## Saturday, 6 February 2021

### A reroll, or a modifier?

Wargaming rules often use die roll modifiers, usually in the context of a die roll needing to hit some target number. Alternatively, some rules allow a reroll when the first roll fails.

Many rulesets these days are designed around the idea of giving special abilities to troop types. Some troops receive a +1 on their attack roll, e.g. in a bucket of dice mechanisms, while other troops are allowed rerolls. These sound like fun variations, and there might be reasons why on one case a special ability gives a +1, an in another case calls for a reroll, but does it also makes a difference from a statistical point of view?

Rerolls

For the sake of simplicity, let us assume we only roll D6s.

One mechanism might call for a roll, needing a 5+ to succeed. It's fairly obvious the probability for succeeding is 1 in 3 (33.33%). If you get a +1 on your roll, the die would only need a 4 or higher, and the probability increases to 1 in 2 (50%). In general, every +1 on a D6 increases the probability for success by 16.67%, irrespective of the original target number.

How does that change when doing a reroll for a failed first roll? Let's assume our die meets the target number with x (x between 0 and 1) probability. We have to consider 3 cases:

1. The original die roll succeeds: probability x
2. The original die roll fails, but the reroll succeeds: probability (1-x), followed by x, or total probability of (1-x)*x
3. The original die roll fails and the reroll fails: total probability (1-x)*(1-x).

So we score a success with a total probability of x+(1-x)*x = x*(1+1-x) = x*(2-x).

We can continue this calculation by now allowing for another reroll: take the probability for success after 1 reroll, and add the probability of failure multiplied by another basic success, and so on. We could do that type of calculation very easily in a spreadsheet, and this gives us the following table or probabilities. for a D6:

 Target number Basic roll 1 reroll 2 rerolls 3 rerolls 1+ 1.00 1.00 1.00 1.00 2+ 0.83 0.97 1.00 1.00 3+ 0.67 0.89 0.96 0.99 4+ 0.50 0.75 0.88 0.94 5+ 0.33 0.56 0.70 0.80 6+ 0.17 0.31 0.42 0.52

It is interesting to realize that these are also the same probabilities for taking the maximum score of a number of dice. Allowing for 2 rerolls is the same as taking the maximum roll of 3 D6’s. After all, it doesn’t matter whether we roll the 3 dice simultaneously, or only after the first one failed etc.

Plotted in a graph, this gives us the following probabilities for reaching a target number (graph made with anydice.com):

We can do a calculation for a D10 including several rerolls as well, and this gives us:

Modifiers

The standard +1, +2, +3 modifier are well-known in wargaming. Computing the probability vs a target number is easy enough, we simply shift the probabilities towards higher numbers, as shown in the table below:

 Target number no modifier 1 2 3 1+ 1.00 1 1 1 2+ 0.83 1 1 1 3+ 0.67 0.83 1 1 4+ 0.50 0.67 0.83 1 5+ 0.33 0.5 0.67 0.83 6+ 0.17 0.33 0.5 0.67

In graph format:

Reroll vs Modifier

So we can ask the question whether it does really matter whether we apply a reroll, or rather a modifier. Let’s assume we only allow a single reroll (the most common case). By plotting the reroll probability vs various modifiers, we see that a D6 with a single reroll produces almost the same probability result as a D6+1. Of course, there are some deviations, but overall, the fit is very close.

We can do the same exercise for a D10, and see that a D10+2 fits closest to a D10 with a single reroll.

This is of course not terribly surprising. The real question is whether rerolls or better, or modifiers are better in a wargames?

This heavily depends on the rest of the rules, of course. A single die mechanic is always part of a larger rules framework. Personally, I don’t mind rerolls, and I don’t mind modifiers. But I do mind if both of them are used throughout the same ruleset without any real consistency. Hence, I prefer either modifiers in a ruleset, or rerolls, but not both. I understand modifiers and rerolls are used in the same game to add some “variety”, but probability-wise, this is often not necessary.

Happy wargaming!