Modding One Hour Wargames, Part 2: Statistical analysis of Attacks and Hits

In the last post, I’ve outlined the basic elements found in OHW’s rules. My goal is to understand how they work and interact before finalizing my rules variants for the eras that most inspire me. The first element I want to analyze in detail is what is usually explicitly perceived as ‘the combat system’, which basically just involves units inflicting a variable number of ‘hits’ on opponents.

From a simulation point of view I see hits not as just representing casualties, but the interplay of all adverse factors hindering a formation’s combat endurance, such as morale erosion, supply depletion, disorder and so on. A typical attack inflicts 1d6 hits; a strong attack inflicts 1d6+2, a weak one 1d6-2. Attacks inflict 200% of the rolled hits in advantageous situations, and are instead reduced to 50% or even 25% (fractions are always rounded up) when rolling for what Kriegsspiel would call ‘bad effect’. I couldn’t find any instance of a double advantage (i.e. 400% hits) in the rules. This means that there are just 12 degrees of attack effectiveness in OHW, and it’s possible to calculate how many hits they will inflict on average:

 

Attack strength	Table 1: Average hits inflicted per attack (possible results in parentheses)
	25% effect	50% effect	100% effect	200% effect
Weak (1d6-2)	0.67 (0,0,1,1,1,1)	1.00 (0,0,1,1,2,2)	1.67 (0,0,1,2,3,4)	3.33 (0,0,2,4,6,8)
Average (1d6)	1.33 (1,1,1,1,2,2)	2.00 (1,1,2,2,3,3)	3.50 (1,2,3,4,5,6)	7.00 (2,4,6,8,10,12)
Strong (1d6+2)	1.67 (1,1,2,2,2,2)	3.00 (2,2,3,3,4,4)	5.50 (3,4,5,6,7,8)	11.00 (6,8,10,12,14,16)

 

A few things become immediately apparent. First of all, only ‘weak’ attacks can sometimes be completely ineffective; all other attacks will put a dent, however small, on their target. Second, there’s only one case in which an attack can result in utter annihilation of a fresh target unit, and it’s quite a rare one: you need to launch a strong attack in favorable conditions, then roll a 6. In most cases, you’ll need several turns to eliminate enemy units – an important consideration when planning a strategy for most OHW scenarios.

 One Hour Wargames has the Fibonacci Sequence at its core!?!

 There’s nothing particularly subtle or elegant in the table above. However, if you instead calculate how many turns a given type of attack will need to dispatch a fresh enemy unit with exactly 15 hits, something special happens. Small differences between certain types of attack disappear and an elegant symmetry emerges. The interesting thing is that you don’t generally get this effect – it’s the fact that units have exactly 15 hits and modifiers being the way they are to make it emerge. I don’t think it is just a coincidence!

 

Attack strength	Table 2: Average number of attacks needed to destroy fresh enemy unit
	25% effect	50% effect	100% effect	200% effect
Weak (1d6-2)	23	15	9	5
Average (1d6)	12	8	5	3
Strong (1d6+2)	9	5	3	2

 

Due to their ‘hidden’ symmetry, OHW’s combat rules imply that the impact of those +2 or -2 attack modifiers some of the units have are statistically equivalent to situational advantages and disadvantages such as cover or flank attacks, AND to numerical superiority! For example, two units rolling 1d6 at 100% effect will dispatch one enemy unit in the same number of turns needed by one unit rolling 1d6+2 (numerical superiority and attack quality have the same statistical ‘weight’). Due to this, you can compress Table 2 into just this sequence of numbers, which correspond to the turns needed to destroy a fresh unit on average:

 “1 – 2 – 3 – (5) – 8/9 – 12/15 – 23”

 How do you use this? Start at number five: a standard attack by just one unit will take 5 turns to beat a fresh enemy. For each additional attacking unit above one, each advantageous situation (e.g. flank attack), and each +2 modifier contributing to the attack – shift by one step to the left. For each disadvantageous contingency (e.g. cover, armor, weak attack) – shift to the right. That’s the estimated number of turns needed to overcome the target unit.

 For example, two (numerical advantage: one shift to left) H&M-era line infantry units (normal 1d6 attacks – no shift) will need 3 turns of their combined efforts to destroy one enemy unit, or 5 if it’s in a town (one shift to the right). One ancient skirmishers unit (weak attack, one shift to the right) would need to throw javelins for the entire battle at a heavy infantry unit (another shift to the right due to armor) to destroy it – which nicely summarizes Iphicrates’ plan at Lechaeum, by the way.

 …All of which might be impressive enough by itself, except that there’s more – the sequence of numbers above is freakingly close to the Fibonacci Sequence! No wonder, then, that OHW rules ‘feel’ somehow right… They literally have the Golden Ratio at the core of its mechanics!

 Possible tweaks

 I don’t think it’s necessary to modify the fundamental mechanisms of inflicting and accumulating hits. In particular, I’ll follow the author’s example and avoid using stacked “x2” hits multipliers resulting in attacks at 400% effect – these would probably be too swingy and would undermine the slow, attritional nature of combat as used in the rules.

 Conversely, changing the amount of hits a given unit can sustain before collapsing (15 in the RAW) seems to be an obvious opportunity for representing different unit sizes/quality/supply/morale. Ideally, I’d like to change it so that the symmetry of Table #2 is preserved. For example, units with exactly 11 hits still largely behave as above, but are defeated 1 turn sooner on average. There are some distortions, but they are relegated to rarely used parts of the table. Some of the other numbers, like 12 or 20, bring the irregularities to the most relevant part of the table instead… but I’m not sure it actually makes a difference during play, we’ll see.

 In general, I would hesitate to give units less than 10 starting hits. That’s because if you do, you’ll start having a whole bunch of attacks possibly eliminating units in one turn – a ‘sudden death’ variant in which hits thus represent a sort of unit ‘survivability’ to attacks, rather than their attritional endurance. This might be feasible for games with much more than six total units per side – where attrition is represented by removal of units. At the opposite end of the spectrum, units with more than 20 hits will be nigh-unstoppable behemots which are probably best avoided except in the most peculiar circumstances…