Necromunda combat stats

enyoss

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By way of explanation...

Before I embark on the thread proper, I'm going to give a bit of introduction.

On Thursday a couple of weeks ago I was sitting at work, slack jawed, contemplating two important questions.

The first, which I was supposed to be thinking about, was how to model uncertainty in catastrophic earthquake risk in the Pacific Northwest. The second, and by far more important question, was how I was going to deal with being raided by Roid Rage in my Necromunda game that coming Sunday (my wife just told me that sounds weird, but I don't care).

My gang currently has the highest gang rating in that campaign, and almost everyone apart from me has a Ratskin Map. I'm not going to be picking scenarios any time soon.

The most likely scenario chosen against me would be an Ambush, with me defending. I took the stopper out of that genie bottle by racking up a big win with one back in the early days of the campaign, and since then it has been the lop-sided scenario of choice for all the underdogs.

But defending a Raid could still be on the cards. And a Raid does exactly what a hand-to-hand gang like mine fears most – it cripples your fighters in hand-to-hand.

Which got me thinking – once their weapon skill has been halved, as it is for sentries in a Raid, the difference between my best combat gangers and my worst is really only a point in weapon skill. The attacks characteristic on the other hand gets by unscathed.

I have a couple of gangers with WS5+ and A1, and I have a couple with WS3 and A2. When the sentry rules are in place this boils down to fighters with either WS3 and A1, or WS2 and A2.

In a Raid I dislike both situations almost equally, but which is better? There was only one way to find out.... FIGHT! No, actually, a much better way was to look at some simple stats :)

Which got me thinking again – what if I could come up with a few simple rules of thumb based on the underlying probabilities and statistics, which allowed me to rank characteristic and weapon combos like this in any situation. Wouldn't that be useful? And wouldn't it be more interesting than thinking about catastrophe risk uncertainty all the time (although, for the record, I really enjoy thinking about that as well).

So that's exactly what I'm going to look into in this thread (along with any other interesting Necromunda stats questions I come across on the way, e.g. how worried should I be about Ambushing my opponent and him/her getting the drop on me...).

First though, a couple of points.
  • I'm not claiming any of this is new. I'm pretty sure it has been done a million times before, and probably in more depth. It might even be in a thread here already. It's not even hard to be honest. I'm pretty confident I've over-engineered a simple problem.

  • One thing I started to notice is that I don't think any of the results here will change our accepted wisdom. I don't even want it to. It became clear to me that players' underlying intuition for what makes a combat a good or bad bet is quite close to what's given by the stats, which is pretty cool.
  • This is for fun, not super-optimizing your gameplay. Even if it were useful for the latter, which I doubt, that's just not Necromunda.

  • I might well get distracted before finishing. You know, life and all that. I already have quite a lot which is waiting to be posted, so progress in this thread will lag what I've done by quite a way.

  • The usual disclaimer that my maths might be wrong. I'm also going to be light on the precise details, but feel free to ask.

  • If you want the code and data used to do calculations just ask.
With all that out of the way, here we go.
 
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enyoss

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Rules of thumb so far

This is a pinned post which will summarize the rules of thumb I've come up with so far (assuming no errors in the maths, and assuming I don't change my mind on weighting mean outcomes against variances). I'm actually quite a bit ahead with getting results than I am with writing them up so sometimes what's here might not reflect what has already been posted. The discussion will be added in time though...

The way to think of the following ranking is like this: given I have two fighters, with all else being equal (e.g. armament, strength etc.), how do I rank these fighters given certain characteristic advances.

Ranking of characteristic advances (from best to worst):
1) A2
2) +1WS
3) High Initiative (I5+)
3) A3+


e.g. With all else being equal, a fighter with WS4 and A2 is "better" than a fighter with WS3 and A3 when it comes to combat (comparatively, the first fighter would be +1WS, and the second would be +1A, relative to each other).

The top spot is held by weird outlier - A2. This can mean your fighter has A2 either on their profile or through using a weapon in each hand - all that really matters is you are rolling two attack dice. The main reason this occupies the top spot is because of the huge impact it has on the uncertainty of your combat score compared to rolling just one dice, but we'll discuss that more later...

