AOS Mathhammer - Judicators v. Vanguard Raptors

I'm creating a Stormcast army and I'm trying to decide between Judicators and Vanguard Raptors with longstrikes.

EDIT - My original calculations did not take into account the shockbolt bow nor the +1 to hit for the judicator prime, both of which will lower the Judicator's scores.  I've updated below.

EDIT 2 - I've fixed two errors in the text below pointed out to me by C4st3r. There happens to be a lot of text removed (but luckily, very little of the math). How embarrassing for me.

Judicators v. Vanguard Raptors

So the initial calculations against no armour is as follows:
Judicators (5) - 2/3 to hit, 2/3 to wound, 1 damage, 4 shots + 5/6 to hit, 2/3 to wound, 1 damage, 3.5 shots = 16/9 + 17.5/9 wounds = 33.5/9 = 67/18
Vanguard Raptors (3) -  4/6 to hit, 2/3 to wound, 2 damage, 3 shots = 8/3 wounds PLUS 1/6 to hit * 2 mortal wounds, 3 shots = 1 MW

Against armour: Judicators have -1 rend, and the VR have -2 rend.

Against 3+ armour, Judicators will do 33.5/18 (=1.86) wounds and VR will do 16/9 + 1 (=2.78)
Against 4+ armour, Judicators will do 134/54 (=2.48) wounds and VR will do 40/18 + 1 (=3.2)
Against 5+ armour, Judicators will do 335/108 (=3.10) wounds and VR will do 8/3 + 1 (=3.67)
Against 6+ armour, Judicators will do 67/18 (=3.72) wounds and VR will do 8/3 + 1 (=3.67)

Unsurprisingly (given their higher rend), VR will consistently do more damage against armoured targets. Against:
3+ armour, they do 1.49 times more damage,
4+ armour, they do 1.29 times more damage,
5+ armour, they do 1.18 times more damage, and
6+/- armour, they do 0.99 times more damage.

They are also 1.25 times more expensive (180 v 160 pts) which means that, in terms of damage output, Vanguard Raptors are a more efficient use of points when facing 3+ and 4+ armour and Judicators are a more efficient use of points when facing 5+ and 6+ armour.

Additional Considerations

Three KEY considerations that aren't included in the above analysis:

1. RESILIENCE - Judicators have 10 wounds instead of VR's 6. So if your squad takes 5 wounds, the Judicators are still going to be pumping out over 75% of their full strength damage and the VR squad will be pumping out only 33% of their full strength damage. If the squad takes 6 wounds, the Judicators are still dishing out approx 65% of their full strength damage and your VRs are wiped out.
2. BATTLELINE - Judicators happen to be battleline in any Stormcast army, whereas you need a Lord Acquilar to make your Vanguard Raptors battleline. Vanguard Raptors are never battleline. This can be a big deal depending on the rest of your army (i.e. you need to fill out battleline slots).

EDIT - The below is not true. The easily accessible +1 to hit bubble in the SE book is the Lord Celestant on foot, but his bubble only applies in the combat phase (and these guys obviously shoot in the shooting phase.  Keeping the text below in case you have access to a +1 to hit from some other source. 

1. SYNERGY - By combining with a +1 to hit bubble, the VRs will do an average of one extra MW per turn (which is pretty awesome). However, they'll do a little less damage through regular wounds (since the Rule of One states that a natural roll of one before modifiers fails). So although the VRs will do an average of 2 MW per shooting phase, they'll do 25% less damage otherwise (because 4/6 to hit becomes 3/6).  Note that under these new percentages (set out below), the VR's do 1.29 times more damage to 5+ armour and 1.08 times more damage to 6+ armour. It's still increasing their damage output as follows:

Against 3+ armour, VR will do 12/9 + 2 (=3.33) (increase of about 20%)

Against 4+ armour, VR will do 30/18 + 2 (=3.67) (increase of about 15%)
Against 5+/6+ or no armour, VR will do 6/3 + 2 (=4) (increase of about 9%)

Lord Celestant (on foot) synergy

So the synergy is pretty great, but is it worth it? I'm going to do a little thought exercise.  Against 3+ armour, the Lord Celestant is worth about 36 pts per phase of shooting at full strength. Against 4+ armour, this becomes 30 pts and against 5+/6+ this becomes 16 pts. A Lord Celestant costs 100 pts. So, approximately, the Lord Celestant justifies his cost (on this alone) if you can boost a full strength squad of VRs through 3 shooting phases against 3+ or 4+ armour (or three full strength squads of VRs through one phase).

We shouldn't ignore the LC's synergy with the Judicators. He won't effect the prime (who already hits on 2+), but the others will do 25% more damage (because a 4/6 to hit becomes a 5/6 to hit) resulting in a total of 37.5/9 instead of 33.5/9, which represents a 12% boost. As Judicators cost 160 pts, that's a benefit of 19 points, which means you'd need to boost a squad through 5 phases to justify the cost.

