The Battle for... somewhere - Provisional Signups

[Nihila]

After some debate about my scouting cost, it has changed(for my game, at least). The minimum is the same, but the cost increases more slowly. It still gets pretty expensive for a Diwigi-scout.
It's:
5+(.5)(Hits+(Attack*Defense*.5)

In other news, Twoy made a Diwigible for the Northern Alliance's game--100 Hits, 150 Attack, 5 Defense, 5 Move, Ranged, Flight, Siege, 2128 Cost.

[BLANDCorporatio]

Hi, no rant for today. In fact, I'll postpone that for Wednesday, by which time I hope to have an actual rules statement, with a (only by the standards of the rants so far) brief justification for the formulas.

I'll just say now however that I just have to go ahead and put a cost on Auto-heal. There's no way to convincingly balance Large vs. Small units otherwise. Well, okay, there's another one, for the Hits term to be exponential in Hits. I trust that seems ludicrous to everyone.

[zilfallon]

Every unit has auto-heal, and every unit NEEDS it. So, if you're going to add a cost for auto-heal which is based on hits, then why not add that straight to formula since it is a must-have? This way, we'd prevent having non-healing units which drop our Comic Accuracy rating , and we'd have nerfed large units.
But really, increasing the cost that much will not be..."balanced" i think.

[Nihila]

I'll just say now however that I just have to go ahead and put a cost on Auto-heal. There's no way to convincingly balance Large vs. Small units otherwise. Well, okay, there's another one, for the Hits term to be exponential in Hits. I trust that seems ludicrous to everyone.

That would depend on how exponential it is. Like [(1.05)^Hits] might not be so bad, while [(10)^Hits] would just be nuts. Unless you meant (Hits^#), which is working fine for the game I'm running. So, it doesn't seem ludicrous to me, it seems reasonably sensible. If a Diwigible(100 Hits, 150 Attack, 5 Defense, 5 Move, Ranged, Flight, Siege) costs 2128 points to pop, I'm not popping a Diwigible when I can get a better bang for my buck popping Dwagons(16 Hits, 24 Attack, 4 Defense, 5 Move, Ranged, Flight) for 150 each.

[Sihoiba]

I'm wondering if perhaps trying to balance everything out in just the pop costs isn't the right way to do it. We know units have upkeep and take time to pop, is it possible to use either of these to help achieve a more balanced state with auto heal.

Something so that while it costs side A X times amount in pop points popping a force of small units capable of killing sides B large unit, those X units only cost a fraction of the large units a turn to maintain.

[BLANDCorporatio]

That would depend on how exponential it is. Like [(1.05)^Hits] might not be so bad, while [(10)^Hits] would just be nuts. Unless you meant (Hits^#)

No, I mean exponential. As in, 3^Hits. So ok, that was a napkin calculation, but still. Wherever the base will be, the result is ludicrous.

Every unit has auto-heal, and every unit NEEDS it. So, if you're going to add a cost for auto-heal which is based on hits, then why not add that straight to formula since it is a must-have?

They may sound like the same thing, but they are quite different in terms of consequences to the game (and the philosophy of balance they impose).

If AutoHeal is free, then a Small unit stack better croak that Large unit, or else they might as well have been playing checkers instead, while waiting to get squashed. This means that you need a very drastic balance criterion, the Cost to Guarantee Croaking. This leads to cost for units that are exponential in Hits and that, I trust, seems ridiculous.

However, once you put costs on AutoHeal, then the situation changes. A small unit doesn't need to croak a large unit, but merely leave a dent that is large enough to cost as much as whatever the Large unit can croak/damage. So you can instead use a more forgiving balance criterion, of trading pop points fairly (or at an advantage to the Small units when they roll better than worst case or whatever). This remains true whatever the cost formula is, as long as you balance the AutoHeal cost according to the previously mentioned principle. Also, unit costs do not explode into ridiculousness.

My approach is to balance AutoHeal costs to favour Small units, but not so badly that Raider attacks become viable against Large units. Fortunately, for Raiders' attacks there's the Hit and Run rule that I can play with, so I have some maneuvering space to get the desired balances out of the system (Small>=Large, Raider>=Small, Large>=Raider).

I'm wondering if perhaps trying to balance everything out in just the pop costs isn't the right way to do it. We know units have upkeep and take time to pop, is it possible to use either of these to help achieve a more balanced state with auto heal.

