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If I had a markerlight on a lightly armored but in cover enemy unit, and it was a very good idea to use it with a squad of fire warriors, my choices are to increase the ballistic skill or lower the cover save.

From some simple math, I kill the same number no matter which choice.

I either give them more 4+ cover saves or less 5+ cover saves. You can see this by working out chance to hit is either 2/3 or 1/2, and chance to fail cover is correspondingly 1/2 or 2/3.

Given that I kill the same, would you give them more 4+ cover saves or less 5+ cover saves?

24 shots:

# 4+ cover saves - probability
0-7 - close to 0
8 - 1.6%
9 - 3.4%
10 - 6.4%
11 - 10.3%
12 - 13.9%
13 - 16.0%
14 - 15.7%
15 - 13.1%
16 - 9.2%
17 - 5.4%
18 - 2.6%
19 - 1.0%
20-24 - close to 0

# 5+ cover saves - probability
0-4 - close to 0
5 - 1.9%
6 - 4.3%
7 - 7.9%
8 - 12.0%
9 - 15.3%
10 - 16.3%
11 - 14.9%
12 - 11.5%
13 - 7.6%
14 - 4.3%
15 - 2.0%
16-24 - close to 0

Remember that you roll to hit before your opponent takes saves. Since these operations are nested, it's better to increase the likelihood of the earlier operations obtaining a preferred result.

Hence it's better to hit with four out of six shots and perhaps force four saves at 4+ than it is to hit with three out of six shots and perhaps force three saves at 5+. In the former case, you have the potential to cause four wounds, while three is the best in the latter case.

Every shot you miss is a wound roll you automatically fail, and an armour save they don't need to make.

That is incorrect. It doesn't matter if I increase the BS or reduce the cover save, I have just as much chance of killing 4.

You probably don't do statistics so I'll make it clearer for you:

24 shots:

Increase BS

# dead - probability
0-1 - close to 0
2 - 1.7%
3 - 4.7%
4 - 9.4%
5 - 14.5%
6 - 17.7%
7 - 17.5%
8 - 14.3%
9 - 9.8%
10 - 5.6%
11 - 2.8%
12 - 1.1%
13-24 - close to 0

Reduce Cover

# dead - probability
0-1 - close to 0
2 - 1.7%
3 - 4.7%
4 - 9.4%
5 - 14.5%
6 - 17.7%
7 - 17.5%
8 - 14.3%
9 - 9.8%
10 - 5.6%
11 - 2.8%
12 - 1.1%
13-24 - close to 0

There has to be a good reason for wanting more or less cover saves. For example, the 4th edition torrent of fire rule. I can't think of any examples in 5th edition, that is why I'm asking.

Nope, my point was correct, although you seem to have managed to miss it. My point wasn't about the probability, which is the same when you abstract it beyond the set of operations indicated by the rules. My point was about what is possible given the probabilities at each step of the hit/wound/save process.

Let me restate it for you:

It's better to increase the likelihood of the earlier operations obtaining a preferred result in a nested set of operations.

You may remember from your basic stats classes that there's a difference between the likelihood of some idealized result obtaining, and what dice actually do.

Given that dice will not follow the idealized distribution, and that bad luck in earlier operations will have a knock-on effect in later operations since a failure to hit is also a failure to wound and an automatic save, you are better off maximizing the odds of hitting your target in order to maximize the number of possible wounds, and hence of possible failed saves.

You can check this for yourself, if you want to perform the weighted expected value calculations for yourself. Remember, it's not simply the likelihood of results obtaining in any operation (or just the aggregate), but also the restriction on possible results from prior operations.

Basically, you're applying a really simplistic statistical model to what is essentially a problem of algebra, and hence you miss the importance of ordered operations and how that plays out in game-theoretic situations such as Marker Lights in the game of maximizing the number of wounds done to some unit in Warhammer 40k.

Had you considered this, you could have then answered your own question, such as why anyone would want to lower the cover saves given that the weighted expected value for doing so is less than that of lowering the threshold to hit. The answer is simple: If you limit your model to considering either raising the cover save or lowering the threshold to hit, then you fail to notice that this operation has its own weighted expected value in relation to the number of marker lights you can take.

While you can only lower the threshold to hit to BS5, using one or two markerlight hits, you can negate a unit's cover save entirely using three or four markerlight hits.

