tl;dr In Shadowrun 5E, if you have ever asked yourself whether to use Breaking the Limit or Second Chance in a critical situation, the definite answer is: If $\frac{e}{n} < \frac{7}{18} = 0.3\bar8$ (where $n$ is your dice pool (without edge) and $e$ is your edge attribute) use Second Chance, otherwise use Breaking the Limit. I have the rest of this post for those of you who don't know what I've been talking about.

I am a big fan of pen-and-paper role-playing games (RPGs). In a nutshell, they are similar to RPGs you might know from video games. All players play a single character, except for one particular player, the game master, who replaces the computer and plays 'the rest of the world.'

Shadowrun is one such RPG. It is set in the near future, where magic has returned, and megacorporations have taken over the world. The games have two major parts: 1. social interactions and 2. fights. To have suspense in the fights, you need some rules and most systems include dice roles to introduce randomness. In Shadowrun, you roll a certain number of 6 sided dice (or d6 for short). The better your character is at a task, the larger your dice pool for a roll. To determine the success of your action, you count the number of 5s and 6s you rolled, and if you pass a threshold, you succeeded.

Now, player characters are of course good at what they are doing and often get out of situations where a regular mortal would simply have to give up. You could say your characters have a certain edge. Coincidentally, this is a resource each Shadowrun character has, and you can spend it to get out of tight corners. There are a few ways to spend it, but I want to compare only two of them for this post. If you have a roll that you really need to succeed, you can decide to spend edge either

There are some technicalities I'm leaving out here, e.g. limits and glitches, and that you could use Breaking the Limit after your roll with some disadvantages. At least in my group, these two are the most common scenarios, so I feel justified in that choice.

Now I'm a mathematician by training and I was asking myself: What is the best choice? I mean, which option should I choose to maximize my number of 5s and 6s? Maybe you have asked yourself the same question. If so, this blog post is for you 😉

First of all, I would like to talk about the intuition and why (at least for me) it is not straightforward to see which choice is better.

In the first case, you can reroll 6s, but on average you only get 1/6 of your dice to be sixes. On the other hand, in the second case, you can reroll all failures, which are on average 2/3 of your pool and 2/3 > 1/6. This might indicate that Second Chance is the better option, but that leaves out two important points. (1) You can reroll 6s infinitely often and (2) you get bonus dice. (1) actually, should not have too large of an impact because your pool of remaining dice should get smaller exponentially fast. Let's assume you start with 12d6. On average you have 2 6s after the first roll, so your new pool is 2d6. Here you expect 1/3 6s, so most of the time, you won't have a 6 to continue rolling at all. (2) might be more interesting and is also easier to understand and quantify. If you get enough additional dice, even if you can only reroll 1/6 of them, it might still be better than the 2/3 of the smaller pool.

However, after all these thoughts, I'm still at a loss to confidently say which strategy is better. So I decided to compute an exact expression for the number of expected successes in both cases and compare them.

No matter which of the two edge options you choose, you can be lucky in your role or not. What I'm trying to say is that for a single roll, you cannot prove or predict any outcome. However, you will probably roll multiple times, and you want to come out on top on average. Hence, we want to maximize the expected number of successes (i.e., 5s and 6s) given a dice pool of $n$ dice. One way to think of this is, if I roll the same pool using the same rules many times and average the number of successes, this is what I get. If you base your decision on which edge strategy to take on the expected value, you should come up on top in the long run.

So I sat down and computed the expected value for both strategies and it turned out that there are really simple formulas for both of them. However, the derivation is a bit technical, so I will defer it to another blog post (here is the one dedicated to Second Chance and here the one to Breaking the Limit).

Let $n$ be the number of dice in my pool and $\mathbb{E}_\text{SC}[k \mid n]$ denote the expected number of successes with a pool of $n$ dice using strategy Second Chance and $\mathbb{E}_\text{BL}[k \mid n]$ the same for Breaking the Limit. Then we have

Now $\frac59 > \frac25$, so this seems to indicate that Second Chance is superior, but this neglects the increased dice pool of Breaking the Limit. Let $e$ denote the bonus pool. Then the actual comparison is

We can solve this equation for the fraction $\frac{e}{n}$ and get

This implies that for $\frac{e}{n} < \frac{7}{18}$ we should use Second Chance and for $\frac{e}{n} > \frac{7}{18}$ we should use Breaking the Limit. In the unlikely event that $\frac{e}{n} = \frac{7}{18}$ , both actions are equally good from an expectation standpoint (one might be better than the other based on the variance, for example, but that would take us too far). The intuition behind the formula is also more accessible because it only depends on the size of the edge pool compared to the regular pool. If you have a small $e$, then the added dice cannot compensate the fact that the expected value scales only with $\frac25$. If $e$ is large, the bonus dice make you beat the better scaling of $\frac59$.

To spare you the trouble of computing the fraction yourself, I have computed a table for you with typical pool and edge values such that you can easily read what the optimal strategy is in your case. Please keep in mind that I have only considered the expected values in my analysis. I haven't talked about glitching and I have ignored the fact that Breaking the Limit allows you to ignore 'your limits' (big surprise), whatever that means. Both points might lead you to favor one strategy over the other even though the expected successes are smaller.

I hope you found this blog post helpful. In any case, if you have any questions, thoughts or comments I would like to hear them. Just reach out to me via Twitter or e-mail.

If you want to continue this thread of posts, I would suggest turning to my post about Second Chance.

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