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Badass Rank is a reward system in Borderlands 2 and Borderlands: The Pre-Sequel. Badass Rank is earned by completing challenges. Each of the challenges, save for location specific challenges which serve as one-time bonuses, give an increasing amount of Badass Rank upon completion. 
The following chart shows how much rank is given upon completion of each challenge level.
|Challenge level|| Rank earned
(Borderlands: The Pre-Sequel)
Earning enough Badass Rank will award a player with a Badass Token, which can then be spent on one of five randomly selected stat upgrades out of a list of 14 possible choices. Less-picked options appear more often, so players are prevented from gaining a wildly unbalanced distribution of tokens. These upgrades affect ALL of a player's characters, present and future.
This means that to each Badass Token will require a greater increase of Badass Rank than the previous one. However, there is no limit to the Badass Rank or the amount of Tokens that can be earned.
In Borderlands 2, completion of all challenges (including those specific to DLC) results in 24578 Badass Rank. When starting from 0 Badass Rank, this will provide a total of 274 Badass Tokens.
In Borderlands: The Pre-Sequel, completion of all challenges (including those specific to DLC) results in 12964 Badass Rank. When starting from 0 Badass Rank, this will provide a total of 192 Badass Tokens.
Badass Tokens can be used to gain stat upgrades. The upgrades can be enabled or disabled at any time.
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
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:
This table covers the bonuses for 0-27 tokens spent.
Although the general formula for 27+ tokens is good for a rough estimate, it's wrong starting at 28 tokens. At 200 tokens, there is a more than 10% difference to the actual value.
- ↑ Values observed in IGN Live Presents: Borderlands 2, link in sources.
- ↑ See talk page for data. The formula fits all known data to within observation error of the progress bars. It is possible that the formula uses round() instead of floor().
- ↑ There are reports of there being fifteen stats; however, only fourteen have been observed in videos.
- ↑ The sources contain a table with values up to the 358th spent token.
- 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 a player's personal play style. All defense stats are reasonably useful. There are a few shields, however, such as the Love Thumper and Flame of the Firehawk, which benefit from a high recharge delay.
- Accuracy can be thought of as an increase in the effective range of a 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 more time may be spent reloading for the same damage increase.
- Critical Hit damage is generally inferior to Gun Damage but can still be useful, especially for sniper Assassins.
- Elemental chance and damage value varies considerably from character to character, as does melee damage.
"Top 10" distribution
Since there are five choices each time a token is redeemed, simply choose four undesired stats leaving ten preferred stats to invest in. Inevitably, the undesired stats will be the first four choices and the bottom choice will always be preferred.
This means 40% more Tokens for each of the preferred ten stats, which is equivalent to 28.71% more bonus on those stats. The disadvantage is a reduction of 8.07% in the sum of all bonuses due to the diminishing returns, although a few points in the undesired stats can greatly curb the disadvantage.
If one 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.