Necromunda combat stats

enyoss

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Before going back to the stats, I'd like to point out that "impertinent questions" are the most interesting way to start thinking about new things to look at, so it's definitely a case of the more the merrier when it comes to them :p.
 

spafe

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But @spafe's impertinent questioning raises an important and satisfyingly distracting side question:
Woo yeah, go impertinent questions!

Next up... Why don't birds have a 3rd 'dorsal' wing so they can function like air fish? ... to far?

but seriously, that's a great write up, and thanks for answering it in such a detailed manner :). Looking forward to the combat stats :p
 

enyoss

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Come, let me celebrate the difference between us...

… before I stave you head in.

Hopefully this update will be briefer than the last one for a couple of reasons.

Firstly, the updates are getting a bit unwieldy.

Secondly, it's taking too long to write everything up, and my wife's patience is already at breaking point with how much time I devote to my spaceman gangs – I've created a diversion by inferring I'm engaging in an online affair, but she's bound to find out the truth sooner or later.

Anyway, it's about time we found our theoretical fighter a friend to fight against, and have a look at what comes out.

So far, we know that a fighter's score, treated in isolation, is best improved by the following:
  • Increasing attacks to A2
  • Increasing WS
  • Increasing attacks to A3+
… in that order.

But how does this survive contact with the enemy? Common sense tells us that maximising your score (while reducing the variance) should, in theory, improve your chances of winning against another fighter. So far so obvious. But what other insights drop out when doing this?

Firstly, to keep things simple, we'll consider our fighter (imaginatively called Fighter1) and his opponent (Fighter2) have the same number of attacks. They also have the same WS and Initiative, and neither has charged. That is, everything hangs on the result of the attack dice – he who scores highest, wins the combat. We'll spice things up and add that extra stuff back in later on.

In general we're rooting for Fighter1 by the way, so when we test out those extra ingredients it will always be to his advantage.

What we are really interested in is the difference between the two fighters' scores. A difference of +1 or more means that Fighter1 has the higher score and has won. A difference of -1 or less means that Fighter2 has the higher score and has won. A difference of zero means the combat is a draw (Boo! We want action damn you!).

For Fighter1 having attacks A1-A3, the following graph shows the probability of each of the possible differences, and basically tells us how many hits each fighter would get if they won. The probability of each fighter winning, and the probability of them getting a draw, is given in the relevant portions of the graph (separated by lines going up from differences of +1 and -1, the win thresholds for each fighter).

CombatScoreDifferences_CdfFALSE_Fighter2Attacks-Equal_EXAMPLE1.png


Interestingly, when Fighter1 faces his doppleganger with the same number of attacks, the probability of him winning the combat doesn't actually change much between A1 and A3. The biggest difference between the different curves is that our fighters are more likely to win or lose by a lot if they both have A1, compared to A2 or A3. This comes back to the whole “rolling just one dice has a massive variation in outcome” thing. We don't really want that - slow and steady wins the race.

Although I've put summaries of how often each Fighter wins on the figure, it would be nice to actually be able to calculate this visually. Short of getting your ruler out, measuring the height of the curve for all differences 1+, and adding them up, that's not so easy to do with just this figure.

So, being a nice kind of guy who likes to make life easy, the following plot shows exactly this information. I'll explain how to interpret it in a moment.

CombatScoreDifferences_CdfTRUE_Fighter2Attacks-Equal_EXAMPLE1.png


This is just another modified cumulative distribution function like the ones we looked at before. So, how to read this?

Basically, if you want to know the probability of Fighter1 winning the combat for a given number of attacks, just look up where the solid-circle curves cross the dashed line going up from +1 on the x-axis. E.g. tracing up the +1 dashed line, the line crosses the black solid-circle curve at 42%, just as we expected. To get the probability of Fighter2 winning, do exactly the same but tracing up the -1 dashed line and seeing where it crosses the dotted-triangle curves (at 42% again, how very boring).

It might look pretty trivial here as Fighter1 and Fighter2 are identical twins, but graphs like this last one can really come into their own when one fighter or the other gains an advantage. For example, here is how things change if Fighter2 has just A1, while Fighter1 can have A1-A3 like before.

CombatScoreDifferences_CdfTRUE_Fighter2Attacks-1_EXAMPLE1.png


But we already know that A2 is a great thing versus A1 with diminishing returns afterwards, so we'll postpone looking into that for the time being. Instead, we're going to start seasoning the combat with those other ingredients: WS, Initiative, parries, and combat skills.

