‘Deterministic’ OHW Rules-As-Written (RAW) Playtest

My first round of OHW playtesting left me with a desire to amend specific aspects of OHW’s RAW. The last few posts were basically preliminary notes in preparation to do just that. However, I think there’s one more, necessary step before I commit to the changes I’ve written so far: more playtesting! I want to really cut down the changes down to the minimum to stay as close to the OHW spirit as possible. So, I want to be sure I fully understand – not the rules, which are extremely simple, but – their interplay and emerging features, which I think are the important part.

I will use Horse & Musket era rules since I think they’re the ones needing the fewest changes to provide tactically interesting and historically-flavored games. I will play several of the scenarios solo (damned pandemic). And I will play diceless, i.e. in a completely deterministic fashion. I’ll try to explain both how and why… but perhaps trying to be just a tad more humble than the Man Himself:

 

"Military science consists in first computing all the probabilities, and then, evaluate precisely, with a nearly mathematical method, what is the share of chance... Chance is a mystery for lowly minds, and it becomes a real thing for superior men." [Napoleon]

 ...ok, that sounded a bit snooty from you Bonnie, but modesty is not a typical curricular skill for Emperors after all.

How to play deterministic OHW

It’s very simple. I will use my hex mod (see this post) instead of free movement, and record the average expected hits of an attack (see this post) instead of rolling dice. For example, a standard infantry attack will inflict 3.50 hits on the target; a cavalry charge on the flank 11.00 hits. That’s it. Everything else will be as in the RAW.

OK, but… why?

I think there are two main styles of solo playing, and I usually refer to them as ‘playing to be challenged’ and ‘playing to learn’. The first one tries to recreate the same experience of a face-to-face game without an actual opponent, presenting a genuine tactical challenge via methods involving randomization, hidden information and/or programmed opponents. The second one is aimed at learning something new from the game, regardless of whether it’s ‘fair’ or ‘fun’ in the traditional sense; this basically just involves the solo player honestly pursuing the best interests of both sides at the same time. The latter approach is the one I actually prefer, and is described perfectly here and here.

My goals for OHW solo play are strictly in the ‘playing to learn’ camp, as I’d like to learn:

·         Implications of the rules: which tactics do they encourage? Are those historical or not? Which historical battlefield behavior is not represented?

·         Are the various unit types balanced? Is some type of unit generally less useful than others?

·         Is the scenario balanced?

·         What is the best Course Of Action for both sides in this scenario? How would a programmed opponent work?

In my view – and for the above goals only – randomization is just added noise. Randomization is surely fun in face to face battles or in ‘challenge’ solo games, where every die roll can dynamically change the situation. But when playing to learn, I think the best pieces of information you can come off with are the average/expected outcomes rather than the specific outcomes of a given randomization sequence. It’s like playing a large number of randomized games and averaging them, if you want. This does turn the game into a somewhat insipid chess-like version of itself, of course; but it serves the ‘learning’ goal very well.

Very basic example of randomless analysis: one-on-one engagement of line infantry units in the RAW

This is what would happen in deterministic OHW if a line infantry unit decided to attack another one (say it’s 15” away, above shooting range; red is the attacker).

One-on-one infantry engagement, average outcome per turn
Turn	Avg Hits on Red unit	Avg Hits on Blue unit
Red 1	(red moves into range…)	(…and cannot fire)
Blue 1	3.5 (blue shoots)	0
Red 2	3.5	3.5 (red shoots)
Blue 2	7.0	3.5
Red3	7.0	7.0
Blue 3	10.5	7.0
Red 4	10.5	10.5
Blue 4	14.0	10.5
Red 5	14.0	14.0
Blue 5	Eliminated	14.0

 This means that per the RAW, 1:1 infantry engagements are long, costly and risky.

Long: they need 5 turns on average to reach a decisive outcome.
Costly: the expected outcome is to leave the defending unit almost completely spent at the cost of one attacking unit. It’s almost like a piece exchange in chess.
Risky: they slightly favor the defender because it shoots first; but the advantage is small, making attacker victories possible.

[Side Note - it’s of course possible to compute just how risky it is for both opponents, when playing with dice… the results largely justify the ‘deterministic’ solo approach, in the sense that the distribution of outcomes as per the RAW won’t be that ‘dicey’ or ‘random’ after all. The real turning point seems to be Turn 4, in which you usually reach an equal chance of surviving or being destroyed. If you reach turn 4 in better shape than your opponent you should probably press forward… If not, there is no great dishonor in a quick tactical reassessment].

One-on-one infantry engagement, chance of elimination
Number of 1d6 attacks received	Per turn	Cumulative
1	0%	0%
2	0%	0%
3	9.26%	9.26%
4	44.37%	49.52% (turning point!)
5	77.85%	87.68%
6	93.92%	98.65%
7	98.79%	99.93%

 All things considered however, the tactical lesson here seems to be “avoid 1:1 engagement at all costs if possible”. How does that sound, Nappy?

 "The Art of War consists in the following: […] you concentrate more troops than your enemy on the point where you attack, or on the point where you are attacked. But this art cannot be learned in the books, or acquired as a habit; it is a way of acting which is the proper genius for war." [Napoleon]

…Well Bonnie, I can't possibly argue with someone in that attire. If you excuse my dumbness however - that’s more easily said than done, since I’ve found that this 'concentrating troops' thing is not that trivial after all. Guess it means I’m not a ‘proper genius for war’, huh?

The next few posts will be AARs of OHW scenarios played as above. Au revoir!

Modding One Hour Wargames, Part 2 bis: Combat Results Variance

In the previous post, I explained why I find OHW’s rules so elegant and robust. This is just a short add-on note I felt urged to write when I considered a few of the design choices in the game.

As you might know, there are two types of combat result modifiers in the game: additive (either -2 or +2 to the rolled result) and multiplicative (the result is multiplied x2, x0.5 or 0.25). An intriguing ‘hidden’ feature of the game’s math is that these two types of modifier turn out to be largely equivalent when considering the amount of turns needed to eliminate an enemy unit (see previouspost).

A normal unit needs 5 turns to eliminate a fresh opponent. This is reduced to 3 turns both in advantageous situations (x2 hits multiplier) AND if the unit has superior attack power (+2 hits). This might prompt the question: then why bother having both types of modifier? Couldn’t strong units just inflict x2 hits, or advantageous situations give a further +2 to the roll? Well, I think Mr. Thomas has it right once again, since I think there two advantages in having both additive and multiplicative modifiers.

Simplicity: Additive modifiers are only dependent on the attacker unit, and nothing else. Multiplicative modifiers are situation-dependent only. There’s no math whatsoever involved prior to the roll, in the sense that you don’t have to add several modifiers, keep the result in mind, roll and apply the calculated modifier. Of course the very notion that this might be problematic may sound ridiculous to us grognards, but I think it’s a rather nice touch that lowers the entry threshold for non-gamers and kids.

Variance: Additive modifiers are considerably more predictable than multipliers, and don’t change how results are spread around the expected value. Stacking positive multipliers would result in far too unpredictable distributions of results – a standard attack with a x4 multiplier could inflict 4-24 hits, which would just throws game (and scenario) balance out of the window. Conversely, stacked fractional multipliers don’t pose the same problem, (they just result in a handful less hits here and there, especially if paired to the ‘always round up’ rule).

So what?

This is just a quick note to myself (and other prospective OHW tinkerers): think twice before messing up with OHW’s hit roll variance. Don’t allow multipliers to go higher than 2x. Additive modifiers larger than +2 seem safer on paper.

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…