Getting the most out of rerolls and modifiers

[Rogue]

I've been pondering how useful the various buffs are when applied to different units. As in, if i have various characters handing out reroll 1s, reroll everything or +1 to hit auras, who gets the most benefit from what.

 

A couple of things are obvious: reroll all is always better than reroll 1s (except when you hit on 2s, when it's a wash); +1 to hit is always better than rerolls 1s (because one face producing an extra hit is better than one face producing the chance of an extra hit). 

 

But are there any situations where it's better to throw buffs on one set of troops over another?

 

Here are basic chances of hitting naturally, rerolling 1s and rerolling everything (all given in 36ths, for ease of comparison).

 

 

Reroll 1s is simple - the better your chance of hitting in the first place, the more you benefit in absolute terms. However, you always gain a 17% over the base roll. That is, 2+ benefits more but was hitting more anyway, whereas 6+ benefits least but even a single extra hit is a significant increase.

 

Reroll everything is more interesting - models that have a better chance of hitting still benefit more from an individual reroll, but are less likely to generate those rerolls in the first place. For example, 2+ to hit with full rerolls gains 5/36 hits, or just 17% more than they would without rerolls (makes sense - full rerolls is the same as reroll 1s to a 2+ shooter). But 4+ with full rerolls gains 9/36 hits, a full 50% more. And while 6+ only gains 5/36 (same as 2+), that represents an 83% increase. Overall, 4+ gains the most potential hits, which might make it a better target for full rerolls than anything else.

 

I've already pointed out that +1 to hit is always better than reroll 1s. But what about full rerolls? Turns out that full rerolls are better than a +1, unless you hit on 6+, in which case it's marginally better to hit on 5+ base (11/36 vs 12/36).

 

So broadly speaking, full rerolls beats +1 to hit, which beats reroll 1s. So far, so good. But this might help us determine where to put various auras. Let's imagine I have a unit hitting on 2s, and another on 4s - which should I buff with my full reroll aura?

 

 

Here, the unit you choose to buff is on the left, and the unbuffed unit is along the top. The number where they meet is the hits/36 for both units combined. To read the table, choose the 'to hit' score for the unit you choose to buff, and follow a row across - a green box tells you it was the right choice, yellow makes no difference, and red was the wrong call.

 

For example, if you chose to buff the 2+ unit, you were wrong - it's always the worse choice (or at best evens with 6+), because a 2+ unit is already accurate, and gains less from the full reroll aura. But if you chose to buff a 6+ unit, you were wrong too, because a 6+ unit is too inaccurate to benefit much from the rerolls.

 

Oddly, it seems that in this very artificial situation, buffing the 4+ unit is always the best option, followed by either 3+ or 5+ (with it making no difference when the two go head to head).  

 

One more comparison. This time, you can buff one unit with a +1 to hit, and the other with reroll 1s. Which way round do you go?

 

 

Here, the +1 unit os on the left, the reroll unit along the top. In every case, it's better to improve the ballistic skill of the less accurate unit, and reroll the more accurate one. Improving BS is a steady progression, worth 6/36 each time. But rerolls 1s is worth as little as 1/36 to 6+ units, and only 5/36 to 2+ units. That being the case, you get best value by maximising the reroll bonus (sticking on the more accurate unit) and putting the fixed bonus for +1 on the other unit.

 

Now, I suspect that all I've done here is tell people what they already knew anyway. But I enjoyed playing round with the numbers, so there's that.

 

[Alcyon]

I think the numbers are very cool, but I've had a few tonight and all I can think about is how abstract it is. Let's see if I can put some models to numbers. 

 

Mostly I think of Space Marines but for most armies you only have one main ballistic skill. It's the B&C so let's say 3+. Then you have some variation, HQs, Venerable Dreads with BS2 but you weren't buffing those anyway. Then you also have units including vehicles that moved with Heavy weapons, and maybe also units you know will be aiming at a target with -1 to hit them. 

The takeaway as I understand it is that re-rolls (whether of 1s or all) are most valuable on units with a BS of 4, and that +1 to hit is best for units with low BS. I take that to mean units that move with Heavy weapons are best with a Chaplain, whereas you're better off keeping your Captains and Chapter Masters back with units that will be firing from a stationary position. Makes sense when you can buy aura Captains for cheap with minimal equipment, and when the Chaplain comes with a power weapon, invulnerable save and combat-relevant litanies.

[Rogue]

Abstract is my thing :)

 

Okay, here's one that feels more practical. Should I stick a captain or a lieutenant with my shooty fire-base?

 

From a reroll perspective, the answer is: doesn't matter, they both have the same effect.

 

The maths is linked to the opening post. I pointed out that reroll 1s always produces a 17% increase in accuracy (theoretically). We can also represent that with the fraction 7/6, showing an improvement of 1/6 over the baseline.

 

The chances to cause a successful wound can be calculated by multiplying two fractions together (to hit and to wound). So hit on 3s, wound on 4s is 4/6 × 3/6 or 12/36 or 1/3.

 

To add the effect of a reroll 1s aura, we just multiply that result by 7/6, meaning that 1/3 improves to 7/18. But because we're just multiplying three fractions together, the order doesn't matter.

 

In other words, it makes no difference whether we modify the to hit or to wound roll. So a captain and a lieutenant have the same effect.

