[Guide] Math Strategies page 2

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Hey Gus, looking into your maths on this, seems to be a few curiosities. Are you accounting for the 50% elemental damage multiplier or Attack Power?

1: With the 50% multiplier for element, then any Defence based build would win out.

For example, optimal 1:2:0 Pegasus against a Simple 1:1:1 – Optimal deals scratch damage against Simple. Simple deals ~6,000 against Optimal.

2: With Attack Power and #hits calculated, even ignoring element, then half your sample group (Ninja Assassin, Twin Diablos, Twin Salamanders and White Tiger) aren’t accurate.

Ninja Assassin with a 0.6 multiplier means Optimal deals 7837 damage before Def, or ~3000 damage a hit, x2 (ignoring elemental multiplier, which again means no damage is dealt). Simple deals 3,300 a hit x2.

Twin Sals? 0.6 again, and Simple wins, again ignoring Elemental multipliers. Same with Diablos.

Optimal White Tigers at 0.4 deal less damage to Simple White Tigers than they receive. Again, before elemental mods.

3: For any hits where they share the same integer, I’d suggest “Draw” is more appropriate. If both sides need one hit to kill the other, it’s a question of who hits first; only where there’s a serious difference is there a serious advantage, so rounding up aggressively to the next integer is probably a more accurate indicator.

So whilst this should be fairly accurate for Void, 1.0 Atk Power units, of which there are none, it seriously undervalues the potency of Def for any other situation barring elemental weaknesses.

4: It’s probably worth noting that HP is quite a bit easier to itemise than Atk or Def for most of the game. Might it be an idea to assume a few loadouts and check then?

Six Galaxy Swords: 7,200/10,800/4,800
Six Infinity Blades: 6,000/6,900/3,000
Six Dragon Steak Barbeques: 48,000/4,800/4,800
Six Invisible Cloaks: 7,800/1,170/2,460
Six Angel Slayers: 4,920/4,350/750

 
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I didn’t explain yet how I did the simulations and I’ll soon. First I need to reorganize / rewrite this guide to be more “understandable” for most of people.

About the skill points ratio: the spreadsheet I made is only a basic thing, to a quick test. I did a real simulation using more than 25 billions of combinations, using another units + hypothetical units and another variables too, to be more accurate.

And yeah, I was kinda surprised too when I analysed the results, but then I understood why HP > Def in general (later I’ll explain with more details).

 
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The problem with calculating ratios on infinite variables is that this does not apply in a closed system, or account for attack modifiers.

For the sake of calculation, assume that the unit in question has 100 in all aptitudes, and Base Stats of 1/1/1.

At level 9,999 they have 10,000 Health, 10,000 Attack, 10,000 Defence before stats.

You have, at most, 49.995 stats to assign. Assume that energy and action are irrelevant.

This means the maximum possible stat is 59,995.

Possible stat loadouts:
A: 1:0:0
B: 1:1:0
C: 1:1:1
D: 2:1:0

A may have:

1: 59,995/10,000/10,000
2: 10,000/59,995/10,000
3: 10,000/10,000/59,995

B may have:

1: 34,997/34,998/10,000
2: 10,000/34,998/34,997
3: 34,997/10,000/34,998

C has:

26,665/26,665/26,665

D may have:

1: 43,330/26,665/10,000
2: 26,665/43,330/10,000
3: 10,000/43,330/26,665
4: 10,000/26,665/43,330
5: 26,665/10,000/43,330
6: 43,330/10,000/26,665

0 Atk isn’t really viable, so let’s look at the top 2/3 of each.

To begin, ignoring modifiers:

Looking at A2 we can see that the absolute highest minimum damage that can be dealt is ~3000 – four hits to kill any target with no points invested into Health. However A2 has the disadvantage that they can be one-shotted in turn by B1, B2, C, D1, D2, D3 and D4, and their units can be killed instantly by any critical hit, so it’s pretty much rocket tag.

Comparatively, B1 against B2 and B3 deals minimum damage: ~1750. Twenty hits to kill B3, six hits to kill B2. Conversely, it only takes two hits to kill another B1. Defence multiplies the value of health. B2 > B1.

Against C, B1 deals 10,000 damage a hit – three hits to kill. C only takes two hits to kill B1 since it hits for 17998 a hit. B1 < C
Against C, B2 deals 10,000 damage a hit – the same three hits, but C only deals 1333 damage a hit – eight hits to kill B2. Twenty-six to beat B3 – but nobody’s saying 0 Atk was very viable. B2 > C

Against D1, B1 kills in two hits, and dies in two hits. B1 = D1. Draw.
B2: Still kills in two hits. Dies in eight hits. B2 > D1.
Against D2, it’s Rocket Tag. B2 and B1 both kill D2 in one hit, D2 kills B1 and B2 in one hit. B1 = D2 & B2 = D2. Draw.
D3: B1 kills and dies in one hit, same for B2. B1 = D3 & B2 = D3. Draw.
D4: B1 deals minimal damage. D4 kills B1 in two hits. B1 < D4
D4 deals minimal damage against B2 as well, so B2’s minimum damage wins the day. B2 > D4

C against D1: C hits for ~18,000, 3 hits to kill. D1 retaliates for 1,333 – ~20 hits to kill. C > D1
C against D2: C hits for ~18,000, two hits to kill. D2 retaliates for 18,832 – two hits to kill. C = D2 / Draw.
C against D3: C hits for 1,333, eight hits to kill. D3 is still a two hits to kill. C < D3.
C against D4: Both hit each other for 1,333, C wins from having more health. C > D4

D1 against D2: Two hits and it dies, two hits before it kills. D1 = D2. Draw.
D1 against D3: Two hits and it dies, still, but eight hits to kill. D1 < D3.
D1 against D4: Three hits and it dies, still eight hits to kill. D1 < D4
D2 against D3: One hit kill on both sides. D2 = D3.
D2 against D4: Five hits to kill D4, dies in two hits. D2 < D4.
D3 against D4: D3 wins; higher scratch damage. D3 > D4.

