Constructed Design Theory [Part 1:Resilience]

[OldTrees]

Literally, what affects it. But that's already in the first post.
But what's really relevant to resilience is its' comparative worth. You can compare it to (broadly) games, ex. MTG, ETG, etc. Or on a different scale, compare it to current resilience, attack (in some ways the opposite of resilience), costs and more.
But even if everything had zero hp which is (ex.) worth a value of zero, they'd all still not be the same due to the other factors.
I'm not sure if this is relevant, but there's also the scale to consider. For example Yugioh is in the thousands while ETG is in the ones and tens.

Exactly relative resilience is more important then objective resilience. Hence I set an arbitrary hp (4) as resilience 1 and compared the other hp to the resilience of that hp.

I think comparing True Vanilla creatures that are also considered exemplary balanced would be a good point once I have a means to account for the effect of casting cost on resilience.

Also 0hp is definitely not a resilience of 0, as long as it survives to make a single attack then it has some resilience.

I was specifically talking about the scale of Nhp vs (N+1)hp in ETG.

[Rutarete]

The zero as zero value was an example.
(off topic) I wonder if we can compare resiliences like different currencies...
Anyway, the problem with with setting an arbitrary hp and comparing is that while resilience is relative, it doesn't have the right 'ways' to be compared more 'correctly'.
Another way of saying that is 1/2 plus 2/3. you have to make them the same scale 3/6 + 4/6 to add them, or in this case, compare them
so that should be a big focus

[OldTrees]

The zero as zero value was an example.
(off topic) I wonder if we can compare resiliences like different currencies...
Anyway, the problem with with setting an arbitrary hp and comparing is that while resilience is relative, it doesn't have the right 'ways' to be compared more 'correctly'.
Another way of saying that is 1/2 plus 2/3. you have to make them the same scale 3/6 + 4/6 to add them, or in this case, compare them
so that should be a big focus

Yes. What I did was make guesses at
1hp/4hp=A
2hp/4hp=B
3hp/4hp=C
4hp/4hp=D
5hp/4hp=E
6hp/4hp=F
10hp/4hp=G
15hp/4hp=H
2hp/1hp=B/A

on a side note. I do not expect a lot of addition with this theory. mostly multiplication.

[Rutarete]

The zero as zero value was an example.
(off topic) I wonder if we can compare resiliences like different currencies...
Anyway, the problem with with setting an arbitrary hp and comparing is that while resilience is relative, it doesn't have the right 'ways' to be compared more 'correctly'.
Another way of saying that is 1/2 plus 2/3. you have to make them the same scale 3/6 + 4/6 to add them, or in this case, compare them
so that should be a big focus

Yes. What I did was make guesses at
1hp/4hp=A
2hp/4hp=B
3hp/4hp=C
4hp/4hp=D
5hp/4hp=E
6hp/4hp=F
10hp/4hp=G
15hp/4hp=H
2hp/1hp=B/A

on a side note. I do not expect a lot of addition with this theory. mostly multiplication.

That means an equation is needed

[OldTrees]

That means an equation is needed

I am not sure an equation more than a set of values is needed for this factor (hp).
After all it is discrete values we are dealing with not a continuous line.

[Rutarete]

That means an equation is needed

I am not sure an equation more than a set of values is needed for this factor (hp).
After all it is discrete values we are dealing with not a continuous line.

Maybe to find a comparative value take an ETG card and see how it's affected/affects the normalcy of a different game. The bigger the difference, the bigger the range of values.