No time for that here, though in the meantime I'll give you a hint by pointing to that big juicy spike in probability which is hanging out in no-man's land, just waiting to be claimed...
 
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Mamutera

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Hey, in the last figure I see that the curves for A2 and A3 have moved to the right but with almost no ¨ vertical ¨ displacement. Does that mean that the difference in attacks does´t affect that much the chances of Fighter 1 to win? It looks like having more attacks affects mostly in that Fighter 1 wins has more ´big wins¨ with a much better attack score difference but the probability of winning hash´t changed that much.
 
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enyoss

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Hey, in the last figure I see that the curves for A2 and A3 have moved to the right but with almost no ¨ vertical ¨ displacement. Does that mean that the difference in attacks does´t affect that much the chances of Fighter 1 to win? It looks like having more attacks affects mostly in that Fighter 1 wins has more ´big wins¨ with a much better attack score difference but the probability of winning hash´t changed that much.
Although it looks like the curve has just shifted to the right, this is not actually the case. It just looks like this because, apart from the Fighter1 A1 case, the individual points on the other solid curves have actually been displaced upwards.

We can convince ourselves of this by comparing the Fighter1 A2 solid red curve on the last graph for x=1 with the previous graph where x=0. If the curve were just shifted these would have the same value, but as it turns out they are slightly different - if we just shifted the graph we would expect a value of 58%, where we actually have 55% when x=1 on the lower graph.

This means the extra attacks for Fighter1 really do mean he wins more often. For example, when both Fighters have A1 we saw Fighter1 winning 42% of the time. But when Fighter1 has A3 and Fighter2 is stuck on A1, this goes up to 60% (this comes from seeing where the Orange solid curve crosses the dashed line coming up from x=1 on the last graph shown).

This actually has implications for assessing true power of WS versus A, which I was going to deal with in a moment, but now you've gone and ruined it. I hope you're proud of yourself :'(.

But yeah, when I get to assessing the power of WS bonuses I'll explain the previous graph properly (rather than just dumping it down and running off :)).
 
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enyoss

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Also, I forgot to say @Mamutera, you are absolutely right on the bit about big wins.

When Fighter2 is stuck with A1, not only does increasing Fighter1's attacks mean he will win more often, he's also more likely to win big. I suspect this is mostly down to how dire the A1 variability is for Fighter2 - when he rolls a 1 or a 2 he's screwed and screwed big, which is not that unlikely with just one dice. I'll look into it and let you know!
 

Fold

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Wow. It's fairly shocking how having treble the number of attacks of your opponent leads to what I see as a fairly marginal improvement in your chances to score a hit or two.
 
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enyoss

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Wow. It's fairly shocking how having treble the number of attacks of your opponent leads to what I see as a fairly marginal improvement in your chances to score a hit or two.
It's not completely negligible, but it's definitely an instance of diminishing returns.

This is why, in my view, once your Heavy rolls up an attacks advance of A2 you should both breath a sigh of relief, but also cross your fingers you don't waste an advance going any higher!

I actually didn't bother plotting A4 as it adds such little power over the A3 situation for Fighter1 it wasn't worth the pixels.

Edit: of course, that was supposed to read "cross your fingers" not "cross your gingers". Stupid autocorrect.
 
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TakUnderhand

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Great analysis :)
I've done a few curves on the "power of parry" before. But there always seems to be something more interesting to do than write it up :)

Looks like you may be covering it anyway :D
 
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enyoss

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A prelude to parrying

Great analysis :)
I've done a few curves on the "power of parry" before. But there always seems to be something more interesting to do than write it up :)

Looks like you may be covering it anyway :D
I suspected I wasn't the first person looking in this :) (I'm going to ignore your sly inslut on my social life and other pastimes... for now :)).

It might take me a while to get to parries, so in the meantime I'm going to post up three of the best strategies I've come across.

I've given them suitably pretentious names to distinguish them, the intention being that the name corresponds loosely to the type of fighter using it. Trademark applications are pending, naturally.

"Optimal", in the sense given in the strategies, means optimising our win rate as per the discussion in my previous post A winning assumption. Rules for each strategy are given in order of priority (e.g. for Orlock optimal, the always parry a 6 rule trumps the never parry if winning rule).