Conclusion - so are Judicators or Vanguard Raptors better?

Ultimately, it depends. It depends on whether your opponent has a horde of lightly armoured guys or few heavily armoured warriors. It depends on how you plan to use them - are they meant to take out the enemy's force multipliers? Are those force multipliers going to only be visible for one turn (in which you want to have the highest output possible for that turn)? Does your enemy have more shooting than you in which case their resilience will matter?

Personally, I already have a unit of Judicators so I'm likely to grab a squad of VRs so I can see which work better with my style. Ultimately, it will probably depend on whether I also end up running a Lord Celestant on foot and the rest of my army's target saturation. There's no point in running VRs if they're the easiest models to kill per point and they always die first.  Anyway, we'll see how it works!

Mathhammer - CHAINAXE vs. CHAINSWORD

I've had the opportunity to get in a few games of 8th edition now (love it so far) and I've found myself wondering if chainswords were better than the chainaxes I paid for. So I thought I would do a bit of mathhammering and figure out what was what.

I've set out my math below (the probabilities are for "to hit", "to wound", then failed armour save) and then drawn conclusions below that. If you don't like math, you can jump ahead to the conclusions heading.

Berzerker w/ chainsword (WS3+, S5, 0 Rend):

vs T4, 2+ save = 2/3 * 2/3 * 1/6 = 2/27 wounds per attack
vs T5, 2+ save = 2/3 * 1/2 * 1/6 = 1/18 wounds per attack
vs T6, 2+ save = 2/3 * 1/3 * 1/6 = 1/27 wounds per attack
vs T7, 2+ save = 2/3 * 1/3 * 1/6 = 1/27 wounds per attack

vs T4, 3+ save = 2/3 * 2/3 * 1/3 = 4/27 wounds per attack
vs T5, 3+ save = 2/3 * 1/2 * 1/3 = 2/18 wounds per attack
vs T6, 3+ save = 2/3 * 1/3 * 1/3 = 2/27 wounds per attack
vs T7, 3+ save = 2/3 * 1/3 * 1/3 = 2/27 wounds per attack

vs T4, 4+ save = 2/3 * 2/3 * 1/2 = 2/9 wounds per attack
vs T5, 4+ save = 2/3 * 1/2 * 1/2 = 1/6 wounds per attack
vs T6, 4+ save = 2/3 * 1/3 * 1/2 = 1/9 wounds per attack
vs T7, 4+ save = 2/3 * 1/3 * 1/2 = 1/9 wounds per attack

vs T4, 5+ save = 2/3 * 2/3 * 2/3 = 8/27 wounds per attack
vs T5, 5+ save = 2/3 * 1/2 * 2/3 = 2/9 wounds per attack
vs T6, 5+ save = 2/3 * 1/3 * 2/3 = 4/27 wounds per attack
vs T7, 5+ save = 2/3 * 1/3 * 2/3 = 4/27 wounds per attack

vs T4, 6+ save = 2/3 * 2/3 * 5/6 = 10/27 wounds per attack
vs T5, 6+ save = 2/3 * 1/2 * 5/6 = 5/18 wounds per attack
vs T6, 6+ save = 2/3 * 1/3 * 5/6 = 5/27 wounds per attack
vs T7, 6+ save = 2/3 * 1/3 * 5/6 = 5/27 wounds per attack

Berzerker w/ chainaxe (WS3+, S6, -1 Rend):

vs T4, 2+ save = 2/3 * 2/3 * 1/3 = 4/27 wounds per attack
vs T5, 2+ save = 2/3 * 2/3 * 1/3 = 4/27 wounds per attack
vs T6, 2+ save = 2/3 * 1/2 * 1/3 = 2/18 wounds per attack
vs T7, 2+ save = 2/3 * 1/3 * 1/3 = 2/27 wounds per attack

vs T4, 3+ save = 2/3 * 2/3 * 1/2 = 4/18 wounds per attack
vs T5, 3+ save = 2/3 * 2/3 * 1/2 = 4/18 wounds per attack
vs T6, 3+ save = 2/3 * 1/2 * 1/2 = 1/6 wounds per attack
vs T7, 3+ save = 2/3 * 1/3 * 1/2 = 1/9 wounds per attack

vs T4, 4+ save = 2/3 * 2/3 * 2/3 = 8/27 wounds per attack
vs T5, 4+ save = 2/3 * 2/3 * 2/3 = 8/27 wounds per attack
vs T6, 4+ save = 2/3 * 1/2 * 2/3 = 2/9 wounds per attack
vs T7, 4+ save = 2/3 * 1/3 * 2/3 = 4/27 wounds per attack

vs T4, 5+ save = 2/3 * 2/3 * 5/6 = 10/27 wounds per attack
vs T5, 5+ save = 2/3 * 2/3 * 5/6 = 10/27 wounds per attack
vs T6, 5+ save = 2/3 * 1/2 * 5/6 = 5/18 wounds per attack
vs T7, 5+ save = 2/3 * 1/3 * 5/6 = 5/27 wounds per attack

vs T4, 6+ save = 2/3 * 2/3 = 4/9 wounds per attack
vs T5, 6+ save = 2/3 * 2/3 = 4/9 wounds per attack
vs T6, 6+ save = 2/3 * 1/2 = 1/3 wounds per attack
vs T7, 6+ save = 2/3 * 1/3 = 2/9 wounds per attack

