Statistical analysis 1: spears vs hw+sh (long post)

Lakissov · Post by **Lakissov** » Tue Jun 20, 2006 5:25 pm

Statistical analysis 1: warriors with spears versus hand weapon plus shield

The question has already been discussed slightly here at druchii.net. In that thread the participant of the discussion were also using statistics. What I view as a disadvantage of that discussion is the fact that only the averages were considered. To me, just the average value of wounds inflicted is not enough, as the deviation from that value can be substantial.

I pursue in this article a different approach. Instead of calculating the expected value of the wounds inflicted, calculate the probabilities that a certain amount of wounds will be inflicted. E.g. if we talk about a frontage of four dark elf warriors charging, then I calculate the probability that it will inflict 0 wounds, then the probability that they will inflict 1 wound, then same for 2 wounds etc.

However, just how many wounds will be inflicted is not yet saying us much. What we really want to know is the probabilities for different values of CR resulting from the wounds inflicted. This will be the subject of this article.

In fact, I will not be doing those probability calculations for all possible values of CR. What I am really interested in is intervals. The first interval is victory for dark elves (meaning that the net result from kills is improving my CR). The second interval is a 1 or 2 CR loss to me. The fact is that I will field the troops in a few ranks plus a banner. This gives me +4 CR (I do not count outnumbering), so if my warriors lose kills CR by 1 or 2, they are still a good addition to the combat as their 3-4 CR from ranks and standard still outweighs the loss in kills and makes my victory more likely. Then I am interested in the probability of loss by three, because it is still a net gain if I have the full ranks before the fight and a standard. Lastly, I need to know the probability of loss by 4 or more, as if it is too big, then the unit should not enter combat at all.

How to proceed then? There is not so much of a point in calculating these probabilities for all possible opponents. Instead I calculate the base case against opponent which is slightly worse than my elves – WS 4, S3, T3, AS 5+, A1, no special abilities, and a front of four models (both elves and opponent). Then I see how the change in some of the individual characteristics influences these statistics. Thus I see the sensitivity of these probabilities to different characteristics.

The raw data resulting from calculations is here: http://img205.imageshack.us/img205/7281/rawdata1mu.jpg

One can see that here I calculate the probabilities for three cases: DE are charged, DE charge themselves, nobody charges. In each of the cases I calculate the probabilities of CR being in certain intervals for both variants: spear plus shield and hand weapon plus shield.

Now we can make the next step and actually start comparing. I calculate the difference between spears and hand weapon + shield for each interval and then make a graphic representation of them (as it is easier to deal with).

Here are the data and the graphical representations for different values of WS:
http://img156.imageshack.us/img156/2672 ... ers5wj.jpg
http://img62.imageshack.us/img62/3914/ws19fp.jpg
http://img79.imageshack.us/img79/6599/ws28zm.jpg
http://img208.imageshack.us/img208/4231/ws37tu.jpg

We can see that in the first and third chart we have positive value of difference for winning and negative for losing. That means that spears have a bigger chance to win the combat and a smaller chance to lose it (meaning at any losing interval). On the charge, indeed, the spears are worse than hand weapon and shield, but the difference there is much smaller than the difference in the second round (so it might still make sense to charge with spears in order to enjoy the spear benefit during the second round of combat). What we can also see is that the abovementioned superiority of spears decreases with the increase of opponent’s WS, but the difference is not very big, so opponent’s WS should not be so important for us (though combined with some other factors it might still influence our choice).

Now let’s see the sensitivity to Strength:
http://img116.imageshack.us/img116/6463/snumbers2gp.jpg
http://img100.imageshack.us/img100/9251/s14sa.jpg
http://img133.imageshack.us/img133/8422/s29vd.jpg
http://img147.imageshack.us/img147/6097/s30dy.jpg

Here is a contradiction to what was pointed out in the previous discussion on the topic. The point was that higher strength will make spears less beneficial, but we can see that it is not really so. Strength 4 and 5 make the shield plus hand weapon combo more likely to result in a worse loss while making the chance of victory more likely. Strength 6 makes the use of hand weapon plus shield much worse than spears. Why is this so? Well, higher strength has two effects: it is more likely to inflict wounds and it is also more likely to kill. Hand weapon plus shield make armor save better, and thus help to fight the first effect. The bigger number of attacks (two ranks from spears) helps to reduce the number of high strength opponents. What does this mean? It means that the sensitivity of our choice to opponent’s strength is itself affected by opponent’s toughness and armor save (the bigger toughness and armor save, the less useful big number of attacks). We shall check that guess later – after we have looked at sensitivity of our choice to other stats.

