Necromunda combat stats

[enyoss]

Weaponskill can save the kill, but a charger lacks for want of attacks...

Ok, I'm postponing dealing with those combat skill strategies. Again. But only for a short while I promise.

Instead, I'm going to post a short (ha!) discussion on the relative effectiveness of Weaponskill and Attack advances.

I did actually address this in one of the first posts in this thread. My argument was as follows: roll one attack dice and your outcome follows a uniform distribution and is therefore very uncertain, but if rolling two dice you get an improvement in score about equivalent to +1WS, but because your outcome is no longer uniformly distributed your uncertainty decreases hugely.

So increasing your attacks from A1 to A2, in any way you can, is the first thing you should consider to improve combat effectiveness.

That is, broadly, true. It's also a massive lie. Ok, not a lie, but definitely misleading. It's actually true for one fighter's actual combat score.

But... if two opposing fighters roll one attack dice, although their individual outcomes are uniformly distributed and highly uncertain, the difference between them is not uniformly distributed at all! So while rolling one attack dice is certainly not as good as rolling two, it's not as bad as you might think.

But @enyoss, you might wail, why lie? Well, it was a convenient way of conveying what is, broadly, the right message. And, broadly, it conveys it in a kind of true and easy to understand way. Plus, I didn't have a load of handy plots to point to, which has now been remedied. So I'm now going to go beyond 'broadly', and explain a bit more.

I'm mainly dropping this post in now as I've seen a few comments in discussions here recently about how WS tends to put all other combat advances, e.g. Attacks, to shame. You might say that increasing your WS by +1 is always better than increasing Attacks by +1. But, as it turns out, that's not entirely true, at least for that primary A1 to A2 upgrade (it is almost always true for A3 and up though...).

Why not? Because, as it turns out, the effectiveness of WS and Attack advances is dependent on the baseline difference in WS between two fighters. And here is the most important bit: their effectiveness is dependent on the baseline difference in WS between two fighters, once it has been adjusted for other bonuses. The kicker here is that most common bonus of all: the charge.

Before going further, I'll point out that the findings are summarized at the end in some Rules of Thumb, for those who don't want to trudge through the detail (shame on you).

Anyway, I'm going to consider three experiments (Cases 1-3). For each case, we consider two fighters. Both fighters roll one attack dice and have equal WS, apart from the following exceptions:

A) Fighter1 has +1WS relative to Fighter2
B) Fighter1 has 2A

Each case then modifies things a bit more:

CASE 1: Fighter1 has an additional disadvantage of -1 to his combat score
CASE 2: Baseline. No further adjustments.
CASE 3: Fighter1 has an additional advantage of +1 to his combat score

For example, let's say both Fighters have WS3.

CASE 3 here might represent a Fighter1 having just charged his opponent, so he gets an extra +1 to his combat score.
CASE 3A would look at what happens if Fighter1 actually had WS4 instead of WS3 (as well as the charging bonus).
CASE 3B would look at what happens if Fighter1 has A2 instead of A1 (i.e. he rolls two attack dice).

Here's where those universal plots I put up in my second to last update come into play. Those, as it turns out, can tell us every outcome of these combats, and are used to get the numbers below.

I'm going to talk in terms of probability that Fighter1 wins combat, P(win), and probability that Fighter1 wins or draws combat P(win/draw). In this instance, I'm considering a draw to be any combat where the fighters are tied in their combat scores... let's worry about initiative later. I'll also show the equivalent numbers when Fighter2 bumps his attacks up to A2 (given in brackets), and give the baseline probabilities when both fighters are identical both having A1 or A2:

CASE 1A: P(win/draw) = 58% (45%), P(win) = 42% (31%)
CASE 1B: P(win/draw) = 55% (41%), P(win) = 40% (26%)

CASE 2A: P(win/draw) = 69% (60%), P(win) = 58% (45%)
CASE 2B: P(win/draw) = 69% (59%), P(win) = 55% (41%)

CASE 3A: P(win/draw) = 81% (73%), P(win) = 69% (60%)
CASE 3B: P(win/draw) = 82% (74%), P(win) = 69% (59%)

BASELINE_A1: P(win/draw) = 58%, P(win) = 42%
BASELINE_A2: P(win/draw) = 59%, P(win) = 41%

There's quite a lot to digest there, so I'll summarize:

1. If you already have a WS modifier disadvantage from your opponent (CASE 1), adding an extra +1WS to your fighter is more effective than going from A1 to A2.
   
   
2. If your two fighters are equally matched after all WS modifiers have been counted, +1WS is more effective than going from A1 to A2, but if you have high Initiative either advance is good.
   
   
3. If you already have a WS modifier advantage over your opponent (CASE 3), going from A1 to A2 is more effective than adding an extra +1WS advantage.

You can take my word for the fact that as the WS modifier gets worse for Fighter1 WS advances just keep getting more valuable, and as the WS modifiers get better for Fighter1 Attacks keep getting more valuable (or you can look at the numbers yourself... they're all in the plots above ).

The thing is, an awful lot of fighters in Necromunda bumble around with WS3 and I3. So, in many instances, WS modifiers, as I've called them, boil down to one thing - charging bonuses. We can therefore modify the above to give us some rough rules of thumb.

Rules of Thumb:

1. If you have been charged by your opponent, or have low WS (<WS3), having an extra +1WS would be more valuable than going from A1 to A2.
   
   
2. In ongoing combats having an extra +1WS is most valuable, but if you have I4+ then going from A1 to A2 is just as good or better (*see below...).
   
   
3. If you have charged your opponent, or already have high WS (>WS3), going from A1 to A2 is more valuable than having an additional +1WS.

The caveat in (2) is thus: although I kind of lied before, increasing your attack dice from one to two does have a beneficial impact. It's better than rolling just one attack dice as it not only makes you more likely to win the combat, but it skews the probabilities of the outcomes such that they become more bunched up into the wins. To see the difference between CASE 2A and CASE 2B, imagine shifting the black curve in the following figure one point to the right (CASE 2A), and then compare with the red curve (CASE 2B).

Although they curves would be quite similar in terms of their mean, the red curve is more asymmetrical and slightly squashes down the losing probabilities and fattens up the winning probabilities. It's definitely a close run thing, but if all else is equal, this gives the A1 to A2 advance the edge here. So in (2), if you have I4+ you could go either way, but I would say A1-A2 is perhaps slightly better.

How does this work in a real game?

It means that your shooting type ganger, who is probably going to get charged more than they charge themselves, wants defensive WS advances rather than having A2 (just as well, because often they can't get the extra attack from wielding two weapons).

On the flip side, combat type gangers, who are probably going be charging quite a bit, want to make sure they get that A2 rather than +1WS (again, just as well, because they often get exactly this from wielding two weapons... and if you're not doing that, I hope you see why you should be).

