Unit Cost Equation

[The_Tauren13]

Ok, new chart
(can you tell I dont have a life)

Code:

---

Morale   Cost

   0     0.450

   1     0.460

   2     0.480

   3     0.510

   4     0.550

   5     0.600

   6     0.660

   7     0.730

   8     0.810

   9     0.900

  10     1.000

  11     1.100

  12     1.195

  13     1.285

  14     1.370

  15     1.450

  16     1.525

  17     1.595

  18     1.660

  19     1.720

  20     1.775

  21     1.825

  22     1.870

  23     1.910

  24     1.945

  25     1.975

  26     2.000

  27     2.020

  28     2.035

  29     2.045

  30     2.050

---

Attatched is another graph of the data.
Its a much smoother spread, too However, the change close to 10 is less severe...

[Saber Cherry]

Hmmm.......

Yeah, it has better endpoints. Is it possible to increase the slope around 10 while keeping the endpoints and the smoother line?

[The_Tauren13]

Saber Cherry said:Is it possible to increase the slope around 10 while keeping the endpoints and the smoother line?

Probably. Lemme give it a go...

And why is the spacing all wacked out in the 2nd chart?

[Saber Cherry]

The_Tauren13 said:

Probably. Lemme give it a go...

And why is the spacing all wacked out in the 2nd chart?

Editing always messes up (code)(/code) sections. Forum bug.

[The_Tauren13]

Code:

---

Morale   Cost

   0     0.500

   1     0.510

   2     0.523

   3     0.541

   4     0.564

   5     0.596

   6     0.638

   7     0.694

   8     0.768

   9     0.868

  10     1.000

  11     1.251

  12     1.439

  13     1.580

  14     1.686

  15     1.765

  16     1.824

  17     1.869

  18     1.903

  19     1.928

  20     1.947

  21     1.961

  22     1.971

  23     1.979

  24     1.985

  25     1.990

  26     1.993

  27     1.995

  28     1.997

  29     1.999

  30     2.000

---

[Saber Cherry]

Perfect!

[sushiboat]

Here is a quick and dirty formula based on a multiple linear regression that
included only units with morale below 30. (AFAIK, the 30 and 50 morale
numbers are only symbolic and represent a qualitatively different state.) I
didn't include movement because the spreadsheet I have needs work to
adjust those numbers into a useable format. (Plenty of other boring
qualifiers omitted.)

Cost = -386.0 + (11.9 * Size) + (2.3 * HP) + (-2.0 * Prot) + (11.0 * Morale) +
(21.8 * MagicRes) + (1.4 * Enc) + (-4.8 * Str) + (2.8 * Att) + (.7 * Def) +
(4.0 * Prec)

The adjusted multiple R-squared is .70, so about 70% of the variance in cost
is explained by the predictors. How does that compare to the formula in the
first post?

In case you are not familiar with multiple regression, don't be fooled by the
size of the coefficients. There is some collinearity -- some of the stats used
as predictors are correlated with each other, so their unique predictive
contribution is affected. For example, strength has a zero-order correlation
of .24 with cost, but in the multiple regression, its coefficient is negative.

Edit: Commander vs. non-commander makes a difference, so here are the
formulae after splitting the two groups.

Non-commander formula (adjusted R-squared = .81):

Cost = -60.1 + (12.3 * Size) + (2.9 * HP) + (.8 * Prot) + (5.3 * Morale) +
(4.1 * MagicRes) + (-1.9 * Encumb) + (-8.0 * Str) + (1.5 * Att) + (.5 * Def) +
(-1.9 * Prec)

Commander formula (adjusted R-squared = .71):

Cost = -459.7 + (16.8 * Size) + (.3 * HP) + (-2.5 * Prot) + (11.0 * Morale) +
(25.4 * MagicRes) + (5.0 * Encumb) + (.8 * Str) + (-1.1 * Att) + (6.2 * Def) +
(-.1 * Prec)

[Saber Cherry]

How on Earth did you come up with these? I assume you were using data compiled in Excel spreadsheets and some automated technique, but...

OK, I admit it, I'm

not familiar with multiple regression

I have to wonder how well a forumla with negative coefficients for strength, encumbrance, and precision will predict the value of future units... but since you out-math me, I'll gracefully bow before you

[The_Tauren13]

50 morale is symbolic, but I think 30 morale is real...