Sensitivity to toughness:
http://img72.imageshack.us/img72/3433/tnumbers7ca.jpg
http://img83.imageshack.us/img83/8595/t11ch.jpg
http://img72.imageshack.us/img72/4967/t22yn.jpg
http://img72.imageshack.us/img72/4524/t34ys.jpg
Toughness seems to have a very serious effect on the difference between spears and hand weapon plus shield combination.

Sensitivity to armor save:
http://img212.imageshack.us/img212/4672 ... ers1qn.jpg
http://img91.imageshack.us/img91/5192/as19pu.jpg
http://img132.imageshack.us/img132/1280/as29zf.jpg
http://img91.imageshack.us/img91/4559/as31al.jpg
Here we have a very similar picture to toughness. Well, it is not surprising in fact. Now one more interesting thing in these graphs is that with higher toughness/better armor save the hand weapon plus shield combo is much more concentrated towards a draw than spears. But as we discussed before, having a draw means a victory for us (the participation of warriors in the battle is justified as they bring static CR and do not lose out in kills). This means that against high toughness/good armor save we will most likely not take spears. As said before, the influence of toughness and armor save on the choice is very big, as it influences the probability distributions a lot (that was the initial purpose of making the sensitivity analysis – to see what has more influence and what has less).

Sensitivity to opponent’s number of attacks:
http://img223.imageshack.us/img223/2327/anumbers1pu.jpg
http://img83.imageshack.us/img83/783/a18mm.jpg
http://img81.imageshack.us/img81/8006/a25xe.jpg
http://img156.imageshack.us/img156/838/a30zl.jpg
Here I would say that when being charged by an opponent with many attacks, we are better off with hand weapon and shield. The advantage of spears over shields in the second round of combat is increased by a bigger number of attacks (though their disadvantage when charging themselves is also increased). Thus, I would suspect that when we charge against high toughness (or good armor save) opponents with many attacks we are better off taking hand weapons and shields, while against low toughness (or bad armor save) opponents with many attacks we take spears on the charge. The effect of many attacks seems to be similar to the effect of strength.

Finally, the influence of frontage (issue raised in the already mentioned previous discussion):
http://img157.imageshack.us/img157/5158/fnumbers4iy.jpg
http://img157.imageshack.us/img157/9484/f13qd.jpg
http://img118.imageshack.us/img118/4346/f20gk.jpg
http://img118.imageshack.us/img118/227/f35go.jpg
Expanding frontage slightly biases us towards spears, but only very slightly.

During the sensitivity analysis a hypothesis was brought forward:
Both high strength and increased number of attacks amplify the effect of toughness and armor save on the difference between spears and hand weapon plus shield combination.

Let us check this hypothesis by altering T and AS at fixed S5, and then doing the same thing at A2.

So here are the toughness sensitivity graphs at S5:
http://img216.imageshack.us/img216/4620 ... ers1yr.jpg
http://img68.imageshack.us/img68/1430/s5t18wh.jpg
http://img63.imageshack.us/img63/1605/s5t29rj.jpg
http://img216.imageshack.us/img216/5620/s5t39cm.jpg

Armor Save sensitivity at S5:
http://img63.imageshack.us/img63/2959/s ... ers4oz.jpg
http://img62.imageshack.us/img62/5434/s5as13rj.jpg
http://img61.imageshack.us/img61/449/s5as27xv.jpg
http://img109.imageshack.us/img109/9963/s5as32uz.jpg

Toughness Sensitivity at A2:
http://img153.imageshack.us/img153/7291 ... ers8fn.jpg
http://img92.imageshack.us/img92/319/a2as11gp.jpg
http://img76.imageshack.us/img76/122/a2as20ce.jpg
http://img117.imageshack.us/img117/2272/a2as32fw.jpg

Armor Sensitivity at A2:
http://img153.imageshack.us/img153/2925 ... ers2nn.jpg
http://img75.imageshack.us/img75/6172/a2as14ac.jpg
http://img148.imageshack.us/img148/888/a2as28of.jpg
http://img148.imageshack.us/img148/9603/a2as39cv.jpg