[Midnar]

At this point I start wondering if anybody read this (http://elementscommunity.org/forum/index.php/topic,16655.msg294132#msg294132). I added the big part in an edit a week ago, so it may have gone unnoticed. The formula I suggest is based on logarithm (sorry for the complicated math). Since it fits this topic better, I'll just copy-paste it:

I tried a formula and was amazed by how well it works with every vanilla creature ! First, the ideas behind the formula:
- The more HP a creature has, the less important it is to give it more hp. Therefore, resilience is not a linear function of hp's, but rather a logarithmic one.
- Many vanilla creatures are X/5 with a cost of X quantas, suggesting that resilience of 5hps creatures is 1. I thus tried the base-5 logarithm, but it was no good. This is because...
- All creatures (except 0hp creatures) have a "base resilience", because it's possible that the opponent will not have any creature control, making hp's uninmportant.
- In other words, resilience = (chance to survive despite CC) + (chance to survive because the opponent lacks CC) = logarithmic part + static part

And now, the (temporary) formula:
Resilience = (log₅(hp)+1) / 2
With only one adjustment to make: Life creatures get one free hp.

Here are the theoretical costs of all unupped vanilla creatures, using this formula:
Abomination: 5
Purple dragon: 10
Flesh Spider: 2.52, rounds up to 3 (not taking Web into account)
Mummy: 4.27, rounds to 4 (not taking it's passive into account)
Bone dragon: 10
Skeleton: .5, rounds to 1 (version 1.27 will boost it with a new ability, making it less UP)
Graviton mercenary: 3 =/= 4
Colossal dragon: 9.39, rounds to 10
Hematite golem: 4.23, rounds to 4
Gnome rider: .72, rounds to 1 (this one will generate quanta with next update)
Horned Frog: 2.15, rounds to 2
Cockatrice: 3.37, rounds to 3
Emerald Dragon: 10
Ash Eater: 1
Crimson Dragon: 10.10, rounds to 10
Blue Crawler: 2.52, rounds to 3
Ice Dragon: 9.51, rounds to 10
Photon: .5, rounds to 0
Golden Dragon: 12.15, rounds to 12
Dragonfly: 0.72, rounds to 1 (same as Gnome Rider)
Azure Dragon: 9.51, rounds to 10
Devonian Dragon: 10
Black Dragon: 10
Phase Spider: 2.86, rounds to 3 (not taking Web into account)
There's only one discrepancy : Graviton Mercenary. Now just look at card usage statistics: Graviton Mercenary is underused. Meaning that it is probably UP, which is what the formula suggests. The only non-rare creatures used less than Graviton Mercenary are Blue Crawler (also slightly UP according to the formula) and Hematite Golem (balanced, but overshadowed by the OP Graboid).

About upgraded creatures: using either -1 or -2 depending what we need to match the actual card cost isn't the good way to do. We could take inspiration from spells. Low-cost spells get a 1-quantum reduction to their cost when upgraded. High-cost spell get a 2-quanta reduction. I'll work on this later, time for dinner now 

[OldTrees]

Wow...
Wow...

That is great. It had gone unnoticed thanks for repeating it.

It has a good constructed theory.
[Resilience = Constant x Probability of no CC + Function of Hp x Probability of CC]
I personally think that the underlined part is incomplete because cheaper cards are more resilient that expensive cards. (currently testing this to see if is significant enough for an official version of the theory.)

Also it has already been tested for accuracy by comparing to creatures already in the game and the general wisdom about those card.

For now the CPR theory has no determined function between unupped and upgraded (aka the CPR theory does not use the -1.5 function yet.)

I owe you karma. You did what I have been waiting for. You beat me to a solution/model that fit better.

I petition everyone who is reading this thread to give Midnar at least 1 point of karma.
Midnar I will add you in the list of authors in the final thread.

For now I will quickly see how this works on the difficult dragons and other true vanilla creatures (non  ).

Once again thank you!

[Midnar]

Wow, my first point of karma ! I'm grateful 
I said I was going to work on the upgraded formula, but finally I didn't because the lack of any response kinda demoralized me. Now I'm really motivated, but tired (it's quarter to midnight in France). I'll work on this tomorrow, promise !

