I've been running with all of my Clicker Heroes gilds on Frostleaf for some time now. I've mentioned it in posts twice, and both times people have commented to say it would be better to have someone else gilded. (The Masked Samurai in one case, Terra in the other.) Both times I plugged some stuff into my spreadsheet and determined Frostleaf was, in fact, the best option for me at those precise moments in time. I was posting a comment to Sky about how Frostleaf was still the best for me so maybe he was wrong when I thought about it a bit more and decided Terra probably was better for him than Frostleaf...
My problem so far (and quite possibly Sky's as well) is that we're looking at specific moments in time and deciding who is the best to gild at that time. Once a hero has been established and you've bought all their upgrades they all scale in exactly the same way. For every 25 levels you buy the cost will have increased by a factor of 5.42 and the damage done will have increased by a factor of 4. Under this scaling system whoever is the best choice will flat out stay the best choice for all time. You should gild whoever is best amongst heroes you can afford. There is one catch... Every time a hero levels to a multiple of 1000 they get a damage boost of 10 instead of a damage boost of 4. Essentially, you get a 2.5x bonus for hitting the multiple of 1000. So if two heroes are close to each other in power level then having one of them get a 2.5x bonus can easily swing the balance of power between them.
In the specific case Sky brought up, with Terra and Frostleaf, my Frostleaf with an equal amount of money spent on them does a little over twice as much damage. But my Frostleaf had recently crossed a 1000 barrier, and my Terra was close to breaking her own 1000 barrier. If I removed 300 levels from each hero Terra will gain 17% more benefit from being gilded than Frostleaf would. This is the state Sky is in, as he mentioned his Terra was around level 1500 which is about the same cost as Frostleaf to 2950. But pay just a little bit more and bump them up to 1550 vs 3000 and suddenly Frostleaf gains 111% more benefit from being gilded.
One other interesting thing to note is the above comparison isn't exactly fair. Terra and Frostleaf don't actually gain precisely 4x damage when you buy 25 more of them. You're also adding on 25 more levels which get multiplied through by all your other multipliers... How much of a boost this is depends on how many levels you already have. For example, taking Terra from 1000 to 1025 actually multiplies the damage done by 4.1 while taking Frostleaf from 2450 to 2475 is only a damage boost of 4.0408. This seems like a small difference, but it is a crucial one. Frostleaf at 2450 is 7% better than Terra at 1000 even though it's the same level difference as the 2950 to 1500 comparison and no extra 1000 level barriers were broken along the way.
This does mean that as you keep leveling and leveling the 'right' choice would (very slowly) shift up the depth chart even if there wasn't the weird quantized breakpoints at the 1000 marks. The difference does start getting really small as you get more of them, though, so it wouldn't be enough to swing a huge gap in any reasonable length of time. The 2x gap that Frostleaf owns at the 3025v1575 mark doesn't go away until sometime after my spreadsheet breaks from numbers getting too big. At the 11000v9550 mark Frostleaf maintains a 26% edge over Terra. But at the 10975v9525 mark Terra is up to a 99% edge over Frostleaf!
So now I have a couple questions... Can I work out a more general formula of some kind to actually rank who I'd want to gild at any given cash level? And how much does it cost to switch between two gilded states?
Assuming I never ascend again Terra will be 'right' for 550 of every 1000 levels when compared to Frostleaf. And she'll be 'wrong' the other 450, and she'll be more wrong than she is right, so I'd stick with Frostleaf if I couldn't switch. I have 173 gilds, and they cost 80 souls each to move, so I'd have to pay 13840 souls to make that swap. I have 111805 souls right now. So I'd have to give up more than 12% of my damage in order to pick up the 17% benefit from having Terra gilded during that window. This is a gain, but it's such a significant amount of my damage that it's a terrible idea if I'm going to keep ascending and ever play in a range where Frostleaf is way better. So I think the back and forth swap is right out. (Unless it actually lets me push on to the point where I earn more than 13840 extra souls? At my current depth I'd need to be able to push through 150 more levels to make that much. Since each level has a boss that has double health that's not going to happen by getting a 2.5% damage boost.)
What about the start of a general formula? I feel like the first number I need for each hero is their self buffed value at level 200. This is a high enough level that you'll have all of their upgrades, and is also where the 4x multipliers start showing up. None of the other multipliers should matter since they'll impact every hero by the same relative factor.
Ok, I get that value, and it isn't very interesting. Each hero is better than the one before it except the truly terrible ones (Betty, Midas, Amenhotep). So then I tried dividing by the base cost of the hero, but that wasn't any better. Now each hero is worse than the one before with a couple exceptions. Mostly around the terrible ones again, though The Masked Samurai performs a lot better than the two before him.
What I really care about is comparing how two heroes would perform if I could spend all my money on just them. Which means I want to be comparing the cost efficiency of each hero at that gold value. So I added a cell to my spreadsheet for each hero to work out how many levels I could get for a given amount of gold using the formula for summing a geometric series.
