, Boston, and Toronto.
For the first subproblem, you can assume that your team wins all its remaining games, and that NY wins all its remaining games against the other teams. Now, with NY out of the picture, figure out if it's mathematically possible for your team to finish in first place.
A: Let's say my team is opposing $n$ other teams, call them $T_1,\dots,T_n$. Let $w$ be the number of wins my team ends up with if it wins all its remaining games. Let $s_i$ be the number of additional games $T_i$ needs to win to tie my team at $w$ wins; i.e., $T_i$ has already won $w-s_i$ games. For each set $I\subseteq\{1,\dots,n\}$ let $r_I$ be the number of games that the teams $T_i,\ i\in I$, have left to play among themselves.
Proposition. It's mathematically possible for my team to finish no worse than a tie for $k^\text{th}$ place if and only if there is an $(n-k+1)$-element set $J\subseteq\{1,\dots,n\}$ such that $\sum_{i\in I}s_i\ge r_I$ for every $I\subseteq J$.
My proof would use Hall's "marriage theorem". I never really studied network flows, but I think they are related.