A lot of draws and trade-offs. We enter a metagame situation. If everyone has a D2 build then the smart player gets a D4 build. If D4 builds become popular, smart players go for B2 builds for a net advantage in this situation. There is no build that isn’t beaten by another of these builds except one (B2 beats or draws against every other viable build listed. Its weakness? 1:1:2 – more health comes into play when both participants have enough Def to set the opposition’s damage to 0.05)

Defence is what makes the difference, not health. Defence exists to multiply the value of health by up to 20x its standard value. In your example, you have a 100/100/5,000,000, and you’re hitting on the exact same thing: Enough Defence effectively multiplies Health by 20. There’s a reason why Defence is consistently the lowest aptitude, and health the highest – Defence is potentially worth more than either.

As a trade-off, you’re quite correct – surplus Defence is wasted, it’s only ever a 20x bonus to health, but up to that point it’s one of the biggest deciding factors.

I assume Kong’s about at its legible post limit, so I’ll start in a new post about multipliers.

 
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So…. Multipliers.

The most obvious and immediate is Aptitude.

Taking an average from all non-hero units, the “average” aptitude is:

75/44/33

If you strip out starter units (anything with nothing better than +25) you get:

100/55/42

So roughly speaking, you get 100% of your Health stat, 55% of your Attack stat, and 42% of your def stat.

Next comes another obvious multiplier: Elements and immunities.

In every case in the above example, it was assumed to be a void element target or two unrelated targets. This is rarely the case, and whilst you should expect a unit to win against its elemental weakness, that’s not the case against equal elemental strength.

So against the same element, A2 loses the use of 29,998 stat points.

This means even with full attack investment any attacker is dealing scratch damage against 29,998 defence.

In the case of same elemental multiplier, this makes C one of the most viable builds – Against D2 it receives ~1100 damage a hit, and deals ~4,000 damage a hit. Against D3 it deals 666 damage a hit (16 hits), and only dies after 24 hits.

Against their immunity, they deal a fifth as much damage.

The more points you have invested into attack, the more you suffer.

Third multiplier is Attack Power. Approximately a tenth of units have an actual 1.0 attack power, if that. For the most popular ones, we have about 0.6.

If you account for a 0.6 Attack Power in Mr Average, the 100/55/42 guy, this means you have an effective 33 aptitude and Attack is the hardest to pump meaningfully. If you account for this average, then investing 5000 into Def (2,100) is enough to induce scratch damage from someone who has invested 6,300 into Attack. That gives you a profit of 1,300 stat points over 5,000.

Finally, you have items. As indicated, Def can only really be added by items in small amounts, health can be added in great quantities, and with the ultimate Galaxy Sword x6, for the standard 100/100/100 guy in the above post this makes base stats of 17,200/20,800/14,800, a huge difference.

With this all in mind, a 1/2/0 pattern is not necessarily the right choice. You can compensate for a large proportion of that with equipment, but you cannot really compensate for a low defence.

 
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I’m really glad when somebody reads a text and then wonder about it, not only accepting the facts as true. You’re welcome to discuss, Kholai.

The problem with calculating ratios on infinite variables is that this does not apply in a closed system, or account for attack modifiers.

Well, I don’t like to talk about my personal life (basically because it doesn’t matter for anyone), but that could help explain some points here: I work as a statistical consultant and daily I have to analyse real problems and figured out some useful data about it. In real world, the concept of “infinity” is not tangible, there isn’t “infinity production”, “infinity money”, “infinity people”, “infinity profit”, “infinity whatever”.

But using the mathematical concept of limit helps us (mathematicians, statisticians, physicists, etc.) to see if there is any patterns or tendency that will allow to modelling into a deterministic system and then analyse the results, like I did in this game. This game has the 9999 level cap, therefore 49995 skill points available, and this is number is high enough to see the convergence (to 1:2:0 ratio, in this case). I just used limit to illustrate the convergence.

For the sake of calculation, assume that the unit in question has 100 in all aptitudes, and Base Stats of 1/1/1.

At level 9,999 they have 10,000 Health, 10,000 Attack, 10,000 Defence before stats.

You have, at most, 49.995 stats to assign. Assume that energy and action are irrelevant.

This means the maximum possible stat is 59,995.

Possible stat loadouts:
A: 1:0:0
(…)

In my first attempt to figure out the best ratio, I did exactly what you described above. I used some hypothetical and real units/stats, some ratios and see what happened. Then, I found the ratio 0:3:2 as the best one, but I know this result didn’t have a statistical significance, I got only about 1-sigma certainty. In statistics, 1-sigma means nothing, basically. Therefore, the result cannot be classified as trustworthy. And that happened cause the sample size was too small and you have fixed a level / total of skill points.

So, I started to code a better simulation. The first step was increase the sample size. In my simulations, I divided the ratio by 75, so I could have ratios like: 75:0:0, 74:1:0, … , 0:75:0, 1:74:0, 1:73:1, …, 0:1:74, 0:0:75. That way, you’ll have 2926 possible combinations.

Next step is simulate each ratio against all others, starting from level 0. For example, play 75:0:0 against 74:1:0, …, 0:0:75, and then 74:1:0 against 74:0:1, …, 0:0:75, and so on. Then, increase the level by 50 and simulate all again, until reach level 10000 (so, you’ll have 200 ties levels). That way, you’ll have more than 25 billions of games simulated in each 50 levels from 0 to 10000 (0, 50, 100, …, 9950, 10000).