Nurglitch wrote:Hence it's better to hit with four out of six shots and perhaps force four saves at 4+ than it is to hit with three out of six shots and perhaps force three saves at 5+. In the former case, you have the potential to cause four wounds, while three is the best in the latter case.

The logic here does not apply. I can hit with three or four shots, whether I'm BS3 or BS4. I start with the same number of shots.

Also, I can't believe you completely missed the increased gain from lower variability.

It is much better to chose the option with lower variability, given the same return. This increases predictability.

However in this case the variability is exactly the same.

Yes, you start with the same number of shots, then you roll to hit, and then you roll to wound, and then you roll to save.

Three separate rolls, with the number of possible successes in the latter two dependent on the number of actual successes of the previous roll.

If you're more likely to hit with BS4 than with BS3, then you're more likely to have more rolls to wound, as well as being more likely to cause a wound.

That's why you need to figure this as weighted expected value, because you're making a decision based not only on the likelihood of any expected outcome, but also on the utility of that outcome.

As I've pointed out that there are three outcomes, the results of three separate rolls, which are nested, that means 'conditional upon the previous result', so the utility of any particular roll is weighted to its order in the nested conditionals.

What you should be getting from this is the futility of relying on the likelihood of aggregate results when you're trying to understand how adjusting the values of the component operations affects which adjustment you should prefer.

You might also want to distinguish between missing something and not addressing it because it's irrelevant... You know, just a thought. But hey, while we're covering basic decision theory:

If one is discussing expected value, particularly weighted expected value, it is better to chose the option with lower variability, given the same utility, if and only if the product of expected value is of equivalent weight.

A to hit roll and a saving throw are not equivalent weight. As I have pointed out, the weighting of a to hit roll to a saving throw is 3:1, since the relevant possibilities are:

[hit, wound, not save] - success!
[hit, wound, save] - fail!
[hit, not wound, save] - fail!
[not hit, not wound, save] - fail!

So, if there is a 10% likelihood of event A happening, and a utility of 10 when it actualizes, and if there is a 10% likelihood of event B happening, and a utility of 10 when it actualizes, but effect A is necessary for effect B, then it follows that increasing the likelihood of event A by 10% instead of increasing the likelihood of event B by the same follows from a simple comparison of weighted expected value if you want to maximize the total outcome.

The conditional probability is factored into in the tables I posted post 4.

They are still the same tables.

Take the tables I posted in post 2, use the corresponding cover save, and you get the tables posted in post 4.

You're not understanding probability.

Just as a test I decided to simulated this situation and look at the distribution of kills. I used onlainri's situation and used Orks as my example targets. This first graph shows the distribution of number of kills by our twelve man firewarriors squad over 10000 iterations. The blue line is reduced cover saves and the red line is increased ballistics skill. As you can see they are almost identical. When I bump the simulations up to 100000 iterations they were identical. This seems to support onlainari's hypothesis. My apologies that zero kills doesn't show up. it was around .2% in each case.

Rosicrucian, you are awesome.

There still may be a reason to force more cover saves however, it's just not "you can kill more".

Maybe it's "you have a greater chance of scoring 10 wounds, thus allocating a save to the special weapon"?

Nurglich's point was that, given that 4 fire warriors is hardly an exhaustive sample, there's a decent chance that they'll deviate from the mean. I.E., there's more chance that your opponent will roll poorly.

Assume you fire 12 shots. With BS 3, you will, on average, hit with 6. With BS 4, you will, on average, hit with 8. Sometimes this will be more, sometimes this will be less. Now, if we assume complete averages, then yes, 4+ cover vs. 5+ cover will reduce the above to equality. But the whole point is to make them take large numbers of saves. There's always the chance that they'll have a below-average set of rolls, causing them to lose more men.

This is why torrent of fire style armies work, particularly Orks - make them take enough saves, roll enough handfulls of dice, and they'll at some point roll below average and more men will die. This is why it's better to increase your hit rolls than to decrease cover.

Lastly, and non-statistically, I prefer to make my rolls easier rather than making my opponent's rolls harder. I want my dice to work for me!

tzeentchling wrote:Nurglich's point was that, given that 4 fire warriors is hardly an exhaustive sample, there's a decent chance that they'll deviate from the mean. I.E., there's more chance that your opponent will roll poorly.