Now for the comparison in each fight activation, a berzerker will have 3 attacks with a chainsword and 2 with a chainaxe, so each model will do the following number of wounds:

vs T4, 2+ save = 6/27 (0.22) w/ CS and 8/27 (0.30) w/ CA - CHAINAXE WINS
vs T5, 2+ save = 3/18 (0.17) w/ CS and 8/27 (0.30) w/ CA - CHAINAXE WINS
vs T6, 2+ save = 3/27 (0.11) w/ CS and 4/18 (0.22) w/ CA - CHAINAXE WINS
vs T7, 2+ save = 3/27 (0.11) w/ CS and 4/27 (0.15) w/ CA - CHAINAXE WINS

vs T4, 3+ save = 12/27 (0.44) w/ CS and 8/18 (0.44) w/ CA - EVEN
vs T5, 3+ save = 6/18 (0.33) w/ CS and 8/18 (0.44) w/ CA - CHAINAXE WINS
vs T6, 3+ save = 6/27 (0.22) w/ CS and 2/6 (0.33) w/ CA - CHAINAXE WINS
vs T7, 3+ save = 6/27 (0.22) w/ CS and 2/9 (0.22) w/ CA - EVEN

vs T4, 4+ save = 6/9 (0.67) w/ CS and 16/27 (0.59) w/ CA - CHAINSWORD WINS
vs T5, 4+ save = 3/6 (0.5) w/ CS and 16/27 (0.59) w/ CA - CHAINAXE WINS
vs T6, 4+ save = 3/9 (0.33) w/ CS and 4/9 (0.44) w/ CA - CHAINAXE WINS
vs T7, 4+ save = 3/9 (0.33) w/ CS and 8/27 (0.30) w/ CA - CHAINSWORD WINS

vs T4, 5+ save = 24/27 (0.89) w/ CS and 20/27 (0.74) w/ CA - CHAINSWORD WINS
vs T5, 5+ save = 6/9 (0.67) w/ CS and 20/27 (0.74) w/ CA - CHAINAXE WINS
vs T6, 5+ save = 12/27 (0.44) w/ CS and 10/18 (0.56) w/ CA - CHAINAXE WINS
vs T7, 5+ save = 12/27 (0.44) w/ CS and 10/27 (0.37) w/ CA - CHAINSWORD WINS

vs T4, 6+ save = 30/27 (1.11) w/ CS and 8/9 (0.89) w/ CA - CHAINSWORD WINS
vs T5, 6+ save = 15/18 (0.83) w/ CS and 8/9 (0.89) w/ CA - CHAINAXE WINS
vs T6, 6+ save = 15/27 (0.56) w/ CS and 2/3 (0.67) w/ CA - CHAINAXE WINS
vs T7, 6+ save = 15/27 (0.56) w/ CS and 4/9 (0.44) w/ CA - CHAINSWORD WINS

CONCLUSIONS

Some interesting results to analyze.  The first (and the most obvious IMO) is that the chainaxe is better against against 2+ armour.  That is because each attack is now TWICE as likely to make it through the armour. The margin of betterness becomes even higher against T5/T6 because of the chainaxes other benefit (the +1S).

When looking at all of the other results, the chainsword tends to do better in the extremes where the chainsword's extra attack outweigh's the aggregate benefits from the chainaxe (being the extra point of strength (which is useless against T4 and T7) and the -1 rend (which is less useful when the enemy's armour is shitter)). After reviewing the math, it made sense to me that the chainaxe upgrade only costs 1 point - of the 16 scenarios where the opponent has 3+ armour or worse, the chainaxe won 8, the chainsword won 6, and they tied on two.

Interestingly enough, once you ignore 2+ armour saves, the decision appears to be more connected to toughness than armour. Look at T4 guys - the chainsword wins twice and they're tied the final time - the axe never wins. Same with T7.  Compare that with T5/T6, where the chainaxe always wins.

Basically, if you think you're going to be facing mostly T5 and T6 (or 2+ armour), it's worth it to use your chainaxe, otherwise keep the chainsword.

Or you could do what I plan on doing for my tournament armies - which is to have a couple of chainaxes (maybe 33% of the unit) and then remove casualties depending on what's on the field opposite me. That way the list can be more balanced and I can adapt to the circumstances.