Assume we have $n$ white points on a line, and that at a certain time a random subset of those points turns black. We have two teams A and B consisting of a finite number of players each of which is betting that at least one point in a certain interval will turn black (no two players can bet on the same interval, no matter from which team). We say that a player wins if indeed some point in his/her interval turns black and award each team points according to the number of its winning players.
Now assume that the players and their intervals are fixed in such a way that in each possible outcome, team A is awarded at most one point more than team B, and that in at least one outcome it is awarded exactly one point more. Prove that at least one player in team A wins whenever his/her team wins.
Edit. Upon a demand in the comments, let me try to put the question a bit more formally. We consider the set $[n]=\{1,\dots,n\}$ for some fixed $n\in\mathbb N$. Let $A,B$ be finite disjoint sets of intervals of the form $[a,b]=\{a,\dots, b\}$ with $1\le a\le b\le n$ and $a,b\in\mathbb N$ such that for any subset $K\subset [n]$ we have $$ \#\{ [a,b]\in A\,:\, K\cap[a,b]\neq \emptyset\} \le \#\{ [a,b]\in B\,:\, K\cap[a,b]\neq \emptyset\} +1,$$ and that for some specific $K$ equality is achieved. Prove that there exists $[a,b]\in A$ such that for any $K$ for which equality is achieved, we have $K\cap [a,b]\neq \emptyset$.