What else? It also means that once your fighter has WS4, if you're charging into combat a lot and still rolling just one attack dice you should probably stop punching the air when you get a further +1WS advance – cross your fingers for Attacks next time, or arm yourself to compensate!

The bottom line: rolling two attack dice is still my most favoured advance, and once you start getting other WS advances it becomes even more important. It is definitely still the case that WS trumps rolling three or more attack dice though, and it is still a great advance for any H2H fighter.

Ok. enough of that. Next time it's definitely doing to be strategies for those combat skills .

 

[enyoss]

The Basics of Good Habits...

As a prelude to discussing the combat skill strategies, I should really mention some of the good habits which I consider to be essential for resolving combats.

Interestingly, when I googled the title of this post to get a nice headline phrase, one of the top things to come back was this: http://www.bladderbowel.gov.au/all/goodhabits.htm. Not sure what that augers for the content of this post, but even if you find the rest of my trite comments useless you can at least glean some useful lessons from there, so it's not a complete waste of time.

A quick warning: there is no maths here at all, it's basically just common sense. So much so that these habits will probably be nothing new to anyone reading this. Whether or not you actually practice them is another thing. Whether or not you actually need me to pontificate to you about it is another thing also, but I'm going to do it anyway as these habits are quite important when following combat skill strategies. So it makes sense to point them out here for clarity.

Another quick warning: the content of this post (and some of the previous ones, actually) does somewhat contradict the overarching principles espoused at the opening of this thread. To paraphrase - “The work presented here is not designed or intended to be used to over-optimize gameplay. That's just not Necromunda”. I still, in the strongest of terms, agree with that, and frown on players using stuff like this as a tool to beat up on casual or easy going players. But there are players out there, like myself, who just get a nice warm feeling inside when they know they are doing it right, even if not overly bothered about winning. Nothing wrong with that I suppose. But if you're going to do it right, remember to do it graciously, and always with a smile on your face – a friendly one, not a predatory one, that is.

Lecture over, but I had to put it in, just in case anything here gets used for evil ends.

Anyway, one of the key things I'm going to dwell on, which links back to my previous rants about the importance of initiative, is the distinction between getting a tie in your combat score with your opponent and getting a drawn combat. Simply put, a tied combat score is not a drawn combat – it could be a win, a loss, or a draw. It's all down to your fighters' initiative. And when you're making combat skill strategy decisions based on whether you've won, drawn, or lost, it's really important to get that distinction right.

So, here are my top tips for habits to develop when applying parry/deflect/counter attack strategies:

1: Always know your own initiative and combat skills before entering combat (seriously)
2: Before rolling any dice, know your opponent's initiative and combat skills
3: Always know what weapons your opponent is carrying, and what weapons they are using
4: Always take your time to calculate combat scores and think about your strategy - making errors is easy


If you want to struggle through and miss opportunities for delivering the killing blow to you opponent, seeing as I'm such a nice guy I've prepared some useful entries for your Necromunda phrasebook:
- “Ok, drawn combat. I'll parry your dice there then. Ok, drawn combat again - what's your initiative?”
- “Give me a second, I just need to look up my guy's initiative.”
- “Curses! A doubled fumble. Oh well, at least my flail cancelled your parry – what do you mean you don't have a sword?”

On the other hand, if you want to make strategy decisions with devastating accuracy, your fighters' names whispered to naughty children to scare them into being good, here are some other lines for your phrasebook:
- “Ok, we're tied. You have higher initiative and would win the tie, so I'll parry your dice there then. Ok, tied combat again, you win the tie.”
- “Ok, we're tied. We have the same initiative so the combat is drawn afterall.”
- “Ok, we're tied. My initiative is higher so I win the tie.”
- "Thanks to my flawless application of parry, deflect and counter attack strategies, another of your guys goes out of action. Let us raise a chalice of his blood in toast to @enyoss."

[Ok, I'm pushing your patience a bit with the last one - especially considering the sheer number of wheels I'm reinventing here - but it's my thread so I can say what I want (I'm clearly from the Trump school of philosophy when it comes to internet ownership paradigms. I also seem to have inherited his penchant for the third person.)]

Anyway, you don't have to be that pedantic about language, it's only there for illustrative purposes, but I hope you get the idea - a combat isn't drawn unless you both have the same score and the same initiative value. Any other 'draw' is just a win or loss waiting in the wings. Don't let these wins and loses surprise you.

It's all pretty simple really, but surprisingly easy to forget it you don't watch it. I'll put myself on the block first, and give you an example of my own combat habits from my last game:

Me: What's this guy's weaponskill? (Correctly, I asked before charging)
Op: WS3
Me: Cool, same as me. I'm charging. What weapon(s) are you using?
Op: Chain and autopistol
Me: Cool (I forget about Disarm. My opponent doesn't ask what I'm using)
** Both Roll **
Me: So you got 554, I got 321. I get +1 for charging. Erm (flustered from elementary arithmetic), that looks like it's a draw (wrong). Erm, I'll reroll my 1 here using Counter Attack, and I'll use Deflect to make you reroll that 5 there.
Op: Done. Rerolled for 542.
Me: Done. Rerolled for 432. Ok, that looks like a draw (yep, it is this time Einstein). What's your initiative?
Op: Initiative 3
Me: Let me look mine up... ok, I got initiative 3 as well. It's a draw after all.

I'm now going to dissect that exchange to point out how my own bad habits nearly cost me dearly:

Mistake #1) Although I did ask what weapons my opponent was using, I completely forgot to use that information with my Disarm skill. I got lucky - I didn't have a sword, so the chain would actually just make my opponent's fumbles worse for him, so I shouldn't be using Disarm anyway. Good job I forgot really considering my current form - if I'd remembered I had the skill I probably would have used it incorrectly anyway.

Mistake #2) Technically not my bad habit – it was my opponent's mistake (his only one... he's a good player but also a nice guy, so this might have been a concession!). He forgot to ask what weapons I was using. If he had asked it would have been obvious I wasn't carrying a sword and had no parry to cancel, so him using a chain only harmed his chances of winning combat.

Mistake #3) I added the scores up incorrectly and wrongly assumed we had tied. I got lucky - I had actually lost, so my Deflect call turned out to be right. I did remember the optimal Counter Attack strategy to always reroll my 1's though (obvious as it is!), so I at least got that right. Adding scores up incorrectly is pretty unforgivable really, so make sure you take your time. That goes for the whole combat process actually. Miscalculating the score by one point or parrying at the wrong time is equivalent to giving your opponent a free WS advance, and will sink you faster than anything.