Other observations:

Stated in dramatic terms...
  1. Though shalt not exceed A4 when fighting with a chain
  2. Though shalt not exceed A3 when fighting against someone with Feint
  3. Though shalt know thy parry strategy and stick to it

Parrying strategies *** Added 06-Feb-2016 ***:

Here are the most effective parrying strategies, in order of increasing difficulty. The optimal strategies optimize your win-draw rate and are therefore the 'best', the max strategies optimize how much you win by (but not your win-draw rate) and can be thought of as the runners-up, and the rest are not optimal in any way yet still effective and easy to remember:

Escher Strategy

A simple yet surprising powerful strategy which is best suited to low strength fighters who are willing to trade an optimal overall win-rate for an increased probability of occasional big wins.
- Always parry

Orlock-optimal Strategy

An intermediate strategy which gives the optimal win rate while leaving draws untouched. This is best suited to middle of the road fighters who would like to win as often as possible, but who are grateful to take a breather on the draws.
- Always parry an opponent's 6
- Always parry if losing
- Never parry if winning
- Never parry a draw


Escher-max Strategy

A modification to the regular Escher strategy which optimally maximizes the difference you win by. As it doesn't optimize our number one priority of simply winning, I'm not calling it 'optimal', ergo the 'max'. Suitable for the same kind of fighter who would use the Escher strategy.
- 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


Goliath-optimal Strategy


An advanced strategy which optimizes your win rate by gambling on turning draws into wins when the odds are in your favour (they can also turn into losses though, hence the gamble). Best suited to high toughness high strength fighters who are comfortable risking a moderate loss if there is a chance of dealing out a high damage moderate win.
- Always parry an opponent's 6
- Always parry if losing
- Never parry if winning
- Always parry a drawn opponent's 5
- Parry a drawn opponent's 4 if he has no 4's on his other dice, otherwise do not parry
- Never parry a drawn opponent's 3 or lower
 
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enyoss

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Expected combat score

The most sensible place to start is at the beginning. That is: for a given number of attacks, what is my expected (mean) combat score. By combat score, I mean the score calculated using the highest number rolled, adjusted for fumble and critical hit penalties/bonuses.

For the first pass at my sentries question the solution is then simple – all else being equal, for my ganger with WS2 and A2 to be better than the ganger with WS3 and A1, her expected combat score using A2 has to be at least one point higher than using A1.

So all I have to do is calculate these expected combat scores and we're done. Luckily it is pretty straightforward to calculate these exactly.

The next figure shows the expected combat score for four combat situations. The x-axis gives the number attacks. The y-axis gives the expected combat score.

Means.png


The situations are:
- D6. We discount fumbles and critical hits and just want the expected highest score when rolling X D6. Just a test case really.
- Knife. The fighter is armed with only a knife, and no other weapons (this situation also represents clubs, power fists, massive axes etc.. Even swords, as the parry doesn't enter at this stage. Due to the involvement of player decision in when to parry they are more complex, and we'll have to derive some kind of optimal strategy for them. We'll look at that with some simulations later).
- Chain. The fighter is armed with one chain.
- Double chains. The fighter is armed with two chains. When comparing with the other situations you actually have to bump their A number up one as this situation benefits from an extra attack (i.e. the other situations essentially become two Knives, Chain and Knife, etc.). Just included for reference really.
- Feint. The fighter is armed with a knife, but his opponent has the Feint skill (i.e. the fighter fumbles on rolls of 1 or 2).

Before delving in too much, what should we expect to see here?

Imagine we have 600 attacks (I did say imagine). We roll our dice. From an expectation (or average) point of view we expect 100 1's, 100 2's, 100 3's, etc. So clearly in this case our highest dice score is going to be one of those 100 6's we rolled. So when we have lots of attacks our expectation for the D6 situation, which is only interested in the high score, is 6. So far so obvious.

Now consider the Knife situation. In the same way, the highest roll score here will also be the first of those 100 6's, so a 6. However, this time we also have all those fumbles and critical hits to think about. We have 100 1's left to worry about, and 99 6's (the hundredth 6 is the one we used for our high roll). We'll take one of those 1's, and apply it to our high score, bringing it down to 5. That leaves 99 1's, each giving a -1 fumble penalty, which are balanced perfectly by the 99 6's we had left, each of which gives a +1 critical hit bonus. All those 99 fumbles and critical hits therefore cancel out. So when we have lots of attacks using a Knife we are just left with an expected combat score of a 5.