A lot of draws and trade-offs. We enter a metagame situation. If everyone has a D2 build then the smart player gets a D4 build. If D4 builds become popular, smart players go for B2 builds for a net advantage in this situation. There is no build that isn’t beaten by another of these builds except one (B2 beats or draws against every other viable build listed. Its weakness? 1:1:2 – more health comes into play when both participants have enough Def to set the opposition’s damage to 0.05)

Yeah, you basically stated the principle of Game theory (it’s the Minimax problem). If this game was a real PvP, it’d be very interesting to play and analyse (mathematically speaking).

I didn’t try to find the minimax estimator in this case, because it’d need a huge amount of analysis to figured out something that few people actually would use, so… maybe one day.



Defence is what makes the difference, not health. Defence exists to multiply the value of health by up to 20x its standard value. In your example, you have a 100/100/5,000,000, and you’re hitting on the exact same thing: Enough Defence effectively multiplies Health by 20. There’s a reason why Defence is consistently the lowest aptitude, and health the highest – Defence is potentially worth more than either.

See by this perspective: if you have a 5,000,000/100/100 unit, how many attack your enemy will need to one-hit kill? See, your enemy will need to have about 4,760,000 attack to one-hit kill your unit. On the other hand, having 100/100/5,000,000, your enemy only will need about 2000 attack to one-hit kill.


Enough Defence effectively multiplies Health by 20


That’s not the point. HP/Atk/Def are directly correlated and we are trying to figure out what is the best ratio to use having N skill points. You said: “There’s a reason why Defence is consistently the lowest aptitude”, and that’s why you’ll need to invest a lot more skill points in defense to have a “enough defense to null the enemy attack”.

And the point is: instead use your skill points to (try to) null the enemy attack, why not try to delay your unit death increasing the HP? As you said, HP has the highest apt and Def the lowest, so that means 1 skill point invested in HP will be higher (not necessarily better, of course) than 1 skill point invested in Def.


About your next post (multipliers, element, aptitudes):

1) Aptitudes: in my simulation, I used almost all units in the game (except the crappy ones), including titans and Heros, and 10 hypotheticals units too (using independent simulations).


2) Multipliers (attack power, elements, immunities): it doesn’t matter, irrelevant. Why?

a) Unit A always will play ONLY against Unit A, but using different ratios. For example, Pegasus will only play against Pegasus. There is no point to test Pegasus against different units, the goal is to find the best ratio, not the best unit. That’s why element multiplier is irrelevant.

Note that elemental multiplier (ie, a unit playing against a weak/strong unit) is a strategical decision made by the players, it’s metagame.

Suppose you have a fire unit with stats 100/100/100, what’s the point to play against a wind unit with stats 100/100/100? To find the best ratio, we have to use the same parameters. In this case, unit fire 100/100/100 playing agaisnt a unit fire using a different ratio.

In the real game, if you spam a unit fire 100/100/100 to play against a wind unit 100/100/100, your enemy could (and should) counter-attack with a water unit, and then you should counter-attack using a thunder unit, and so on. See? To find the best ratio, there is no point to use the elemental multiplier, because it’s based on metagame.

b) Because a unit will always play against the same unit, the attack power and immunities are mathematically redundants.



With this all in mind, a 1/2/0 pattern is not necessarily the right choice. You can compensate for a large proportion of that with equipment, but you cannot really compensate for a low defence.

1:2:0 (for levels > 3000) is the optimum ratio, which means it’s the ratio with highest win rate against all other ratios. And it doesn’t mean it’s the right choice, as you said, because if you’re level +4000 (for example), there is no point at all to invest your skill points in HP or Defense; at this level, your HP/def base stats will be high enough to beat any quests in the game, so it’d be a smart choice to improve attack only.


P.S.: Considering the ratio using the equipaments, I’ll try to check later. But I guess it won’t change too much.

 
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- See by this perspective: if you have a 5,000,000/100/100 unit, how many attack your enemy will need to one-hit kill? See, your enemy will need to have about 4,760,000 attack to one-hit kill your unit. On the other hand, having 100/100/5,000,000, your enemy only will need about 2000 attack to one-hit kill.

True, but if you have 5,000,000 stat points to throw around, then if you put 1,000,000 into Defence and 4,000,000 into Health, then against ~4,760,000 attack you’re ultimately the same as if you had 5,000,000 Health. Meanwhile how do the two fare against 1,000,000 Attack? 2,000,000 Attack? 3,000,000 Attack? Instead, if you have 2,500,000 Health and Def what then?

Against anything with less than 1,000,000 attack, having 1,000,000 Defence multiplies Health by 20. Against 2,000,000 attack, it’s effectively multiplied its worth by two. So if they have invested less than twice as many points into Attack as you have into Defence, you’re getting at least double the mileage out of your health points.

If you have 2,500,000/200/2,500,000 then you’ve got the same five million health you would have if you put it all into health. Except you also have 50,000,000 health against anyone with less Attack than you have in Defence.

- And the point is: instead use your skill points to (try to) null the enemy attack, why not try to delay your unit death increasing the HP? As you said, HP has the higher apt and Def the lowest, so that means 1 skill point invested in HP will be higher (not necessarily better, of course) than 1 skill point invested in Def.

If you invest 1,000 stats and get 1,000 HP, compared to 500 Def.
If that 500 Defence means that your Def is > 1/2 their HP then you’re doubling the value of all the Health points you have. If you have 2,000 Health and 500 Def, then your health will be effectively 4,000 against someone who’s invested 1,000 or less into Attack (pretending that Attack is a 1:1 exchange, which as I’ll go into below is not the case).

If you invested that into Health instead, your health is effectively 3,000 against everyone. You’ve got all 1,000 points, but you haven’t got the conditional 1,000 points. This has good points – You can now take a hit from someone who invested 2500 into Attack, and bad points – You cannot survive four hits from someone who has 1,000 in Attack.