Mistake #4) I didn't even know my own initiative, never mind my opponent's. Even if that first roll had been a tie, if my opponent had I2 I could well have ploughed on with the wrong Deflect strategy without knowing I had already won. Although my opponent also didn't know my girl's initiative he didn't really need to due to having no parries or deflect of his own (although, to be clear, while not strictly a mistake for him I think it's a good habit to get into asking anyway).

That all happened in a few seconds, as is the case with close combats usually. Despite my negligence miraculously not making things worse, I vowed to tighten things up for next time. Combat is hard enough to win in the first place without flushing it down the toilet with your own sloppy habits. I suspect this is partly why so many people find it more difficult to use CC gangs - slight mistakes here and there are punished far more brutally than for shooting, and they're much easier to make.

Good habits are good habits. And good habits are essential for combat skill strategies. If you practice them even when you don't need them, you'll remember them when you do. A bit like getting into the habit of rolling serious injuries for fighters in decreasing order of importance - no use dithering over whether to use Medic to re-roll that lowly ganger's death sentence or save it for your leader who's next.

In hindsight, I suppose this entire post was just an electronic hair-shirt for me to wear until I get my own house in order. Still, it should prove useful if anyone intends on following the parry/deflect strategies discussed earlier.

Now, on to those combat skill strategies (seriously, I fear they are becoming my very own “Petrocelli's house”...)

 

[sebwiers]

In the games we played, you weren't required to reveal what a stat was (or what skills you had, or anything not expected to be WYSIWYG represented) until it was time to use it. When use of skills / stats / weapons happened at the same time, the 'attacking' player (usually the one who's turn it was) had to show thier cards first. So habit 2 & maybe 3 would work differently. Had that happened in our campaign, and I was the op, the conversation would go like this:

En: What's this guy's weaponskill? (Correctly, I asked before charging)
SW: I'll tell you that when its time to calculate combat score. You might not even be in charge range!
En: I'm charging. What weapon(s) are you using?
Sw: That depends. What are you using? As the defender, I might choose to use a flail or a knife, depending on whether you are using that sword the model is holding.
...(rest might well have gone the same)

 

[enyoss]

Habit number 2 definitely does depend on whether your gang rosters are open or not. Then again, I think for most (or all) campaigns hosted here that is the case, so I'm taking a punt on that being the norm.

Weapons, with the exception of pistols and grenades, should always be known before you charge as you should either follow WYSIWYG or have open rosters for that. So really that habit should be a given - asking just makes sure you get it right (although it might be interpreted as an insult to their modelling capabilities!).

I kind of assumed habit number 3 would play out exactly as you described, That is, both players declare their available weapons, and the exact combination which is finally used comes from iterating to one of the game theory equilibria for the possible selections. If players can't decide for some reason (no equilibria exist) then it would probably have to come down to a roll off, with the losing player declaring which weapons they are using first. In your group it looks like the attacker always declares first, although I'm not sure how common that is.

That still doesn't change the habit though, it's just a further step when you're doing it right .

 

[sebwiers]

Due to wysiwyg, weapons would be known, but that doesn't mean you would know what weapons the fighter was opting to use in HtH. In the example you gave, the target of the charge has an autopistol, chain... and knife. He doesn't have to use to use the chain to get that second attack, he can use the knife instead. But yeah, either way it still makes sense to ask what weapon a model HAS, because of proxy models, swords that look like knives, etc. That last one is very common - people often have models with really large knives that could go either way, and it's nice to know which is which before charging (hell, even before deploying).

Its been a while, we might have done the equalibria / roll off thing a couple times, and then decided to use 'attacker declares first' to avoid those cases. The idea being, if a guy is charging towards you, he already has his weapons drawn, and you probably still have to ready yours (one reason for the bonus to the attacker).

 

[enyoss]

Yeah, in the example I gave, I was actually fully expecting the following exchange:

Me: What weapons do you have?
Op: Autopistol, chain, and knife. You?
Me: Power axe, stubgun (with DDs), and knife. Whatever weapon you're using, I'm using my power axe and stubgun
Op: Ok, I'm using my autopistol and knife.

The equilibrium here is my opponent uses knife + autopistol (the chain is his worst response to any of my options), and then I'm indifferent to any of my weapons (in terms of winning combat) so I just use the ones which help most in the to-wound stage.

It would get a bit tricky if we had the following:

Me: What weapons do you have?
Op: Autopistol, chain, and knife. You?
Me: Sword, stubgun (with DDs), club, and knife
Op: If you use that sword, I'm using my chain.
Me: Fair enough. If you use your chain, I'll use my club and stubgun (with DDs).
Op: Ok, I'm not using my chain theh.
Me: Fair enough. I'm using my sword.
Op: Fine. I'm back to using the chain then...
...
There is no equilibium here (I would never actually say this... I'd sound like a tool), but we clearly have two options which either can settle on.
1) Me sword + club, opponent chain + autopistol/knife.
2) Me stubgun (with DDs) + club, opponent autopistol + knife.

(1) isn't an equilibrium as I could improve my situation by swapping my sword for the stubgun (with DDs) and be better off for it.
(2) isn't an equilibrium as my opponent could improve his chances of winning by swapping his chain for a knife.

These are the two solutions we will wobble between though. Solution 1 is better for me, solution 2 is better for my opponent. Neither is ideal, but neither is as bad as it could get either. Ideally I would take it to a roll off (or some other way of choosing), and let the winner choose which solution we stick with. In practice though, that means correctly identifying all such solutions in the heat of battle. Not so tough here, but not so easy when you're playing.

I think your solution, or something similar, is probably best @sebwiers: one person somehow declares their weapons first, and the other responds.

However, most importantly, even if I lose that roll off I can be happy that my eventual equipment usage is the best possible response given my opponent's choice, because I'll be in the habit of making these decisions quickly and accurately .

 

[enyoss]

Counter Attack: the Escher-Max strategy

So, I've posted up a some Parry strategies already. And one of those, the Escher-Max strategy, is the same whether you are Parrying or Deflecting, so although I might not have explicitly mentioned it that gives us one Deflect strategy as well.

I've not even mentioned Counter Attack so far though. So, while I'm finishing off my first in-depth post which assesses the performance of the Escher-Max Parry/Deflect (it's becoming something of a leviathan update to write), I'll drop in the corresponding Escher-Max strategy for Counter Attack:

Counter Attack: Escher-Max
- Never re-roll a 6 (duh!)
- If you have one attack, always re-roll a 3 or less
- If you have more than one attack, always re-roll your lowest dice (even if it is a double of your highest dice)

Like the Escher-Max parry/deflect, this strategy tells you to re-roll whenever it is mathematically sound to do so (i.e. whenever you are more likely to improve your score than you are to make it worse). There are other families of strategy (Orlock and Goliath) which optimize slightly different things, but I'm going to leave those for now as they're not quite as easy to analyze cleanly when it comes to the mathematics (which will follow for this strategy in the post after next).

