MathHammer: New Helblaster Volley Gun w/ Engineer (UPDATED)

[Eirik]

The 120pt Helblaster Volley Gun does an average of 4.2 hits at 24'', or 8.5 hits with the addition of a 65pt Master Engineer
UPDATES:Determined optimal strategy for cases of 0 or 1 misfires, reduced to 3 optimal strategies, updated my results and discussion to reflect this.

Abstract: I simulated one million shots of the Helblaster Volley Gun using matlab. I assumed the weapon to be firing at long range, although this does not affect the best 'firing strategy' for the weapon. I considered firing with or without an engineer, and three Risk Strategies depending on how willing to score triple MISFIREs the empire general was. When fired without an engineer, I found the gun to do about 4.2 hits on average. When fired with an engineer, I found that the best strategy was to reroll sometimes when no MISFIREs are rolled, always reroll when 1-2 MISFIREs are rolled, and never reroll when 3 are rolled. This resulted in a doubling of the number of hits to about 8.5.

Introduction: The new empire army book brings a much needed sense of balance to the army's war machines. Unfortunately, most warriors have little experience against the new lineup. High Elf generals prefer reliability and generally consider the unknown to be dangerous. The new HelBlaster volley canon can reach dangerous levels of damage with the assistance of a Master Engineer who not only allows the weapon to fire at BS4, but also allows the reroll of one artillery die which mitigates MISFIRES. Here I attempt to aid my fellow Asur by determining what can be expected from this piece of hardware.

Methods: Without an engineer the firing of the Helblaster is quite straightforward (and dangerous). Simply roll three artillery dice and sum the result to determine the number of attacks made. These attacks are made at BS3, S5 at a range of 24'' and benefit from the Armour Piercing special rule. If one MISFIRE was rolled then halve the number of attacks. If two were rolled then no attacks are made and the machine actually MISFIRES. If three are rolled the machine makes a full 30 attacks and then explodes.

When an engineer is included the weapon's BS increases to 4, so you are hitting with 1/2 instead of 1/3 of attacks (or 2/3 and 1/2 at close range). The weapon also becomes safer, and able to inflict more damage, thanks to the option to reroll one of the three artillery dice. This introduces an aspect of strategy to the weapon as well. Under what cases should you reroll when you throw three dice and get no misfires? And how reckless should you be with regard to trying to obtain three MISFIRES and get a full 30 shots?

There is an advantage to rolling many misfires, namely in getting to make 30 shots if you score three misfires. I outlined three Risk Strategies with regards to this issue: Risk Averse, Risk Neutral, Suicidal. A Risk Averse general will always reroll a MISFIRE. A Risk Neutral general will always reroll MISFIREs except when they roll three. In this case they keep their three MISFIREs, take the 30 shots, and lose the war machine. Finally a Suicidal general does not reroll a triple MISFIRE, and in the case of a double MISFIRE rerolls the NON-MISFIRE die in an attempt to get 3 MISFIRES. Otherwise they reroll as normal.

When one MISFIRE is rolled, it is always best to reroll that MISFIRE. There is no risk of a worse outcome and a good probability of an improved outcome. When no misfires are rolled, we can determine the average outcome of rerolling the lowest die. If the average outcome is better than the current outcome, then we reroll. I have determined the conditions under which rerolling is advantageous. Set Y to be the lowest value showing on the three artillery dice. Set X to be the sum on all three dice minus Y. If 5 > X/12 + Y then you should reroll your lowest artillery die. The Calculations section below demonstrates the reasoning behind this.

Threre are three total strategies obtained by combining one of three Risk Strategies to deal with situations of 2 or 3 MISFIREs, and the optimal strategy outlined above to deal with situations of 0 or 1 MISFIREs. I simulated 1,000,000 fires of the Helblaster and applied each of the 3 strategies to these data. I computed the average number of 'hits' scored under each scenario. I also tested a Helblaster with no engineer for the same data.

Results:
With no engineer the Helblaster scores about 4.2 hits on average at long range (6.3 close range). With an engineer, the Helblaster performs best using the Risk Indifferent and Suicidal strategies. These techniques gives about 8.5 hits on average at long range (11.3 at close range). The average number of hits for each Risk Strategy are given below.


The Suicidal and Indifferent strategies result in almost the exact same number of hits. More detailed results, including histograms showing the frequency distribution of the number of attacks/hits, are available in the Notes below.

Conclusions: Although the Suicidal and Indifferent strategies performed similarly in terms of number of hits, the Indifferent strategy is obviously superior because the risk to the war machine is strictly less. You can expect a competent enemy general who wants to maximize damage to reroll misfires unless they roll 3 of them. The benefits of rerolling with 3 misfire dice are very small, as the machine is still guaranteed to misfire and is likely to do no damage if you do.