Something so that while it costs side A X times amount in pop points popping a force of small units capable of killing sides B large unit, those X units only cost a fraction of the large units a turn to maintain.

This is a good point and suggests yet another approach at this. In any case, there should be some kind of upkeep cost, whether it's justified by healing your units or simply keeping them well fed. So in a rules system I'd propose, I'd use the general name "Schmuckers" for that resource that you can use to buy and heal (or simply keep) units.

As I said, I plan to try and write a rules doc for Wednesday, but it'll be some time until we can play those, so making two sets will be ok. When the current games finish, we can try these as well.

[Nihila]

No, I mean exponential. As in, 3^Hits. So ok, that was a napkin calculation, but still. Wherever the base will be, the result is ludicrous.

Well, [(1.05)^Hits] is actually very easy on higher Hits units--by 30 Hits, it reaches 4.321...(And I'm serious. (1.05)^30 does start 4.321).

Which is ludicrously tiny. (1.15)^Hits is a bit more reasonable. With the Hits values on the left, and (1.15)^Hits on the right(rounded to 2 decimal places), a listing up to 30 Hits:

Spoiler: show

1-->1.15
2-->1.32
3-->1.52
4-->1.75
5-->2.01
6-->2.31
7-->2.66
8-->3.06
9-->3.52
10-->4.05
11-->4.65
12-->5.35
13-->6.15
14-->7.08
15-->8.14
16-->9.36
17-->10.76
18-->12.38
19-->14.23
20-->16.37
21-->18.82
22-->21.65
23-->24.89
24-->28.63
25-->32.92
26-->37.86
27-->43.54
28-->50.07
29-->57.58
30-->66.21

Okay, the values run away at the end, but overall, it makes hits pretty cheap. Much, much cheaper for small unit than big units, though.
I will, however, list three more values for the above chart to make it clear how sharply things start to rise.

Spoiler: show

50-->1 083.66
75-->35 673.87
100-->1 174 313.45

So, starts off not so bad, escalates to about 1 million cost for a 100 Hit unit. This is just an idea, make of it what you will.
I don't really like the thought of putting a cost on Auto-Heal, though. It feels less "Erfworldian."

[zilfallon]

What i meant in my previous post, was:

If we want to be accurate to comic, then we must NOT enable units to not have auto-heal. If you really want to fix the hp problem that way, it is just another variable in the formula of every unit, since every unit MUST have auto heal. So in the end, it has nothing to do with auto heal, it is just a hits variable with a constant base.

[WaterMonkey314]

I'm inclined to go with the prohibitive hits cost formula instead of putting a cost on Auto-Heal - in the comic, we don't see nearly as many huge units as we've seen floating around in our games (we've generally seen a large or juggernaut unit for maybe at most 1 stack of normal infantry - much more than in the comic).

[BLANDCorporatio]

Oh well, I've been running into a spot of inconsistent constraints when trying to put a cost on AutoHeal AND impose certain (im)balances, so since that solution proves not popular I won't pursue it.

There's one other thing I will suggest, but first a brief explanation of why I said that the cost formula (if we make a few assumptions) will be exponential. Once we see that calculation, I'll give you my proposed alternative. So here goes:

Spoiler: show

Let there be a large unit, L, and a small unit S, and they have H_l and H_s Hits respectively. To guarantee croaking the large unit, I need k_n small units; I need to guarantee croaking the large unit, or else it will autoheal and the attack rendered useless. I also want the cost of the small units I use to be less or equal to the cost of the large unit. So therefore,

Cost(L) >= k_n*Cost(S).

What's k_n? Well, the large unit actually may have as much as H_l*8/(8-Defense of large unit) hits, while the small units may make as little as 0.5*2*H_s damage. This works out to

k_n = (H_l/H_s)*(8/3).

One more thing. I'd like, say, to impose that this cost inequality be valid as you increase Hits for the large unit by 1. It's easier to see the exponential character this way, because the inequality become

Cost(H_s + 1) >= ((H_s + 1)/H_s)*(8/3)*Cost(H_s).

Notice that I said that the cost of the unit becomes merely a function of its Hits. It's a reasonably safe approximation, if we agree to consider that both Large and Small units are in a way similar- maxed out in Defense and Attack, and of the same Move.