- Multipliers (attack power, elements, immunities): it doesn’t matter, irrelevant.

Then this is the fatal flaw in your calculations.

1: Unit A will play only against Unit A.

Unless Unit A is void, your simulation has now effectively doubled unit A’s attack stat. This is inaccurate.

2: Because a unit will always play against the same unit, the attack power and immunities are mathematically redundant.

This also couldn’t be further from the truth, because Defence is almost a binary state of utility.

Why? Because Attack Power and Elemental Modifier are applied before damage is calculated, not after.

In your simulation, if you have A (2500/3000/1000) facing B (1000/3000/2000) then A will win – 3000 * 1.05 -2000 = 1050, instant kill.

In reality however, if you include Like vs. Like elemental, or a 0.5 Attack Power then B wins, despite having lower stats. 3000 * 0.5 = 1500. 1500 * 1.05 – 2000 = 75. 1500 * 1.05 – 1000 = 575. B wins in five hits.

To not modify any of the existing formulae, I’ve duplicated a page into your Simple Simulator to incorporate Attack Power only. Check the results:

https://docs.google.com/spreadsheet/ccc?key=0AjLI06iDaPmMdDdTNEdaUUxqUlhPSmlZdWFzUlJmbUE#gid=2

I’ve set it for 9,999 1:2:0 against 0:1:1 with Draws (same # of hits) noted.

The only one where A consistently wins against B is Dark Knights (which is why I keep saying they’re so great).

Whilst you might dismiss Elemental Modifiers for simplicity (assuming all equal relationships), you cannot dismiss Attack Power. The fact that the attack stat is consistently reduced prior to the attack calculation for most units in the game and expect to get accurate results; attack power is a direct modifier to base stats and aptitudes that must be accounted for.

 
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Unrelated to our ongoing discussion, would you mind checking the Damage mechanics?

Whilst checking damage for Thunder Shield versus Thunder III, I was using Shield against 10-1.
My Atk: 6587, 0.4 AP = ~2634 damage.
Since its defence is 4733, this means I’m dealing the minimal 131-132 damage per Thunder Shield hit.

So far so good, but if the expected damage from a crit is 1.65 Damage + [1.5 Damage – Def] then the minimum Damage that comes out the end should be 1.65x 2634. As it is, Crits from the Shield are actually ~400.

Going against Reaper in 9-2, 3380 Def, so the regular damage is the same (132) but the crits were 1533, 1108, 1090, 1602, 1129, 1295…

So… yeah, any ideas? I’m leaning towards the idea that crits now only have a 15% minimum with a ~1.5 multiplier as my starting estimate.

 
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Originally posted by Kholai:

True, but if you have 5,000,000 stat points to throw around, then if you put 1,000,000 into Defence and 4,000,000 into Health, then against ~4,760,000 attack you’re ultimately the same as if you had 5,000,000 Health. Meanwhile how do the two fare against 1,000,000 Attack? 2,000,000 Attack? 3,000,000 Attack? Instead, if you have 2,500,000 Health (…)

I didn’t say that 5kk/100/100 or 100/100/5kk was a smart choice to make, of course it’s not. The point was to compare these stats using big numbers (infinity) and see how it goes.

If you invest 1,000 stats and get 1,000 HP, compared to 500 Def.
If that 500 Defence means that your Def is > 1/2 their HP then you’re doubling the value of all the Health points you have. If you have 2,000 Health(…)

Again, you’re using one example to make a point. In math, to prove a theorem / proposition, you must demonstrate that a statement is always true (you might list all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. One method of proof is use the “Statistical proof” (as I did making the simulations), and I got 4.3-sigma significance (which is very good). Of course, every math proof requires a fundamental statements (ie, axiomas), and if you don’t choose the right parameters, it’s very likely you’ll have a non-true conclusion.


Then this is the fatal flaw in your calculations.
1: Unit A will play only against Unit A.
Unless Unit A is void, your simulation has now effectively doubled unit A’s attack stat. This is inaccurate.
2: Because a unit will (…)

Attack power and elemental multiplier are only a constant variables! They decrease/increase the attack effectiveness, but they won’t change the convergence. See the limits below:





You can multiply x by any constant (attack power, elem. mult.) and still won’t change the result. Using any constant between 0 to 1 will slow down the convergence, but won’t stop it.

Btw, in my simulations I used the attack power and the 0.5 elem. multiplier for non-void units. But after analyse the results, it was clear to me that they’re irrelevant parameters to find the best ratio.


In your simulation, if you have A (2500/3000/1000) facing B (1000/3000/2000) then A will win – 3000 * 1.05 -2000 = 1050, instant kill.

In reality however, if you include Like vs. Like elemental, or a 0.5 Attack Power then B wins, despite having lower stats. 3000 * 0.5 = 1500. 1500 * 1.05 – 2000 = 75. 1500 * 1.05 – 1000 = 575. B wins in five hits.

To not modify any of the existing formulae, I’ve duplicated a page into your Simple Simulator to incorporate Attack Power only. Check the results:

https://docs.google.com/spreadsheet/ccc?key=0AjLI06iDaPmMdDdTNEdaUUxqUlhPSmlZdWFzUlJmbUE#gid=2

I’ve set it for 9,999 1:2:0 against 0:1:1 with Draws (…)

Again, enumerate many confirmatory cases doesn’t make it a proof.


Originally posted by Kholai:

Unrelated to our ongoing discussion, would you mind checking the Damage mechanics?

Whilst checking damage for Thunder Shield versus Thunder III, I was using Shield against 10-1.
My Atk: 6587, 0.4 AP = ~2634 damage.
Since its defence is 4733, this means I’m dealing the minimal 131-132 damage per Thunder Shield hit.

So far so good, but if the expected damage from a crit is 1.65 Damage + [1.5 Damage – Def] then the minimum Damage that comes out the end should be 1.65x 2634. As it is, Crits from the Shield are actually ~400.