It is also a mistake to assume that any one of these three strategies is always the optimal solution. The situation at hand is an important determining factor in how much to risk when firing a Helblaster. If your opponent is targeting a high priority unit, such as a dragon-mounted prince, and rolls triple misfires, they are likely going to take the 30 shots and try to kill your expensive piece. If they are targeting a unit they could not realistically destroy and it is their sixth turn, they might choose to reroll one of those 3 misfires in an attempt to salvage the 120pts the piece represents.

Assuming your opponent knows what they are doing, expect about 4.2 hits without an engineer and 8.5 hits with an engineer. I would expect them to adopt the Indifferent strategy most shooting phases and perhaps fall back on the Averse strategy when the late game arrives and enemy targets have lower priority. I do not recommend worrying much about the prospect of triple MISFIREs, since this will only happen in one out of 216 shots.

Calculations:
Determining the optimal reroll strategy when no MISFIREs are rolled originally:
Assume the lowest die shows the number Y, and the total on the other two dice is X.
The reroll can be MISSFIRE, 2, 4, 6, 8, or 10. If it is 2-10 then your total number of shots is that number + X. If it is MISFIRE, then your total number of shots is X/2. If you do not reroll, then your total number of shots is X+Y. Then the average number of shots on a reroll is
(X/2 + 2+X + 4+X + 6+X + 8+x + 10+X)/6
and we want to reroll when this result is better than
X + Y
Then we solve the inequality
(X/2 + 2+X + 4+X + 6+X + 8+x + 10+X)/6 > X + Y
(5.5x + 30)/6 > X + Y
(11/12)X + 5 > X + Y
5 > X/12 + Y
So we reroll if X/12 + Y is less than 5.

Notes:
Histogram detailing frequency of number of attacks/hits without an engineer: http://imgur.com/MAm6I
Histograms detailing frequency of number of attacks/hits with an engineer for 3 strategy combinations: http://imgur.com/fdOX4

[Brewmaster_D]

You, sir, are a saint.

Thanks for crunching these numbers, I'm sure we'll be seeing this piece fielded a lot more now!

D

[RobTheWise]

Amazing article - thanks!

[Sturen]

That is some serious mathhammer, very helpful!

One thing I can't get my head around is why, in the 'with engineer' chart, Risk neutral is slightly higher than suicidal. Surely suicidal is always either equal or higher (with the exception of rerolling the 4 in MM4 and scoring a 2) than neutral?

Your final point is important I think, the 1/3 chance of losing half the other dice is significant. Would the threshold value for which rerolling the 2 becomes too risky be easy enough to calculate?

[Eirik]

That is some serious mathhammer, very helpful!

One thing I can't get my head around is why, in the 'with engineer' chart, Risk neutral is slightly higher than suicidal. Surely suicidal is always either equal or higher (with the exception of rerolling the 4 in MM4 and scoring a 2) than neutral?

You are thinking that a roll of 2 misfires always results in 0 shots for a Risk Neutral player and sometimes results in 30 shots for a Suicidal player. This is not the case. The Suicidal player has a 1/6 chance of turning 2 misfires into 3 and gaining 30 shots, so they gain about 5 shots on average. The Risk Neutral player rerolls one of the misfires which will become a non-misfire with probability 5/6. That's a 5/6 chance of getting a number of shots equal to half the value showing on two non-misfire artillery dice (this works out to 12.5 shots on average for Indifferent players, higher than the 5 shots average for Suicidal players).

Your final point is important I think, the 1/3 chance of losing half the other dice is significant. Would the threshold value for which rerolling the 2 becomes too risky be easy enough to calculate?

It's a 1/6 chance of misfireing (1/3 is for direct hits on a scatter die). It turns out that deciding whether to reroll 2s, 4s or only misfires was the wrong way for me to do it. The better way to do things is to take both the lowest die and the total on all three dice into account. I have modified the strategy for rerolling when no misfires occur to reflect this. It turns out there is only one 'best' strategy, and it takes the risk of rolling a misfire into account.
So basically the answer to your question is to look at the Calculations section I added above. It's easy enough to calculate the risk analytically, without even needing to simulate all the possibilities. Thanks for the questions

[Sturen]

You are thinking that a roll of 2 misfires always results in 0 shots for a Risk Neutral player and sometimes results in 30 shots for a Suicidal player. This is not the case. The Suicidal player has a 1/6 chance of turning 2 misfires into 3 and gaining 30 shots, so they gain about 5 shots on average. The Risk Neutral player rerolls one of the misfires which will become a non-misfire with probability 5/6. That's a 5/6 chance of getting a number of shots equal to half the value showing on two non-misfire artillery dice (this works out to 12.5 shots on average for Indifferent players, higher than the 5 shots average for Suicidal players).

