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?
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:
- The original die roll succeeds: probability x
- The original die roll fails, but the reroll succeeds: probability (1-x), followed by x, or total probability of (1-x)*x
- 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 |
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:
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 |
1.00 |
1.00 |
1.00 |
2+ |
0.83 |
1.00 |
1.00 |
1.00 |
3+ |
0.67 |
0.83 |
1.00 |
1.00 |
4+ |
0.50 |
0.67 |
0.83 |
1.00 |
5+ |
0.33 |
0.50 |
0.67 |
0.83 |
6+ |
0.17 |
0.33 |
0.50 |
0.67 |
In graph format:
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!
I think I mostly prefer modifiers - provided the number of them is small - to re-rolls or rolling several dice and selecting the best - for the following reasons:
ReplyDeleteModifiers are contained in a list of significant factors which can be altered easily by crossing out or addition of new ones in draft.
Re-rolls take more time.
Rolling several dice obviously requires more dice (cost), a container so they don't scatter over the battlefield and/or damage the troops and take time to collect and check.
OTOH, the option of re-rolling up to a number of times equal to the chances in one's favour, but having to accept the last roll, as used Strategos: An American Kriegsspiel, is simple to use and gives the player a decision to make, which could be regarded as representing committing more troops to a stalled attack - "One more charge, boys, and the day is ours!" - with all the attendant risks.
Yes rerolls can take many different forms. I also feel rerolls as a mechanic are not as versatile as modifiers, esp when rolling vs a target number. But I can see games designers include rerolls for "variety" and adding more "special abilities" into the mix.
DeleteI love this blog. This stuff matters in games, and it's so well explained even an unmathematical type like me gets it.
ReplyDeleteThat's the idea :-)
DeleteInteresting!
ReplyDeletePersonnaly (but that's me) I dislike re-rolls because they take more time, and also they tend to break the narrative (unless a realistic reason explains why a same action happens twice).
My choice between a modifier, or rolling two (or three) dice at the same time, would also depend on the story. A positive modifier gives a feeling of strength & superiority; rolling two or three dice gives the impression of 2 or 3 very fast blows, or of multiple missiles.
I know that it has nothing to do with probability...
Very interesting.
ReplyDeleteThe difference with a re-roll of course is that the decision as to whether to apply the modifier doesn't need to be made until after the first die roll (so it's not the same as rolling two dice together?). Might not need it if you are lucky the first time.
Really enjoying reading through your posts. I'd love to hear your thoughts on your favourite combat systems/mechanics you've come across in games over the years. Cheers, and keep up the great work!
ReplyDeleteThanks!
DeleteA bit difficult to answer, since combat mechanics do not live in a vacuum, but are connected to other game procedures. But overall, I like mechanics that:
- involve both players, e.g. an opposed die roll, or a to hit/to save mechanism. Anything that makes both players roll a dice, because it reduces downtime and increases involvement.
- mechanics that allow players to apply modifiers regarding the status of their own troops to their own die rolls. E.g. cover should be a + on the defender's roll, not a - on the attacker's roll.