Similarly using a chain, expect that the first fumble we apply to our high roll of a 6 counts as a -2 penalty, instead of -1. The remaining 99 fumbles and critical hits still cancel each other (fumbles after the first still count as -1), so for the chain our expected combat score using lots of attacks is 6-2 = 4. The same reasoning puts the double chain expected combat score on 3.

The Feint situation is bad news for the fighter. Unlike the other situations, there are now twice as many fumbles as critical hits (1s and 2s both count as fumbles), and the two no longer balance. Fumbles start to dominate. Therefore, no matter how many attacks you have your expected combat score will never stabilize to a nice value. In fact, once you get past A4, it just keeps plummeting until you are well and truly buggered.

Looking at the figure, it's obvious that each situation does indeed follow this behaviour, although it takes about 15 or so attacks before they really settle down to their many-attacks averages.

Also, apart from the knife situation, all other curves have a peak around A4, before their expected combat score starts to reduce. That is, from an expected combat score perspective, taking your attacks beyond A4 makes you worse in combat. Fortunately such dilemmas are rare in the underhive (you can't say I didn't tell you this wisdom might be useless).

But next time that daemon possessed A3 knife and chain-wielding Totem Warrior on spook is reducing you to red mist with his ten attacks, you can be smug in the knowledge that the joke is on him – he would have wailed on you even more efficiently if he was only A1.

Back to our original question though: what are the expected combat scores for A1 and A2. For a knife, reading up from 1 and 2 on the x-axis we see that the expected combat scores are around 3.3 and 4.2 respectively. (I should point out at this point that when we can only have whole number outcomes, dealing with expected combat scores which are decimal numbers is something of an abstraction. It's perhaps better to deal in Median scores, which are usually whole numbers. We'll come back to that in a minute).

So although having A2 increases your expected score, it doesn't quite manage to increase it by +1. This holds for all attacks where the difference is 1, e.g. A3 versus A2 etc.. In fact, the expected benefit you get by adding an extra attack just gets smaller the more attacks you have (and is always <1), while a +1WS bonus always adds +1 to your combat score.

Therefore a WS3 fighter with A1 will, on average, get a higher combat score than a WS2 fighter with A2. The WS3 and A1 fighter is the better sentry. Problem solved.

More generally: All else being equal, an extra point of WS is more valuable than an extra point of A.

Or not...


One mathshammer habit is to only deal in expected outcomes. It's by far and away the approach used most often to assess whether a combat is in your favour or not. And for good reason too – it's straightforward to do, and often gives a simple wounds dealt minus wounds taken metric which roughly indicates if a combat is in your favour.

However, the games designers for 2ed 40K were smart. Rick Priestly himself describes it best: the 2ed 40K combat system is elegant. And being elegant, we can go beyond the expectation viewpoint and flesh things out a bit more...
 

enyoss

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It's all about the distributions

(Two rules of thumb given at the end, but in summary: trade overkill for certainty. It's a good trade to make)

Sticking with just the one fighter, rather than looking at how the expected (or mean) combat score varies with number of attacks, we now look at the whole distribution of possible values.

The exact probability distributions for a fighter with attacks A1 through to A6 is shown below. Each curve shows the probability of a particular combat score outcome (i.e. the highest value rolled when adjusted for fumbles and critical hits) for a given number of attacks. For example, for a fighter with two attacks, looking at the red curve we see that a score of 5 (e.g. from a roll of 6 and 1, or a roll of 5 and 3) will occur with probability 0.25, or around 25% of the time.

CombatProbabilities_CdfFALSE_ChainFALSE_FeintFALSE.png


The median score on each curve is marked by the crossed box. I've roughly grouped scores into Loser, Middling, and Winner categories. If you are fighting and get a Loser result then you'd better hope your opponent fluffs their roll. Similarly, if you get a Winner result you would have pretty good odds of coming out on top in the combat, unless you had seriously bitten off more than you could chew.

Ignore the dashed line which represents the ludicrous situation of a fighter with 30 attacks, we'll come to that later.

The obvious (and most important) thing to notice from the plot is the odd one out – the fighter with A1 (the black curve). This fighter has exactly the same chance of getting a result of zero as she does a result of five (or any of the other possible values), i.e. 0.167. This is terrible! The distribution is uniform which leads to huge uncertainty in the outcome. This is fine is you like flying by the seat of your pants and scoring decent victories just as often as crushing defeats, but most players prefer a little more stability when trying to stack the odds in their favour.