So anyway, the formula above, let's make it an equality. It defines via recursion a function, and that function is

Cost(H) = H*((8/3)^Hits)*Cost(1)

There's now two options. The obvious one is to

Spoiler: show

Relax a bit the requirement of small-vs.-large efficiency. Maybe a small unit is not that effective against a large unit that has one more Hit than it does, but it may be efficient vs. a large unit that has n more hits. We get the following recursion formula:

Cost(H_s + n) = ((H_s + n)/(H_s))*(8/3)*Cost(H_s)

Which means that the function itself is

Cost(1 + a*n) = (1 + a*n)*((8/3)^a)*Cost(1) for Hits = 1 + a*n (Hits: 1, 1+n, 1+2*n ...),
Cost(2 + a*n) = (2 + a*n)*((8/3)^a)*Cost(2) for Hits = 2 + a*n (Hits: 2, 2+n, 2+2*n ...),
Cost(3 + a*n) = (3 + a*n)*((8/3)^a)*Cost(3) for Hits = 3 + a*n (Hits: 3, 3+n, 3+2*n ...),
...
Cost(a*n) = (a*n)*((8/3)^a)*Cost(n) for Hits = a*n (Hits: n, 2*n, 3*n ...).

So in other words we limit the exponent a bit for 8/3 to values where it doesn't explode.

Then there's solution number two. The exponential nature was produced by the k_n factor (see the first spoiler), so here's one way to combat that. Increase the Attack Cap, so that

Spoiler: show

k_n = H_l/H_s.

(For H_s = H, H_l = H + 1) If we have a recursion

Cost(H+1) = ((H+1)/H)*Cost(H),

then solving the recursion yields

Cost(H) = H*Cost(1), which the current formula allows.

Now to actually make it so that k_n = H_l/H_s, we need the damage cap for the small unit to be

(8/3)*(1/(0.5)) or in other words 16/3.

So at the end of that we have a cap for Attack of 16/3 for small units (16/3*Hits, or 12*Hits^(2/3), whichever is lower, to keep the current property of the attack cap that it starts to get worse after 9 Hits).

That means a unit can do slightly more than 5 times its Hits in Attack.

Ok, what does that mean? What chances does a larger unit have to survive this attack? For this comparison, let's say we have 1 Small unit with 9Hits and 48Attack, and a large unit of 10 Hits or more. Running once through the calculation for the 10Hits unit:

- first, let's see what are the odds for the small unit to roll x% or worse. The dice roll as (2d6+8)/20 which gives

Spoiler: show

1 in 36 cases to roll 50%.
3 in 36 cases to roll 55% or worse.
6 in 36 cases to roll 60% or worse.
10 in 36 cases to roll 65% or worse.
15 in 36 cases to roll 70% or worse.
21 in 36 cases to roll 75% or worse.
26 in 36 cases to roll 80% or worse.
30 in 36 cases to roll 85% or worse.
33 in 36 cases to roll 90% or worse.
35 in 36 cases to roll 95% or worse.
36 in 36 cases to roll 100% or worse. Duh.

- next, a unit with X hits may have as many as X*8/3 combat hitpoints. A unit with 10Hits actually has 26.(6) combat hits. From this, we see that if the 9H unit rolls 50%, or even 55%, the 10H unit survives. The 10H unit therefore survives in 3 of 36 cases.
- Moving on, the chances of survival for units with more Hits are:

Spoiler: show

6 in 36 for 11H
10/36 for 12H
15/36 for 13H
21/36 for 14H (better than fair)
26/36 for 15H
30/36 for 16H
33/36 for 17H
35/36 for 18H
and a unit with 19H will always survive being attacked by one maxed out 9H unit.

So. Yeah. The odds, themselves, seem fairish to me. However, as consequences of that attack cap we have
- all units better have max def, unless they, because of other specials, do not expect to be on the front
- carnage, motherfucking carnage. The piles of bodies will reach the skies, and while Erfworld has autoheal, it kinda has this too somehow.

[Nihila]

- carnage, motherfucking carnage. The piles of bodies will reach the skies, and while Erfworld has autoheal, it kinda has this too somehow.

Heh. Perfect. What we need more of is carnage. Well, like you said at an earlier point in this thread, wolverines are notoriously fierce, even without stacking bonuses--erm, traveling in packs. So, I'm comfortable with INCREDIBLE, COSMIC Attack scores, itty bitty Hit points.

And, the bodies disappear when the allied turn starts, so they won't ever pile up too much.

[BLANDCorporatio]

Allright!

Well there's more ways to try and tame the exponential character of a cost formula that favours small units. We can mix the two options I mentioned in my post:

- raise attack cap, but less drastically, so that units of the same number of hits have x odds of walking away (this reduces the base for the exponential), AND
- balancing so that small units are efficient against units larger by n hitpoints. This makes the exponent increase slower.