The chances of rolling poorly (or very well) are already factored into the equations. That's probability for you. Now if you want to bet on outlying possibilities, that's something you can always do, but the chances of it working out that way can be found on the tables that Onlainari already posted.

In either case, there is some benifit to chosing the improved BS over the reduced cover save, and that's the ability to force saves on special models in the squad. This is something that the math-hammer doesn't show but is an important consideration. In the end however, the number of kills is going to average out to be the same in both cases (as Rosicrucian has clearly demenstrated).

@Onlainari: I don't think it's just maybe - it's definitely better to get that greater chance of forcing the heavy or special weapons guys to take saves.

So long as the weapons you're shooting are all the same (fire warriors) and the target has special/heavy weapons, you want more hits to force them to save. This is definitely a matter of "you miss 100% of the shots you don't take" since you have to force them to take a save in the first place, regardless of how good their save is going to be.

If you don't care, or if there is nothing special to kill, I'd still go with more hits because I enjoy it psychologically.

lambadomy wrote:@Onlainari: I don't think it's just maybe - it's definitely better to get that greater chance of forcing the heavy or special weapons guys to take saves.

Lambda has it -- or, at least, pretty close.

Since you have the choice, you want to optimize the most damaging result. This isn't as interesting if you're shooting at, say, Necrons, or other models with all identical units, but if there's variation in the squad and you can apply resources (in this case, markerlights) to either increase wounds or decrease saves, the ideal situation is going to be fairly close to increasing wounds until one wound per enemy model is expected, and beyond that point decrease the saves.

Why? At less than one wound per model, due to wound allocation, the most valuable models in the unit won't take wounds, so one wound per model forces saves to be taken on every model type in the unit. If you have more wounds than models in the opposing unit, you run the risk of losing excess wounds to overkill (as in, one model takes two saves, and fails both -- you generated an extra failed save that didn't do anything since the model was already dead). Realistically, the chances of significant overkill are fairly small until you're talking significantly more wounds than models as the excess saves are most likely to go to the "cheap" models which you're unlikely to get overkill on anyway due to sheer numbers.

As for Nurglitch, his argument is fundamentally flawed. In brief, while his argument that early rolls in sequence weight the later rolls is essentially correct, he fails to take into account that the weighting goes both ways and in reality exactly cancels out. A 3+ 4+ sequence is exactly identical to 4+ 3+ in expected, modal, and median results. It doesn't actually make any difference which order the tests are taken in since all tests have to succeed.

Again, though, the probability and averages do cancel out... if you have a large enough sample. As Rosicrucian's plots show, when the sample size is large, things will average out to the most probable numbers. 40K does not have that large sample. Here, deviations from the norm dominate over the norm. I imagine if Rosicrucian redid his plots for sample sizes of 10,50, and 100, the comparison would not be as close, and the plots would probably differ from run to run.

tzeentchling wrote:Again, though, the probability and averages do cancel out... if you have a large enough sample. As Rosicrucian's plots show, when the sample size is large, things will average out to the most probable numbers. 40K does not have that large sample. Here, deviations from the norm dominate over the norm. I imagine if Rosicrucian redid his plots for sample sizes of 10,50, and 100, the comparison would not be as close, and the plots would probably differ from run to run.

Well, yeah, the plots would differ from run to run. And there wouldn't be an advantage for one or the other. There is absolutely no difference in doing a 3+ 4+ sequence and a 4+ 3+ sequence (beyond the differences in wound allocation mentioned above).

Nope, my argument is exactly on the money because the hit/wound/save set of operations are non-commutative and thus weighted for the appraisal of expected value (likelihood x utility) in the order in which they are nested.

As I have mentioned, for somewhere around the fourth time now, there are three rolls and the possible number of dice rolled in the second two rolls depends upon the results of the first two. You can't roll to wound if you haven't hit!

So it is not the case that we are merely dealing with a commutative set of probabilities, such that (2/3x*1/2y) ≠ (1/2y*2/3x).

Hence it is not the case that the number of dice rolled to hit depends on the number of successful saving throws rolled. which would be the case if the hit/wound/save operations were commutative.