As for your equation, while it may be perfect for all units already in the game, it doesnt do much for modding, which is what I was interested in; I ran the Size 1, 5 HP, 3 Prot, 10 Mor, 5 MR, 2 Enc, 5 Str, 11 Att, 10 Def unit Im including in my mod through your equations and he came out costing -133 from your combined euation, and 20 from the unit only one. So, it isnt really a global equation that can be applied to any unit a modder might come up with, which is what I was hoping to eventually get. Not that Im claiming any equation can fully encompass this complicated game I just want to make a stab at it...



Now, I have a couple questions: Can the same 'diminishing returns' idea be applied to attack and defence? That is, is the difference between 10 and 13 attack greater than between 20 and 23? How much attack or defense skill would be required to double the effectiveness of a 'normal' unit (that is, 10 HP, 0 Prot, 10 Att, 10 Def, etc.)? Is defense really worth more than attack? For some reason I have gotten that into my head, but then, Im a n00b at Dominions...

In my initial equation, should the magic resistance be pulled out of the square root? Should it be wieghted more? Does low magic resistance make an otherwise powerful unit worthless?

new equation: (thanks Saber Cherry )
sqrt( ( ( HP * ( 10 + Prot ) * Def * MR ) / 1000 ) * ( ( Str * Att ) / 10 ) ) * ( ( 3 + Move ) / 5 ) * ( ( 40 + AP ) / 50 ) * ( 9 / ( 6 + Enc ) ) * f( Mor )
f( x ) = 0.5 + ( ( 0.75 ) ^ ( 10 - x ) ) / 2 | x < 10
f( x ) = 2 - ( ( 0.75 ) ^ ( x - 10 ) ) | x >= 10


Heres another (crazy) idea:
sqrt( HP * ( 10 + Prot ) * f( Def ) * f( Str ) * f( Att ) ) * ( ( 3 + Move ) / 5 ) * ( ( 40 + AP ) / 50 ) * ( 9 / ( 6 + Enc ) ) * f( Mor ) * f( MR )
f( x ) = -sqrt( 10 - x ) / 3 + 1 | x < 10
f( x ) = sqrt( x - 10 ) / 3 + 1 | x >= 10

This is assuming that morale, magic resistance, attack, strength, and defense all follow the same cost distribution...
Attatched is a graph of this f(x).

[sushiboat]

How did I do it? Many statistics programs exist that can do a multiple
regression in a few seconds. The tricky part is recognizing problems and
interpreting the output.

What use is it to a modder? Use it as you will. It is an imperfect description of
the way that the unit stats are related to unit costs in the original game.

Below are some new formulae. I added some stats, and I removed stats post
hoc that didn't make a significant unique contribution. Something I might do
in the future is separate mages from non-mages.

Non-commander (adjusted R-squared = .81)

Cost = -51.7 + (11.2 * Size) + (3.1 * HP) + (.9 * Prot) + (5.9 * Morale) +
(4.5 * MagicRes) + (-1.6 * Encumb) + (-8.6 * Str) + (-1.8 * Prec) +
(.5 * TacMove)

Commander (adjusted R-squared = .82)

Cost = -269.6 + (1.6 * HP) + (-1.7 * Prot) + (8.9 * Morale) +
(12.4 * MagicRes) + (3.6 * Def) + (2.2 * TacMove) + (.6 * Ldr_Norm) +
(1.5 * Ldr_Undead) + (3.3 * Ldr_Magic)



Edit: OK, here are separate formulae for non-mage and mage commanders.

Non-mage Commander (adjusted R-squared = .77)

Cost = -125.6 + (9.2 * Size) + (1.4 * Prot) + (10.9 * Morale) +
(7.6 * MagicRes) + (-6.6 * Encumb) + (-3.5 * Str) + (-2.0 * Def) +
(2.3 * TacMove)

Mage Commander (adjusted R-squared = .79)

Cost = -233.0 + (7.7 * Morale) + (16.0 * MagicRes) + (4.1 * Str) +
(-10.7 * Att) + (11.5 * Def) + (-5.7 * Prec) + (.9 * Ldr_Norm) +
(1.1 * Ldr_Undead) + (3.5 * Ldr_Magic)