My MMO have a heavy focus on PvP, so the combat mechanics have to be pretty balanced or it will die a horrible death. But, as with the vast majority of social MMO's, in order to offer the game for free you have to sell the items in the game for real money to keep the servers up and running. And very few people will purchase in game items unless it gives them an advantage over people without those items. But the people without the means to buy those items will quickly get frustrated if they can't compete with the 1% who do. So they get frustrated and quit. Which is bad for business.
So I came up with a logarithmic equation which I thought would solve my problem, and it does to a certain extent. The equation is based off of the sigmoid function, P(t) = 1/(1+e[sup]-t)[/sup] with t being a function of the proposed attack power. This would be multiplied by the actual damage of the attack (based off of a Guassian function, most likely) and added as a bonus to the attack. This would have the result of an attack being 80% effective at around 55% of the max potential attack power. So even if you had the best of the best, you would only be 20% better than the average player. Furthermore, if a 100% effective player scored a hit with an average base amount of damage, the lesser geared player still has a chance to beat him or her if they achieved a hit with a high base amount.
However, this has led to some unfortunate scenarios. First, there is a limit to the max amount of power one can get. When you get to where e[sup]-t[/sup] = -1, you get an asymptote. Any higher numerical value will return the same amount as a number just less. So if my max limit is 1000 power, a person with 999 and a person with 9999 will get the same benefit. This paints me in a corner in terms of expansions and continued progression.
Secondly, I think 55% of max is too low. I would ideally like to see around 65%-70% of max return an 80% efficiency. So while some people might buy their way to the top, it will still give many players opportunity for improvement solely within the game, which is key to player retention. But, try as a I might, I can't get the function to behave in the way I want it to. It's most likely a property of the function I used, but I'll most likely toss it out on the Math and Physics forum to see what other, more mathematically inclined people feel about it. Plus, I'm not sure if the people who purchase in game items for out of game dollars expect more bang for their buck, so to speak.