I think the problem here is that people are ignoring all the other game theoretic considerations at play and relying on a abstracted assessment of the total likely outcome, rather than the weighted expected value assessment of each operation as would be indicated when trying to decide which operation would most benefit the whole by having the likelihood of success increased.

These other considerations, to really hammer a dead equine, are utility and weight. Utility is obvious: in Warhammer the point of combat is to cause wounds, and only whole wounds can be caused.

Weight, for reasons that escape me, seem to be less obvious to people even though I have provided a handy chart showing how missing the to hit roll narrows the potential utility of increasing the likelihood of failing saving throws.

The order of the tests is important because if you fail a to hit roll, then you automatically fail the following to wound roll and your opponent passes the save. That makes a difference even when each of the tests needs to succeed, because the possibility of success in the last two tests is conditional upon the previous test in the series.

The likelihood of rolling 2/3 on six dice (dice = utility, fyi) will not change the likelihood of rolling 1/2 on any other number of dice. So what? We know the likelihood is the same because likelihood abstracted from order and utility is communtative, and thus uninformative where our model includes order and utility. What's important is the utility of increasing the likelihood of rolling whatever, and since that utility will be radically greater when you will be most likely to roll the most dice, you should increase your BS before reducing your opponent's cover saves.

Still, I like the idea of telling my opponents that a (3+, 4+) sequence is the same as a (4+, 3+) sequence, and roll to wound with my rending claws before rolling to hit.

Ok, Nurglitch.

Quantify the difference.

(I'm working on an exhaustive analysis of 6 "attacks", 3+ 4+ sequence vs. 4+ 3+ sequence to show they're identical. Feel free to beat me to the punch demonstrating that there's any form of difference between the two.)

Sure.

Weighted Expected Value
Weight = success or fail, expected value = the product of likelihood and utility.

Our operations are ordered:

To hit -> To wound -> Save

So the weight will be 3:2:1, as indicated in a previous post.

Let's call it 3+ to hit, 4+ to wound, and 4+ to save in the 1st case and 4+ to hit, 4+ to wound, and 3+ to save in the 2nd case. Six dice for each set of operations.

1. (2/3*6d) + (1/2*(2/3*6d)) + (1/2*(1/2*(2/3*6d)))

2. (1/2*6d) + (1/2*(1/2*6d)) + (2/3*(1/2*(1/2*6d)))

tzeentchling wrote:Again, though, the probability and averages do cancel out... if you have a large enough sample. As Rosicrucian's plots show, when the sample size is large, things will average out to the most probable numbers. 40K does not have that large sample. Here, deviations from the norm dominate over the norm. I imagine if Rosicrucian redid his plots for sample sizes of 10,50, and 100, the comparison would not be as close, and the plots would probably differ from run to run.

Sure if you look at smaller sample sizes, you will get some slightly different results based on variability. However I fail to understand how that is really relevent. The chances of getting those outlying possibilities of killing everything you shoot at or killing nothing at all are still there and the graphs that Rosicrucian posted show this. Still, if you are rolling some dice (regardless of the number) your rolls are going to tend towards the average (as everyone here has shown). So while the outlying possibilites will show up, they will show up rarely. The probability of rolling those outlying possibilities doesn't change when you shrink the sample size.

I mean, if I were to roll 2d6 and have to guess what the total was going to be every time (craps anyone?), I would guess 7 every time. While I know full well that unless they are loaded, 7 isn't going to be the total every time, it's going to be the total far more often than any other single number. And even if there is only one roll, 7 is still the most likely number to show up, so why wouldn't I pick it?

Ok, I see your problem. For probabilities p, q, andr, you're calculating the results as p + pq + pqr. The actual expected value is pqr.

For things to work out how you're claiming they do, you have to get a wound for every step, not just one at the end after all tests.

Let's look at your equations -- both are D for dice, p for hit probability, q for wound probability, r for save failure probability.

You're therefore giving:

Dp + Dpq + Dpqr

In which, clearly, increasing p instead of R is advantageous. Unfortunately, that's not how it works.

Look at what you're adding up -- to get your result, you're adding expected hits (Dp -- dice rolled, times probability of hit), expected wounds (Dpq -- Dice rolled times probability of hit, times probability of wound), and then expected failed saves (Dpqr -- Dice rolled times probability of hit times probability of wound, times probability of failed saves).