Ok, but I still don't have a simple formula for who to gild... But now that I have all this stuff plugged into my spreadsheet I can just enter in an amount of money to see what hero is best at that money level. And because I'm still obsessed with Frostleaf I also want to look at how much better the best hero would be compared to just leaving all my gilds where they are on Frostleaf.
And now that I did that, I don't think it's very useful to post. The problem is there are so many different values I could use, and the 'best' actually changes a lot between them. I tried doing every multiple of 10^20 and then every multiple of 10^5 and things would swingly wildly. 1x10^125 had The Masked Samurai on top, but 1.5x10^125 has Orntchya on top with Frostleaf almost equal. Once Lilin becomes a reasonable option she seems to stay on top, but that's likely just because her breakpoints line up well with the ones I randomly decided to look at. Even by 1x10^130 Lilin is almost 4 times as good as Frostleaf, and she runs almost exactly 3000 levels behind him so they're never going to get offsync on the 2.5x multiplier. So if I ever reach the point where I can have Lilin up after I ascend it'll be right to gild her up for sure.
What does this all tell me? Well, I've been pretty lucky to always have Frostleaf math out to be the best when I've checked since he doesn't seem to be showing up as the best in my random samples. It's entirely possible he's just frequently the best in the little chunk of the leveling curve where I've been playing the most. Which certainly implies that gilding him up was a good choice.
What confuses me a little is I've been trying, and I can't find any times when Dread Knight, Atlas, or Terra are the best. Even at the 1500 Terra mark where Sky rightly pointed out she's beating Frostleaf... She's worse than Orntchya, Ivan, Treebeast, Aphrodite, and Brittany. Ok, I found one... At 4.2x10^129 Terra gets to buy level 2000 and marginally jumps ahead of Orntchya to take first place. But Orntchya is at level 979 at that point, so she'll take the lead back by 1.8x10^130. And while Terra was barely ahead at her 2000 mark, Orntchya has a _commanding_ lead at her 1000 mark.
I think the main thing that's going on is some of the earlier heroes are just really good and the new heroes like Dread Knight and Atlas are stuck competing with those heroes when they're breaking into their primes. By contrast when Orntchya gets up to 1000 everyone from Natalia and earlier has wasted multipliers thanks to the 4100 cap.
Ok, I fiddled with it a bit more and now I can easily have it give me the amount needed for a given hero to hit a given level. Like, say, 1000. Some heroes are the best at the precise moment they hit 1000 - Treebeast, Ivan, Brittany, Fisherman, Samurai, Forest Seer, Frostleaf, and then every hero from Phthalo on up.
The 1000 level mark feels like it has to be the time when a given hero has hit their power apex relative to the others. It has an edge compared to the 2000 mark because there's no one further up the food chain close to hitting the 1000 mark and blowing you away. Terra is the best at 2000 as mentioned above, but only for a _very_ brief period of time. And then it tails off even more at 3000 where she gets passed by Cadmia, whoever is after Cadmia, Orntchya, and Lilin.
Anyway... What conclusions can I draw from all this rambling and numbers? I think if you're not even the best when you're at level 1000 you should never be the primary gild focus. As time ebbs and flows it feels like quite a few heroes are reasonable choices, and they happen to be the ones who are the best at 1000. I can see a case being made for any of Treebeast, Ivan, Brittany, Fisherman, Samurai, or Forest Seer when you're at the stage where buying Frostleaf is non-trivial. I think I'd lean towards Brittany myself, but Samurai is also a fine choice. Perhaps the best plan is to spread out your gilds amongst all of those bottom heroes to smooth out your boost from gilds. You'd need to level them all up, but that's fine. I do think once Frostleaf becomes an easy buy he becomes the dominant choice, mainly because I like only having to buy one hero on an entire ascension. (I click once to kill a unit and advance to level 140 thanks to Iris. Then I click the holiday trinket to make some money, buy some Frostleafs, and win. (I do buy 200 of everything just to make sure I get all the relevant buffs.) It's easy, and while Frostleaf does get passed for top spot by different heroes as time goes on he always stays near the top in power. Or at least, he did. The new heroes shake things up a bit. But the cheapest of the new champions are stuck competing with previous good heroes, including Frostleaf himself, and I don't think you can justify switching off of him to one of them. Phthalo might be where things change. Certainly if you're reaching the point where Frostleaf is running over level 4100 you have to make a change to something further up the chain. Even at something like 3900 Frostleaf he's still the 5th best option and only off the best (Cadmia) by a factor of 5.
So, I think the right strategy is to gild a bunch of the low level heroes. Then switch to Frostleaf. Then switch to something higher up when you reach the point you can afford that higher up hero pretty trivially. And I think I'm going to wait until at least Phthalo and maybe even Orntchya before I made a switch.