Some examples of how to use the Escher-Max Counter Attack strategy:

• If you have one attack and roll a 3, you should re-roll
• If you have one attack and roll a 4, you should not re-roll
• If you have two attacks and roll two 5's, you should re-roll one of those 5's

My guess is that the last one is counter intuitive to most players, and that very few people would make that call. The reasoning works as follows though:

1: If you have two 5's, your high score is 5
2: If you reroll one of those 5's, your possible outcomes are:
(a) a 1, which reduces your high score from 5 to 4 (a change of -1), and occurs with probability 1/6
(b) a 2-5, which leaves your high score unchanged (a change of 0), and occurs with probability 4/6
(c) a 6, which increases your high score from 5 to 6 (a change of +1), and occurs with probability 1/6
3: So the expected change to your current high score is -1 * (1/6) + 0 * (4/6) + 1 * (1/6) = 0. That is, you are indifferent to re-rolling that dice. If you're indifferent we might as well take the re-roll as not take it (it makes the rules easier too ).

By the same reasoning, having a high score of 5 with a second dice of 4-2 would also give you an expected change of 0. If your second dice is a 1 then you actually have a lot to gain by trying to get rid of the fumble (although I doubt anyone wouldn't re-roll a 1 anyway).

By the same reasoning again, having a high score of 4 with a second dice of 4-2 which you then re-roll actually gives you an expected improvement to your score of -1 * (1/6) + 0 * (3/6) + 1*(1/6) + 2*(1/6) = 1/3, so we're not even indifferent when our high score is a 4 – we actively have something to gain by Counter Attacking no matter what is on our other dice.

Anyway, there it is, our first Counter Attack strategy.

By the end of this week I'm aiming to post up a full assessment of how much advantage you can expect to gain in combat by using this strategy, along with the parry and deflect strategies. After that I'll post up various other Counter Attack strategies from the Orlock and Goliath family (following the same logic of the corresponding families of parry strategies) and show which one you might pick depending on your type of fighter or appetite for risk in combat.

In the meantime, it would be interesting to hear if the Escher-Max Counter Attack strategy above is at odds with how people already play this skill (or not!)... particularly given some of the discussions in the Community Edition thread (@Anthony ). Thanks!

 

[sebwiers]

I'd say if you don't expect a net statistical benefit from the reroll (as when your high roll is already a 5 or a 6) then there is no reason to re-roll, especially if doing so would risk a draw or loss. Conversely, when you are loosing combat and getting +1 would allow you to tie / win, its probably a good idea.

 

[enyoss]

Agreed, absolutely. But I wanted to keep any appeals to win/loss out of this particular strategy, simply because that makes it impossible to generalise the results to any WS/I situation.

Extending out to win/loss considerations and optimisation is covered in the Orlock & Goliath strategies I've already covered (and what you mention is really the Goliath strategy in reverse, as it were).

I guess you could specify that since in many situations a score of a 5 is quite likely to lead to a win, if you score a 5 you should not counter attack on the assumption you have won. Then it still generalizes well.

I really gave the double 5's situation as the most extreme (and surprising) case though. You should definitely always reroll the similar 4's situation as those are not even neutral, and I doubt many people do that as it is (mainly through a disproportionate fear of the fumble I would think).

 

[enyoss]

Parry and Deflect. Part I: Basics of the Escher-Max strategy

So, after many digressions, here it is (finally) – a full discussion of the first of our parry and deflect strategies.

Well, Part I of it anyway. There is actually so much to cover that, eventually, this is going to spread to several parts.

By the end of it, we'll have a full framework for:

• when to parry
• how much advantage it will give you
• what types of fighters stand to gain most from parrying
• how much mistakes under pressure will cost you
• and, eventually, other strategies.
• And then Counter Attack. Phew!

I've already posted up a few of the best parrying strategies strategies, but have been delaying any discussion around their performance until I had enough time to do it properly. So, I'm afraid this is going to be a long one (although you probably didn't need a crystal ball to figure that one out)...

Of the various strategies I posted already I'm actually only going to discuss one for now: the Escher-Max strategy. I will go on to discuss all the other strategies in a later post, but right now Escher-Max will introduce most of the core concepts and things to think about when evaluating strategy performance.

I pick the Escher family of strategies as they have a nice feature the other strategies don't – they can all be easily generalized to see what happens if either fighter has a WS/charge/initiative advantage over the other, in exactly the same way I have discussed before. The other strategies (Orlock & Goliath) can't do this, as their decision process involves seeing if you've won or lost, which already depends on these advantages – circular arguments suck.

And out of all the Escher strategies, Escher-Max is the best. So it's makes sense to start there.

Anyway, here is the strategy again:

Escher-Max

• Always parry an opponent's 6
• Always parry an opponent's 5
• Parry an opponent's 4 if he has no 4's on his other dice, otherwise do not parry
• Never parry an opponent's 3 or lower

Mathematically, this optimizes the following philosophy: always force the parry if you are more likely to improve the difference between your combat scores than you are to make it worse. And it optimizes this perfectly (as in, exactly).

To see how our strategy actually performs, I'm going to compare it against two baselines:

Baseline1) No parry
Baseline2) No parry, but with +1WS

Mostly we are interested in comparing with Baseline1, which just tells us how much a parry increases our chances in combat. This is what people are mostly care about.

Baseline2 is pretty important to keep an eye on as well though, because it helps us understand the value of a parry compared to the pretty common and well understood +1WS advance. For example, it helps us evaluate if a fighter with WS3 and a parry is as good as a fighter with WS4 and no parry.

Ok, I suppose it's time to show you the money and put some results out there.

As usual, we going to consider two fighters, Fighter1 and Fighter2, who are dopplegangers (for now). We are interested in the difference between their combat scores – that tells us how many hits each fighter gets. Positive differences mean Fighter1 has won, negative differences mean Fighter2 has won.

First, we'll look at the probability of getting each score difference. As always, we're rooting for Fighter1, so he's the one with the sword doing the parrying. In this case, Baseline2 just assumes our fighter drops his sword and picks up a +1WS bonus instead (maybe he charged, who cares, he just has a +1 bonus to his score instead of having a parry).

A few other quick points.
- One, for the next plot we'll look at what happens when each fighter has A2. This is simply because the curves are smoother so it's easier to see what's going on – it's the same story for all cases really.

- Two, the numbers shown here (and everywhere) are exact – we look at the 6^4=1296 possible outcomes from two fighters with A2, and then multiply that by six again to get possible all 7776 possible outcomes if we re-roll one of those dice.

- Three, although I've talked about parrying so far, the mechanics are essentially the same for a fighter using Deflect, so I'll discuss this at the same time. The only difference is that we can parry only when our opponent's highest dice is higher than our own, whereas we can Deflect whenever we want – the actual strategy we use if exercising either option is exactly the same (and given above).