The problem is the result you're calculating is total successful die rolls and not total failed saves -- the Dpqr term is the expected total failed saves, but you're adding in the other terms you use to get there. And, really, we don't care about total successes (beyond the point above about optimal wounds to roll saves on due to wound allocation), just failed saves.

Lowinor:

No, you're still missing the point. I would have thought the addition signs would have tipped you off about how I'm calculating the expected value of each operation and then aggregating them.

Maybe it's because you're calling some formulae "equations"?

Since the utility of wounding depends on the result of hitting, the expected value for each of these operations must factor in the result the result of the previous operation, which is also the expected value of that operation. That's the weighting.

The expected utility of wounding is the expected value of hitting, and the expected utility of failing saves is the expected value of wounding.

So you get the algebra of my formula wrong, because you're confusing the dice rolled in hitting to be the same dice as are rolled in wounding and saving. Perhaps you were confused by the '6d' nomenclature, thinking it to be the number of dice rolled, rather than as the set of potential utility.

So yes, increasing the expected value of hitting is preferable to spending the same amount to increase the expected value of failing saves.

My point, to reiterate it ad nauseum, is that we care about the choice between increasing the likelihood of hits and increasing the likelihood of failed saves where the aggregate of the likelihood is the same.

By applying the weighted expected value formula as I have done, we can see the relationship between the individual parts, the relationship we should care about: the relationship between utility and order rather than the relationship between raw probabilities which we have known from the outset to be otherwise identical.

If I have made a mistake then it is, as you have pointed out, that I have understated the comparison by writing the likelihood of passing a save rather than failing a save in #2. The choice becomes even more obvious when corrected:

1. (2/3*6d) + (1/2*(2/3*6d)) + (1/2*(1/2*(2/3*6d)))

2. (1/2*6d) + (1/2*(1/2*6d)) + (1/3*(1/2*(1/2*6d)))

That is, at least, if you don't make the mistake of assuming the results of the dice rolled to be the same in each operation!

If that happens, then the weighted expected value breaks down into a mere expected utility comparison, and improving the likelihood of any operation is beside the point where we are only concerned with the result and not its likelihood of occurring.

Still, I think the expected utility is a good thing to try and get people's attention away from the likelihood of any utility occurring, and towards the importance of ordering and utility when it occurs.

The fact is that if you roll six hits, you are not going to roll more than six wounds, and your opponent is not going to fail more than six saves. Hitting puts a limit on the utility of wounding, and the utility of causing your opponent to fail saves.

Sigh.

So, what's the unit of your formulae? What do they actually calculate? You're multiplying numbers and then adding them when adding them is meaningless. You still don't seem to understand what your calculations actually yield. The only thing they actually do calculate is successful die rolls, which as I've said isn't particularly useful.

Just to explain that your calculations are pointless, allow me to give you two examples:

1) 6 pulse rifle shots, first against ork boyz in the open, second against terminators.

(1/2*6d) + (2/3*(1/2*6d)) + (1*(2/3*(1/2*6d))) = 7d

(1/2*6d) + (2/3*(1/2*6d)) + (1/6*(2/3*(1/2*6d))) = 5.33d

You're telling me that the difference between ork boyz and terminators, in your "utility" calculation is 7 to 5.3? In reality, you kill six times as many boyz.

2) A more abstract example; a sequence of 1+ 4+ compared to 4+ 1+, using "6d" again:

1+ 4+: (1*6d) + (1/2*(1*6d)) = 9d

4+ 1+: (1/2*6d) + (1*(1/2*6d)) = 6d

Please note -- in the above situation, you roll the exact same dice in both cases, but by your math the utility is different.


For extra credit figure out what happens in your formula if you put in an automatic fail anywhere except the first term; you end up with non-zero "utility" for a sequence of rolls that will never provide results. The reason for this is you're calculating total successful rolls instead of something actually useful.

Let me put it another way:

There's no difference in effect between "roll a die, and if it's 3 or higher, roll another die, and record a wound if it's 4 or higher" and "roll a red die and a blue die, and if the red die is 3 or higher and the blue die is 4 or higher record a wound".

The difference between 3+ 4+ and 4+ 3+ is changing which color you look at when you roll the dice in the second example. Statistically it ends up the same. The expected result is exactly the same. The distribution of successes is exactly the same. There's no difference.

