Badass Rank

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. ^[1]

The following chart shows how much rank is given upon completion of each challenge level.

Challenge level	Rank earned (Borderlands 2)	Rank earned (Borderlands: The Pre-Sequel)
1	1-10	1
2	5	5
3	10	10
4	50	25
5	100	50

Badass Tokens[]

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.

Conjecture:^[2]

$\text{Badass Rank} = \lfloor \left(\text{Badass Tokens}\right)^{1.8} \rfloor$

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 and Badass Ranks marginal — A graph of how much additional Badass Rank is required for the next Badass Token, given how much has been earned. (The graph is "bumpy" because Badass Rank can only be earned in discrete numbers.)

Rewards[]

Badass Tokens can be used to gain stat upgrades. The upgrades can be enabled or disabled at any time.

Stats[]

There are fourteen known stats that can be upgraded using Badass Tokens.^[3]

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:

$\text{Bonus in percent} = \text{round}((\text{Badass Tokens spent on stat})^{0.75} \times 10) \div 10$

Badass Tokens and bonus — A graph of bonuses by the number of Badass Tokens spent on that stat.

Badass Tokens and bonus marginal — Another graph showing the additional bonus from the next Badass Token spent on an stat, given how many have been spent on that stat already.

This table covers the bonuses for 0-27 tokens spent.

Tokens spent	Magnitude	Increase
0	0.0%	-
1	1.0%	+1.0%
2	1.7%	+0.7%
3	2.3%	+0.6%
4	2.8%	+0.5%
5	3.3%	+0.5%
6	3.8%	+0.5%
7	4.3%	+0.5%
8	4.8%	+0.5%
9	5.2%	+0.4%
10	5.6%	+0.4%
11	6.0%	+0.4%
12	6.4%	+0.4%
13	6.8%	+0.4%
14	7.2%	+0.4%
15	7.6%	+0.4%
16	8.0%	+0.4%
17	8.4%	+0.4%
18	8.7%	+0.3%
19	9.1%	+0.4%
20	9.5%	+0.4%
21	9.8%	+0.3%
22	10.2%	+0.4%
23	10.5%	+0.3%
24	10.8%	+0.3%
25	11.2%	+0.4%
26	11.5%	+0.3%
27+	(11.5+.3n)%	+0.3%

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.^[4]

↑ 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.

Strategy[]

Stat choice[]

Gun Damage is arguably the best all-around offensive choice except for heavily melee-focused characters.
Defense is split between Shield Capacity and Max Health. Recharge delay and recharge rate may be even more valuable depending on a player's personal play style. All defense stats are reasonably useful.
- However, lower recharge delay may be detrimental to some play styles relying on empty shields, e.g. with a Maylay Shield or Flame of the Firehawk.
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 or low reload times. It is less useful for weapons with small magazines and high reload times because it doesn't speed up reloading. It is useless for most Jakobs weapons that "fire as fast as you pull the trigger".
- In most cases, gun damage is preferable to fire rate because of reload time and because higher gun damage lowers ammo consumption. However, fire rate trumps gun damage when elemental damage over time is desired.
- High fire rate is a double-edged sword when using Hyperion weapons with the "sustained fire increases accuracy" tag, because more rounds of the magazine will be wasted before full accuracy kicks in.
Critical hit damage is usually inferior to gun damage, because it only increases critical hits while gun damage increases both critical and non-critical hits. There are only a few cases where critical hit damage applies and gun damage does not, like melee attacks or Amplify Shields' amp damage.
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.

Uniform distribution[]

If one wishes 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.

32-bit Limit[]

429496729 is the absolute limit to Badass Rank, and obtaining any more rank after this point, will cause it to become -429496729. This is the result of overflow, caused by the game being 32-bit. This puts a limit at 62530 Tokens. 62530^1.8=429496047.6, which gets rounded down to 428486047, the final Badass Rank to obtain the final token. But to understand why 429469729, is not equal to 4294967296, is because you must first divide by 10; (2^32)/10=429496729.6, and that's where the 32-bit limit is. 4466 Tokens, with a remainder of 6, can be put into all 14 bonuses, or 6253 tokens can be put into 10 bonuses; magnitude of 546.3% or 703.2% respectively.

Sources[]

[1] Values observed in IGN Live Presents: Borderlands 2, link in sources.

[2] 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().

[3] There are reports of there being fifteen stats; however, only fourteen have been observed in videos.

[4] The sources contain a table with values up to the 358th spent token.

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