How to prove domination in baseball?

Question

Several teams played a baseball tournament (as a reminder, there are no ties in baseball); each team played every other team exactly once. We say that team $A$ is dominating team $B$ if either $A$ beat $B$ heads up or if there exists a team $C$ such that $A$ beat $C$ and $C$ beat $B$. (Notice that it is entirely possible for team $A$ to be dominating team $B$ and team $B$ to be dominating team $A$). Team Baseball collected the most points and won the tournament. Prove that Team Baseball dominated every other team.

Answer 1

For three teams $A,B,C$, there are $3$ games, therefore $3$ points to score. Without loss of generality, let $A$ be the Team Baseball. Then it must score $2$ points, otherwise it can not be the Team Baseball. Indeed, if $A$ scores $1$, then either $B$ or $C$ must score $2$ OR both must score $1$ each. It implies team $A$ won both $B$ and $C$, hence dominate both.

For four teams $A,B,C,D$, there are $6$ games, therefore $6$ points to score. Again assume $A$ is the Team Baseball. Then it must score $3$ points and others less. Indeed $A$ can not be called Team Baseball with less than $3$ points, for example $2$, otherwise other team must score $2$ implying deuce. Again $A$ is dominant with score of $3$ implying it won all other teams one by one.

For $n$ teams, there are $P(n,2)$ games and Team Baseball must score $n-1$ points to be called such. Hence it is dominant.

Answer 2

I assume team B has at least as many wins as any other team, i.e., it is at least tied for first place. Let A be any other team; we want to show that team B dominates team A.

If team B beat team A, there is nothing to prove. So we assume that team B beat team A. Since team B scored at least as many wins as team A, it follows that B must have wone more games than A against the other teams. Thus team B must have beaten some team, call it C, which team A did not beat. So B beat C and C beat A, i.e., B dominates A.