Here are a couple of curious related results about a generalized 2-player 1-set tennis game: the winner of the set is the first player to win $n$ games, and the winner of each game is the first player to win $k$ points. Let $p>1/2$ be the probability that the stronger player wins a point, and let $P(n,k,p)$ be the probability that the stronger player wins the set. Then
- $P(n,k,p)>P(nk,1,p)=P(1,nk,p)$ if $n,k>1$, and
- $P(n,k,p)>P(k,n,p)$ if $n>k>1$.
The way the paper goes about proving this is pretty clever, but I'm still wondering if there is a more direct combinatorial proof. Say, for a rational $p$, after clearing denominators, we are counting more of something on the left than on the right because of such-and-such an injection.