All the other curves belong to the same family – a nice bell shaped (ish) curve, which is really what we want. Just one additional attack over the A1 case gives the red curve, and not only a good increase in the median (an increase of almost +1 to the median score, roughly equivalent to +1WS) but a serious improvement in reliability.

The only major difference between A2 and higher is the skewness of the curve. The red A2 curve is quite lop-sided and drops off quickly after peaking at 5, but as A goes up the curves become more symmetrical around their mean at 5.

Still, you can see that most of the benefit of extra attacks is gained by going from A1 to A2, and by the time we are at A3 or A4 we have gained most of the benefit we can hope to gain. The median stabilizes at 5, and the shape stays roughly the same. Going to higher attacks values, i.e. A5 and more, doesn't really do much.

In fact, uncertainty is minimized when A4, and then starts to increase again for A5 and higher (you can show this by taking the standard deviation divided by the mean for each of the distributions, also known as the coefficient of variation, and showing it is smallest for A4). An illustrative example of this is shown by the A30 curve. While the curve becomes more symmetrical and has the same median at 5, we have re-opened the door to uncertainty as both extreme wins and loses occur more often. For most fighters, winning the combat by a 'reliable' +2 or so is preferable to an equal chance of winning by +6 or losing by -4. We trade overkill for certainty. It's a good trade to make.

Note that if just looking at the expectation, in this case approximated by the median value, we can't actually tell the difference between A3 or A30. We really have to dig into the distributions themselves to figure this out.


The results are shown as a cumulative distribution function (CDF) in the figure below.

CombatProbabilities_CdfTRUE_ChainFALSE_FeintFALSE.png


This is one minus the cumulative sums of the y-axis values in the previous plot for each curve, and is just another way of telling the same story.

When interpreting this plot, trace each curve from the left hand side to the right - the longer the curve can stay close to the 1.0 line at the top, the better the situation for the fighter.

For example, reading up from an attack score of 4, we see that a fighter with A1 can expect to get a 4 or more 50% of the time, whereas a figher with A2 gets a 4 or more 70% of the time, and A3 gets it 78% of the time. A4+ does do that much better than A3, and we see that the biggest jump was really when going from A1 to A2.

In summary, we have two rules of thumb from this:
  1. When factoring in uncertainty, the jump from A1 to A2+ is critical.
  2. A4 is the optimal number of attacks, but A3 is almost just as good, and A2 isn't that much worse either.
 
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spafe

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This is some impressive math... however I have one eeny weeny teenie tiny thing... Aren't you better off taking fighters with high T and W to survive and raise the alarm, rather than worry about winning your combat?
 
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enyoss

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This is some impressive math... however I have one eeny weeny teenie tiny thing... Aren't you better off taking fighters with high T and W to survive and raise the alarm, rather than worry about winning your combat?
Absolutely! There is a very strong emphasis on 'all else being equal' in the above analysis (which in this case would include T and W). Mainly because it serves as a simple starting point we can branch out from.

There are lots of other things which could be considered though.

For example, if you have very high T and very low S, you might actually want those extreme win/loss situations to crop up as if you lose by a lot you can still survive, but unless you win by a lot you're not going to cause any appreciable damage. Or as you quite rightly point out, sometimes there are other factors which are much more important to the job in hand (i.e. the job of simply surviving in the Raid example).

My hope is to formulate look-ups for strategies to use in such situations along with some mathematical reasoning, although I that's probably a bit fanciful.

I guess the Raid motivation was just what got me thinking about it all, rather than being the best application. After all this work my opponent went and picked an Ambush anyway! I have some thoughts on that I'll post up soon...
 

enyoss

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Feints and Chains

Just for completeness, I'm dropping in plots of the probability distributions shown above adjusted for situations where (a) the fighter is armed with a chain, (b) the fighter's opponent has the Feint combat skill. These two situations cropped up earlier when discussing the expectation view of things.

The first two plots are for the Chain situation (probability distribution function PDF, and cumulative distribution function CDF), and the second two are for the Feint situation (PDF and CDF again).

CombatProbabilities_CdfFALSE_ChainTRUE_FeintFALSE.png


CombatProbabilities_CdfTRUE_ChainTRUE_FeintFALSE.png


CombatProbabilities_CdfFALSE_ChainFALSE_FeintTRUE.png


CombatProbabilities_CdfTRUE_ChainFALSE_FeintTRUE.png


I'm not going to go into these in much detail, but the following observations might be useful:

(a) When fighting with a chain, your combat effectiveness peaks at A4. Looking at the CDF plot (second from the top), this is because the A4 curve performs best when trying to maximise the probability of exceeding a losing score (i.e. getting 3 or more), while also giving one of the highest probabilities of achieving a winning score (i.e. getting 5 or more).

