I understand taking rebuys and add-ons into consideration, but I don't get why the initial buy-ins matter. For example, assuming no rebuys and add-ons, if the league has some $20 tournaments and some $40 tournaments, the winner of a 16-player $40 tournament would be awarded 41% more points than the winner of a 16-player $20 tournament. If it's mostly the same players in every game (as in a league) and they like to vary the buy-ins for whatever reason, why should more points be awarded for playing a higher buy-in tournament with the same field of players? I'm not criticizing the practice, but I would like to understand the rationale -- or is it "just because"?
Generally speaking, higher buy-ins equal a higher skill level. Players will take the game more seriously if they invested more money. Though this isn't true for every player, on average I think it's true. It's easier to go all-in on the first hand in a $1 buy-in tournament, than in a $1,000 buy-in tournament. You don't risk as much.
Playing in a larger field also means you need to outlast/outplay more players. Making finishing 5th more challenging when playing against 19 players than when your playing against 9.
Of course these numbers lose their relevance when you're always playing for the same buy-in and with the same number of players.
I think a ~slightly~ higher points reward based on higher buy-in amounts (or for no-rebuy events) may be justified, but not to that extreme. Awarded points differing on field size are usually over-rewarded, too. I've yet to hear a rational explanation as to why the amounts are usually so extravagant.
Whether the increase in skill factor is the square root is another matter entirely. This is an arbitrary value in the formula. You can use any power value between 0 and 1 you like. Different powers will change the relative difficulty level of higher buy-ins and more players. A higher power will bring them together, a lower one will widen the gap, with a power of 1 giving equal points regardless of buy-in or players, and a power of 0 giving a 1:1 relation between points and number of players and/or buy-in amount (twice the players = twice the points).
The square root (or power of 1/2) is smack in the middle and an obvious choice for a formula. One shouldn't read more into it than that.
In this formula, changing the power value will also affect the importance of rebuys and add-ons. Here too, a higher value (closer to 1) will decrease the importance, while a lower value (closer to 0) will increase the importance.