Going against Reaper in 9-2, 3380 Def, so (…)

Sure, no problem. The files are at my home, I can check it later.

But to make it more organized, can you make a list of all pendent matters that you wanna know? Something like that:
List:
1) Find the Attack power of these units (Reapers, black dragon, etc).
2) Verify the Thunder shield damage
3) etc.

I ask this cause I know there are some pendent infos to verify, but I don’t know which one I should get. And I’ll have to verify by parts (the source-code is huge), so I’ll be more organized that way.

 
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Although if someone wishes to demonstrate this position mathematically, they’re welcome to do so, I’m not so much interested in proving any theorem as resolving the issue that the given one does not work when applied in a real system given the data provided. To this effect, the fact that multiplying Attack is converging on infinity eventually is not relevant to my concerns.

The best ratio must surely be one that actually works in-game at level 9,999, with the absolute maximum possible statistics available in-game, to handle as larger percentage of the opposition than another possible build. 1/2/0 does not accomplish this, and I’m not convinced there is one in practice (though with Ranged Attacks being what they are, full attack is as close to it as we can expect).

The fact that an attack power of 0.6 means that x Attack Power is Scratched by 0.6(x) Defence means that at that point defence is worth more_, because 0.6(x_) defence is as valuable as 0.95x.

It’s harder to demonstrate this relationship, because 50,000 Defence is either worth 5 (if your opponent has 6 Attack), 47500, if your opponent has 50,000 Attack, or the full 50,000 if they have more than that.

Bring in variable aptitudes for Defence, and 50,000 Defence could be worth 5,000 (10% aptitude), 14,500 (10% aptitude against an Aeon with 14,500 Attack), all the way up to 45,000 at 90% aptitude, or 900,000 with Stat Immune/Elemental Modifier.

Likewise, whilst health is only ever useful once, Defence is always useful at least once.

This is why CoT3 Popo with 12,000,000 Health and 3,000 Defence is easier to beat than he would be at 4,000,000 Health and 1,000,000 Defence against all opponents with less than 1,000,000 Attack, and this Health isn’t as valuable as Defence.


For the things that haven’t been discovered yet…

1: Most importantly, I believe we need to recheck the Critical Damage calculation. It’s dealing a minimum of 0.15 x, not 1.65 x, and if it were dealing 3.15 x before Def, then 8791 modified damage should translate to crits over Scratch on both targets.

Eyeballing it I’d guess 0.15 + ~1.5 instead of 1.65 + 1.5.

2: We don’t actually “know” the attack modifier on spells. I’ve calculated Tornado at 0.2, Thunder Shield and Fiery Fist at 0.4, Thunder 3 at 0.8 and so on, but it would be good to have the information for the wiki, though for something like Tornado we don’t even know how many times it hits.

3: The attack power for the twelve new units as of last version. I’ve calculated Reapers to be 0.8, Fafnir and the ranged titans to 0.6 along with guessing the others, but code confirmation is always helpful.

4: Just remembered – how the heck does Lily’s Charm work? Is it keyed off another status effect or is it its own unique thing?

Cheers.

 
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Originally posted by Kholai:

For the things that haven’t been discovered yet…

1: Most importantly, I believe we need to recheck the Critical Damage calculation. It’s dealing a minimum of 0.15 x, not 1.65 x, and if it were dealing 3.15 x before Def, then 8791 modified damage should translate to crits over Scratch on both targets.

Eyeballing it I’d guess 0.15 + ~1.5 instead of 1.65 + 1.5.

The critical damage changed and your estimation is right:


X = 0.15*X + [1.5*X – 0.8*def – 0.2*def*RANDOM_(0~1)]
If [1.5*X – 0.8*def – 0.2*def*RANDOM_(0~1)] < 0, then replace it with 0 (zero).


Besides, there is 10% chance for units and 5% for Heroes.

 
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Originally posted by Kholai:

2: We don’t actually “know” the attack modifier on spells. I’ve calculated Tornado at 0.2, Thunder Shield and Fiery Fist at 0.4, Thunder 3 at 0.8 and so on, but it would be good to have the information for the wiki, though for something like Tornado we don’t even know how many times it hits.

Here the attack multipliers (AM), elemental class (EC) and status types (S) on spells. See if this values make sense, I’m not sure if the multipliers are right because the physics spells (like tornado) are kinda complex:


1) Name: Charming smile; EC: void; AM: 0.01; S: charm
2) Name: Sword of judgement; EC: void; AM: 1; S: stun
3) Name: Ice blast; EC: water; AM: 1.2; S: freeze
4) Name: Flame fist; EC: void; AM: 1; S: burn
5) Name: Poison cloud; EC: void; AM: 0.8; S: poison
6) Name: Dragon roar; EC: void; AM: 0.6; S: none
7) Name: Meteor; EC: void; AM: 0.3; S: none
8) Name: Rain of Arrow; EC: void; AM: 1; S: stun
9) Name: Thunder shield; EC: thunder; AM: 0.4; S: shock
10) Name: Fist; EC: void; EC: AM: 1.5; S: none
11) Name: Thunder; EC: thunder; AM: 0.8; S: shock
12) Name: Tornado; EC: wind; AM: 0.2; S: none

There are some spells that I didn’t find yet. I know the Giant Fist is missing, I remember it’s coded in another place apart. Can you tell me what is still missing?


4: Just remembered – how the heck does Lily’s Charm work? Is it keyed off another status effect or is it its own unique thing?

The Charm is a new status effect, it can affect all units except Heroes. More technical info: 30s duration (game time) and 50% chances to “convert” the enemy units in ally.