(b) Applying the same logic to the Feint CDF (bottom figure), we see that combat effectiveness peaks at A3. Also, looking at the Feint PDF (second from bottom) we can really see the drift in the median/mean as the peak of the curve shifts to lower values as the number of attacks increases.

Two rules of thumb come out of this:
  1. Though shalt not exceed A4 when fighting with a chain
  2. Though shalt not exceed A3 when fighting against someone with Feint
 

enyoss

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A winning assumption

From here on (well, for the next couple of bits anyway) I'm going to make a fundamental assumption – ideally, fighters want to maximize their probability of winning a fight, rather than the probability of winning by a lot.

That is, if we had two choices:

a) Win combat 70% of the time, and always score 1 hit when we do win.
b) Win combat 40% of the time, and always score 2 hits when we do win.

… we would rather have (a) than (b).

Obviously there is some trade off here, and exceptions do exist – for example, if the percentage in (b) was 69%, that would obviously be the better choice. In practice though, such nice clear cut counter examples are not so easy to come by.

The main reasoning behind this assumption is that in the unforgiving world of Necromundan close combat, where even the slightest loss can quickly escalate into a curb-stomping roll on the injury table, it is almost always safest (and wisest) to score a single furtive blow which bounces off harmlessly as long as you live to fight another day.

Again, exceptions do exist. One example might be a poor fighter with high T and high W, who can maybe shrug off the increased chance of losing while waiting for that big win. Such a poor yet resilient fighter might be employed to simply hold up the enemy, but the questions still remains – if his purpose is to soak up damage rather than deal it out, wouldn't you rather he soaked it up slower at the expense of getting those occasional massive wins which you didn't even really need anyway?

Poor fighters are poor fighters, and trading your chance of winning in the vein attempt to make up for it with the occasional big win won't make them much better. Good fighters would rather just win as often as possible. So in both cases our assumption is a reasonably good strategy.

So as a general rule, this is the assumption we will make...

Assumption: our fighter's optimal strategy is to maximize the probability of winning a combat, irrespective of how much he wins by.

Now we have made this assumption we can come up with a corresponding optimal strategy for when to parry, and unlock the secrets of one of the most powerful abilities available to a fighter...
 
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enyoss

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Ambushes. Bloody Ambushes

Ambush, my faithful Ambush. Why hast thou foresaken me? You made an excellent slave, but a cruel master.

So, I got Ambushed. Again. This time I had the lower gang rating and was rubbing my hands in glee at the prospect of Ambushing my hapless* opponent, and then it all went wrong. May all Ratskin scouts and drinking holes rot in hell.
[* I say hapless - he is actually a really nice guy, and knows what he's doing, so although there is one hapless player at the table I don't think it is him].

Predictably, I lost. Which begs the question – how predictable was this? Ambushes have been a dominant feature in both the campaigns I'm playing in right now, so it's about time I looked into this a bit more.

Ambushes are a strange creature. The defender places a few clueless gang members in the centre of the board, and the attacker then deploys their entire gang hiding and in cover around them. The defender then splits their remaining gang into groups of two or more models, and rolls a D6 for each in turn. On a 1-5 the group sets up within 4” of a friendly model already on the table. But on a 6, the group can be deployed anywhere on the battlefield. I'll call this a surprise deployment from here on.

Once all this is done, the defender rolls a D6 for each group which managed to get a surprise deployment, and on a total of 6 or more he gets the first turn. Any other result and the attacker goes first.

In 20 years of Necromunda, I never played an Ambush until 4 months ago. The whole ambushed becoming ambushers thing was a tricky beast to figure out, so no one ever risked it. And it's heartening to read from one very popular (and long!) campaign blog here on t'internet, that I'm not the only one who felt this way.

So I started to think about the best way to defend against an Ambush, and how, off the bat, I could roughly figure out my chances of winning if my opponent adopted this best defence.

Firstly, we have to think about those surprise deployments.

In my view they are a red herring. Well, not quite a red herring, but a conduit to something far more important.

They are a red herring, in that defenders often think the point of surprise deployments is about getting great positioning on the attacker. In my view, this is a secondary purpose. Sure, it's pretty important, but it's not their crucial purpose.