[Alcyon]

I love that you've got the math down. Ok, so Captain and Lieutenant auras are the same - but their implementation is still different, right? You want a Captain with your gunline when you're facing an army that's tough to hit (or if overcharging plasma, though the Chaplain's +1 to hit is best there.) But I think you will usually prefer a Lieutenant especially when your gunline is full of Bolt weapons. It seems like there's much more variation in enemy toughness, and a Lieutenant could help your Boltguns wound enemies more effectively. Almost all of them are S4 so if you're shooting at other marines you'll need a 4+, making the re-roll 1s most effective there.

 

And then you also can look at number of shots, since each additional shot a model fires makes re-rolls more useful per point spent on the aura. But then many units with lower shot counts do more damage or are more important to hit; but then they also often have their own buffs. For example, you probably don't need a buff character near Eliminators because of the Sergeant's ability, and buffing your Aggressors instead of your Intercessors is probably a better plan - particularly because the Captain and Lieutenant buffs apply in both the shooting phase and the fight phase.

Edited June 10, 2020 by Alcyon

[Rogue]

Yes and no. The effect of the captain and lieutenant can be different, but almost entirely due to their other characteristics - the reroll ability is a mathematical constant, so it makes no difference at all what you're firing at or with (outside of outlying cases like plasma, where the specific overheating rule interacts differently with reroll 1s).

 

It's all to do with the fact that a series of multipliers can be arranged in any order to produce the same outcome. For example, 2x3x4=24; but so does 3x4x2, or 4x2x3. However you arrange that sequence of numbers, it will always come to 24.

 

if we apply that to the maths of the attack sequence, we get the same effect. Let's say that you have a venerable dreadnought firing an assault cannon at a T8 target. 2+ to hit (5/6 chance), then 5+ to wound (2/6 chance). So the basic chance to cause a wound [ie force a saving throw] is 5/6 x 2/6, or 10/36.

 

Now let's apply the captain's reroll 1s aura. This adds 7/6 into the maths, just after the roll to hit: 5/6 x 7/6 x 2/6, or 70/216.

 

What if we use a lieutenant instead? Now it would be 5/6 x 2/6 x 7/6, which is still 70/216.

 

The same applies to your boltgun example. Marines firing at marines is 3+ then 4+, which is 4/6 x 3/6, or 12/36 (or 1/3). With a captain: 4/6 x 7/6 x 3/6, or 84/216 (7/18). With a lieutenant: 4/6 x 3/6 x 7/6, or 84/216 (7/18). Exactly the same.

 

Volume of fire doesn't change things much either - it's just another number to add to the sequence. 10 marines rapid-firing bolters gives 20 shots. Which gives us either 20 x 4/6 x 7/6 x 3/6 or 20 x 4/6 x 3/6 x 7/6. Same thing either way.

 

I think a lot of the problem here comes from a more intuitive approach. As you suggested above, a captain feels like a better fit for a unit throwing out a lot of shots; a lieutenant feels more useful with a unit that needs help making the wound rolls. But that's not supported by the numbers.

 

One more example. Let's say i have 36 dice. Whenever I roll them, they always produce an exactly even distribution - 6 6s, 6 5s, 6 4s and so on. And let's use them to roll out 36 bolter shots again other marines, supported by either a captain or lieutenant.

 

Captain first.

• I start with 36 dice. I roll, needing 3s, and 24 are hits, 12 are misses, of which 6 are 1s. I reroll the 1s (which produce one of each number), gaining a further four hits, for a total of 28 hits.
  
• I roll those 28 dice for wounds, needing 4s. 14 come up as wounds. 
  

Now the lieutenant.

• I start with 36 dice again. I roll, needing 3s, and get 24 hits. I have no reroll to hit this time, so take those 24 dice forward to the wound roll.
  
• I roll 24 dice to wound, needing 4s. 12 come up as wounds, and 4 come up as 1s. I reroll those 4 dice, gaining two extra wounds, for a total of 14 wounds.
  

Either way, we end up with 14 wounds from 36 shots. The captain carries more hits forward, but the lieutenant does more work with the hits that he has. It balances out in the end.

 

At the extreme, 2s to hit and 6s to wound, the captain sees 30 dice hit, rerolls the 6 1s and ends up with 35 hits. Of those 35, he gets almost 6 wounds (5 and 5/6). The lieutenant gets only 30 hits, producing 5 wounds and five 1s, which on a reroll generate almost another wound, for 5 and 5/6 again.

 

Going the other way, 6s to hit and 2s to wound, the captain sees 6 hits, rerolls 6 dice and gets another hit, for 7 hits. of those seven dice, we can average out five wounds over the first six rolls, plus a 5/6 chance of an extra wound off the last die, for 5 and 5/6. The lieutenant gets just the six hits, which produces 5 wounds and a reroll, which generates a 5/6 chance of a bonus wound, for a total of 5 and 5/6 again.

 

Whatever we do, it's just permutations on 5/6, 1/6 and 7/6, and so produces the same outcome every time.

 

The big difference, as you pointed out, is when other rules come into play. Plasma guns are a great example, because the captain's reroll affects their chances of blowing up (reducing it from 1/6 to 1/36), whereas the lieutenant has no effect on this. With plasma, the captain is a much better option. [in a single round of shooting, the outcome in terms of wounds caused would still be the same. However, the captain would have kept more plasma gunners alive for future rounds. Which is why he's better.]

 

All of this math is also predicated on you using either the captain or the lieutenant on the same unit. If you're choosing between different units to support with a single character (such as aggressors or intercessors in your example), then the decisive factor is now the differing output of the units. But that's a whole other kettle of numbers.