[Midnar]

After doing the calculations for the upgraded cards, I found that the resilience formula I suggested overrates high-hp creatures and underrates low-hp creatures. I changed the formula slightly, giving more weight to the base resilience; this mostly fixed the problem.
New formula: Resilience = (log₅(hp)+2) / 3

Recalculation for unupped vanilla and semi-vanilla creatures using the new formula:

Creature	Theoretical cost	Actual cost	Notes
Abomination	5	5
Purple Dragon	10	10
Flesh Spider	2.68	3	Ignoring web
Mummy	4.47	4	Ignoring the passive
Bone Dragon	10	10
Skeleton	.67	1	Ignoring the passive
Graviton Mercenary	3	4	Discrepancy
Colossal Dragon	8.96	10	New discrepancy
Hematite Golem	4.15	4
Horned Frog	2.43	2
Cockatrice	3.58	3
Emerald Dragon	10	10
Ash Eater	1.33	1
Crimson Dragon	10.73	10
Blue Crawler	2.68	3
Ice Dragon	9.34	10
Photon	.67	0
Golden Dragon	11.44	12
Azure Dragon	9.34	10
Devonian Dragon	10	10
Black Dragon	10	10
Phase Spider	3.24	3	Ignoring Web

Graviton Mercenary was already discussed. Colossal Dragon is also UP imo (i was actually surprised when my first formula assessed it as being balanced).


Upped vanilla and semi-vanilla creatures, always accounting upgrade as a 1-quantum cost reduction:

Creature	Theoretical cost	Actual cost	Notes
Micro Abomination	.91	1
Amethyst Dragon	11.45	11
Flesh Recluse	4.37	3	Ignoring web; Discrepancy
Elite Mummy	3.47	3	Ignoring the passive
Ivory Dragon	10	10
Skeleton	.62	1	Ignoring the passive
Graviton Guard	2	2
Massive Dragon	10.79	11
Steel Golem	5.73	4	Discrepancy
Giant Frog	3.05	2	Discrepancy
Elite Cockatrice	3.77	3
Jade Dragon	12.17	12
Ruby Dragon	11.15	12
Abyss Crawler	5.23	4	Discrepancy
Arctic Dragon	12	11	Discrepancy
Light Dragon	13.18	13
Sky Dragon	12.91	12
Silurian Dragon	11.40	12
Black Dragon	11.45	12
Phase Recluse	4.67	4	Ignoring Web

All 5 discrepancies can be fixed by accounting upgrade as a 2-quanta cost reduction, as initially suggested by OldTrees. However, I currently do not understand why some creatures should get a 2-quanta bonus from the upgrade, while most of them only get 1.

[OldTrees]

After doing the calculations for the upgraded cards, I found that the resilience formula I suggested overrates high-hp creatures and underrates low-hp creatures. I changed the formula slightly, giving more weight to the base resilience; this mostly fixed the problem.
New formula: Resilience = (log₅(hp)+2) / 3

All 5 discrepancies can be fixed by accounting upgrade as a 2-quanta cost reduction, as initially suggested by OldTrees. However, I currently do not understand why some creatures should get a 2-quanta bonus from the upgrade, while most of them only get 1.

As you said in the other thread HP should be a factor in the Power variable due to catapult and acceleration.
I assume these calculations were under the old Power=attack model.

The 1-2 cost reduction was the old formula. This formula uses a different basis and thus most likely will have a different resulting upgrade factor. As you said before that factor will probably proportional to the original total Cost.

[Midnar]

As you said in the other thread HP should be a factor in the Power variable due to catapult and acceleration.
I assume these calculations were under the old Power=attack model.

The 1-2 cost reduction was the old formula. This formula uses a different basis and thus most likely will have a different resulting upgrade factor. As you said before that factor will probably proportional to the original total Cost.

Indeed, I used the Power = Attack (+ HP/50) model I mentionned in the other thread. HP/50 is only added for gravity creatures (for all other creatures it would barely make a difference of .1 or so).
I tried to make the factor proportional to the original cost, but this definitely doesn't work. For some reason, the creatures that get the best upgrades are the small ones (e.g. Flesh Spider gets +3|+0 for free, while Crimson Dragon gets +3|-1 and +2 CC).