 
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1) Name: Charming smile; EC: void; AM: 0.01; S: charm
– Looks about right, 16k Atk deals 156 damage to 1-1.
2) Name: Sword of judgement; EC: void; AM: 1; S: stun
– Definitely correct.
3) Name: Ice blast; EC: water; AM: 1.2; S: freeze
– Tested and correct for #1, incorrect for Ice Blaster II.
4) Name: Flame fist; EC: void; AM: 1; S: burn
– Does not match either Kentaro’s or Lily’s damage multiplier. Does the spell number match theirs?
5) Name: Poison cloud; EC: void; AM: 0.8; S: poison
– Checked and confirmed correct on all levels.
6) Name: Dragon roar; EC: void; AM: 0.6; S: none
– Confirmed.
7) Name: Meteor; EC: void; AM: 0.3; S: none
– Confirmed with Meteor III as well.
8) Name: Rain of Arrow; EC: void; AM: 1; S: stun
– Incorrect.
9) Name: Thunder shield; EC: thunder; AM: 0.4; S: shock
– Dead on.
10) Name: Fist; EC: void; EC: AM: 1.5; S: none
– I believe this is Giant Fist, and eyeballed it to be correct on 1-1.
11) Name: Thunder; EC: thunder; AM: 0.8; S: shock
- Confirmed correct.
12) Name: Tornado; EC: wind; AM: 0.2; S: none

– Correct.

Thanks for these.

The oddball ones are Fiery Fist and Rain of Arrow, both of which fire multiple missiles at fractional damage.

Ice Blaster II is missing (It’s not a 1.2 multiplier)
Volcanoz is missing. (Should be around 0.6 I think)
Giant Fist X is missing (depending on whether Fist = Giant Fist, that may be missing too).

It’s a slightly different thing, but the Heal spells are missing as well.

 
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3: The attack power for the twelve new units as of last version. I’ve calculated Reapers to be 0.8, Fafnir and the ranged titans to 0.6 along with guessing the others, but code confirmation is always helpful.

I’m not sure if it’s missing some unit/hero/boss, the list is huge.

Just a quick explanation about the list below, see the example:
Name: Omni Knight; AM: 0.2 (1x), 0.3 (2x), 1 (1x), 0.3 (3x); S: none

It means:
Name = the unit/hero/boss Name (do’oh), AM = attack multiplier, (Nx) = number N of hits, S: status type.
The first hit of Omni’s attack deals AM = 0.2, then the next two hits deals AM = 0.3, then next hit AM = 1, and the last 3 deals AM = 0.3.


Units:

Name: Thunder Crystal; AM: 0.3 (x6); S: shock
Name: Wyrm; AM: 0.5 (3x); S: none
Name: Reaper; AM: 0.8 (1x); S: drain
Name: Lamia; AM: 0.4 (1x), 0.8 (1x); S: freeze (2x)
Name: Ork Bomber; AM: 1 (1x); S: burn
Name: Aeon; AM: 0.6 (5x), 0.3 (3x), 0.6 (1x); S: none
Name: Sea Serpent; AM: 0.5 (3x); S: none
Name: Terror Bajaj; AM: 0.3 (8x); S: none
Name: Masked Brawler; AM: 0.8 (2x), 0.4 (1x); S: none
Name: Black Dragon Zero; AM: 0.7 (1x), 0.8 (1x), 0.5 (3x); S: poison (5x)
Name: Mermaid; AM: 0.6 (3x); S: none
Name: Fafnir; AM: 0.6 (2x); S: shock


Enemy units:

Name: Ninja Dog; AM: 0.6 (3x); S: drain
Name: Gold TurdZilla (Ex); AM: 0.4 (3x); S: shock
Name: Dark Guardian (Ex); AM: 0.7 (2x); S: none
Name: Popo Clone (Ex); AM: 0.8 (1x); S: none
Name: Sir Kholai Clone (Ex); AM: 0.4 (3x); S: shock
Name: Shadow Goblin (Ex); AM: 1 (1x); S: none
Name: Viegraff Clone (Ex); AM: 0.6 (1x), 0.9 (1x); S: none, knockdown
Name: Divine Angel (Ex); AM: 0.8 (1x), 0.3 (1x), 0.6 (1x); S: shock (3x)
Name: Lilith (Ex); AM: 0.2 + Random(0 to 0.4) (7x); S: burn
Name: Gold Paladin (Ex); AM: 1 (1x); S: none


Heroes (special 1-mana attack):

Name: Khobbit; AM: 0.6 (1x), 0.7(1x), 0.9 (1x); S: none (1st atk), knockdown (2nd & 3rd atk)
Name: Dragonzord; AM: 0.6 (x4), 0.9 (1x); S: none
Name: Power Go Goblin; AM: 0.3 (6x), 0.4 (4x), 0.5 (10x); S: shock, burn, freeze
Name: Rhino; AM: 0.7 (1x), 0.9 (1x); S: none
Name: Jasmine; AM: 0.6 (3x); S: burn
Name: Jack; AM: 0.6 (10x); S: none
Name: Omni Knight; AM: 0.2 (1x), 0.3 (2x), 1 (1x), 0.3 (3x); S: none
Name: Viegraff; AM: 0.6 (1x), 0.9 (1x); S: none, knockdown
Name: Kholai; AM: 0.4 (1x), 0.6 (1x), 1 (1x); S: shock
Name: Steel Armored Popo; AM: 0.8 (3x); S: none
Name: Oni; AM: 0.7 (1x), 0.9 (1x); S: burn (2x)
Name: Frei; AM: 0.8 (1x), 0.2 (3x), 0.8 (1x), 0.2 (4x); S: none
Name: Lutea; AM: 0.6 (1x); S: poison
Name: Gatotkacha; AM: 0.3 (8x), 1 (1x); S: none
Name: Sir George Lancelot; AM: 0.9 (1x), 0.2 (5x); S: none
Name: Grullborg; AM: 0.7 (1x), 0.9 (1x); S: none
Name: Lilith; AM: 0.2 + Random(0 to 0.4) (7x); S: burn
Name: Joey; AM: 0.7 (1x), 0.9 (1x); S: shock (2x)