Edited June 11, 2020 by Rogue

[Rogue]

Bonus thought: if you have exploding 6s on your to hit roll, go with a captain - maximising the number of hit rolls you make works to your advantage.

 

But on the other hand, if you have exploding 6s on your to wound roll, it makes no difference if you have a captain or lieutenant - the captain gives extra to wound rolls (because more hits went through), and the lieutenant gives the rerolls; the two have the same effect overall.

[Alcyon]

My mind's a little blown. This is like the Monty Hall problem almost, the intuitive approach completely fails just like you said. Very cool, thank you! Now all I see is numbers, you know, blonde, brunette, redhead...

[Rogue]

this one might be a bit more practical. I've been thinking about charge distances (although the maths will work just as well for any 'roll two dice and add them together' situation, like psychic tests).

 

Here, I've looked at four options: the base odds of making a charge at any distance; the odds if you have +1 to charge; if you can reroll one of the dice; if you can reroll both dice. The results are all presented as percentages.

 

 

The yellow box in each row indicates your best option (assuming you have any kind of choice). The box with a border in each column indicates the point at which the odds go over 50%, meaning that you're more likely to make a charge of that distance than fail it.

 

It throws up a few interesting things:

 

As a general rule, a +1 to charge is less useful than a reroll. However, a +1 is most useful when looking for either a 12" charge (because trying to score an 11 is easier than fishing for multiple sixes) or a 3" charge (because the +1 makes it a guaranteed thing).

 

[side note: a +2 to charge is always better than rerolling one dice, and almost always better than rerolling both - turns out you're more likely to get a 6" charge when rerolling both than you are to get a natural 4" charge, so for a 6" charge, keep the rerolls. +3 to charge always beats any kind of reroll.]

 

Rerolling one die is more useful for long charges - it gives the best odds of making anything from 8" to 11", and is better than a full reroll at 12". Conversely, rerolling both dice is better for short charges - it's best for anything in the 4" to 7" range, and fractionally better than rerolling one die at 3". (However, it's important to note that this is a statistical generalisation, with a big caveat that I'll come to in a minute...)

 

So, from a deep strike perspective, the ideal option would be finding a way to reroll one of the charge dice (such as a CP reroll). This gives you a better than evens chance of making even a 9" charge, compared to the base odds of little better than 1 in 4. In fact, rerolling 1 die is the only modifier that achieves better than 50% for the 9" charge. [Practical example: genestealer cults tend to like the clamavus (+1 to charge) for deep striking; turns out the Augur warlord trait (reroll charges) is better - 42% vs 48%.]

 

But here's the big caveat. In real life, optimal rerolling depends on your actual dice roll. For example, the option to reroll one die gives a 52% chance of landing a 9" charge, which is the best odds available. But if you roll a double 1s, then rerolling just one die is no help at all. In that specific situation, rerolling both is way better, because it has at least a chance of still coming good (a 28% chance, but still).

 

Or, as another example, you're looking for a 7" charge but roll a 5 and a 1. Overall, a full reroll of the dice is the best odds for a 7, but once the dice have fallen, it's now better to just reroll the 1 - all you need is a 2, which is an 83% chance on one die, and so is better odds than the 58% chance of rolling a 7 with both.

 

Here's how that plays out:

• if you can reroll one die needing a 2 (83%), that's your best option for charges of  6" and higher; it's break even on 5", and you're better to roll both again for 4" or less. 
• if you can reroll one die needing a 3 (67%), that's your best option for charges of 7" and higher, but for 6" or less, it's better to reroll both.
• if you can reroll one die needing a 4 (50%), that's your best option for charges of 8" and higher, but for 7" or less, it's better to reroll both.
• if you can reroll one die needing a 5 (33%), that's your best option for charges of 9" and higher, but for 8" or less, it's better to reroll both.
• if you can reroll one die needing a 6 (17%), that's your best option for charges of 10" and higher, but for 9" or less, it's better to reroll both.

For example, let's say you're looking for a 10" charge but roll 4 and 4. in this case, you should reroll one of them, as your chances of a 6 on one die (17%)  is better than your chances of rolling a 10 or better on 2 dice (16.7%). However, if you only needed 9" and rolled 4 and 4, your 28% chance of getting a 9 with both dice is the better option. And of course, if you roll a 4 and 4 when looking for a 12" charge, then a full reroll is the only option. 

 

Obviously, this only matters if you have the choice (ork players, looking at you) - but as has been pointed out already, this is more about playing with numbers than providing practical advice.

[Lukoi]

I don't think you're considering the strength of weapon  vs toughness of target issue.  My ballistic skill of 2, or 3, or even 4 is much more impacted by the wounding phase due to that on average than the flat, level playing field model I believe you are using here.

 

Given the variety of targets you typically face (ranging in strengths, dramatically impacting your wound rate), I find that rerolls of WR1 are more valuable, than HR1 rerolls BUT if I have the ability to take a CM, that definitely superseds the WR1 reroll of the LT.  For me, that translates to -- my smash captain moving forward is a CM, and my LT stays with my marine gun line.  

 

Obviously other factions have their own version of these dynamics.

 

If I was a pure gunline army and could fit EVERYONE in a 6" (or 9" if your faction has the ability to grow the bubble), I'd definitely take one CPT, one LT - but given that's hard to pull off, I'd use two LT's in most situations across the breadth of my positions.

