Knockout tournaments are often used in sports (or experiments where preferences are registered by comparisons instead of measurements) to determine the champion of an event. A knockout tournament plan (KTP) for n players is a rooted binary tree with n leaves to be labeled by the n players. Each subtree of two leaves represents a match between the two players labeling the two leaves; the winner of the match then moves on to label the root of the subtree. While there are many KTPs to choose from for a given number of players, in the real world an almost balanced KTP is usually chosen. One reason could be the perception that a balanced KTP is “fair” to the players, in the sense that, given a random labeling of leaves by players, a stronger player has a better chance to win the tournament. Surprisingly, it has been shown that not all KTPs have this property, and it is difficult to prove this property for any general class of KTPs. So far the property has been shown to hold only for balanced KTPs. In this paper we extend it to some classes of almost balanced KTPs.