Two more things. Firstly, we always assume that you parry your opponent's highest dice, although feel free to fall on your sword and do otherwise if you like (why are you even reading this?). Secondly, if a strategy tells you to parry it is assuming you can actually do so, i.e. your opponents highest dice roll beats yours. No cheating please.

Here are the strategies, in order of increasing difficulty:

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

That was easy. Now onto something a bit more interesting

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


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


*** EDIT: New strategy added 06-Feb-2016 ***
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 a 3 or lower


There you go. Feel free to try them out and/or pick holes where they may exist (pretty sure it's airtight).

My recommendation for most people would be Orlock-optimal, although if you want to speed up your gameplay the Escher strategy is almost as good. Goliath-optimal, while being my favourite, is a bit trickier to employ quickly during the game at first, although with a little bit of practice it just becomes second nature.

I myself use Goliath-optimal... with my Escher. I like the gambling I do.

A full set of results showing how these improve the difference in combat score plots, similar to those shown above, will follow when I have time :).
 
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sebwiers

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I actually played a sword heavy Goliath gang, and pretty much used the Goliath parry strategy (I even had a couple guys with two swords). The whole idea was bingo, not to loose close combat, over killing. Being in close combat at all meant I'd win the game eventually, and taking out somebody I just charged was actually a bad thing.

And yeah, my leader with t4 / w2 / 4+ save was quite willing to suck up a few hits to give his guys a chance to get into position to use those (short range) weapons; as the graphs show, the risk was pretty low when the bulk of enemy weapons were str 3. He almost never actually got to charge in with his sword / chainsword combo, but he sponged up bullets because $%@# what a threat. He also spent a lot of time plinking around with his bolt pistol for multiple turns at long range, enough that I actually regretted switching to a plasma pistol.

THB though, it was my mass of str 4 shooting weapons and +2 to hit with pistols that won battles for me. My ammo rolls stunk, but it didn't matter because by the time I was out, I either had pinned / downed much of the enemy, or was in range to charge, or both.
 

enyoss

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THB though, it was my mass of str 4 shooting weapons and +2 to hit with pistols that won battles for me. My ammo rolls stunk, but it didn't matter because by the time I was out, I either had pinned / downed much of the enemy, or was in range to charge, or both.
There is something immensely satisfying about the inexorable march of a Goliath gang backed up by the roar of bolt pistols.

You're absolutely right though regarding close quarters shooting. I actually think the most successful close combat gangs are the ones who embrace being a shooting gang who can dare to get close enough that they get great modifiers to hit. What's the worst that can happen - they'll charge you? Oh, please.

:D
 
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Eldarin

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Should you not decide based upon your likelihood of bettering your result? Reroll if you are equal or better in terms of attacks, combat score modifiers and parry
 

enyoss

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Well, it all depends on how your are interpreting 'bettering your result'.

...

I actually wrote a long and master crafted response to your message which would have convinced you completely that these strategies do precisely what you're suggesting.

But my stupid iPhone logged me out and when I came back that first sentence was all that remained (seriously, does anyone else know why the iPhone 6 won't save passwords even if I tell it to - it's been driving me nuts for months and this is the last straw. I'm fuming... that message really was mastercrafted ).

If I have time tomorrow I'll try and reproduce it. Using my Linux PC. Yes Apple, it's a PC not a Mac, so although I don't skateboard down escalators and wear a fedora at least it bloody works. Rant over.
 
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TakUnderhand

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Maybe, just maybe, it's not Apple's fault this time...

A while back, I had an issue on this site with certain key words giving a 404 forbidden error. That word beginning with G, ending in amble, seemed to do it.
Now I believe malo fixed it. But it could have got broken again.
Or perhaps you hit a similar word, still on the blacklist?
 
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enyoss

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Thanks for the pointer @TakUnderhand.

Unfortunately though it's not just this site either. It automatically logs me out of every website after a few minutes (although it does store my username and password for rentry here on yaktribe - hoorah for small mercies!).

Not usually a problem with this thread as I have it all written up in a text document offline, but I just got a bit cavalier this time and paid the price.

Interesting to hear of a word blacklist. I wonder how much and what kind of unimaginable filth Malo had to read to ensure its comprehensiveness?!
 
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Eldarin

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By better your result I just meant less than 50% chance of losing - I've got a spreadsheet somewhere showing the win/draw/loss of different variations of +1 combat score, 2A and Parry for/against that is quite useful.