[Rogue]

I am considering it. If you back up to my post on 11th June, I go through it in detail. But more specifically, the issue of weapon strength versus target toughness is addressed in the wound roll - that's entirely what the wound roll is for, after all. Whatever the weapon, and whatever the toughness, it all boils down to a single roll on a d6, with a success probability of 1/6 to 5/6. In that sense, it's the same as a to hit roll. It's just that a to hit roll is broadly a given (BS 3+, for example) which is then modified by various unit abilities; whereas a wound roll is determined on a case-by-case basis. Either way, the re-roll 1s ability has the same effect.

 

It's not about a level playing field or otherwise. It's about reducing both rolls (to hit and to wound) to their simplest form, which is always n/6.

 

I think you're making the same intuitive mistake that most of us make - that the value of a to hit or to wound re-roll varies depending on the situation and the numbers required, but the maths is that it doesn't. Hitting on 3s, wounding on 4s produces the same number of wounding hits with a captain as it does with a lieutenant. As does hit on 4s, wound on 3s, for that matter.

 

You're correct to say that a Chapter Master out-performs a lieutenant, but that's an unbalanced comparison. Full re-rolls always beats re-roll 1s (unless you only needed 2s, in which case they're at parity), so a chapter master is always a better option than a lieutenant (which is why you pay more for them).

 

And yes, if you're looking to bolster a gun-line, scattering lieutenants along the line is cheaper than captains, and has the same effect re-roll-wise (although that's ignoring any additional benefits the captains might bring, but if they just hide in ruins and do nothing but give re-rolls, they're effectively the same). And it's also true that having a captain and a lieutenant is better than having either one alone - re-rolling to hit and to wound will produce more successes than only re-rolling one or the other.

 

 

Here's some maths to back me up, using my imaginary dice that always roll with a perfect distribution.

 

Hit on 3s, wound on 4s, captain (re-roll 1s to hit)

36 dice produce 24 hits and six 1s, which produce 4 more hits for a total of 28 hits. Those 28 hits produce 14 wounds.

 

Hit on 3s, wound on 4s, lieutenant (re-roll 1s to wound)

36 dice produce 24 hits. Those 24 hits produce 12 wounds and four 1s, which produce 2 more wounds for a total of 14 wounds.

 

Hit on 4s, wound on 3s, captain (re-roll 1s to hit)

36 dice produce 18 hits and six 1s, which produce 3 more hits for a total of 21 hits. Those 21 hits produce 14 wounds.

 

Hit on 4s, wound on 3s, lieutenant (re-roll 1s to wound)

36 dice produce 18 hits. Those 18 hits produce 12 wounds and three 1s, which produce 2 more wounds for a total of 14 wounds.

 

The underlying maths here is 4/6 x 3/6 x 7/6 (where chance to roll 3+ is 4 in 6, chance to roll 4+ is 3 in 6, and effect of a re-roll 1s is 7/6 (maths up-thread)) - no matter which order you put those three fractions in, the result is always the same: 84/216 or 14/36.

[Lukoi]

Here's where you and I arent in synch and why I discuss value.

 

You are more likely going to see increased use of the wound reroll due to the strength differences.

 

My ability to land a hit is uniformly dependant on my bs. My ability to wound however, is predicated on the added variable of strength v toughness.

 

Yes they both allow for identical 1/6 improvements, but the timing actually does matter.

 

Go look at your actual matches and look at opponent s and aggregate some data on toughness of targets, and you'll likely come to your own conclusion.

 

Mathematically, at least for me, reroll WR1s for me has a greater impact based on the opponents I've faced thus far.

 

I definitely like the math exercise in the abstract, I'm just trying to apply it practically to boot.

 

Admittedly, I'm a SM player and our 2, 3 BS skills generally mean that we are largely in good shape to hit anyway. Maybe it would matter more if I were an IG player for example. Or maybe it wouldnt matter at all, given that to hit is largely kept to just the variables of BS vs buffs/debuffs.

[Daynga-Zone]

Maybe I’ve missed the math along the way, but wouldn’t reroll 1’s be more valuable on whichever characteristic is more likely to hit or wound? For example if you have a model with BS 2 that rerolls 1’s any 1’s have a 5/6 chance of converting to a hit whereas let’s say that weapon is S 5 vs T 7 so it wounds on 5’s meaning reroll 1’s to wound would only convert to a success 1/3 of the time.

 

Generally it’s easier to find higher BS weapons than S weapons so rerolling 1’s to hit would be more valuable on a more likely to hit weapon and reroll 1’s to wound would be more valuable on a high S weapon with a lower chance to hit?

[Rogue]

Here's where you and I arent in synch and why I discuss value.

 

You are more likely going to see increased use of the wound reroll due to the strength differences.

 

My ability to land a hit is uniformly dependant on my bs. My ability to wound however, is predicated on the added variable of strength v toughness.

 

Yes they both allow for identical 1/6 improvements, but the timing actually does matter.

 

Go look at your actual matches and look at opponent s and aggregate some data on toughness of targets, and you'll likely come to your own conclusion.

 

Mathematically, at least for me, reroll WR1s for me has a greater impact based on the opponents I've faced thus far.

 

I definitely like the math exercise in the abstract, I'm just trying to apply it practically to boot.

 

Admittedly, I'm a SM player and our 2, 3 BS skills generally mean that we are largely in good shape to hit anyway. Maybe it would matter more if I were an IG player for example. Or maybe it wouldnt matter at all, given that to hit is largely kept to just the variables of BS vs buffs/debuffs.

