Professor Halfbrain has spent his entire weekend watching the games at the local chess tournament. As usual, every two players played exactly one game against each other. A win yields 1 point, a draw yields half a point, and a loss yields nothing. At the end of the tournament, Professor Halfbrain carefully analyzed all the games and compared them to the final tournament ranking of the players.
To his excitement, Halfbrain realized that there were games in the tournament in which the winner had reached fewer points (in the final ranking) than the loser (in the final ranking). Professor Halfbrain decided to call such games nonsensical, and to call all other games (including all the ones that ended in a draw) sensical games. The professor managed to prove the following two deep theorems on sensical and nonsensical games.
Professor Halfbrain's first theorem: In every chess tournament, at least $0$ percent of the played games are sensical.
Professor Halfbrain's second theorem: There exist chess tournaments, in which at most $100$ percent of the games are sensical.
This puzzle asks you to improve the two theorems of professor Halfbrain and to make them even deeper. Find an integer $x$, so that the "$0$ percent" in the first theorem may be replaced by "$x$ percent", and so that the "$100$ percent" in the second theorem may be replaced by "$x+1$ percent" (again yielding true statements, of course).