Here, solid black represents Baseline1 (no parry), dashed black is Baseline2 (no parry but +1WS instead), Escher-Max Parry is in blue, and Escher-Max Deflect is in turquoise.

So what's the story?

Well, compared to our vanilla Baseline1 fighter, using Escher-Max Parry, Escher-Max Deflect, or having no parry but +1WS advantage, are roughly equivalent when it comes to reducing how often the bad outcomes occur - that is, they all make the negative differences much less likely to occur (the curves sit on top of each other when the attack score difference is less than zero, and are lower than the 'no parry' curve)

Things start to change a bit when we look at tied or won combats. For Escher-Max Parry (blue), its main effect is to pile up probability in the tied and marginal win outcomes – once you get to attack score differences of +3 or higher there's not a huge difference between Escher-Max Parry and having no parry at all (Baseline1). Compare this with Baseline2, which is clearly always better than Baseline1 (as we would expect). This shows that while parrying Escher-Max is a big improvement on having no parry at all, it isn't as good as a straight +1WS advantage, at least when it comes down to how often you're getting big wins.

For Escher-Max Deflect though (turquoise), instead of just piling up probability in the tied and marginal win outcomes it spreads it out amongst all of the winning outcomes. Therefore it's always the case that Escher-Max Deflect is better than having nothing at all, even when it comes to big wins. In fact, there's not much difference between Escher-Max Deflect and having no parry with +1WS – there's a little blip at differences of +1, but we'll show in a moment how that doesn't really impact things much.

Anyway, looking at the probability density function (PDF, the figure above) is useful in that we now understand what's going on, at least when it comes to how often we expect to get each individual difference.

That doesn't tell use much about our overall win/draw/loss probabilities though so, as always, we are going to look at the cumulative distribution function (CDF) from now on. This figure, as described in my previous posts, tells us everything we need to know about the parry (in fact, it tells us several Parts worth...). Note that although the last figure was for both Fighters having 2 attack dice (for convenience), we now look at what happens when they have just one attack dice each.

I'll spare you the hassle of having to go back and re-read my previous post on how to interpret this thing (assuming you read it to begin with...), so here is how you get win/draw/loss probabilities from this:

Probability(F1 win) = look up from Attack Score Difference +1 and read off the value of each curve
Probability(F1 win or draw) = do the same for Attack Score Difference 0
Probability(draw) = Probability(win or draw) – Probability(win)
Probability(F1 loss) = 1 – Probability(win or draw)

I'll also spare you the hassle of actually having to do that, and just give the numbers here (along with numbers for when Fighter1 and Fighter2 have 1 or 2 attacks, stats are given below the figure they belong to):

Fighter1 A=1 & Fighter2 A=1 (figure shown above)
F1 wins: 42% (Baseline1), 58% (Baseline2), 50% (EM Parry), 54% (EM Deflect)
Draw: 17%, 11%, 22%, 17%
F1 loses: 42%, 31%, 27%, 29%



Fighter1 A=2 & Fighter2 A=1
F1 wins: 55% (Baseline1), 69% (Baseline2), 63% (EM Parry), 69% (EM Deflect)
Draw: 15%, 12%, 18%, 11%
F1 loses: 31%, 18%, 19%, 20%



Fighter1 A=1 & Fighter2 A=2
F1 wins: 31% (Baseline1), 45% (Baseline2), 41% (EM Parry), 45% (EM Deflect)
Draw: 15%, 15%, 19%, 14%
F1 loses: 55%, 40%, 40%, 41%



Fighter1 A=2 & Fighter2 A=2
F1 wins: 41% (Baseline1), 59% (Baseline2), 50% (EM Parry), 58% (EM Deflect)
Draw: 18%, 15%, 22%, 15%
F1 loses: 41%, 26%, 27%, 27%

If we take a moment to look at how often Fighter1 loses in the above statistics, it is exactly as we predicted before: When it comes to how often you lose, parrying or deflecting Escher-Max has exactly the same effect as having +1WS (Baseline2). One exception is when both fighters roll just one attack dice - in this case parrying or deflecting Escher-Max is better at stopping you losing than simply having +1WS and no parry. In any case, all three are a damn sight better than having no parry at all.

When it comes to how often Fighter1 wins, Baseline2 and Escher-Max Deflect go toe-to-toe in every situation apart from when both fighters roll just one attack dice. Parrying Escher-Max isn't that much worse, but bleeds a bit of probability from the wins into the draw category compared to the other two. This feature of the parry, that it tends to increase the number of tied combats you get, will be pretty important when we come to evaluate the relative importance of the parry to high Initiative fighters (more on that in Part X... that's not Part ten BTW, it's just Part 'I fear this might go on for longer than I expected oh God I have RSI and perma-squinting screen eyes'.).

Anyway, that's it for now. I was going to wallow in some mean, variance and skewness statistics to convince you that Escher-Max Deflect is in every way as good as the no parry +1WS case, but I think that can wait until later.

I'll leave you with a summary though/.

Summary:

• Escher-Max Deflect is almost identical to having no parry but +1WS.
• Escher-Max Parry, Escher-Max Deflect, and having no parry but +1WS, are functionally identical when it comes to how often you lose.
• Escher-Max Parry makes drawn combats more likely, at the expense of having fewer big wins than Escher-Max Deflect.

 

[crimsonsun]

Your posts are a perfect demonstration as to why scalies are the most ultimate of badasses with a sword and a club, plus discus. Then run the math on the effect that a toughness increase and a single point of ws bring. With two wounds a 5+ save t5 ws5 3 a and a parry hitting at str5/6 your the ultimate combat monster.

Scavvies love ambushes btw bit hate hate defending reacies.

 

[enyoss]

I've never played against Scavvies, so I hadn't noticed just how good scalies are. Base WS4 and A2 is pretty awesome, especially early in a campaign.

 

[enyoss]

Parry and Deflect. Part II: Weaponskill Advantages

And so here I find myself – on a very delayed and very long train ride into London. What better time to make a start on Part II of our parry and deflect series?

So far, we have looked at the fundamental behaviour of having a parry or deflect compared to not having a parry at all (Baseline1), and compared to not having a parry but having a +1WS advantage (Baseline2).

Now, we're going to look at how the parry benefits high and low WS fighters. To do this we only need to look at the plots I've already posted up – as I've said before (ad nauseam), those plots encapsulate almost every shred of mathematical parry/deflect related information you could wish for. We just need to look for it.

In Part I, we gave the win/draw/loss probabilities for Baseline1, Baseline2, Escher-Max Parry, and Escher-Max Deflect, assuming our fighters were equally matched in WS. We allowed attacks to vary (each plot shown corresponds to a different Fighter1 Attacks/Fighter2 Attacks combination), but that was it.