Well, if you ask me, the graphs show clearly that the hypothesis was correct. In sum, we can conclude the statistical analysis we made in the following way:

1. The higher the opponent’s weapon skill, the more beneficial it is to choose hand weapon and shield; however, the influence of weapon skill is rather small
2. The higher the opponent’s toughness, the more beneficial it is to choose hand weapon and shield; toughness has a very strong effect on the difference between the two equipment variants, so should influence our choice a lot
3. The better the opponent’s armor save, the more beneficial it is to choose hand weapon and shield; armor save has a very strong effect on the difference between the two equipment variants, so should influence our choice a lot
4. Both the opponent’s strength and the opponent’s number of attacks per model make the effect of toughness as well as the effect of armor save stronger, so against opponents with high S or A toughness and armor save should have even more weight for making the choice of equipment
5. The wider the frontage we face, the more beneficial it is to take spears
6. The abovementioned considerations are only applicable when we are charged by the enemy or when we expect to stay in combat for more than one round; if we expect to break the opponent during our charge, we should always take hand weapon and shield

Also based on the graphs, I would make a set of rules of thumb which more or less define the border between spears and hand weapon plus shield combination:
S3-4, AS5+ T3 – spears
S4, AS4+ or T4 – flip a coin
S5 or A2 and AS4+ or T4 – hand weapon and shield
S3, AS4+ and T4 – hand weapon and shield

Well, I hope I did not bore you all here too much with my graph-heavy article…

PS: I wonder why it is not possible to simply attach a rar-archive to the post in this forum, with all the pictures...

Waerik · Post by **Waerik** » Tue Jun 20, 2006 6:04 pm

A nice read...

It could also be noted that your analysis on the strength sensitivity of armour also aplies to corsairs, e.g. corsairs will die more (points wise) vs shooting that reduces armour by two or more...

Sleekdd · Post by **Sleekdd** » Tue Jun 20, 2006 6:19 pm

Lakissov wrote:PS: I wonder why it is not possible to simply attach a rar-archive to the post in this forum, with all the pictures...

Giving a link to the excell-file would help a lot too.

Anyway, the calculations you made seem to be based on a few simple guidelines coupled with our higher Initiative:

- If the enemy attacks increase in number or quality, you're better off trying to kill them before they attack. Spears are the better option.
- If the defensive capabilities of the enemy increases, your attacks score less effect so the defensive bonus becomes better.
- Spears benefit more from an expanded frontage since they can cram in more attacks in the gained space. Other than that, the odds of winning/losing combat become reinforced. If you have an edge, that edge will increase and vise versa.

As a side question, side probabilities can by definition not become negative but your numbers do show this every now and then. Maybe I don't quite understand what they're representing.

Post by **Dyvim tvar** » Tue Jun 20, 2006 6:24 pm

Wow. Great job. Stickied and will likely move to the HOF later.

Waerik · Post by **Waerik** » Tue Jun 20, 2006 6:34 pm

The negative probabilities (i.e. along the y axis) represents the probability for the enemy coming out on top...

Lakissov · Post by **Lakissov** » Tue Jun 20, 2006 6:49 pm

SleekDD wrote:Giving a link to the excell-file would help a lot too.

Absolutely true! But i have no idea where i could upload an excel file, while i was able to find a place to upload pictures...

SleekDD wrote:coupled with our higher Initiative

totally forgot to list that assumption alongside the other ones

SleekDD wrote:If the enemy attacks increase in number or quality, you're better off trying to kill them before they attack. Spears are the better option

not exactly. if in addition the enemy is tough and armored, then shield and hw is better.

SleekDD wrote:Spears benefit more from an expanded frontage

however, what i find out is that this effect is very small compared to, say, toughness. the main contribution of this post is not listing the influencing factors, but ranking them

SleekDD wrote:side probabilities can by definition not become negative but your numbers do show this every now and then. Maybe I don't quite understand what they're representing.

they are showing the difference between the probability that spearelves will end up in a certain CR interval and the probability that hw+shield elves will end up in it. the negative difference implies that the probability for shield elves ending up in tne given interval is greater than the corresponding probability for spearelves

Lakissov · Post by **Lakissov** » Tue Jun 20, 2006 6:54 pm

Waerik wrote:The negative probabilities (i.e. along the y axis) represents the probability for the enemy coming out on top...

no-no! i am not comparing our units with the enemy units. i am comparing the probabilities for speareves and shield+hw elves getting certain CR from kills.