Of course, that makes sense. 12.5 seems high... if the average on each of the dice is 6, two dice gives 12. This halved gives 6, and only improves 5/6 of the time for an increase of 5 on average, no?

It's a 1/6 chance of misfireing (1/3 is for direct hits on a scatter die). It turns out that deciding whether to reroll 2s, 4s or only misfires was the wrong way for me to do it. The better way to do things is to take both the lowest die and the total on all three dice into account. I have modified the strategy for rerolling when no misfires occur to reflect this. It turns out there is only one 'best' strategy, and it takes the risk of rolling a misfire into account.
So basically the answer to your question is to look at the Calculations section I added above. It's easy enough to calculate the risk analytically, without even needing to simulate all the possibilities. Thanks for the questions.

The new stats are helpful, who'd have thought it would be so complex for one warmachine! I much prefer our reliable Asur contraptions.

The real question is, can multiple engineers provide multiple rerolls?

[Eirik]

Of course, that makes sense. 12.5 seems high... if the average on each of the dice is 6, two dice gives 12. This halved gives 6, and only improves 5/6 of the time for an increase of 5 on average, no?

Oops, not sure how I made that mistake. Given two dice with no misfires, the average roll is (2+4+6+8+10)/5=6 each, for a total of 12. This is halved then multiplied by 5/6 as you said, so the total is 5. This means there is no difference in the average number of shots for the Indifferent and Suicidal strategies.
If you check out the new graph, you can see the average as well as an error bar. The circle represents what the average number of hits is for this sample, the error bars represent the range over which we can say the true average lies with a high degree of certainty. Since the error bars for the two strategies overlap, we cannot be highly certain that one is greater than the other. So our empirical and theoretical findings confirm each other.

The new stats are helpful, who'd have thought it would be so complex for one warmachine! I much prefer our reliable Asur contraptions.

The real question is, can multiple engineers provide multiple rerolls?

I'll take the bait. I think the answer is yes, though it might get FAQd. Keep in mind they have to choose separate dice to reroll, no one die can be rerolled multiple times.

[Sturen]

Oops, not sure how I made that mistake. Given two dice with no misfires, the average roll is (2+4+6+8+10)/5=6 each, for a total of 12. This is halved then multiplied by 5/6 as you said, so the total is 5. This means there is no difference in the average number of shots for the Indifferent and Suicidal strategies.
If you check out the new graph, you can see the average as well as an error bar. The circle represents what the average number of hits is for this sample, the error bars represent the range over which we can say the true average lies with a high degree of certainty. Since the error bars for the two strategies overlap, we cannot be highly certain that one is greater than the other. So our empirical and theoretical findings confirm each other.

Intriguing! Is it sad that I am this entertained by the optimal strategy for a miniature warmachine I'll never use? (Feel free to finish the sentence at any point after entertained).

I'll take the bait. I think the answer is yes, though it might get FAQd. Keep in mind they have to choose separate dice to reroll, no one die can be rerolled multiple times.

Yes, I assume the choice of which two dice to reroll would have to be made immediately; there is no waiting until you can see the result of the first: even thinking about it makes my head hurt. However, I would be willing to bet that its more beneficial to leave the second engineer to work on a different warmachine than attempt to combine forces on one...

[Eirik]

Yes, I assume the choice of which two dice to reroll would have to be made immediately; there is no waiting until you can see the result of the first: even thinking about it makes my head hurt. However, I would be willing to bet that its more beneficial to leave the second engineer to work on a different warmachine than attempt to combine forces on one...

I would disagree with this. Since there are no rules as to which die is rerolled first, it falls to the player whose turn it is, namely the Imperial General. Situations in which the result of one reroll would influence your decision with respect to the second reroll would occur, though not too frequently. I could do the mathhammer on the second reroll but I won't. I expect you would get some sort of diminishing returns.

[possumman]

Incredible, thanks!
At first I was disappointed that the three strategies you give yield (almost) exactly the same output (8.45 - 8.5 hits), but on reflection I like it; because there isn't really a 'optimal' way of firing it, a general can pick any option depending on the situation and how lucky he feels, thus increases the variety of playstyles you can use with it.

[Marwynn]

Now THAT is how you MathHammer!

Good to have these numbers in your head. Most Empire Generals are saying it's best to get two HBVGs instead of one with an engineer since that guy's more than halfway to a second one already.

Greater potential, but if we just double 4.2 to 8.4 it seems that the engineer comes out a bit ahead at a cheaper cost with his 8.5 hits. That's assuming there's room for an engineer, however.

Hmm...

[Sturen]

Double the shots, more likely to hit, only 65 points, the engineer seems a steal!