To look at fighters who are unmatched in WS, we analyse the plots as described in my earlier post “Weapon skillz pay the bills, but playaz use their initiative...” (as I retype that I am making a mental note – less of the corny titles please).

I'm going to consider two test cases of unmatched fighters:

Case A: Fighter1 has a +2WS advantage
Case B Fighter1 has a -2WS advantage

Neither case is that rare – it arises every time a fighter with WS4 charges an opponent with WS3, for example.

As before, for each case we consider the following:

Baseline1) Fighter1 has no parry or deflect)
Baseline2) Fighter1 has an extra +1WS advantage, in this case giving her a mighty +3WS advantage in total)
Escher-Max Parry) Fighter1 can parry, and uses the Escher-Max strategy to determine when to do so
Escher-Max Deflec) Fighter1 can deflect, and uses the Escher-Max strategy to determine when to do so

So, how does this change our win/draw/loss probablities? I'll deal with each case in turn, varying each Fighter's attacks between 1 and 2. The story is exactly the same if we go up to 3 attacks for each fighter, but I'm leaving that out here for brevity (we'll deal with how attacks influences the value of a parry in a little while).


Case A: Fighter1 has a +2WS advantage

We get the following numbers in almost exactly the same way as Part I, and using the same plots. The only difference is that instead of looking up to see where our curves cross Attack Score Difference = +1 to get our win probability, we look up from Attack Score Difference = +1 – (Fighter1 WS Advantage) = +1 – 2 = -1

Similarly, instead of looking up to see where our curves cross Attack Score Difference = 0 to get our win or draw probability, we look up from Attack Score Difference = -2.

Fighter1 A=1 +2WS & Fighter2 A=1
F1 wins: 69% (Baseline1), 81% (Baseline2), 80% (EM Parry), 79% (EM Deflect)
Draw: 11%, 8%, 8%, 8%
F1 loses: 19%, 11%, 12%, 12%

Fighter1 A=2 +2WS & Fighter2 A=1
F1 wins: 82% (Baseline1), 90% (Baseline2), 90% (EM Parry), 89% (EM Deflect)
Draw: 8%, 5%, 4%, 5%
F1 loses: 10%, 5%, 6%, 6%

Fighter1 A=1 +2WS & Fighter2 A=2
F1 wins: 60% (Baseline1), 73% (Baseline2), 72% (EM Parry), 72% (EM Deflect)
Draw: 13%, 11%, 10%, 10%
F1 loses: 27%, 16%, 18%, 18%

Fighter1 A=2 +2WS & Fighter2 A=2
F1 wins: 74% (Baseline1), 85% (Baseline2), 84% (EM Parry), 84% (EM Deflect)
Draw: 11%, 7%, 7%, 7%
F1 loses: 15%, 8%, 9%, 9%

What do we notice here? Look at the win/draw/loss probabilities between Baseline2, EM Parry and EM Deflect. They are almost always within 1% or so of each other.

That is, when you have higher WS than your opponent, parrying Escher-Max and deflecting Escher-Max are basically equivalent. Furthermore, their effects are almost identical to having an extra +1WS and just not having a parry.

And why does it work like this? It's all because the part of the curve we read our win and win+draw probabilities from when we have a WS advantage over our opponent is towards the left of the plot, which (as mentioned in Part I) is where the Baseline2, EM Parry and EM Deflect curves are indistinguishable from each other. Take another look. To make it easier to visualize I've shifted the curve over two places to the right, rather than making ourselves look up from Attack Score Difference two places to the left - but it's all the same:

Pretty cool huh? Sadly, the news isn't so rosy if you have a parry while at a WS disadvantage...


Case B: Fighter1 has a -2WS disadvantage

To get these numbers we look up where each curve crosses Attack Score Differece = +3 and Attack Score Difference = +1, in exactly the same way as already described for CASE A.

Fighter1 A=1 -2WS & Fighter2 A=1
F1 wins: 19% (Baseline1), 31% (Baseline2), 23% (EM Parry), 29% (EM Deflect)
Draw: 11%, 11%, 14%, 14%
F1 loses: 69%, 58%, 63%, 57%

Fighter1 A=2 -2WS & Fighter2 A=1
F1 wins: 27% (Baseline1), 40% (Baseline2), 30% (EM Parry), 40% (EM Deflect)
Draw: 13%, 15%, 15%, 15%
F1 loses: 60%, 45%, 55%, 45%

Fighter1 A=1 -2WS & Fighter2 A=2
F1 wins: 10% (Baseline1), 18% (Baseline2), 13% (EM Parry), 19% (EM Deflect)
Draw: 8%, 13%, 10%, 11%
F1 loses: 82%, 69%, 77%, 70%

Fighter1 A=2 -2WS & Fighter2 A=2
F1 wins: 15% (Baseline1), 26% (Baseline2), 17% (EM Parry), 27% (EM Deflect)
Draw: 11%, 15%, 14%, 15%
F1 loses: 74%, 59%, 69%, 58%

What's going on here? Well, aside from the overall drop in win probabilities across the board, performance of the Escher-Max Parry is nothing like what we saw in Case A

Look at our win and loss probabilities. While Baseline2 and Escher-Max Deflect are still tracking one another quite nicely, Escher-Max Parry really starts to lag behind. In most cases, the Escher-Max Parry results are actually closer to the results when we don't have a parry at all, rather than Baseline2 or Escher-Max Deflect.

Again, this comes from the fact that the bit of the curve we read our win and win+draw probabilities from when we are at a WS disadvantage is towards the right of the plot, which is where the EM Parry and EM Deflect curves start to diverge from one another. Again, here is a tinkered version of the plot to convince you (tinkered with by shifting the curve two places to the left):

What does this mean? Simply put:

• A parry is most valuable if you already have a WS advantage over your opponent.
• If your opponent has a significant WS advantage over you, a parry brings almost no advantage at all.
• Deflect is unaffected by either mismatches in WS, and will always contribute a bonus roughtly equivalent to having +1WS.

Why the asymmetry for the poor old parry? It comes from the fact that you can only parry your opponent's highest dice if your own highest dice is lower.

For either high or low WS fighters, if they have rolled lower than their opponent they are likely to have lost the combat (certainly in fact, for the low WS fighter), but at least they can force the parry to try and claw back the win.


What happens when they roll higher than their opponent though? For high WS fighters this is not a problem: in cases where they have a higher dice than their opponent they are unable to parry, but they have already won anyway. Low WS fighters are in trouble though. It's entirely feasible that they have a higher dice roll than their opponent and are therefore unable to utilize the parry, but could have still lost the combat! In this case not being able to parry is a significant hurdle, and one which comes up surprisingly often for our poor low WS numpty.