Boss

Name: Boss Shadow; AM: 0.6 (x8), 2 (1x); S: drain (9x)
Name: Boss Clown; AM: 0.6; S: burn
Name: NIN-Dog Master: spawn Ninja Dogs (see Unit Ninja Dog)
Name: Boss Omega Popo; AM: 0.8; S: none
Name: Boss Leviatan; AM: 0.6 (3x); S: none
Name: Boss Dragon; AM: 0.6 (3x); S: none
Name: Kentaro The God Fist (spell name: Ryuseiken); AM: 1; S: none
Name: Lord Of Hell; AM: 0.9 (1x), 0.4 (1x), 0.1 (1x); S: burn (3x)
Name: Tiger God; AM: 0.4 (6x); S: none
Name: Raging Thunder Demon; AM: 0.6 (4x), 0.9 (1x); S: none
Name: White Tiger God; AM: 0.6 (6x); S: none
Name: The Death (boss Reaper); AM: 0.8 (1x); S: drain
Name: Phantom Beast Lord; AM: 0.6 (4x); S: poison
Name: Boss Frei; AM: 0.8 (1x), 0.2 (3x), 0.8 (1x), 0.2 (4x); S: freeze (9x)
Name: Boss Flame Devil; AM: 0.8 (7x); S: burn
Name: Metraton; AM: 0.7 (1x), 0.6 (2x); S: none (1x), knockdown (2x)
Name: Boss Aeon; AM: 0.6 (5x), 0.3 (3x), 0.6 (1x); S: none
Name: Gold Dragon; AM: 0.7 (1x), 0.8 (1x), 0.5 (3x); S: poison (5x)
Name: The Witch of Forest (boss Lutea); AM: 0.6 (1x); S: poison
Name: Barbatos The Giant Elder (boss Popo); AM: 0.8 (1x); S: none


About the spells: I’ll check later.

P.S. 1: There is a spell called Arrow, I thought it was the Rain of Arrow.
P.S. 2: The Fire Fist and the Kentaro’s fist are different. I’ll recheck them.

 
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Originally posted by gust4v3:

Awesome, awesome stuff, thanks Gus. Looks I’ll have to update the wiki when I’m not feeling so lazy.

 
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Now the list is (almost) complete and it’s right:


Name: Fire Volcano I; AM: 0.6 (3 hits per volcano, 1 fire volcano); E: fire; S: burn
Name: Fire Volcano II; AM: 0.6 (3 hits per volcano, 3 fire volcanos); E: fire; S: burn
Name: Fire Volcano III; AM: 0.6 (3 hits per volcano, 5 fire volcanos); E: fire; S: burn

Name: Ice blast I, AM: 1.2 (2 hits); E: water; S: freeze
Name: Ice blast II, AM: 3.6 (2 hits); E: water; S: freeze

Name: Thunder storm I, AM: 0.8 (3 hits per thunder, 1 thunder); E: thunder; S: shock
Name: Thunder storm II, AM: 0.8 (3 hits per thunder, 3 thunders), E: thunder; S: shock
Name: Thunder storm III, AM: 0.8 (3 hits per thunder, 5 thunders), E: thunder; S: shock

Name: Meteor I; AM: 0.3 (6 hits per meteor, 1 meteor); E: void; S: none
Name: Meteor II; AM: 0.3 (6 hits per meteor, 3 meteors); E: void; S: none
Name: Meteor III; AM: 0.3 (6 hits per meteor, 5 meteors); E: void; S: none

Name: Dragon roar; AM: 0.6 (5 hits); E: void; S: none

Name: Poison I; AM: 0.8 (2 hits per poison, 1 poison); E: void; S: poison
Name: Poison II; AM: 0.8 (2 hits per poison, 3 poisons); E: void; S: poison

Name: Swords of Judgement; AM: 1 (1 hit per sword, 3 swords total); E: void; S: stun

Name: Rain of Arrows; AM: 0.3 (1 hit per arrow, 50 arrows total); E: void; S: stun

Name: Flame fist; AM: 0.2 + Random(0 to 0.4) (1 hits per fist, 20 fists total); E: void; S: burn

Name: Charming Smile; AM: 0.01 (1 hit); E: void; S: charm

Name: Thunder shield; AM: 0.4 (3 hits every 24 frames (30 total hits), duration: 240 frames = 10s); E: thunder; S: shock

Name: Fist (Popo); AM: 1.5 (1 hit, 1 fist); E: void; S: none

Name: Giant Fist X (1 hit per fist, 10 fists total); E: void; S: none
1st fist; AM: 0.3
2nd fist; AM: 0.15
3rd fist; AM: 0.45
4th fist; AM: 0.3
5th fist; AM: 0.45
6th fist; AM: 0.6
7th fist; AM: 0.9
8th fist; AM: 0.75
9th fist; AM: 1.05
10th fist; AM: 1.5


Pendent:

1) Heal I; multiplier: x (x unknown)
2) Heal II; multiplier: x*1.5 (x unknown)
3) Heal III; multiplier: x*2 (x unknown)

Name: Tornado I; AM: 0.2 (??? hits, duration: 144 frames = 6s, 1 tornado); E: wind; S: none
Name: Tornado II; AM: 0.2 (??? hits, duration: 144 frames = 6s, 2 tornados); E: wind; S: none

 
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bump.:)

 
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This is one of those threads that just makes me love silly internet games and the people who love them. Fun to read through your guys’ exchange!

 
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Great info and very useful.

