Badass Rank

Badass Rank is Borderlands 2's replacement for weapon proficiency. Badass Rank is earned by completing challenges. Challenges give increasing amounts of badass ranks according to their level, while some are just one off bonuses.

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. Less-picked options appear more often, so you cannot have a too unbalanced distribution of Tokens. 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.



Rewards
Badass Tokens can be used to gain stat upgrades. You can enable or disable these upgrades at any time.

Stats
There are fourteen known stats 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 Badass Tokens on a particular stat has diminishing returns. The formula for the magnitude of the bonus by the number of Badass Tokens is:

Magnitude = (Badass Tokens spent on stat)0.75



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

Stat choice

 * Gun Damage is arguably the best Badass stat for anyone but high level melee and stealth focused Assassins (and it's decent for them too).
 * Defense is split between Shield Capacity and Max Health. Recharge Delay and Rate may be even more valuable depending on your personal play style. All defense stats are reasonably useful.
 * Accuracy can be thought of as an increase in the effective range of your weapon.
 * Fire Rate is most useful for automatic weapons with large magazines, such as SMGs and Support Machine Guns. It is less useful (but decent) for shotguns and rifles. It is generally inferior to Gun Damage, as you will spend more time reloading for the same damage increase.
 * Critical Hit damage is generally inferior to Gun Damage but can still be useful.
 * Elemental chance and damage value varies considerably from character to character. So does melee damage.

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

"Top 10" distribution
Since there are five choices each time you redeem a Token, if you pick ten preferred stats, at least one of them is guaranteed to show up every time. Therefore, if you wish you can invest all your Tokens in ten preferences and none in the other four. This means 40% more Tokens for each of those ten stats, which is equivalent to 28.71% more bonus on those stats. The price is a reduction of 8.07% in the sum of your bonuses.

Uniform distribution
If we wish to have a uniform distribution (which is optimal if our valuation for each option is exactly the same), it suffices to just repeatedly hit the confirm key.