Deflect, on the other hand, suffers no such constraints for either fighter. Even if the low WS fighter has rolled the same or higher than his opponent (e.g. both she and her opponent have a 6... the nightmare outcome for low WS fighters with a parry, but a total peach for fighters with Deflect), she can still force the reroll if it is expedient to do so (in the above case of both her and her opponent rolling a 6, she would definitely want to force that reroll!).

What does this mean for gang selection and evaluation?

Basically, if you have a WS2 juve and are hoping to turn them into a WS3 ganger by giving them a sword, you're going to be disappointed. It's not exactly a waste of credits - it does bring some advantage afterall, and if you get the charge against a WS3 opponent you eliminate your WS disadvantage altogether. However, there is a force multiplier between the parry and WS, so giving that sword to a fighter with WS3 instead might be a good idea.

And because I like a summary at the end, I'm going to repeat what we stated earlier:

Summary:

• Not all parries were created equal
• Fighters with a WS advantage over their opponent gain most value from a parry. In this case, the parry is roughly equivalent to having +1WS and no parry/deflect.
• Fighters at a significant WS disadvantage stand to gain much less from having a parry.
• Deflect is always roughly equivalent to having +1WS and no parry/deflect.

 

[enyoss]

Parry and Deflect. Part III: Initiative Advantages

This part will be brief(er), and will draw entirely on numbers we have presented in Part I and Part II.

Overall there is a story developing here by the way – going from matched fighters, to fighters with WS advantages, to fighters with Initiative advantages, and then on to fighters with different Attacks, we're building towards a framework which determines the value of parries and combat skills to almost every fighter you might encounter in a gang. This is obviously pretty handy when it comes to choosing equipment, or deciding which skill table to roll on to maximise you close combat effectiveness. At least, that's my (somewhat ambitious) aim.

Anyway, on with looking at Initiative.

As always, we're going to talk in terms of:

Baseline1) no parry/deflect
Baseline2) no parry/deflect, but with +1WS
EM Parry) Fighter1 has a parry, using Escher-Max strategy
EM Deflect) Fighter1 has deflect, using Escher-Max strategy

I'll break this one down into High WS fighters, Matched WS fighters, and Low WS fighters.

High WS Fighters

As mentioned in Part II, the win/draw/loss probabilities are just about the same for Baseline2, EM Parry and EM Deflect when your parrying fighter has a WS advantage over your opponent. We can also deduce the following:

• Both Parry and Deflect decrease the number of draws relative to Baseline1

One other observation is that the probability of a draw for EM Parry and Deflect also becomes quite small– your win or draw category is really dominated by just winning outright. Also, and most importantly, the probability of you getting a draw is always smaller than it would be if you had no parry at all (Baseline1).

What does all this mean?

Well, if you have high WS and a parry/deflect the difference in Initiative between you and your opponent, which will decide who wins draws, becomes less important than it would be if you had no parry at all.

This is good news if you have high WS and low Initiative: not only does the parry act like having a +1WS bonus, it also mitigates your Initiative disadvantage. Not entirely mind you. You'll still lose 5-10% of combats you would have won if you had high Initiative, but that's better than the 10-15% you would have lost if you had no parry at all.

If you have high Initiative then you're sitting pretty as always – you just won't have to rely on it as often as you otherwise might.


Matched WS Fighters

Here we go back to the numbers presented in Part I, and just look at the draws:

• Deflect decreases the number of draws relative to Baseline1
• Parry increases the number of draws relative to Baseline1

So if you Escher-Max Deflect, you will slightly reduce the impact Initiative plays in the combat (mainly by just turning draws into outright wins). Again, good news if you have low Initiative – it will hamper you slightly less often, to the tune of losing you 3-5% fewer combats (I did say slightly).

If you Escher-Max Parry, the story is a bit different. Just as EM-Deflect slightly reduces the probability of getting a tied combat relative to the Baseline1, EM-Parry slightly increases it (by the same amount). So if you EM-Parry, Initiative will play a marginally bigger role in combat. Not such great news if you have low Initiative, but better if you have high Initiative.


Low WS Fighters

Again, looking at the numbers from Part II when Fighter1 has a WS disadvantage, we can deduce the following when Fighter1 is at a WS disadvantage:

• Both Parry and Deflect increase the number of draws relative to Baseline1

So if you Parry of Deflect Escher-Max and are at a WS disadvantage, you will get more drawn combats, so Initiative will play a bigger role. Although the effect is only slim, typically a few percentage points, this means that low WS fighters with low Initiative are compounding their disadvantage.


To summarize:

• Escher-Max Parry/Deflect decreases the number of drawn combats if the fighter has higher WS than his opponent.
  
  
• Escher-Max Parry increases the number of drawn combats if the fighter has the same WS as his opponent, while Escher-Max Deflect decreases the number of drawn combats.
  
  
• Escher-Max Parry/Deflect increases the number of drawn combats if the fighter has lower WS than his opponent.
  

Or stated in more practical terms:

• Escher-Max Parry/Deflect decreases the impact of low Initiative if the fighter has higher WS than his opponent.
  
  
• If a fighter has the same WS as his opponent, parrying Escher-Max increases the impact of low Initiative, while deflecting Escher-Max decreases the impact.
  
  
• Escher-Max Parry/Deflect increases the impact of low Initiative if the fighter has lower WS than his opponent.
  

 

[Zustiur]

Are you going to be continuing this analysis? It is very interesting even if most of the variables are out of the player's control. I'm hoping you will expand into wounding because the choice of weapons is where we have the greatest control. When is it a good idea to have a club instead of a sword for example.



sent via tapatalk

 

[enyoss]

I do intend on continuing .

I have a ton of stuff which I have looked at and haven't written up yet. This includes the last parts of analyzing the parry.

In answer to your question though, no, if you have one weapon I don't think it is ever a better idea to take a club instead of a sword. If you analyze combat options using game theory and ranking weapon choices by either win rate or loss rate, the sword always wins out over other options. No matter what your opponent is carrying, it is always your best response (I think... can't remember for sure, but I'm almost certain this is the case).

There is one exception - if your opponent is using a chain your sword is completely nullified, so you might as well take a club instead. They could always switch back to use a sword or a club then though, and then you switch back to your sword etc. This game goes on forever - there is no equilibrium situation for it to settle down to.

Interestingly (for stats nerds like me at least), if you remove the option to take a club from one of the fighters, and remove the option to take a sword from the other (perhaps mimicking limitations due to House Weapon Lists), a Nash Equilibrium does pop into existence and the little game above settles down to one fighter using a sword and the other a chain.

If you ranked combat effectiveness on wound rate it would be different, but I think many people adhere to the wisdom that a high chance of wounding isn't very helpful if you don't hit in the first place. I use weapon strength as a tie breaker when the odds of winning/not-losing are close.

