Badass Rank

Badass Rank is a reward system in Borderlands 2. 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
1	1-10
2	5
3	10
4	50
5	100

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.

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:

$\text{Bonus in percent} = \left(\text{Badass Tokens spent on stat}\right)^{0.75}$

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%

↑ 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.

Strategy

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 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.

Uniform distribution

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.

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.

[1]

[2]

[3]