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Monday, 30 April 2018

An analysis of combat resolution in Rommel

Recently we played our first games of Rommel (see my regular wargaming blog), a WW2 ruleset published by Honour.

The procedure for resolving combat felt a bit unusual to me, so I decided to analyze the procedure from a mathematical point of view. Note that this post is not a review of the ruleset as a whole - which I like very much for some its original concepts and clever ideas.  But I am always interested in analyzing specific procedures, how they work, and whether we can gain some additional insights from running through the numbers ;-)

Combat resolution in Rommel

First, let's take a look at how combat resolution in Rommel works. Rommel is a grid-based ruleset, and when resolving combat, the combat factor of all the units that end up in the same gridcell are added together. A D6 is then rolled, and cross-indexed on the table shown below. One then has to count how many "yellow boxes" have a number equal or lower compared to the total combat factor, resulting in the number of hits on the opposing units.


E.g., suppose we have a total combat strength of 15, and I roll a 3. Looking at the "3" column, I count 2 yellow boxes (9 and 14), whose value is less or equal than 15. Thus, I inflict 2 hits on the enemy.

Note that since 3 units can occupy a single gridcell, and the combat factor per unit is typically 3, 4 or 5, we might have a combat factor in the 9-15 range. This can be modified due to tactical factors, artillery support, etc., but those are typical numbers (at least in the scenarios we played).

When I first played Rommel, I felt this "user interface" for determining the number of hits was a bit strange, and as a good DIY wargamer, I always wonder whether I can replace it with something more to my own liking, without compromising the initial outcomes too much.

Expected number of hits per combat factor

Counting numbered boxes seemed a bit non-transparant to me. It is difficult to judge whether the number of hits we can expect for a given combat factor in a gridcell goes up linearly, whether there are certain "clicks" that suddenly give an advantage, etc.

So the first thing to do is to compute the expected number of hits per combat factor. The expected value of a random process is simply the average value one can expect when repeating the process an infinite number of times, and is computed by averaging all outcomes, weighted by the probability that each outcome occurs. For our procedure, we have 6 outcomes per combat factor, and each has an equal likelihood of occuring.
E.g., let's compute the expected number of hits for combat factor 15. We could roll a 1 on the die, resulting in 1 hit, or we could roll a 6, resulting in 4 hits. Averaging over all possible outcomes we get:

expected hits = (1+2+2+3+3+4)/6 = 15/6 = 2.5 hits.

 Or course, we will never score exactly 2.5 hits, but this is an average taken over all possible rolls for combat factor 15.

In order to compute the expected value for all combat factors for 1 up to 40, I simply made a spreadsheet, listing the possible outcomes per combat factor as given in the original resolution table, and simply let the spreadsheet compute the average. Since the minimum and maximum number of hits could also be of interest, I plotted these in additional columns as well.



Hits on Die roll






Combat Factor 1 2 3 4 5 6 Expected Min Max
0 0 0 0 0 0 0 0.00 0 0
1 0 0 0 0 0 1 0.17 0 1
2 0 0 0 0 0 1 0.17 0 1
3 0 0 0 0 1 1 0.33 0 1
4 0 0 0 0 1 2 0.50 0 2
5 0 0 0 1 2 2 0.83 0 2
6 0 0 0 1 2 2 0.83 0 2
7 0 0 1 1 2 2 1.00 0 2
8 0 0 1 1 2 3 1.17 0 3
9 0 1 1 2 2 3 1.50 0 3
10 0 1 1 2 2 3 1.50 0 3
11 0 1 2 2 3 3 1.83 0 3
12 1 1 2 2 3 3 2.00 1 3
13 1 1 2 2 3 3 2.00 1 3
14 1 2 2 2 3 4 2.33 1 4
15 1 2 2 3 3 4 2.50 1 4
16 2 2 2 3 3 4 2.67 2 4
17 2 2 2 3 4 4 2.83 2 4
18 2 2 2 3 4 5 3.00 2 5
19 2 2 3 3 4 5 3.17 2 5
20 2 2 3 3 4 5 3.17 2 5
21 2 2 3 4 4 5 3.33 2 5
22 2 2 3 4 5 5 3.50 2 5
23 2 3 3 4 5 5 3.67 2 5
24 2 3 3 4 5 5 3.67 2 5
25 2 3 4 4 5 5 3.83 2 5
26 3 3 4 5 5 6 4.33 3 6
27 3 3 4 5 5 6 4.33 3 6
28 3 4 4 5 5 6 4.50 3 6
29 3 4 4 5 5 6 4.50 3 6
30 3 4 4 5 6 6 4.67 3 6
31 3 4 5 5 6 6 4.83 3 6
32 3 4 5 5 6 6 4.83 3 6
33 4 4 5 5 6 6 5.00 4 6
34 4 4 5 5 6 6 5.00 4 6
35 4 4 5 5 6 6 5.00 4 6
36 4 5 5 6 6 6 5.33 4 6
37 4 5 5 6 6 6 5.33 4 6
38 4 5 5 6 6 6 5.33 4 6
39 4 5 5 6 6 6 5.33 4 6
40 5 6 6 6 6 6 5.83 5 6

The number of expected hits goes up by combat factor (as we might expect), and the minimum and maximum number of hits go up as well. Note that these are the "raw results" before applying any modifiers after the roll, which in Rommel depends on tactical cards being played by one or both players.

