Probability for wargamers 9 - d20 vs 3d6

While driving in this morning, I was listening to the latest View From The Veranda podcast with Neil Shuck and Henry Hyde (yes, yes, pedants - by which I mean they were on the podcast, not in the car - not that it wouldn't enliven my commute!). One of the topics that came up was the Beyond The Gates Of Antares Kickstarter, and the fact that it uses d10, not d6, from which the discussing drifted (as VFTV often does) into designers' choice of dice for such things as Phil Hendry's Augustus to Aurelian.

Somewhere about then it started to click that this series of posts is as much, if not more, about understanding probabilities for game design as to allow you the wargamer to understand (and, heaven forbid, mini-max) the odds. And I got to thinking afresh about the issue I mentioned in my Principles of War battle report the other day, namely the d20 morale roll.

Essentially (and I may have this slightly wrong since I'm working from memory, but the core concept is about right), in PoW units have a strength, which is typically a number around 10-12. Combat casualties reduce this strength, and a morale test is basically taken by rolling strength + modifiers vs a d20, as follows:

Roll	Effect
1	always succeed
<= Str	succeed
<= 2 x Str	shaken
<= 3 x Str	retire shaken
> 3 x Str	rout
20	always fail, one row worse than what the roll would be otherwise

My issue with this is that because it's on a d20, it's more prone to extreme results compared to a 3d6 roll. Go back and read up on the odds with 3d6, and then let me demonstrate with a graph.

The chart represents the odds (up the side) of rolling greater than or equal to a target number (along the bottom) on both 3d6 (the green curve) and d20 (the blue line). Key things to note: 

• the 50% point is the same for both rolls;
• the 3d6 is a fairly smooth curve that makes the extremes harder.

Take a look at the graph a different way, and you'll see the latter more clearly.

For the stats and math heads amongst you, the green curve (the 3d6) is what's called a normal distribution, Gaussian distribution or bell curve. For games designers, it has the useful feature that extremes are rarer, compared to a linear distribution (the blue curve for the d20). Note that the lines cross at about 6 and 15.  Particularly, note there's a 1 in 20 chance, or 5% of an automatic fail, which seems to me to be a bit vicious on a strong unit.

So, what would happen if we swapped the morale roll for PoW to be 3d6, rather than d20, and for the sake of completeness made 3 an automatic pass and 18 an automatic fail?

Well. The key thing is that auto pass and auto fail become much harder - 0.45% rather than 5%. I think this might be a bit too hard. So - what if we make that 3 and 4 always pass, 17 and 18 always fail. The odds are then a hair under 2% (4 chances in 216) of automatic failure, which feels about right as a compromise, and if we were feeling vicious we could make an 18 two column shifts from the actual result.

The other thing that happens (and I'll leave you to work out the numbers if you're interested) is that weaker units become a bit more likely to be shaken, and stronger ones less so. For example, if you look at the first graph, a unit of strength 5 is more likely to roll over 5 on 3d6 than d20. Interestingly, it's less likely to roll more than 10 (and thus retire) and considerably less likely to roll more than 15 and rout. It does make it a bit harder to rout units, in other words. 

I'm not convinced this is perfect by any means, as it seems to make routing a bit harder for units that probably should rout. But it does remove that nasty 5% chance of a perfectly sound full strength unit retiring shaken, which I think is a bit unrealistically excessive. Students of the Napoleonic era may disagree. Perhaps one approach to this would be to also change the table so that instead of the steps being at x2, x3, they're at +4, +8??

Thoughts?

Battle Report - 29-Feb-2012 - The Battle Of Cuidad Rodrigo

The final group of reinforcements on both sides turn up.

Apologies that this is somewhat short - I'm currently down with a rotten cold, and eyeing the Saxon lead mountain and two approaching deadlines for it with trepidations.

Not at the club, for once - this was the first real battle in Gavin's Peninsular campaign, held in Matt's rather sumptuously appointed games room.

As Marechal Victor, I and half the French army were late (and completely lacking in cavalry, as I'd managed to lose them on the map!), so Rich got to run the opening few turns until my forces got there. He did rather get my best infantry unit cut to bits, but we survived... unfortunately I had to leave with a couple of turns to go, so Gavin finished the end off, and ... to be honest, the details are rather lost in a haze of Lemsip at present. So I'll redirect you to Gavin's report on the battle, with far better pictures and a good deal more coherence :D

Vive l'Empereur!

Wednesday marked the start of what promises to be an interesting campaign, courtesy of Gavin and a mix of folks I know from the club and folks I've not met before, but are part of his regular gaming circle.

I'll not steal his thunder, as the superb web-based campaign system is his doing and he's blogging about it in detail: but I will note that this promises to be fun, though I'm a little nervous about being the overall french commander of eight whole Corps.

My first set of orders went in today, and the master map (at least, as far as I can see it from my viewpoint) is ready to be updated with reports from the field. The plan is, I gather, that battles will be fought using variously Principles of War and Grand Armée in 6mm, and possibly, if we have a brigade-sized skirmish, I might petition to see if we can give Napoleon At War a crack.

All in all, I'm looking forward to this. Watch this space, although updates will be, perforce, a little behind, partly because it's Gavin's show and partly because, well, les Anglais might be reading!

[As a note - these posts will be tagged with 'vive l'empereur' if you want to find them!]

Battle Report - 24 Jan 2011

Been looking forward to this for a bit - a tryout of the "Principles of War" Napoleonic rules at the club, courtesy of G.

As it was a tryout, and neither of the two of us running the French knew the rules, we went for a fairly classic and simple set up - the opposition were a mix of British and Portuguese set up on the side of a hill (one assumes, traditionally, in the pouring rain where two maps join). The conventional approach was adopted - wind up the infantry in column of attack, point at brow of hill, support with artillery and light cavalry, and leave the heavy cavalry off to one side to lend tone to the otherwise vulgar brawl.

Things went much as expected: the cavalry on the French left disappeared in a huge cloud of hooves and dust to meet with their Allied counterparts, and the French infantry weathered a storm of withering fire from the line of British (who were unlucky to be shaken twice at speculative long range by a gun battery I really should have moved closer) and Portuguese.

As expected? It boiled down to 'will the French break before they make contact?' Luckily for R and I, they didn't, and the upshot was that one line of Portuguese fell back, shaken, as did one slightly over-keen unit of artillery, and the column followed through onto the British line. 

We called it a night there, as the end result looked pretty predictable from there one.

On the whole? Great set of rules that didn't get in the way - some of the mechanics were interestingly different, with a bit of the feel to Blitzkrieg Commander about the orders/actions allocation.

But ultimately? They felt right. Units did what I expected they would do in a given situation, and the only flaws in the game were really due to R and I not fully absorbing the rules beforehand. And we didn't really need that loitering unit of light dragoons, as it turned out. :D