 

I don't think you can go with 'Mathematically, at least for me...'. The maths is either right or it isn't - it's not subjective. At that's sort of what I'm getting at here. Our opinions, our subjective views, tend to push us in one direction or another. Your view is that wound re-rolls are more useful. Anecdotally, you might have found this to be true, because very few of us play exactly average games all the time. But is doesn't change the fact that, mathematically, any use of a single re-roll 1s aura during the hit and wound process has the same overall effect. 

 

Take a look at the numbers I ran at the end of my last post. In all four cases, regardless of the ordering of the 4+ and 3+ rolls, and regardless of whether I factored in a captain or lieutenant, 36 dice produced 14 wounds. This isn't a carefully selected fluke - it holds true for any three numbers multiplied together, which is what we're doing here.

 

You're also making the intuitive mistake with the difference between hitting and wounding. You seem to think that to hit is a fixed point, whereas to wound is variable, making re-rolls to wound more valuable. But both are, mathematically, just chances to succeed out of six. Marines, with their 3+ BS, usually hit 4 in 6. But that can vary - a signum will make it 5 in 6; an aircraft will reduce it to 3 in 6. That's the same range as a krak missile - put it into infantry, wound on 5 in 6; light vehicles (T6/7), it's 4 in 6; heavy armour (T8), it's 3 in 6. As a player, you may have the skill and tactical awareness to set things up in your favour, but that's player ability, not maths.

 

Strength versus toughness isn't an added variable - it reads as more complicated because you have to compare the two to produce the number you need to roll, but in the end, it reduces to a single variable, just as the hit roll does. I need to roll one number to hit, and one number to wound.

 

Fair enough, I tend to work in the abstract. But this does have a practical application, which is: stop worrying about which re-roll 1s aura you have - it doesn't matter. 

[Rogue]

Maybe I’ve missed the math along the way, but wouldn’t reroll 1’s be more valuable on whichever characteristic is more likely to hit or wound? For example if you have a model with BS 2 that rerolls 1’s any 1’s have a 5/6 chance of converting to a hit whereas let’s say that weapon is S 5 vs T 7 so it wounds on 5’s meaning reroll 1’s to wound would only convert to a success 1/3 of the time.

 

Generally it’s easier to find higher BS weapons than S weapons so rerolling 1’s to hit would be more valuable on a more likely to hit weapon and reroll 1’s to wound would be more valuable on a high S weapon with a lower chance to hit?

 

The math for that one is the very first thing at the top of the the thread. The curious property of re-rolling 1s is that it has the same chance of kicking in whatever you're trying to roll. That is, if you were to roll 36 dice, you'd expect six of them to rolls 1s (on a perfect distribution) - that's true whether you're looking for 5+ or 2+ to hit. Those six dice produce bonus hits in proportion to your original chance to hit. So models hitting on 2+ get the most benefit (expecting five of those 1s to become hits second time round), whereas models hitting on 6s only expect one additional hit. 

 

Oddly, it means that everyone expects to improve by 17% or 1/6. the 2+ guy gets 30 of 36 hits, then adds five more (a one in six improvement). The 6+ guy gets 6 of 36 hits, then adds one more (also a one in six improvement). Proportionally, the increase is consistent.

 

For a one-off set of rolls, you're right, using your re-roll 1s aura on more accurate models is advantageous - that 2+ shooter gains 5 extra hits, compared to the 6+ guy who just gets 1. But the problem I've been addressing above isn't about one roll in isolation. It's about the relationship between to hit and to wound (which makes sense, hopefully, because we have to do both whenever we shoot or attack). 

 

Let's push it t the extremes:

 

Hit on 2s, wound on 6s, captain (re-roll 1s to hit)

36 dice produce 30 hits and six 1s, which produce 5 more hits for a total of 35 hits. Those 35 hits produce 5.83 wounds.

 

Hit on 2s, wound on 6s, lieutenant (re-roll 1s to wound)

36 dice produce 30 hits. Those 30 hits produce 5 wounds and five 1s, which produce 0.83 more wounds for a total of 5.83 wounds.

 

Hit on 6s, wound on 2s, captain (re-roll 1s to hit)

36 dice produce 6 hits and six 1s, which produce 1 more hit for a total of 7 hits. Those 7 hits produce 5.83 wounds.

 

Hit on 6s, wound on 2s, lieutenant (re-roll 1s to wound)

36 dice produce 6 hits. Those 6 hits produce 5 wounds and one 1, which produce 0.83 more wounds for a total of 5.83 wounds.

 

As always, the ordering of the three elements doesn't matter (because multiplication). What we see each time is that the captain generates more hits, but they're less effective. The lieutenant gets less hits, but does more with them.

 

Put aside the re-roll for a moment. if we need a 2+ then a 6+, it's 5/6 x 1/6, or 5/36. But if we need 6+ then 2+, it's 1/6 x 5/6, or 5/36. Same either way. All the re-roll does is add a 7/6 multiplier, and because it's a multiplier, it doesn't matter where it goes in the order. So it has the same effect whether you apply it to the hit roll or the wound roll.

 

I'm genuinely happy to keep talking about this for ages, but I'm worried that my explanations just aren't very clear. So feel free to come back at me if I'm not making any sense.

[Alcyon]

Rogue is still correct here. Re-rolls to wound don't improve low Str weapons performance against high toughness models any more than re-rolls to hit. I was skeptical too but I checked the same math against wounding on 5s, as you'd have shooting Bolters against Inceptors or something.

 

Hit on 3s, wound on 5s, captain (re-roll 1s to hit)

36 dice produce 24 hits and six 1s, which produce 4 more hits for a total of 28 hits. Those 28 hits produce 9 wounds.

