If so? Where's the cut-off point? At approximately what level or AP does AP eventually overtake the Banner?
I like the design of the item...sort of. But god damn is there not enough information.
That...and a Psychology major isn't very great at math.
This item does different things per champion. Here is a link to the Wiki:
http://leagueoflegends.wikia.com/wiki/Banner_of_Command
Essentially, it only benefits PET auto attacks, like 
's Turrets and 
's Voidlings. Clones, like 
's Child of the Grave, don't receive a bonus. You can see some math examples on that page, but the gist is that it increases values that straight AP won't touch.
For example, I will quote the math for 's turrets below:
Turrets deal damage based on their AD and deal magical damage.
Gain a bonus 1.8 / 2.7 / 3.45 / 4.35 / 5.1 (+ 2.25% AP) attack damage
At rank 5 with 400 ability power, Heimerdinger's turrets go from 34 + (400 × .15)=94 to (34 × 1.15) + (400 × .15 × 1.15)=108.1 AD. Remember: Heimerdinger can have up to 3 turrets.
This means that the base damage is amplified by 15% as well as the AP ratio getting amplified by 15%. So the base damage of 34 is amplified to 39.1, while the Damage from the AP ratio goes from 60 to 69. This means that the overall damage per shot goes from 94 to 108.1. Keep in mind this is only calculating the cannon shots, NOT the beam attack, using 400 AP as our starting point.
In order to compare the difference to straight AP, you would have to work your way back from 108.1. I'll do that below.
108.1 - 34 = 74.1
This means that to obtain the same amount of damage, I would need enough AP to get 74.1 from the ratio.
74.1 / .15 = 494
This means that at 494 AP, you will do the same damage per cannon shot as if you went with 400 AP with Banner of Command. This means that at 400 AP,
prices it at, you can determine the value of the passive.
94 X 21.75 = 2044.5
This means at 400 AP,
is pretty efficient considering that the passive AND active are only priced at 27 gold when you factor in the other stats it gives you. I would recommend picking it up on Heimerdinger for sure, but I would suggest you test it on other champions that have pets to see if the passive is worth it.
Consider:
Outside of a few specific instances like Void Staff vs. very high MR targets, 80 AP + 15% will almost always be better than another AP item for Turret/Plant damage specifically. But you just have to consider how much is coming from what source.
I compare it with the Morellonomiromonomicon. It already is an 80 ap/20% cdr item, meaning you mainly give up grievous wounds, 200g and mana regeneration.
I will just take the grievous wounds as the same value as promote, ignoring them both after here because they are situational passives due to the prevalence of ignite and having to stand next to a cannon minion to promote it.
So you give up 600g worth of mana regen (mana is expensive) and 200 raw gold. 800g is about 38 ap, so you'd need to get stats worth 38 ap from the 15% increased minion damage. Consider a Heimer turret shooting three autoattacks then hitting a laser and that's it. The ap ratio of this is 1. This means you'll need 253 ap to make it efficient. Considering you have two other abilties with good ratios (totalling 1.52 if you hit all rockets, not counting ultimate) and turrets are vulnerable, you'll probably need more like 500 ap to make it worth it. However, if you don't care about the mana anymore for whatever reason ahem Grail ahem and you have minions (Heimer, Morde etc), banner of command is worth the 200g extra over Morellonomirominomicon.
How much of a difference does the upgraded siege minion make in terms of pushing power? I've never used it in-game and don't own Heimerdinger, but once remember a Morgana solo split-pushing and those damn cannons were tough to kill.
I feel that because every 'pet champion' has other non-pet sources of damage, it's not worth building on most of them for most situations. That being said, as soon as you have 2 or more pet champions, you're helping their pets as well as your own. I've built it on Zyra support when we had Yorick top, Elise jungle and Himer mid.... The siege man, the siege.
Yes.
That's just how stats in LoL work. Once you get, say, 10,000 AD - 5% crit chance can offer a substantially larger %DPS bonus than 100 AD.
I could go more in depth - but as far as LoL is concerned you're much better off getting a mix of stats over dumping all your focus into one stat. Tri Force causes tons of damage because it has tons of stats that all enhance their deceptively low values to be considerably more effective than they appear at first glance.
As for the specific maths - I'd check LoL wiki for the ratios, maybe use the spreadsheet offered by Google to punch that data in and figure out the breaking point. Or just wait for someone who likes doing maths and doesn't currently have a headache to crunch dem numbers.
suppose your current AP is x and heimer does 34 + .15 AP. buying a deathcap is 34 + .15 ((x + 120) * 1.3) = 57.4 + 0.195 x buying a banner is 1.15 ( 34 + .15 ( x + 80 ) ) = 51.75 + 0.1725 x regardless of what your AP is, you should always buy a deathcap for more damage. i was unable to verify both 400 and 180 as cutoffs for ap verses banner. for example, at 400 AP zhonyas would be 34 + .15 * 520 = 112 boc would be 1.15 ( 34 + .15 * 480 ) = 106 analogously, 180 also holds the same values.
zhonyas would be 34 + .15 * 300 = 79 boc would be 1.15 ( 34 + .15 * 260 ) = 73 in fact, the cutoff for setting them equal to find the cutoff yielded -2.5e2 AP. since negative values are impossible, dcap is better.
boc is better after a deathcap, however. a boc + deathcap will yield more than deathcap + zhonya. but only marginally. zhonyas + deathcap = 80.8 compared to boc + dc = 83.95
in other news, the '*' character seems to cause italics, so some of the multiplication is implied. none of the italics were intentional for emphasis.
This is a placeholder. Please comment on it, so I'll get a notification and remember to come back to this when I have the time to do math for you ;)
Cheers, Hyrum Graff
the aura buffs minions' damage by 15% including plants, turrets, voidlings, etc. from champions
tl;dr read the lol wiki