The crucial purpose of the surprise deployments is the boost they give the defender in stealing the first turn. As the defender, it really doesn't matter where you fighters are if you have a whole turn to run and hide them into cover before the attacker gets to shoot. Once you have done this the game (almost) devolves into a gang fight, and the attacker loses almost all of their advantage. I say almost, as the attacker still has his models all hidden, but then again the defender can also do this if they get the first turn, so everything evens out in the end.

I've played a few Ambushes now, and I never get stressed if I'm attacking and the defender gets the drop on me with one surprise deployment. As long as it's just one group, even if their positioning is great I just scoot out of the way in the first turn and still take down the chumps hanging out in the middle of the table. Two groups or more though? Just how worried should I be?

Firstly, I'm going to state, boldly, the optimal strategy for the defender:

Ambush defenders: split your gang into as many groups as possible

This is the only way to optimize your chance of getting the first turn. Sure, you could plonk all of your gang into a single group and hope for one huge surprise deployment, but it's still not likely you'll get the first turn, so it probably won't do you much good.

Given this, we can actually be more specific with this commandment:

Ambush defenders: thou shalt split thy gang into groups of two. If thy gang has an odd number of gangers, thy first deployment group shalt have three.

Placing the spare gang member (i.e. there should be only one if you followed the commandment) into the first group you get to deploy (ignoring the two poor sods you placed in the middle of the table to begin with) is a good idea because even if you don't get to surprise deploy them anywhere you like, you can spread them out more to improve your options when deploying your next groups.

So, as a defender we now know how many groups we'll get to deploy, so let's look at how many of these we expect to be surprise deployments. This is shown in the probability distributions in Figure (A). What does this figure show? Well, for various gang sizes, if you follow the rule above it tells us the probability of getting a certain number of surprise deployments. For example, if I have a defending gang with 9 members, this figure tells me there is a 58% chance I will get no surprise deployments at all, a 35% chance I will get one, a 7% chance I will get two, and almost zero chance I will get more than that. Each colour represents a different defender gang size.

AmbushA.png


For reference, it's also useful to express these as cumulative distribution functions. Figure (B) tells us the probability of getting at least a certain number of surprise deployments. This is useful because often we don't care if we get 1, 2, or 3, surprise deployments – we might only care whether we get 1 or more. For example, if I have a defending gang with 9 members, the chance of getting at least one surprise deployment is around 42% (note: this is 1 minus the probability of getting zero deployments in figure (A), i.e. 100% – 58% = 42%).

AmbushB.png


But, really, it's not the number of surprise deployments we care about. As defender we really care about stealing the first turn.

So, let's say I'm the defender. I have finished deploying, and I got two surprise deployments (lucky me). How likely am I to get the first turn (remember, for each of my surprise deployments I roll a D6, and if the total if 6 or more I go first). This chance of getting the first turn, given a number of surprise deployments, is shown in Figure (C) (Note: this figure doesn't care about gang size, just how many actually surprise deployments you actually got):

AmbushC.png


So, given a number of surprise deployments I know how likely the defender is to get the first turn. Figure (C) tells me that. And given a certain gang size, assuming the defender follows our golden rule above I know how likely he is to get each number of surprise deployments. Figure (A) shows me that.

So... for each coloured curve in Figure (A), if I take the percent chance for each value 0-9 on the x-axis, and I multiply it by the corresponding percent chances for 0-9 on the x-axis in Figure (C), and them sum all those products, that will tell me the overall probability that a gang of size X will get the first turn.

This is shown in Figure (D), where I've coloured the points to show which of the curves in Figure (A) (i.e. which of the defender gang sizes) they correspond to. This defender gang size information is also shown on the x-axis in Figure (D) – redundancy might be the mistress of idiocy, but indulge me.

AmbushD.png


So, for my example defender with gang size 9, when I pick an Ambush scenario as attacker I now know there is a 11% chance my opponent will get the first turn. Taking into account how crucial, nay brutal, getting the first turn is an Ambush, I'm going to be even more bold and say that this represents an 11-20% chance of my opponent winning. Which means I have at worst an 80% chance of winning the game should I pick this scenario as the attacker. This is probably the best chance you will ever get in a game of Necromunda.

Admittedly, the 11-20% estimate above is something of a brown number (although the underlying statistics are concrete), but in my anecdotal experience it's not too far from the truth.