Rasputinii · Post by **Rasputinii** » Tue Jun 20, 2006 7:19 pm

Dyvim Tvar wrote:Wow. Great job. Stickied and will likely move to the HOF later.

Agreed.

I have added this to the Druchii Tactics HoF sticky.

Well Done Lak!

Lakissov · Post by **Lakissov** » Tue Jun 20, 2006 8:25 pm

RasputinII wrote:Well Done Lak!

thanks. i plan to use a similar approach to a couple more questions in th nearest weeks to come

btw, i prefer not being called Lak. better the whole surname or the name (Sergei it is)

Sleekdd · Post by **Sleekdd** » Tue Jun 20, 2006 9:19 pm

Lakissov wrote:
If the enemy attacks increase in number or quality, you're better off trying to kill them before they attack. Spears are the better option

not exactly. if in addition the enemy is tough and armored, then shield and hw is better.

Well, you have two effects in play here now. What I intended to say is that if your opponent is good at attacking but relatively vulnerable, it is better to take the initiative and dish out as many attacks as possible to kill them. The proverbial 'the best defence is a good offence'.

If, you combine improved attacking power with better defences, the extra attacks coming from using spears may make little difference and so one has to take his own survival as priority.

Lakissov · Post by **Lakissov** » Wed Jun 21, 2006 4:48 am

SleekDD,
right, something like that.

I just wanted to stress that in this case toughness and armor should have more influence on your decision making than strength and number of attacks.

Post by **Dyvim tvar** » Wed Jun 21, 2006 5:07 am

What it boils down to then is that if you have a pretty good chance of hurting the enemy, you should generally use spears since it will both increase your numebr of kills and it will decrease the number of atatcks that are coming back.

Here's another factorto consider -- multi-wound enemies such as ogres. With spears, you might do more wounds, but you might not cut the number of enemy atatcks coming back if you don't do enough damage to kill an enemy model.

Lakissov · Post by **Lakissov** » Wed Jun 21, 2006 5:41 am

Dyvim Tvar wrote:Here's another factorto consider -- multi-wound enemies such as ogres.

I'll try to incorporate that in my calculator, most likely by tomorrow. And then I'll add the corresponding charts.

Tenebrae · Post by **Tenebrae** » Wed Jun 21, 2006 6:20 am

First of all, my compliments to a job done right.
Second of all, this was a highly informative study, and I do plan to make some use of it. It fully deserves to into the HoF. Congratulations Sergei.

Third: that's a rather cute avatar you have, your kitten?

Waerik · Post by **Waerik** » Wed Jun 21, 2006 8:36 am

no-no! i am not comparing our units with the enemy units. i am comparing the probabilities for speareves and shield+hw elves getting certain CR from kills.

Ok... I thought that you were putting the two combos vs eachother...

Lakissov · Post by **Lakissov** » Wed Jun 21, 2006 8:45 am

Waerik wrote:Ok... I thought that you were putting the two combos vs eachother...

Well, that would not be so useful. After all, they are not meant to fight against each other. They are meant to fight other enemies and to achieve certain aims while fighting. I just compare which combo is better capable of doing that against different kinds of enemies.

Waerik · Post by **Waerik** » Wed Jun 21, 2006 8:50 am

Yea I got that now... I said "thought" (past tense), not think (present tense).

Again, really well done, would really be loking forward to more of this...

I usually do this cind of calculations in my head, when I'm out walking, but they ain't as acurate, and I'm not that good at visualizing data in the form of graphs without drawing them...

And your avatar really is cute...

Not as cute as the Ligers, but close...

Lakissov · Post by **Lakissov** » Wed Jun 21, 2006 8:55 am

Waerik wrote:I usually do this cind of calculations in my head, when I'm out walking, but they ain't as acurate, and I'm not that good at visualizing data in the form of graphs without drawing them...

Yeah, it's always best to have an excel file do both the calculations and the visualizations. I'm not so good at calculating otherwise, prefer just putting together the formulas. And even then, when I have the numbers, I do need them to visualize in graphs to make any judgements - thay just look so much more intuitive.