I would disagree with this. Since there are no rules as to which die is rerolled first, it falls to the player whose turn it is, namely the Imperial General. Situations in which the result of one reroll would influence your decision with respect to the second reroll would occur, though not too frequently. I could do the mathhammer on the second reroll but I won't. I expect you would get some sort of diminishing returns.

Good point, you're right. I don't blame you for not wanting to deal with the maths though...

[Eirik]

Given two dice with no misfires, the average roll is (2+4+6+8+10)/5=6 each, for a total of 12. This is halved then multiplied by 5/6 as you said, so the total is 5. This means there is no difference in the average number of shots for the Indifferent and Suicidal strategies.

I just realized we already know what one of the die rolls is if we have 2 misfires. If the non-missfire is a 2 or 4, then you get more shots on average if you go for the triple misfire. If the non-misfire is 8 or 10, you get more shots on average if you reroll one misfire. If it is 6, you get an average of 5 shots either way.
Of course the differences are tiny and your machine has a much higher chance of survival and future utility if you reroll the misfire. It should probably come mostly down to the situation, target, what turn you're on, etc.

[Tsunami]

Thanks for a great job on the maths

I would like to add to the conclusion, since it can easily be improved even further. If you roll no misfires at all, and you ask yourself "should i re roll the lowest die?" you can do some quick calculations and realize that you would only ever consider re rolling a 2 or a 4. lets start with a 2.

When re rolling a dice (and you had no misfires) you risk losing some shots. depending on the two other dice score you are risking to lose up to 10 shots (plus the 2 from the dice you re rolled), so the case where you would be most unsure if to re roll a 2 or not would be when you rolled 10, 10, 2. if you re rolled the two, the possible outcomes would be [2, 4, 6, 8, 10 and -10(that's the misfire)] this averages out to 3,33. this is larger than two, and hence if you play the odds, you should always re roll the lowest result if it is a 2.

The next case is when the lowest result is a 4. the case where you risk the LEAST amount of lost fire-power is when you rolled 4, 4, 4. the possible outcomes of the re roll would then be [2, 4, 6, 8, 10 and -4(the misfire)] this averages out to 4,33 hence even in this case it would be statistically justified to re roll that 4, though the margin is very slim. The same is true if you roll 4, 4, 6 but not if you roll 10 or higher on the two other dice.

the next consideration you should make is "do i really need more shots?" if you have rolled 4, 4, 4 you must ask "would 14-18 shots solve a problem that 12 shots can not?" in many cases the answer will be "no", and since the average would be 12,33 if you re rolled compared to the 12 you already have i for one would deem it insanity to re roll a 4 if it is not one of those "do or die" situations that happens once every 20 games or so. hence a 4 should only be re rolled if you KNOW that 12 shots will not be enough, but 14-18 shots could secure victory.

This reasoning about "do i really need more shots?" holds true even for the case of re rolling a 2. if you did roll 10, 10, 2 and you are shooting at a unit of 10 fast cavalry you can feel sure that 22 shots will dispose of them, but if you re rolled the two and got a misfire, 10 shots might not be enough. in this case you should not re roll. if you rolled 2, 4, 6 and you are shooting at a unit of infantry on the other hand you know that 12 shots will not kill them (and neither will 20) so it might be worth the gamble of going down to 5 shots for the chance of going up to 20. statistically its a sound option, and it is unlikely to make any huge difference strategy wise, so go with the re roll.


TLDR:
If you do not roll any misfire you should re roll a 2 (if more shots will be helpful). You should not re roll a 4 unless it is a "do or die"-situation.

ps. i know this is covered by the formula you posted. this is just a translation of said formula to real terms that every one might understand

[Eirik]

Thanks for a great job on the maths

TLDR:
If you do not roll any misfire you should re roll a 2 (if more shots will be helpful). You should not re roll a 4 unless it is a "do or die"-situation.

ps. i know this is covered by the formula you posted. this is just a translation of said formula to real terms that every one might understand

Thanks. You remind me of one of the profs in my applied math department. He's always asking "what's the biological interpretation of your model" when we get too into the calculations.

In regards to the 'do I really need more shots' rule.
I suppose the best thing to ask is which decision gives you the best chance of accomplishing your goal. If 10 hits is enough to reliably kill the target then dont reroll your 2 when you get 2-4-4. If you need 15 to reliably kill the target, then reroll.

You could actually do this if you needed to. Design a spreadsheet that has your three artillery dice, the targets toughness/armor/ward and the number of wounds you need to inflict as inputs. Carry this on a laptop to all your games. It could calculate a simple set of binomials to tell you the % chance of getting the needed number of wounds with or without the reroll. Probably overkill, but worth the looks you would get from your opponent.