To better understand this table, I also plotted these results in a graph:


As you can see, the expected number of hits goes up pretty much linearly, but there are a few places where the line could have been made smoother. E.g. there's a sudden jump for combat factor 26, which can be "smoothed out" by adjusting the table above if desired (see also appendix 1 below).

The dotted blue line is the linear "trend", as computed by the spreadsheet. You can observe that our expected value follows this trend fairly well, except near high combat factors. This is due to the maximum number of hits being 6. If 7 hits would be allowed near the end of the table, the expected value would increase slightly for those higher combat factors. However, since such large combat factors do not regularly occur in the game, we will not consider this effect any further.

The graph below shows an additional line, plotting the 0.15 times the combat factor. You can see that the blue trend line matches this 0.15*combatfactor very closely, except for a little offset near the origin.


A different combat resolution mechanic?

The interesting observation about the 0.15 line, is that it is very close to a slope of 0.1666... which is exactly 1/6. This number is promising, because 1/6 is exactly a single "chunk" of a probability step on a D6. Thus, can we design a procedure that produces as its expected value a number that is exactly 1/6 of the combat factor?

There are any number of different mechanics that can do this. A very simple straightforward one is to take as many D6 as the combat factor (thus, roll 15 dice for combat factor 15), and count any 6's as a hit. This produces an expected value equal to 1/6th of the initial combat factor. However, its variance is also very high. The number of hits could range from 0 to 15. See also Buckets of Dice mechanics for further exploring such a procedure.

So let us look at another procedure, and I suggest the following: Take as a fixed number of hits the multiple of 6 just below the combat factor, and any remainder left is used as the target number on a D6 to score an additional hit. E.g., for a combat factor of 15, we would score 2 hits (2*6 = 12); 15-12 = 3, so we need a 1,2,3 on a D6 for an additional 3rd hit. If our combat factor is 11, we would score 1 hit, and we roll a D6 with a 1,2,3,4, or 5 scoring another hit. If our combat factor is 19, we score 3 hits, and we score an additional hit of we roll 1 on a D6, and so on.

It is rather obvious that the spread for any given combat factor is 1 hit , but the chance for this additional hit goes gradually up for each additional unit of combat factor. The graph below plots the expected value for this new procedure, along with the minimum and maximum number of hits.


So, what can we see on this graph?
  • The blue line is the expected result for our new procedure, which follows pretty close the expected value for the original procedure.
  • The dotted blue lines show the minimum and maximum number of hits, which define a much more narrow interval compared to the original minimum and maximum values. I don't think there's a "right" or "wrong" aspect about this, it does depend what you like better: a higher  variability in results (a difference of up to 3 hits between die rolls) or a more narrow variability, with results being only 1 hit apart.
  • The "leveling off" of the blue line for high combat factors is due to the maximum number of hits being set at 6, as per the original procedure. Otherwise, a combat factor of 37 or over could possibly result in 7 hits according to our new procedure. But since such high combat factors do not show up in the game, we ignore this effect.
  • Our new procedure might be more user-friendly to resolve. The players can simply work out the number of hits without having to consult a table.
Other alternatives are possible as well, keeping the same expected value, but altering the minimum and maximum outcomes. E.g. if you like a Bucket of Dice mechanism, you could also divide the combat factor by 3, roll that many dice, and count any 4,5,6 as a hit. Or you could use a variation of what I suggested above, using D12's, etc. There are many possibilities, and in the end, it strongly depends on your personal preferences.

Appendix 1: Adjusting the original combat table

In hindsight, it isn't that surprising that the slope of the expected value is very close to 1/6. After all, this implies one additional hit for one die roll result when the combat factor goes up by 1. You can also see this in the original graph, when the slope of the expected value runs exactly parallel to the 1/6 line, e.g. in the range 14-19.

It is therefore rather easy to play around with the number of hits a little bit, to get an expected value line that runs exactly along this 1/6 slope. All it requires is to have the number of hits go up by 1 for one die roll result when the combat factor goes up by 1. There are different degrees of freedom to do this. E.g., if we start from the current line for combat factor 7, we have for die rolls 1-6: 0, 0, 1, 1, 2, 2 hits respectively. When we go to combat factor 8, the current table indicates 0, 0, 1, 1, 2, 3 hits (the number of hits for a die roll of 6 have gone up from 2 to 3). But you could also put it at 0, 1, 1, 1, 2, 2, which would result in the same expected value for combat factor 8, although with a slighter lower variation in results.

Actually, you could alter the table in such a way that it produces exactly the same outcomes as our modified mechanic. But I'll leave that as an exercise to the interested reader. You can play around with the combat results table yourself by downloading my original excel file.

Appendix 2: Fantasy Warlord

When I was doing the analysis for the new suggested procedure, I remembered I had seen a similar mechanic before. The fantasy wargaming ruleset Fantasy Warlord (Folio Works, 1990) uses % numbers for one figure hitting another figure. E.g. an Orc would have a 40% probability of hitting an Elf. If you have a unit of 8 orcs attacking, that would result in 8*40% = 320%, meaning 3 hits and a 20% chance of inflicting another hit.