Analytics Check

# Rolling for stats in practice: how bad luck protection inflates ability scores

2021/07/17

Rolling for stats in DnD 5e always carries the risk of ending up with a weak character. Despite recommendations to use the standard array, point buy, or any number of homebrew rules for generating stats, rolling rules-as-written nevertheless remains common. For these tables, poor rolls inevitably occur, and when that happens, many DMs implement bad luck protection by allowing bad stats to be rerolled.

In this post, we look at the mathematical consequences of rerolling your stat array. By setting a limit on how bad a player can roll before allowing a reroll — we call that limit the bad luck threshold — the underlying distribution of possible outcomes changes and biases ability scores higher than they would otherwise be.

## Reviewing the standard array and rolling RAW

Before we dive into how rerolling changes the odds, let’s review how rolling rules-as-written compares to the standard array. As a reminder, the standard array is [8, 10, 12, 13, 14, 15]. Rolling for stats with 4d6 dropping the lowest generates a wide range of possible outcomes.

The chart below compares the two methods. The top-left shows the distribution of possible outcomes for the lowest value in your rolled array; typical values are somewhere between 6 and 11. The red line at 8 indicates the corresponding standard array values. Moving right, we see the distribution for the next highest stat in the array. This continues until the bottom-right, showing the possible outcomes of the highest stat in the array.

The standard array more or less selects the average from rolling, but not always! Interestingly, the largest deviation is the highest stat — rolling more often than not gives 16 or higher, compared to the standard array’s 15.

Seemingly small differences result in the fact that rolling will on average give higher ability scores than the standard array. By looking at the total ability score of the array, which is sum of the whole array, we get a better idea of how they compare. The figure below shows the bell-shaped curved of the total ability scores from rolling compared to the red line of the standard array.

The standard array’s red line is slightly left-of-center of the potential outcomes from rolling. Compared to the standard array’s 72 total ability score, rolling has an expected value of about 73.5. An extra point or two isn’t earth-shattering, but it is a difference worth noting.

## Bad luck protection and the upward bias of rolled stats

Armed with an understanding of how different rules generate differing ability scores for a character (point buy is beyond this post’s scope), we shift our focus to how these rules work in practice.

As mentioned in the intro, many groups rightly prioritize the enjoyment of the players over some arbitrary standard of mathematical fairness by rerolling bad arrays. This biases the total ability score upwards by cutting out extreme lows.

One can imagine that a player who has the misfortune to roll significantly under the standard array may want a reroll. This has the effect of chopping off the left side of the total ability score distribution and increasing the odds of the remaining outcomes.

The chart below shows how that shift happens — the underlying yellow-orange bars represent the full possible outcomes of rolling, whereas the slightly higher overlaid blue bars show how the distribution changes when it gets cuts off. The bad luck threshold shown here is the point where we say “any stats that are worse than this should be rerolled”.

The relatively modest bad luck threshold shown above, where a player rerolls when their total ability score is 60 or below (which are effectively commoner stats!), has the same shape as before, but with the left tail cut off. Although the difference seems minor, it is enough to raise the expected total ability score from 73.5 to 74. By allowing players to reroll commoner-tier stats, they expect to gain half of an ability score in the process.

Every bad luck threshold we pick gives a different expected boost to your ability score.

The figure below shows just how that bonus varies. The horizontal axis represents the different thresholds we can set. Higher bad luck thresholds correspond to being more generous with rerolls. The vertical axis is our expected total ability score when we roll using this bad luck protection rule.

As we increase the threshold, we see our expected stat total increase (as represented by the blue markers). Our previous threshold of 60 puts us right at a full ability score increase above the standard array. Increasing it to a higher value of 68 or so is equivalent to two full ASIs above the standard array.

When phrased in terms of ASIs, it’s clear that rolling RAW already gives a notable average boost over the standard array. Adding even a small amount of bad luck protection to that pushes the disparity even further.

As mentioned previously, a bad luck threshold of 60 — which corresponds to an average ability score of 10 — is enough to push rolling to a full ASI above the standard array. A more generous threshold of 66, or an average ability score of 11, doubles the expected stat bonus of rolling from 1.5 to 3 above the standard array.

Of course, a single number like the total ability score cannot capture all of the nuances that come into play when determining what is a “good” stat array. Having a very strong primary stat can more than make up for a poor dump stat or two, and none of that nuance is appropriately captured by this post. Regardless, when looking at different methods for generating ability scores, it’s important to explore how they are used in practice for an honest comparison between them.

## Some thoughts on fairness

There is nothing wrong with ensuring everyone is happy by allowing players to rerolls stats, but tables should be wary as to how this method can affect players that use different stat generating methods. If some players want to roll and others prefer to use the standard array, bad luck protection can lead to a lopsided power balance in favor of the rollers.

On one hand, it is something that could easily go by unnoticed and (in all honesty) likely wouldn’t have too much of an effect on the game. On the other hand, it could feel bad to effectively give an ASI to some players at character creation and not to others. Encouraging everyone to use the same method for generating ability scores ensures an even playing field, but if a table does split, buffing the standard array by an ASI isn’t a half-bad way to go.

Another point that often comes up in these discussions about rolling for stats is the opposite problem — good luck protection. Players who roll far above their party members may feel guilty about their greater character power and choose to reroll to get a more average array. This has a counteracting effect, pushing the expected ability scores lower. One strategy to constrain extreme ability scores is to reroll on both overly weak and overly strong characters. If the reroll rules are symmetric, it leaves the expected stats unchanged from the original method.

The goal of character creation should always be to create a party of adventurers that your table is excited to play. Keeping in mind that something as simple as allowing a player to reroll their stats will inflate them is worth keeping in mind, so that you can ensure balanced table where everyone can shine.