Unit Cost Equation

[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)

[PDF]

Sorry, but I'm pretty sure these regressions are mostly useless (and I've some stat background..). Namely because they do work *only* if the relationships between attributes and the attributes scales themselves are in the range of the sample studied.
So if I make yet another jav totting Lt Inf it will be ok, but if I make something unusual, for example a 1-hp Ethereal elephant, a 100-hp rock-hard militia or some nasty B3W3?3 mage (called "Ice Devils Factory" ), they would be totally out.
Thay also don't take at all into account the equipment of the unit (the values are averaged, difference going to the error margin) : yet these are pretty important...

[Verjigorm]

Equipment is generally accounted for in the resource cost whereas skill is accounted for by the unit cost. The value of units such as mages is extraordinarily difficult to determine because their spells and combinations of spells (even if you leave out item bonuses and communion) is subjective and certain combinations of spells have greater "value" that the sum of the "values" of the spells alone (e.g. Body Ethereal combined with Bone Grinding > Body Ethereal + Bone Grinding).

Unit cost should be a measure of their statistical efficacy versus the "average" unit whose cost could be arbitrarily assigned. The "average" unit's stats are not all 10's and his cost is not 10gp... Once you have a sample space of units, you can determine the most effective ones. The combat simulator is a good tool for this, but it doesn't produce excel-like results. Any unit whose effectiveness against "Average" is 50% should cost the same as the "Average" unit. I'm not sure how you'd scale up the value as the percentages vary....How much more should a unit with 95% effectiveness cost?

[BigDaddy]

PDF said:
Sorry, but I'm pretty sure these regressions are mostly useless (and I've some stat background..). Namely because they do work *only* if the relationships between attributes and the attributes scales themselves are in the range of the sample studied. . .

With all due respect, in my profesional coursework, I was taught that linear regression could be used for forecasting. With a time constraint values will go beyond the range studied (ideally, your model should be updated with each new point). So, the model is a good one, HOWEVER, I'm pretty sure the nice linear lines aren't appropriate for exquisetly powerful or unusual units. Mainly, because A)I doubt that that is how the devs did it and B) power increases dramatically for higher skill values.

A better model for these powerful units might be to do a similar study of summonable units and convert from gems to gold (using ermor, 10-15 gems us ~equal to 200 gold). Unfortunately you can't use the same equation for commanders and regulars, but seperate units would be available.

An even better model would be to use best fit nonlinear equations for the summonable units, then convert to gold. Cheap regression software can only check your guesses for nonlinear equations.

Probably the best method would be to use an expert system (engineering term - artificial intelligence for programmers) to compute costs. Using this method, you could convert summonable unit cost to gold first (or use the system to do it accurately), then input ALL of the units. Then, by pumping your unit through the system you would get an accurate price. The system itself would be somewhat complex, with nodes for each aspect (sacred, etheareal, etc) of the unit. You can get a good quality student expert system program for "free," with the purchase of a US$100-150 expert system text. It MAY allow enough nodes for this system. You may be able to download one for free, but I haven't looked. It could take you a while to learn how to use it. . .

SUMMARY:
Use the linear model or 1gold per point model for units within normal "human" unit ranges (maybe up to 14 skill) with typical aspects (not ethereal). This is how devs appear to have done it. . .

For powerful units, make a model using summonable units and convert to gold, using ermor income to gold (10-15 gems per 200 gold). Use of linear lines here will cause barely powerful units to be too expensive and truly powerful units to be too cheap, but mid range guys should be right on. Models for extreme units should use a nonlinear or expert system model.

OR you could "just" shake the info out of the devs. . .

[sushiboat]

Prediction is not what you might think. Of course, I can't predict what a
modder is going to do. A modder could change Militia to be 10,000 gold. The
regression equations describe how stats are related to price for existing
units. To the extent that you generally like what Illwinter has done with
pricing units, the equations might give you a good starting point. If you want
to make drastic changes, you may be less interested.

My guess is that the Illwinter guys don't use equations. One of them probably
makes a unit and assigns a price based on an intuitive understanding of its
usefulness. After playtesting, the price is modified.

I can put things like magic paths and special abilities (e.g., ethereal) into the
mix. I already did so with flying, which turned out not to be uniquely
important enough to keep in. Keep in mind, however, that the equations
above are already explaining about 80% of the variance. That is the equivalent
of a correlation of .90. Whatever gains come in the future will be very modest
at best.

As for using nonlinear equations, I have looked at polynomial regressions for
some of the predictor variables. Morale, for example, is pretty much linear.
The squared term adds a little, and the cubed term is negligible. In other
contexts, I have played around with log, square root, and reciprocal
transformations. My experience has been that they are most useful when the
vanilla linear regression isn't working well. Here, vanilla works like
gangbusters, so the room for improvement is small.

[Scott Hebert]

sushiboat said:
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)

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.

Er, um, I don't doubt this, but may I ask why Size is a positive value? With the sole exception of Trampling, Size is almost invariably a negative in Dominions, isn't it?

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)

Do you mind if I use this to calculate values for the commanders? Also, is this ignoring other factors (magic, etc.)?

[sushiboat]

Why is size a positive value? I guess the ultimate answer would have to come
from Illwinter. If I remember correctly, size, HP, and strength are highly
correlated with each other, so having more than one in the equation makes
interpretation tricky.

You are welcome to use the equations in any way you like. At the moment,
they cover only national units, not independents. (Maybe next weekend I'll
add the indies.) I have not included magic paths for mages yet. The only
special ability I have looked at is flying. I would like to add other special
abilities and magic paths later. The three equations below are the ones to use
for now.

Non-commander

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)

Non-mage Commander

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

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)

[Scott Hebert]

sushiboat said:
Why is size a positive value? I guess the ultimate answer would have to come
from Illwinter. If I remember correctly, size, HP, and strength are highly
correlated with each other, so having more than one in the equation makes
interpretation tricky.

Ah, that would answer it. Sort of like my idea to 'prorate' Strength and HP gains by the size of the unit?

You are welcome to use the equations in any way you like. At the moment,
they cover only national units, not independents. (Maybe next weekend I'll
add the indies.) I have not included magic paths for mages yet. The only
special ability I have looked at is flying. I would like to add other special
abilities and magic paths later. The three equations below are the ones to use
for now.

Oh, well, that's good, since I don't have the independents in there either.

If I could give you values for commanders without any 'specials', could you calculate a regression formula based on that?