Grinding made an art :)

 
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I don’t like people discussing infinity. It rarely works out well. For example, it doesn’t matter what the final convergence is if you’re dealing with a finite set. You can say the constant modifiers “will slow down the convergence, but won’t stop it,” but so what? The rate of convergence is the most important aspect if stats are capped (they are). If no unit is going to have infinite stats, no unit will reach the final convergence.

Consider, for example, what would happen if the cap on stats was 500. If you used a constant .1, would it give you different results than if you used a constant .9? Of course! It doesn’t matter that if the stats reached infinity they’d converge to a particular value. At 500, they haven’t converged, and thus the rate of convergence is essential. Not only should that be obvious, it can be confirmed with ease. The sequence given by f(x) = x will converge to infinity. The sequence given by f(x) = 1,000,000/x converges to zero.

So which of these is better? If you think the rate of convergence doesn’t matter, the former is better by far. However, what happens if stats are capped at 500? We get f(500) = 500 and f(500) = 1,000,000/500 = 2,000. The equation that converges to zero turns out to be better than the function that converges to infinity!

I don’t think this would have bothered me much except look at what happens with Twin Salamanders facing themselves. You start with X = ATK. You then multiply it by ATK Power: X = ATK * 0.6. It’s facing its own element, so you then halve the ATK: X = ATK * 0.6 * 0.5. It’s attack type is burn, which it has an immunity to, so you then multiply it be 0.2: X = ATK * 0.6 * 0.5 * 0.2. That means the value used in the DEF calculation is 0.06*ATK.

One unit can have 10 times as much ATK as another unit has DEF and still deal scratch damage. It’s going to take a lot more than references to simulations I cannot see and discussions of convergences that may well never be reached to convince me of the merit of any conclusion found here. Especially since I haven’t seen anyone point out when ATK and DEF values are equal, your average damage won’t be the same as scratch damage.

 
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I’ve already replied about what you mentioned, I’m not sure where is it.
Besides, you didn’t understand the point: if you have N skill points, what is the optimum/best ratio you can use?

You’ve analysed attack/defense/HP individually, and it doesn’t answer the question above.

 
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Originally posted by gust4v3:

I’ve already replied about what you mentioned, I’m not sure where is it.
Besides, you didn’t understant the point: if you have N skill points, what is the optimum/best ratio you can use?

Are you seriously going to dismiss everything I said with nothing more than “you didn’t understant [sic] the point”? I raised a specific, legitimate issue, and you… wave your hands at it?

You’ve analysed attack/defense/HP individually, and it doesn’t answer the question above.

Did you even read my post? I didn’t analyze any stats individually. Heck, I didn’t analyze any stats at all! How could you say something so obviously untrue about my post? And if you can’t even get what I said right, how could you possibly know you’ve responded to it?

 
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No, he is seriously going to dismiss everything you said with “Kholai made your point already, only he made it so thoroughly that he needed more than one post. Try reading the rest of the topic before commenting.” Granted, that only really applies to your last two paragraphs, whereas the rest… is actually also answered, in the form of “there’s enough points available that the convergence does happen,” combined with the implied “You didn’t read my post, so I’m not going to read yours.”

 
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there is a lof of useful information in this thread.. thx for that

But that optimal “hp/attack/defense” ratio is surely not true

your “25 billions of games simulated” not cover the whole game… because equipments, actual unitstats, and even the way they attack, can change the result..
and even if you would calculate every case.. i would say it is still not the result we need, as there are a lot of case in that calculations what peoples would not use..

and if you “work as a statistical consultant” and graduated from math [like i did] then you should also agree that any proof that is not complete, is not a proof.. you cant base anything on that

when you mentioned that “Minimax problem”.. that is really important.. i agree that you dont try to calculate that.. because that would be a waste of time for only an online game… but it can totally change your statistically calculated result.

I agree with all other peoples.. it would be better to remove that infinity defense vs infinity HP thing. it is a perfect way to scam peoples… because those limits has nothing to do with the conclusion…
it would be same true to say “Conclusion: That’s why we state that Attack < HP < Defense.” :)

and in practical use.. to complete some of the harder quest at lower lvl [=with low stats], will need you to focus on defense… they are just examples.. but they are sure points in the game…

 
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When you’re a very high level, your goal is to finish 10-6 in a few seconds for your epic stone and gold farming. For that, you only need attack. You get +1 to every stat on level-up anyway.

I’m also not sure that “100% attack” is optimal. If it isn’t a speed-farming situation, other considerations matter. You have elemental advantages and disadvantages. You don’t summon the exact same units as the enemy. You can wield element+ items. The aptitude for HP is greater than the aptitude for other stats.

 
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Originally posted by ForceofLight:

No, he is seriously going to dismiss everything you said with “Kholai made your point already, only he made it so thoroughly that he needed more than one post. Try reading the rest of the topic before commenting.” Granted, that only really applies to your last two paragraphs, whereas the rest… is actually also answered, in the form of “there’s enough points available that the convergence does happen,” combined with the implied “You didn’t read my post, so I’m not going to read yours.”

In other words, you’re going to mindlessly repeat his points as though they somehow answer what I said despite there being absolutely nothing to indicate so. To demonstrate the failing of your post, I’ll point out you choose not to address the fact gust4v3 flat-out made **** up about what my post said. If he makes things up about what I said, why should anyone believe he has actually responded to what I’ve said? How could he know he has responded to my points if he doesn’t even know what those points are?

And I’ll provide a more specific example. You claim my point about convergence is answered in the form of “there’s enough points available that the convergence does happen." This is not a refutation. It does not show I am wrong. All it shows is someone claims I’m wrong. It’s claiming convergence happens without doing anything to prove so. In other words, you are mindlessly claiming things to be true without any proof for them being true.

If you want to mindlessly repeat things because some guy tells you they are true, you can. It just won’t make the claims more believable or contribute anything to the discussion.