In any case, and I'm going to bold this to make it clearer:

Next time you play an Ambush and want a rough idea of how likely you are to win, look it up in this plot.

It might just save your skin when choosing those scenarios...

Now for a couple of observations about Figure (D). The main thing is that the increase in the chance that the defender takes the first turn is linear with respect to the defender's gang size. That is, it's a straight line - if you increase your gang size by 2 models into the next bracket, you add 5% to your chance you getting the first turn, and this holds whether you go from 8-10 models, or from 18-20 models.

As a mathematician, I find this remarkable. This is very unusual. For probabilities involving several dice rolls, it's freakily unusual. I like it. I'd like to say it is a stroke of genius on behalf of the game designers, although I'm not sure they were aware of it (then again, those guys really saw the matrix with this stuff, so maybe they were).

What it means is that this scenario scales really well with defender gang size. A reasonable change in the chance of getting a surprise deployment (e.g. say the defender gets a surprise deployment for each group if they roll a 4+), or the threshold on the total the defender needs to steal the first turn, can easily break this scaling. This could mean the scenario becomes exponentially easier, or harder, for the defender, as their gang size increases (well, not exponentially, but you get the point). But no, it's linear, which is nice.

What it also means is that if gaming groups think the Ambush scenario is too lop-sided, it's quite easy to recalibrate. You just have to decide on how often you would like a particular gang size to win (e.g. “currently defending gangs of size 8 are winning 11-20% of the time, we would like this to actually be 25-30%”), and it's then possible to adjust the chance of getting a surprise deployment and/or the threshold for the defender getting the first turn to tune the scenario to this desired win/lose probability. This can largely do away with incremental playtesting for rebalancing.

Anyway, that's all I have to say on Ambushes (which, looking back on it, was quite a lot actually). Apart from the fact that I should have won my last one, but didn't. And now I had a gang member die from it, that 22% chance of winning was probably the best I'll see for a long time...
 
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catweazle77

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Very interesting read. One question though: in figure C it looks like there is a 100% chance of getting the first turn if you have 4 surprise deployments, although I'd say you need 6 surprise deployments to get to 100%. Did I Miss something?

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spafe

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Having a little zoom in on the graph... I think its just really really high (like 99% or something), as you can just (just!) see a little bit of the dashed line marking the 100% line above the graph line from 4 to 5 dice (meaning the 5 to 6 is also probably ever so slightly below 100%, but the odds of rolling 5 1's are tiny)
 

enyoss

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Yep, spafe has the right of it.

The probabilities in that plot are:
[number of surprise drops, probability of first turn]
0, 0.000
1, 0.167
2, 0.722
3, 0.954
4, 0.996
5, 1.000 (rounded)
6+, 1.000 (exact)

As spafe also points out, the only way of not getting the first turn with 5 surprise drops is if you roll 5 ones, and the chance of that happening is 1 / 6^5 = 1 / 7776 = 0.00013 (or 0.013%), so in practice you're almost guaranteed the first turn.
 

enyoss

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Thanks guys :). The natural bore in me is quite fired up now, and have quite a bit of stuff left which I've been thinking about, so I'm going to keep going for at least a short while longer.

Knowing my luck, after 20 years of silence GW's renewed interest will probably revamp the rules next week *sob*.
 
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enyoss

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Toughness or Wounds? Back to the Raid...

So, a few posts back @spafe made a very salient point. To paraphrase:

“Why bother doing calculations as to which of WS or A upgrades are best for sentries in a Raid, when we are probably more interested in them having T or W advances”.

The answer was clearly obvious – I did it for fun. Plus, as I have hopefully shown above, and will definitely show in a short while when we look at parries, it's also useful when looking at combat in general, not just in Raids. Plus it was fun, remember?

But @spafe's impertinent questioning raises an important and satisfyingly distracting side question: if I then have fighters with T or W advances, which of these make the best sentries in a Raid. Or, more succinctly, which is the better advance, +1W or +1T?

Most people already have a gut instinct answer to this one, and if most people have answered what I think they have then most people are right, but I'm going to go ahead and prove it anyway.

The key (and pretty obvious) thing about toughness (T) and wounds (W) is that they improve your fighters' survivability. The way I'm going to look at this is as follows:

Given a fighter takes a number of hits in a game (hits which would then need to roll to wound), what is the probability that my fighter will survive the game without being reduced to zero wounds.