Though thinking about it, if you've managed a draw from a losing position you wouldn't be able to parry and so your rules would still work
 
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enyoss

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By better your result I just meant less than 50% chance of losing - I've got a spreadsheet somewhere showing the win/draw/loss of different variations of +1 combat score, 2A and Parry for/against that is quite useful.

Though thinking about it, if you've managed a draw from a losing position you wouldn't be able to parry and so your rules would still work
Thanks for the clarification :). Not still not entirely sure I get what you mean – I'm guessing you mean it's best to take the parry when you have an odds on (>50%) chance of improving your current situation?

I agree this is a good strategy to take, but again, it depends on how you define “improve your current situation” (and of course, it all depends on my selective interpretation of your question :D).

As discussed in my previous (and pompously titled) post A winning assumption, I think the outcome you should optimize first is just straight up whether you are winning or losing - I'll call this Priority I. A secondary concern is to maximise the amount you win by, and minimise the amount you lose by, but only if you have done everything you can to try and win – I'll call this Priority II.

The strategies I gave above actually do both of these. Here's how:

1) Let's first focus on Priority I.

Imagine a situation where the dice are rolled, and you have already won combat, but the dice scores are such that you could parry if you wanted. Should you? According to satisfying Priority I, the answer is no.

Why? Well, as things stand you have an expected 100% chance of winning combat (you already won, right). There is literally no way you can improve your situation. Forcing the parry, while perhaps letting you win by more, will in almost all circumstances open up the possibility that you might lose. So your expected chance of winning has to be <100% if you take the parry. On average, you would guarantee making your situation worse in terms of our primary concern – the simple business of not losing combat.

Similar reasoning extends to the case when you have already lost. Your current situation has a 100% expected chance of losing, but if you parry there will probably be at least one outcome, no matter how improbable, where you will win. So if you parry your chance of losing is <100%. Your expected outcome is better than your current situation, so you take the parry.

Draws? Well, they could go either way... leave them as they are for now.

This is precisely the reasoning behind the Orlock-optimal strategy, so named as it embodies this optimization perfectly.


2) Let's now ignore Priority I, and just focus on Priority II. We are now gamblers.

Here, we define improving our situation as winning by more, or losing by less. This then becomes a simple question: if we force the parry, do we expect our opponent's score to be, on average, higher or lower than it currently is. Note: the important detail here is the word score rather than roll. They might re-roll low, but one of their other dice might then step in to become the highest roll, so we have to look at what's going on with our opponent's other dice as well as the one we are re-rolling.

This is, broadly speaking, embodied by the Escher strategy. That strategy isn't optimal for the purposes of Priority II, but it's as close as it comes in practical purposes that it makes almost no difference.

For completeness I'll point out that the optimal strategy for Priority II, Escher-optimal, is actually the following generalization of bits of the Goliath-optimal strategy:

- Always parry an opponent's 6
- Always parry a drawn an opponent's 5
- Parry
a drawn 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

I'm not going to go into the exhaustive reasoning behind this (although I'm happy to do so, if anyone is interested) but the principle is simple:

Always parry if your expected outcome, in terms of combat score difference, is higher than your current situation.

I'm guessing this is exactly what you were looking for @Eldarin?


3) Now let's blend Priority I and Priority II.

First, we try and optimize Priority I when possible. So we follow the Orlock-optimal strategy.

Second, as a fall back, if we have done as much as we can to satisfy Priority I, we optimize Priority II. In practical purposes, this means we try and maximise our combat score difference whenever Priority I shrugs it's shoulders and becomes indifferent – i.e. whenever we have a draw.

This is precisely what the Goliath-optimal strategy goes. How ironic that the Goliath strategy should embody the elegant harmony of brains and brawn, but I digress.


4) As for +1WS, +1I, +1A and their impact?

Well, these (along with charging bonuses, or whatever other combat score modifiers you want to look at) are already built into the curves shown earlier. They just modify where you draw the dashed lines coming up from the x-axis! In my next update I'm going to discuss how to read the curves properly to do this.

Parrys are not included on those curves for one reason – we have to know which strategy we are using before we can measure its impact on combat outcomes.


In summary:

My logic was thus – if I can deduce the optimal parrying strategies, then we can see how those affect the curves in my previous posts. Then, by induction, any other strategies will give worse results. So by just looking at 3 or so parrying strategies we can understand the impact of all parrying strategies (and there are a lot of them – I've been testing 20 or so, and that's not exhaustive).

Anyway, I hope that actually answered your question @Eldarin :D.