Though I might consider applying some more serios analysis to whfb one day - making some regressions and stuff. If I find it useful, that is

Waerik · Post by **Waerik** » Wed Jun 21, 2006 9:02 am

There is no substitute for grahps... It's one of mankinds greatest inventions...

Hell, I'm into economics, we ofthen do not find it worthwhile to say anything without drawing a graph, a picture do say more then a thousand words...

But I really like your aproach, i.e. it is werry worthiwile to concider the distribution, since it tells us a lot of the capabilites of a unit, luck do once agian play it's part...

Though I might consider applying some more serios analysis to whfb one day - making some regressions and stuff. If I find it useful, that is

It do keep your math skills alive...

So, what are you thinking of doing next Sergei?

Lakissov · Post by **Lakissov** » Wed Jun 21, 2006 9:07 am

Waerik wrote:So, what are you thinking of doing next Sergei?

I think i'm going to mess with the issue of how many attacks to direct against characters/champions in units in different conditions. i'm not yet completely sure how to proceed with that though...

Waerik · Post by **Waerik** » Wed Jun 21, 2006 9:10 am

I think i'm going to mess with the issue of how many attacks to direct against characters/champions in units in different conditions. i'm not yet completely sure how to proceed with that though...

Characters are an issue...

But you should always alocate the maximum number of attacks on the champ, since they carry over to normal troups...

Post by **Dyvim tvar** » Wed Jun 21, 2006 1:36 pm

Waerik wrote:But you should always alocate the maximum number of attacks on the champ, since they carry over to normal troups...

Actually that is no longer the case. That language was removed when GW issues an correction 2 or 3 years ago. Extra wounds on a champion do not carry over to rank and file troops.

Fr0 · Post by **Fr0** » Thu Jun 22, 2006 4:00 am

Holy mackrel! :shock:

I was looking for something like this recently. Excellent work, thanks for posting it!

Fr0

Waerik · Post by **Waerik** » Thu Jun 22, 2006 8:00 am

Actually that is no longer the case. That language was removed when GW issues an correction 2 or 3 years ago. Extra wounds on a champion do not carry over to rank and file troops.

hahahahahahhaah, that's how we used to play it, then I re-read the core book and figured that we had been playing it wrong...

It was such a sucky rule, yay

Thanks Dyvim, you made me happy

Lakissov · Post by **Lakissov** » Sat Jun 24, 2006 2:05 pm

All right, I gave this thing a little more thought and made some additions.

First, I rebulit my calculator slightly so that it now allows multiwound models when calculating the probability distributions of CR resulting from kills.

Here are the charts exploring the way in which the comparison of spears to hw is affected by opponents' multiple wounds:
http://img153.imageshack.us/img153/6471 ... rs16hv.jpg
http://img153.imageshack.us/img153/3628 ... rs26be.jpg
http://img153.imageshack.us/img153/8652/w14yb.jpg
http://img148.imageshack.us/img148/8706/w25hu.jpg
http://img148.imageshack.us/img148/299/w39ql.jpg

We can see from these that the effect of wounds is very similar to the effect of armor and toughness - not a surprising result in the least.

Now then, we have three very important characteristics that should define our choice of the combo - T, W, AS. But here comes an interesting question - which of the them are more important? I made a few more calculations to find out.

I decided to see what the choice should be when the opponent has a bad armor save but many wounds or high toughness (the reverse is not possible - I know of no enemies with T below 3 but better than 5+ AS)

So here are the combinations I tested:
T5, AS 7+ (meaning no AS)
T5, AS 6+
W2, AS 7+
W3, AS 7+
W2, AS 6+

Here are the results:
http://img148.imageshack.us/img148/6485 ... rs18jy.jpg
http://img153.imageshack.us/img153/9537 ... rs24jw.jpg
http://img148.imageshack.us/img148/6350/comp12ec.jpg
http://img148.imageshack.us/img148/7391/comp23oa.jpg
http://img148.imageshack.us/img148/2104/comp33ra.jpg

What can we see here? We can see rather clearly, that the combinations favoring hw combo most are T5 AS 6+ and W3 AS 7+. However, what we can also see is that all the combinations tested bias us in favor of spear combo. The interesting result is that many wounds or high toughness are not enough to make up for bad armor save. Armor save is THE most important factor determining the choice between spears and hw+sh