In short, what's my fighter's chance of surviving the game without having to roll to see if they suffer a flesh wound/down/out of action (I'll just refer to this as the chance of surviving the game without going down from now on).

First, an example. I'm going to assume all these hits are strength 3. It doesn't matter where the hits come from, whether shooting, falling or from hand-to-hand combat. I'm then going to look at a few cases. The first cases are fighters with T3 or T4. On top of that, we'll also give these fighters 1, 2, or 3 wounds to squander. All this is shown in the following figure, where W1/W2/W3 cases are denoted by the black/red/green lines, and the diamonds/triangles denote the T3/T4 cases.

ToughnessWounds_Example.png


How to interpret the plot?

Let's say our noble fighter takes 2 hits in the game (S3 hits for our purposes here). Tracing a line up from 2 on the x-axis and seeing where it crosses each of the curves, we see that if we take this number of hits our fighter would have a 25% chance of surviving the game without going down if she has T3 and W1 (black diamonds), a 45% chance of surviving with T4 and W1 (black triangles), a 75% chance of surviving with T3 and W2 (red diamonds), and around a 90% chance of surviving with T4 and T2 (red triangles). Her chance of surviving two hits without going down if she has W3 is 100% no matter what her toughness is - the worst case scenario is that each hit causes a wound, but then she would still have one left and lives to fight another day.

So, to find the advantage of having +1W versus +1T, for each “Number of times hit in battle” value we can simply trace up to a given reference curve (it can be any of them), and see how jumping to either the equivalent curve with +1T or +1W will improve our survivability.

In almost all cases, we see that +1W leads to a better survivability than +1T. There are just two exceptions.

The first exception is where the T4W1 curve crosses the T3W2 curve at around the (x, y) = (8, 5%) mark. That is, if your fighter takes more than 8 hits during the battle, they will actually have better survivability if they have T4 and W1 than if they have T3 and W2 (for the record, with this number of hits both give you a miserable chance of surviving anyway, less than 5% :().

The second, and probably more useful exception, is where the T4W2 curve crosses the T3W3 curve at around the (x, y) = (6, 35%) mark, and the interpretation is the same as the T4W1/T3W2 case.

Still, these are exceptions rather than the rule, which just so happens to be:

Rule of thumb: +1W is a better advance than +1T for fighter durability.

It's also handy to use this plot more generally.

For example, let's say I want to stick my W3 juve out as a bullet magnet to protect my heavy. And let's say I've done my homework, and I think that around 3 shots, each requiring a 4+ to wound, will be poured into his poor prone form (it could be S3 weapons versus a T3 juve, or T4 weapons versus a T4 juve, all that matters is the required to-wound roll). I'll also be conservative and assume that all the shots fired at him will hit (I'll deal with that assumption in just a moment).

What's the chance he will survive without going down?

The following figure gives these look-up values for all such instances. Here, the "Number of times hit in battle" is interpreted as "Number of times hit in the next turn", and "surviving game" in the title becomes "surviving next turn", but it doesn't really change anything.

So using our bullet-sponge juve example above, I read up from 3 hits on the x-axis, look at the 4+ curve for W3 (green diamonds), and see that my juve has a 88% chance of making it through with at least one wound remaining. Not a bad gamble!

ToughnessWounds_Reference.png


[Note that the black/red/green curves used in our first example above reappear in this plot as "4+" and "5+" as we assumed S3 versus T3 and T4 in that example.]

This is your worst case scenario – armour or beneficial to-hit modifiers (for you, not the people shooting at you) just improve this chance. For example, you can roughly factor in the to-hit modifier by adjusting the number of shots by the percent chance of success for hitting. In our example, if those brutes shooting my juve needed a 3+ to even hit him, I multiply the 3 by 2/3 (i.e. their 3+ to hit probability) to give me a new rough estimate of 2 times hit, which now means almost a 100% chance of surviving the fusillade with at least one wound remaining. A very good gamble!

I've put most of the curves anyone could possibly hope for on that plot, so you can look up your chance of surviving a turn in almost any situation (given a few simple assumptions). Just squint and concentrate on the curve you actually need when reading it off.

So, to answer @spafe's question properly:

Yes, wounds and toughness are crucial to sentry selection during a Raid. And all things being equal, fighters with W increases should always be selected in preference to fighters with T increases.


Anyway, that's enough of these distractions. Back to seeing how our fighters cope in combat...
 
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