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.

*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.

I also took apart the Rommel CRT. An issue I have is that it is a linear progression, whereas irl combat outcomes relative to force ratios are anything but linear, but instead suffer from diminishing marginal returns.

ReplyDeleteI tend to agree. OTOH, the "workable" range of the table is limited to the values between 5 and 15 (at least in our games ...). Whether using something different than a linear model actually makes a difference in gameplay is questionable ...

DeleteThe diminishing margin returns Martin Rapier refers to is why I adopted for some of my war games a combat system similar to the one proposed by Charles Grant for his 18th century war games. It does suffer from 'blips' at certain points, but I would have to explain the system at length to demonstrate where they were.

ReplyDeleteYears ago I worked out a relation in which a number of missiles hit a target comprising several individual elements. How many of those individual elements are actually struck? For instance, suppose 3 missiles strike a target comprising a group of 6 individuals. How many of those individuals are struck? Clearly, no more than three, but it might be two, or even just the one.

The thing can be extended to asking oneself, if 300 (and that itself might be an expected fraction of the number of missiles sent off) shots strike a 600-man target, how many of the 600 get hit? It is very unlikely that 300 men each take one hit. You'd expect probably somewhere fairly close to 250, with a few individuals taking two or more hits.

The system I adopted was an approximation only, but it served to give results that to me were not extreme, those close to expectation happened much more often than not, but the occasional anomaly, being occasional only, could be tolerated as the vagaries of chance.

Did you use a binomial or Poisson distribution in your approximations? Because that's essentially what you are trying to model from a mathematical point of view.

Delete