Badass Rank

Badass Rank is Borderlands 2's replacement for weapon proficiency. Badass Rank is earned by completing challenges. Some challenges are repeating, which means that there is theoretically no maximum to the Badass rank a character can achieve.

Badass Tokens
Achieving enough Badass Ranks will award the player with a Badass Token, which can be spent on one of five randomly selected stat upgrades out of a list of possible choices. These upgrades affect all of the player's characters--past, present and future.

Conjecture: Badass Rank = floor((Badass Tokens)1.8)

This means that each Badass Token will require more new Badass Ranks than the last, but there is no limit to the number of Badass Tokens that can be earned.



There will be some option to opt-out of the Badass Rank system entirely on a per-character basis; it is unclear whether this option will toggle off the system entirely for the character, or if it will just restrict the character to use only the Badass Ranks earned by that character.

Options
There are fourteen known options that can be upgraded using Badass Tokens.


 * Critical Hit Damage
 * Elemental Effect Chance
 * Elemental Effect Damage
 * Fire Rate
 * Gun Accuracy
 * Gun Damage
 * Grenade Damage
 * Maximum Health
 * Melee Damage
 * Recoil Reduction
 * Reload Speed
 * Shield Capacity
 * Shield Recharge Delay
 * Shield Recharge Rate

Magnitude
Spending points on a particular option has diminishing returns.

Conjecture: Magnitude = (Badass Tokens spent on option)0.75



This table covers the bonuses for 0-20 tokens spent.

Least uniform distribution
If you have a ranking of preferences for the reward options, the following table gives:


 * Probability: The probability that the best (highest preference) out of the five randomly-selected choices will be your nth preference.
 * Cumulative probability: The probability that the best out of the five randomly-selected choices will be at least as good as your nth preference.



Optimal distribution
Suppose we make the approximation that your valuation of a 1% bonus in each option is constant. The marginal benefit of spending one more Token in an option is then proportional to vx-0.25 where x is the number of Tokens you have spent in that option and v is the valuation of a 1% bonus in that option. For an optimal distribution, this should be constant across all options, which means that x should be proportional to v4, or equivalently, the bonus in each option should be proportional to v3. For example, if you value one option twice as much as another, you should spend sixteen times as many Badass Tokens on that option, gaining eight times the bonus.