 

Hit on 3s, wound on 5s, lieutenant (re-roll 1s to wound)

36 dice produce 24 hits. Those 24 hits produce 8 wounds and one 1, which produces 1 more wound for a total of 9 wounds.

 

It applies vs. toughness 8, too. No matter how you slice it, re-rolls means more chances to hit, which then successfully become wounds, or fewer chances to hit but those hits are more likely to wound.

I wonder if this makes more sense if you think about it in terms of the homogenized stat lines of this edition, and the fact that we increasingly see people firing more shots (i.e. taking Onslaught Gatling Cannons and Aggressors) instead of shelling out for higher Strength or AP weapons. For example take the Redemptor's weapons:

 

Gatling fires 12 shots, making 8 hits, causing 5 wounds. With re-rolls, you get one additional hit and one additional wound for a total of 6 wounds.

Plasma fires D6 shots, making 2 hits, causing 2 wounds. With re-rolls, you get slightly closer to 3 hits, causing very slightly more than 2 wounds.

 

It seems like whichever re-roll you choose, it's more effective with more shots, and higher Str and even AP doesn't seem to quite offset that. The gap narrows against units like Centurions but the tiebreaker isn't big enough to make a difference.

 

In fact the more I screw around with the Mathhammer calculator  the more I'm surprised by just how similar the results are between weapons; 3 Las-Fusil Eliminator shots is basically the same as 10 Hellblaster shots against a vehicle but they do not cost the same in points. It's been a while since we saw that Space Marine Weapons Chart (back in 2017!) and I think that really showed the strengths and weaknesses of stats vs. what our intuitive beliefs were. 

[Lukoi]

Yup you guys are right. As I am combing thru some match stats, I realize I left debuffs in there and those definitely skew the results (in my case) towards the WR1's being a better option.

 

I think I need to look at how many debuffs/buffs there are and try to ascertain how, generally, common they might be to employ to inform my decision on this sort of thing.

 

The only edge that I can see now is cost, and other pertinent roles...with a gun line buffer, seems like LT wins, as it's slightly cheaper an option. For a melee pack, seems like smash captains value makes that the pick.

[Alcyon]

Now that we've established the Captain and Lieutenant auras are functionally identical, I can't escape the question of whether there might be cut-and-dried situations where re-rolls are more or less useful, based on equivalent point totals. Like say you have a few equivalent units points-wise (using 8th ed totals here):

 

10 Intercessors vs. GEQ, 20 shots resulting in 13.3 hits, 8.8 wounds, 5.9 go unsaved. Re-roll 1s gives you 0.988 more wounds.

5 Hellblasters vs. MEQ, 10 shots resulting in 6.6 hits, 5.5 wounds which go unsaved. Re-roll 1s gives you 0.981 more wounds.

3 Centurions w/ Grav/Hurricanes vs. MEQ, 30 shots resulting in 20 hits, 11.3 wounds with 6.4 unsaved. Re-roll 1s gives you 1.119 more wounds.

 

What conclusion could we draw here? Obviously there are a lot more factors at work, the intercessors have 20 wounds vs. 10 and 12, Centurions have T5 and 2+ saves, but is there a trend in terms of weapon effectiveness? It's not more shots = more effective re-rolls, but improved AP does seem like it maybe has an effect?

At the same time, it's approximately one additional wound per entire unit per shooting phase, which makes me feel like maybe I'm overrating the impact of re-rolling 1s. Perhaps we should all be playing Chapter Masters and Chaplains only rather than mucking about with Captains and Lieutenants - they might not really be doing enough to justify their points costs.

Going back to Rogue's original post, let's try a different example. Rather than comparing Captains and Lieutenants, let's compare Chapter Masters and Captains across a few units:

 

10 Intercessors vs. GEQ, we have 5.9 unsaved wounds or 6.914 with re-rolling 1s. Full re-rolls give us 7.9, plus two wounds!

5 Hellblasters vs. MEQ, we have 5.5 unsaved wounds or 6.481 with re-rolling 1s. Full re-rolls give us 7.4, plus two wounds.

3 Centurions w/ Grav/Hurricanes vs. MEQ, 6.4 unsaved wounds or 7.519 with re-rolling 1s. Full re-rolls give us 8.59, plus two w.

 

Seems like Chapter Masters consistently grant about two full additional wounds. What about Chaplains with Recitation of Focus?

 

10 Intercessors vs. GEQ, 5.9 unsaved wounds becomes 7.4 with +1 to hit, an increase of 1.5.

5 Hellblasters vs. MEQ, 5.5 unsaved wounds becomes 6.94 with +1 to hit, an increase of 1.4.

3 Centurions w/ Grav/Hurricanes vs. MEQ, 6.4 unsaved wounds becomes 8 with +1 to hit, an increase of 1.6.

 

Kind of in the middle, which makes sense given the comparable points costs of these HQs, except that you have to remember that Lieutenants don't have a 4+ invuln, and the Chaplain trades a rifle for a power mace and heavy bolt pistol.

 

Rogue, you see anything interesting in this? Something I'm missing? There are some spicy articles on similar questions on Goonhammer but they similarly lack the type of conclusions or practical advice that would convert these stats to in-game action.

 

Edit: I guess for clarity, the kinds of conclusions I'd love to see would be things like: if you have X number of shots/are shooting at Y type of target, you always want to pick Z type of HQ to buff the unit. Like, is it points-efficient to buff Intercessors versus GEQ, or do Hellblasters need more help versus MEQ? Are there certain weapons that particularly struggle against their ideal targets moreso than others, which a buff might make effective? To know this I feel like we have to get a fresh copy of that 3plusplus chart. I messaged them to ask if the author can re-release it.