Well done Lowinor I have never liked statistics much, but I thoroughly enjoyed that explanation!

You and Lambadomy also hit the nail precisely on the head, where the nail in question was what was bouncing around in my head: the fact that it doesn't matter if you have a homogenous target unit. If every wound counts as much or as little as the next, it makes no difference. If the wounds are special, ie. e.g. killing a HW man or instakilling a multi wound model, then volume of wounds matters most of all. I wasn't certain why that seemed to be so, so I greatly appreciate you spelling out why my conjecture was accurate

Lowinor:

I'm not arguing the statistics. I'm simply making the point that given the algebraic structure of the shooting operation in 40k that there's more to it than just the statistics.

If we ignore the fact that saving depends on hitting and wounding, and we ignore the fact that wounding depends on hitting, then sure, there's no difference between the cumulative result if we only consider the likelihood of the cumulative result obtaining.

If we consider those facts, then there is a difference between increasing the likelihood of hitting and decreasing the likelihood of saving. The difference is simple and two-fold: the order and the utility of each operation (each of the three sub-operations and the overall operation).

Increasing the likelihood of hitting may increase the number of possible saving throws, but decreasing the likelihood of making those saving throws will not increase the number of possible hits.

Your assertion that "There's no difference in effect between "roll a die, and if it's 3 or higher, roll another die, and record a wound if it's 4 or higher" and "roll a red die and a blue die, and if the red die is 3 or higher and the blue die is 4 or higher record a wound"." is true if and only if we're talking about the likelihood of a wound obtaining.

It is false if we are talking about maximizing the number of wounds obtaining by maximizing either the expected value of hits or the expected value of failed saves.

Still, I gotta admit, I do like the idea of rolling all my hit, wound, and save dice at once and cherry picking the results!

Basically, Nuglitch is saying is that more hits is better than fewer hits.
This is true if the chance to wound and chance to save is fixed. More hits is better.

In THIS comparison, though, improving the chances on the first roll while decreasing the chances on the last or vice versa? The effects of improving the first instead of the second, or the second instead of the first, cancel out and end up the same.

The average, mean, median, standard deviation on the number of unsaved wounds is precisely the same.
That is all that matters. Unsaved wounds. Casualties. Kills. Same in both situations.

There's no magic theory that invalidates the hard data that is so clearly tabulated graphically by Rosicrucian. They may feel better through some people's intuition and gut feeling perhaps, but there is no actual difference in unsaved wounds caused.


The only difference between the two is, as Onlairi noted, is that the new wound allocation rules give you a chance to take out the squad leader or Special/Heavy weapon wielders in a squad. In that, there is an advantage to increasing the odds on beating the To Hit and To Wound rolls rather than the Save. This

Nuglitch wrote:Still, I gotta admit, I do like the idea of rolling all my hit, wound, and save dice at once and cherry picking the results!

That's either a dishonest strawman right there, or it belies the lack of understanding.

More shots is better because you may kill a special weapon or sergeant character.

Although the unit as a whole will die at exactly the same time on average whether you pair high hit with high save or low hit with low save, dumping more dice into the squad and forcing more saves increases the likelihood of the heavy bolter gakking.

Beyond that I agree with Lowinor.

tzeentchling wrote:Nurglich's point was that, given that 4 fire warriors is hardly an exhaustive sample, there's a decent chance that they'll deviate from the mean. I.E., there's more chance that your opponent will roll poorly.

Assume you fire 12 shots. With BS 3, you will, on average, hit with 6. With BS 4, you will, on average, hit with 8. Sometimes this will be more, sometimes this will be less. Now, if we assume complete averages, then yes, 4+ cover vs. 5+ cover will reduce the above to equality. But the whole point is to make them take large numbers of saves. There's always the chance that they'll have a below-average set of rolls, causing them to lose more men.

This is why torrent of fire style armies work, particularly Orks - make them take enough saves, roll enough handfulls of dice, and they'll at some point roll below average and more men will die. This is why it's better to increase your hit rolls than to decrease cover.

Lastly, and non-statistically, I prefer to make my rolls easier rather than making my opponent's rolls harder. I want my dice to work for me!

I never actually used average in any of my posts.

Average is a rather poor man's mathematical tool.