EDIT: Worth noting that the above is a very simplified (and somewhat wrong) example (before someone comes along and points it out). For the sword and chain situation, for example, an equilibrium does exist where fighter1 alternates between sword/club with probability p1, and fighter2 alternates between chain/club with probability p2. That is, a Nash equilibrium does exist as a kind of optimal "keep them guessing" strategy.

If you only have one weapon per fighter in your gang though, those same p1 and p2 then tell you roughly how many of each fighter you should have on your roster, rather than how often an individual fighter should swap between weapons. I'll come to that when I have time though.

 

[Nitirat]

I'm pretty terrible at maths, but you made it (fairly ) easy for me to understand, and there's some solid tactics to be had out of the analyses, so thanks for taking the time to make these posts

 

[enyoss]

Now for something completely different...

(Summary at the end of the post)

Not that we really need it after such a long break between updates, but now it's time for the Intermission .

I'll be forcing some wisdom on Counter Attack strategy down your throats some time soon, but in the meantime I'm going off piste to look at the chance of going Out of Action (OOA).

First, let's think about how likely any fighter is to go OOA once they've suffered their last wound. Simple you say! - a six on the injury roll does it. So 1/6.

And you'd be right... for that first roll. But what about all the rolls after that?

Let's think about that first roll a bit more:

• On a D6 roll of a 1 you get a Flesh Wound (FW) and get back up (a probability of 1/6).
• On a D6 roll of a 6 you go OOA (also a probability of 1/6).
• And on a D6 roll of a 3-5 you go down (probability 4/6), which is essentially a 'push' – nothing happens, and we just postpone our FW/OOA decision for a turn.

Most importantly, the chance of getting a FW or going OOA when taking any given injury or recovery roll is exactly the same, i.e. 1/6.

Next recovery phase we do the same. We either get a FW or go OOA, or push the decision to the next turn. And so on. After n turns, the probability we have pushed the decision every time and are still Down is:

Prob(Down after n turns) = (4/6)^n = (2/3)^n

This means that as quickly as your first recovery roll (i.e. your second injury roll overall, the first being the roll you take when first suffering your last wound) the odds of you being Down after taking that roll are just (2/3)^2 = (4/9) = 44%.

So you are more likely to have suffered a FW or have gone OOA after your first recovery roll than you are of still being Down. That took me by surprise a bit – I always assumed that Down fighters tended to stay writhing around on the floor for a few turns. But that just ain't the case... their situation tends to resolve itself much faster.

And there's an even more interesting result (to me at least) to come out of this. Over many turns, that (2/3)^n probability of being Down gets smaller and smaller. After three turns it's 30%, after four turns it's 20%, etc.. After an infinite amount of turns it's zero, and that's pretty logical really – you're fighter will eventually get a FW or go OOA, you can't keep avoiding those outcomes forever.

And as we said before, the probability of getting a FW on any given injury roll is exactly the same as the probability of going OOA.

So after infinite turns, we have to have either a FW or OOA, so:

Prob(OOA after infinite turns) + Prob(FW after infinite turns) = 1.

And as P(OOA) = P(FW) on every recovery roll, we must have our eventual chance of either a FW or OOA after infinite turns of rolling for recoveries as:

Prob(OOA after infinite turns) = Prob(FW after infinite turns) = ½.

Of course, a game doesn't go on for infinite turns. So when the game is called, and you still have fighters who are Down, you should assume that they recover to a FW with a probability of ½, or they go OOA with probability ½.

... Which is the same as saying fighters who are Down at the end of the game go OOA on a D6 roll of a 4+.

... Which is exactly how it works in the rules! - who would have thought that GW accurately closed out the infinite summation problem when putting that little rule in there .

That's the simple way of proving that. A more complex way (and some might say the Von Neumann way*), is to solve the infinite sum:

Prob(OOA after infinite turns) =
Prob(OOA turn 1) +
Prob(Down turn 1) * Prob(OOA turn 2) +
Prob(Down turn 1) * Prob(Down turn 2) * Prob(OOA turn 3) + …

= sum_{n=1}^{\infty} Prob(OOA) * Prob(Down)^(n-1)

= a + a r + a r^2 + a r^3 + …

where

a = Prob(OOA) = 1/6, r = Prob(Down) = 2/3,

which is a geometric progression with infinite sum

Prob(OOA after infinite turns) = a (1 – r^n) / (1-r) = (1/6) * (1 – 0) / (1 – 2/3) = 1/2

[* This follows a famous story where the great mathematician (and war-monger ) Von Neumann, when asked to solve a similar problem at a cocktail party, supposedly instantaneously summed the infinite sum in his head rather than spotting the easy solution. Show off.]

I'm only putting that in there because that's how I first tackled the problem (rather than seeing the much simpler solution). And also to prove that A-Level mathematics is useful after all .

In fact, the probability of going OOA overall after going to zero wounds is slightly higher than that thanks to the other way of closing out the infinite sum for fighters who are Down - a swift boot to the face from an enemy fighter in close combat .


So in summary:

Despite the 17% (i.e. 1/6) probability of a fighter taken to zero wounds going OOA on any given injury or recovery roll, the probability of them going OOA overall is at least 50%. Take care of them!


Next up, looking at how True Grit changes this... .

 

[enyoss]

True Grit...

A warning: I'm going to end this post with a suggested rules change for the NCE. The audacity .

So now we know that a fighter who loses their last wound has at least a 50% chance of going OOA, how does True Grit change those odds?

Well, simple enough it just changes the injury roll and recovery probabilities to:

Prob(FW) = 2/6 = 1/3 (1-2 on a D6)
Prob(Down) = 3/6 = 1/2 (3-5 in a D6)
Prob(OOA) = 1/6 (6 in a D6).

So we are now twice as likely to get a FW on any given injury or recovery roll as we are to go OOA. Feeding that into the logic in the previous post (for either the simple or the complex solution) gives:

Prob(FW after infinite turns) = 2/3 (67% chance)

Prob(OOA after infinite turns) = 1/3 (33% chance)

And, importantly, this should in theory change the end of game roll to see whether a down fighter goes OOA from a D6 roll of 4+ (50% chance), to a 5+ (33% chance). Otherwise, fighters with True Grit are being short changed.


In summary:

If a fighter has True Grit, if they lose their final wound the probability they will go OOA is 33%, compared with at least 50% for fighters without True Grit.

Suggested NCE rule change: Fighters with True Grit who are down at the end of the game should go OOA on a D6 roll of 5+, rather than 4+, in addition to the existing effects.


Next up will be my most ambitious analysis yet, but it's going to take a while to write it up in a digestible format, so it'll have to wait...

 

[cardyfreak]

Man u gud at mafz