We could have two sets of rules started on this basis, one that is very lethal, one less so but units cost more. But then, later this week. Not today, alas, sorry.

[BLANDCorporatio]

For "kicks" (I have a bizzare definition of fun, apparently) here's yet another way to try and balance small-vs-large units-

Spoiler: show

Suppose you want units to be efficient, if not against units that have just a few more hits than them, then at least against units that have, say, twice the hits. So a 9 hit unit might not be a better investment than a 10 hit unit, but it's a better investment than an 18 hit unit- while who knows, the 10hit unit might not be as effective against that 18 hit unit!

Anyway, in general, the condition you impose is that the cost of a large unit, that has k_h times the hits of your small units, is larger than the cost of the small units needed to guarantee its croaking. This, in math, is

Cost(k_h*H) >= k_n*Cost(H).

One important observation about k_n- it's a constant now, and doesn't depend on H anymore (unlike the previous times). Let's pick the equality to define our recursive function:
Cost(k_h*H) = k_n*Cost(H).

So, Cost(1) is Cost(1). Cost(k_h) is k_n*Cost(1), Cost(k_h^2) is k_n^2*Cost(1) and so on. This suggests that the function, in general, is

Cost(k_a^n) = k_n^n*Cost(1).

However, k_a^n is the H value, so we have that n is the logarithm (in base k_h) of H. After converting log in base k_h to natural logarithm, which we can calculate in Excel, we have:

Cost(H) = k_n^(ln(H)/ln(k_h))*Cost(1),

where ln(x) is the natural logarithm of x, aka the power to which you need to raise a certain constant (famously known as e in math circles) to get the number x.

This remains an exponential formula, but on the plus side the exponent doesn't have to explode quite so quickly. Those ln-s take care of that. In fact, by this formula cost is "almost" proportional to Hits (but obviously, not quite).

The downside, it's very weird. While it'll be easy to calculate in Excel, it will be harder to just look at a unit's stats and guess where it's cost should be. Not without a quick reference table of costs for maxed-out units of various Hits.

Anyway, so far I know of three ways to "tame" AutoHeal. In summary, the pure ways are the following (mixes between 1 and 2, or 1 and 3, are also possible):

- increase the Attack Cap. Can be used on the same cost formula we've had so far. Keeps the math simple. Will result in very lethal combats. Equal forces encountering each other will likely wipe each other out. Favours High-Def units. (EDIT: or does it? Since everyone's gonna croak anyway ...)
- impose a "small unit of x hits is more efficient than a unit with x+n hits or more". Creates a nice circle of power, if you will (something like 10H>9H>11H>10H for example). Inserts an exponential in the cost formula (more expensive units).
- impose a "small unit of x hits is more efficient than a unit with k_h*x hits or more". Creates a circles of power as the above (only wider as Hits increase). Inserts exponentials and logarithms in the cost formula, which while more complex will result in costs staying more or less as they were before.

[Nihila]

- increase the Attack Cap. Can be used on the same cost formula we've had so far. Keeps the math simple. Will result in very lethal combats. Equal forces encountering each other will likely wipe each other out. Favours High-Def units. (EDIT: or does it? Since everyone's gonna croak anyway ...)

Well, just for my thoughts, in Sihoiba's game, my Aimursa Hunters are designed around the principle of "Well, these'll croak fast, but they can take down a bunch of your units with them." So, I think that a high Attack Cap would decrease defense. Whether this is good or bad is up for debate.

- impose a "small unit of x hits is more efficient than a unit with k_h*x hits or more". Creates a circles of power as the above (only wider as Hits increase). Inserts exponentials and logarithms in the cost formula, which while more complex will result in costs staying more or less as they were before.

I will pass on doing logarithms to work out unit costs. I'm the kind of person who'll design random things on the back of a napkin, and I simply cannot, no matter how hard I try, do logarithms in my head. Except for log{base a} of a^n. That I can do fine.

- impose a "small unit of x hits is more efficient than a unit with x+n hits or more". Creates a nice circle of power, if you will (something like 10H>9H>11H>10H for example). Inserts an exponential in the cost formula (more expensive units).

Well, if we must have more expensive units to have a circle of power that doesn't involve logs, I'll take more expensive units any day of the week.

Well, any day of the week that I have calculus work to do, at least.