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Badass Rank is a reward system in Borderlands 2. Badass Ranks are 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 Ranks upon completion. ^[1]

The following chart shows how many ranks are given upon completion of each challenge level.

Challenge level	Ranks earned
1	1
2	5
3	10
4	50
5	100

Badass Tokens

Achieving enough Badass Ranks 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 acquire a new Badass Token, each Badass Token will require a greater number of Badass Ranks than the previous one. However, there is no limit to the number of Badass Ranks and Tokens that can be earned.

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:

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

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.

Optimal distribution

Suppose one makes the approximation that the 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 one has 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 one values one stat twice as much as another, one 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 a token is redeemed, if ten preferred stats are desired, at least one of them is guaranteed to show up every time. Therefore, players may wish to invest all 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 the bonuses.

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

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

[1] 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().

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

[1]

[2]

[3]

Wikia

Badass Rank

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