Edited July 2, 2020 by Alcyon

[Daynga-Zone]

Ah, thanks for explaining it again. It clicked much better that time.

 

One annoying note on the Chaplain are the limitations he brings with needing a 3+ to pass his litany which may not be able to be rerolled in 9th. That +1 would only be happening 66% of the time without the warlord trait. ~88% with a reroll.

 

How does chapter master compare to having captain and lieutenant rerolls?

[Rogue]

If you were to run a captain and lieutenant buffing the same squad, their combined benefit would be 7/6 x 7/6, or 49/36, which is just over a 1/3 increase in successful wounds overall (36%).

 

A chapter master is more difficult to quantify, as his benefit varies depending on the BS of the buffed unit (it's the first diagram at the top of the thread).

 

With a BS2+ unit, like a venerable dreadnought, the chapter master is less effective than the captain/lieutenant combo (because the CM is effectively providing just captain-eqivalent re-roll 1s in this case).

 

At BS3+, the CM gives a 33% increase, which is marginally worse than the captain/lieutenant combination (36%), but only just (48/36 vs 49/36).

 

It's from BS4+ and worse that the CM takes over, giving a 50%, 67% or even 83% increase in the success rate. However, marines only fire at 4+ when faced with penalties to hit.

 

Overall, the captain/lieutenant combination is superior at BS2+, and just edges it at 3+, but the CM takes over from 4+.

 

So we should be using the CM if we expect our critical units to be suffering penalties to hit (against Eldar flyers, for example), but stick with the dynamic duo the rest of the time.

 

[For completeness, the CM is always at least as good as a captain, and usually better. And a CM with a lieutenant is always the best option - it'll always out-perform a lone CM due to the extra re-roll, and is never a worse option than the captain/lieutenant pairing.]

[Lukoi]

With the new terrain rules in 9th, isnt there an increase to hit penalties more often now?

[Rogue]

@ Alcyon

 

Those practical examples are useful. They reinforce that original bit of maths, that full rerolls (Chapter Master) is better than +1 to hit (Focus), which is better than re-roll 1s (captain/lieutenant). They also show that, for those units at least, the points are pretty well balanced, given how consistently even the results are against targets of choice.

 

Talking of consistency, we can notice that the centurions always get the most wounds, then the intercessors, then the hellblasters. The order is fairly arbitrary, and unimportant in itself. What's interesting is that whichever buffs we apply, we don't don't elevate one unit over another. In other words, the buffs themselves are consistent. We'd expect that, given that they can generally been reduced to a single integer (7/6 for re-roll 1s, for example), but good to see it in practice again.

 

Which suggests to me that looking for the 'best' buff to apply in any given situation may not be the way to go. Any unit will benefit most from the full re-rolls, then the +1, then the re-roll 1s.

 

Instead, perhaps we should look at the potential damage output. Think of it like this - if unit A will cause 30 wounds base, and unit B will cause 15 wounds base, then any buff applied to unit A will generate twice as many additional wounds. So if both units are firing into a screen of pox-walkers, we're better off buffing unit A (assuming BS3+ here, a chapter master gives A an extra 10 wounds, and B a extra 5, so we choose A to buff). How do we increase damage output?

 

Going back to something that Lukoi raised earlier - strength versus toughness. Whilst I disagreed with him that it was more of a variable from a mathematical point of view, he's right that it's the part of the shooting process that has the greatest variability. BS is linear - a +1 to hit increased your success rate by 1/6. Likewise AP - each -1AP increases your success rate by 1/6 too. But +1 strength might not make any difference at all (strength 5 and 6 both need 5+ against toughness 7, for example). So this is the area where we have the most tactical decisions to make. 

 

Back to the pox-walkers. Let's imagine that there are two units in the screen. One is near enough to Typhus to get the toughness buff; the other isn't. Whichever unit I target, the to hit roll is the same, and the save is the same. But we'll get more kills (and more bonus kills) by putting fire into the T3 squad. It's obvious, but I'm rubbish at generating practical examples. 

 

Maybe this is a better example. We have two baseline identical units to fire at - let's say plague marines. One has a toughness buff, from 4 to 5; one has a -1 to hit. I'm using a strength 6 weapon, AP-2. Which do we shoot at? The save is consistent, so we can ignore that. The -1 to hit reduces our chance to hit by 1/6. But the toughness increase make no difference - our S6 weapon wounds both squads on 3s. So in this instance, shooting at the tougher squad gets us more wounds (and again, more advantage from any buffs).  

 

I'm not sure if this is answering your question, Alcyon, but maybe it's a start.

[Alcyon]

Thank you, Rogue, and again let me share my appreciation for you taking the time to work this out and share it with us.

 

I think you and Lukoi are honing in on the best way to extract conclusions from this work. You said that whichever unit starts out inflicting the most wounds is going to benefit more from the buffs. From this we might be able to conclude that you're better off buffing units shooting at things they are already-well suited to shooting at, but would be unlikely to wipe without a buff, and in contrast it's a bad investment to try using HQ auras to buff units that are unsuited to their targets.

 

For example, you could try running a CM with your 10 Bolt Rifle Intercessors to make them more effective against MEQ, but that nets you less than 1 wound improvement. You're better off shooting those Intercessors at GEQ because that improves their effectiveness by almost 3 wounds. 

 

This seems kind of obvious when I say it but it's worth repeating I think that you should always be buffing the units in your army with the most potential damage output for the target. You get a better return on your investment in both the HQ and the unit you're buffing. Therefore if you have no choice but to include buff auras in your army (mandatory HQs) you might decide it's more effective to have a range of weapons that covers all potential targets, and for the most expensive weapons to be buffed, than it is to have a basic weapons that can ping wounds off different types of targets through a high rate of fire.

 

Your comments about Str vs. T also refocus list-building decisions on the comparative Str stats of weapons. This is one area where I think plasma suffers quite a bit as either the baseline stats wound on 3s at best, or even overcharging doesn't help you down to 2s.

[Rogue]

No worries - thanks for making it a conversation instead of a monologue :)

 

I'm coming to the same conclusions, I think. Re-rolls aren't enough to make a unit into something it isn't. But they will make a unit more efficient; better at doing what it can do. And if we can maximise the efficiency, we're ahead of the game. Marines putting bolters into tanks won't achieve much; a captain will improve things slightly, but because it's a proportional increase, it's not a big improvement. But the same marines putting the same bolters into neophytes will see a much greater benefit from that same captain.

 

This cuts the other way too - lascannon are ideal against tanks, as they usually wound on 3+, remove most of the save and deal multiple damage. Turn them on light infantry, however, and we lose efficiency. True, they'll kill things; they'll wound guardsmen on 2+ with no save. But so will a S6, AP-2, D1 weapon. Much of the value of the lascannon is lost, as you're paying for +3 strength, -1 AP and a d6 damage you aren't using. And re-rolls won't save that.

 

Plasma is in an interesting position. S7 is 2+ to wound against guard, 3+ vs marines, 4+ with rhinos and 5+ against a russ. If we overcharge, we're still 2+ against the guard, but are now 2+ against marines too (and killing a primaris every shot), 3+ on the rhino and 4+ vs the russ (with double damage on both tanks, which is handy too). It's toughness 5 and 6 that's in the plasma blind-spot, with both firing modes wounding on 3+. But even then, overcharging means one-shotting most primaris marines, or forcing multiple FNP rolls, so it can still be useful.  

[Alcyon]

One thing I thought of just now - when comparing Captain re-rolls to Chapter Masters when overcharging plasma, I wonder if there's appreciably increased risk in re-rolling all misses rather than just ones. Say you roll to hit on 3+ for 10 plasma shots; about 1.6 should roll ones, right? Then those re-rolls each have another 1 in 6 chance of killing your Hellblaster/Inceptor. But another 1/6th of the initial shots rolled as 2s, so you now re-roll 3.2 dice each of which has another 1 in 6 chance. So a Chapter Master actually makes overcharging slightly riskier, right? Is that enough to make us consider a Captain instead?

[Rogue]

This is true. A captain gives a 1/36 chance of overheating - it's pretty much the ideal buff for plasma. A chapter master has the option to re-roll all fails, so can choose to only re-roll 1s for plasma shots (effectively being a captain). However, if we do elect to re-roll all failed hits, the odds of overheating increase to 2/36 for BS3+ (and, as it happens, 3/36 for 4+; 4/36 for 5+; and 5/36 for 6+, which is somewhat satisfying).

 

But if we stick with BS3+ for a moment, the chapter master is also generating an additional 8/36 hits (compared to the captain's 4/36). So it's a trade-off. Across 36 overcharged plasma shots, the base rate is 24 hits and 6 deaths; a captain gets you 28 hits and 1 death; the chapter master gets 32 hits and 2 deaths. 

 

So a fully re-rolling chapter master is clearly better than no re-rolls at all, in both metrics (hits and overheat deaths). But compared to a captain, you have a decision to make - do you want to maximise hits, at the cost of more dead marines; or will you sacrifice some accuracy to preserve your troops (and maintain future firepower capacity)? 

 

[This isn't really a narrative-type thread, but I do like the idea that a captain is focused on preserving the men under his command, whereas a chapter master can see the bigger picture, and knows when to risk those men to drive home an attack.]

 

 

A second interesting question is about to become irrelevant. As I understand it, in 9th edition plasma will only overheat on unmodified 1s; but in 8th, it was 1s after modifiers, so -1 to hit meant that 2s would overheat too. How does that affect our re-roll bubbles?

 

Let's say we're now firing our 36 (perfectly distributed) plasma shots at -1 to hit. Without re-rolls, we score 18 hits and 12 deaths, with only 3s being a safe miss; that's our baseline. With a captain, we can re-roll the natural 1s, gaining 3 more hits but still racking up two deaths; we can't do anything about all the 2s, which leaves us with a total of 21 hits and 8 deaths. And a chapter master, re-rolling everything, gains 9 more hits at the cost of six deaths, for a total of 27 hits and just those 6 deaths. Suddenly, the full re-rolls of the chapter master are better all round - more hits, less overheating.

 

But chapter masters can be clever, so we could instead choose to only re-roll the 1s and 2s. If we do that, we gain 6 extra hits and just four deaths, for a total of 24 hits and 4 deaths. It's less hits and less death, so it's another tactical decision. It also gives us a 6:1 hit:death ratio, which compares favourably with the 4.5:1 for full re-rolls, 2.6:1 for re-roll 1s, and 1.5:1 for nothing. It's the ability to re-roll the 2s that makes the chapter master more effective here, coupled with the option to play safe and keep the 3s, or gamble them for more hits at the cost of more overheats.