Unit Balance Theory

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Unit Balance Theory

U-boat
I am working on a little idea that probably isn't new, but to many it might be helpful.

[Part 1 - Tiers (vertical differentiation)]

There is a near-direct ratio between the cost and power of a unit stacked up against other units of different costs and powers.

Land and Air Units:

Infantry 1/2/1/3 would have a total of 4
Artillery 2/2/1/4 would have a total of 5
Tanks 3/3/2/5 would have a total of 8
Fighter 3/4/4/10 would have a total of 11

As you can see, the stat total is 1 higher than the cost of the unit, excluding the tank. As the stat total increases by 1, so does the cost. I would call this a 1:1 ratio. It makes you wonder why the tanks total is three higher than it's cost.

Sea Units (transport and carrier are excluded because of their non-combative uses):

Submarine 2/2/2/8 : 2+2+2=6
Destroyer 3/3/2/12 : 3+3+2=8
Battleship 4/4/2/24 : 4+4+2=10

A similar story can be said about these ships. Their stat totals increase by two and the cost increases by 4. This is a 1:2 ratio. The battleship doesn't follow this though. This is probably due to two-hit and bombarding battleships. Thus, because subs and destroyers have a cost increase of 4, a battleship without special abilities would cost 16. This means 8 points are valued to being two-hit and having the ability to bombard. This same 8 can be added to other units if the ratios are the same. As of now the ratio is still 1:2. Which means 8 PUs should be added to a unit that can bombard and is two-hit. This could probably be broken down to 4 for each ability. Thus when the ratio is 1:2, bombardment is a 4 PU increase from the base price.

So should different ratios be used for different types of units?

[PART 2 - Subunits (horizontal differentiation)]

There is an indirect connection between the cost of a unit and the sum of the attack, defend, and movement values which help create a balanced unit. Let's look at a few examples:

(Attack/Defend/Movement/Cost)

Infantry 1/2/1/3
Banzai 2/0/2/3

These two units are statistically speaking, balanced. In this case if you add up attack, defend, and movement and subtract 1, you get 3. Thus I could create any combination of those three categories, but as long as that add up to 4, it should be fine. This is one way to create a varied amount of units without stretching costs.

Artillery 2/2/1/4
Halftrack 2/1/2/4
Recon 1/1/3/4

[Part 3 - The linkage]

As you can see there are two relationships: the tiers of units, and their subunits. Any unit in a tier is balanced against each other, but still weaker than the tier above and stronger than the tier below. I think if I were to take more data, and plot it on a graph somehow, there would be a global maximum for the amount of units versus the size and complexity of the map. There is a point where too many tier and too many subunits in said tiers just become nuisances. Maybe this theory can be applied to maps such as NWO, which have multiple units, in order to either make sense of the unit stats, to consolidate some units, or to balance.

I do find it funny how the fighter and bomber both have the same stat totals (11) but the bomber's cost is 5 more than the fighter. This may be because of SBR, heavy bombers, or just because 4 in one attack has better odds.

I do think that this theory is flawed. A unit that is 4/0/0 definitely should cost more than a 1/2/1 because rolling on a 4 carries much more power than a 1 or 2. This would be like a bunker and an infantry. So maybe there is a threshold for how different the numbers can be before it should be a new tier or before it is just irregular to the point where playtesting should be used to find a cost.

I think there is some backing to think, maybe a way to balance things out initially when trying to put your units, incomes, and map complexity into perspective. By no means to endorse this as a solo method of appropriating unit costs and values because things just don't always have a number behind it, and common sense and trial and error must be must instead. It's still kinda neat to think about though.
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Re: Unit Balance Theory

beelee
interesting ... could be used as as a starting point and fine tuned from there
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Re: Unit Balance Theory

grizzly
In reply to this post by U-boat
Movement is more important than attack or defense, we cant just add the values. Even in games with no tech bombers are still purchased, because 6 movement gives them possiblities fighters can never have. I only do strat bombing runs if there is nothing else to do that turn.

There are two stats not considered here - hit points and production cost.
Cheaper units get more hit points because we can buy more of them, while a smaller army of stronger units takes less production.

12 subs costs the same as 8 destroyers, and has the same combined attack and defense, but wins with 5.333 units left. Simply because the subs take more hits to destroy. Still your idea is interesting.
Just my 2 cents...
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Re: Unit Balance Theory

Veqryn
Administrator
In reply to this post by U-boat
the balance of units HIGHLY depends on the map in question.

take for example the difference between ww2v2 and ww2v3 and the new europe 1940 map, and nwo.

in ww2v2, having an infantry (a 1 movement unit) in west russia is basically equal to having a tank in west russia.  The extra movement doesn't really matter for the tank, because you can hit everything from west russia, and there are no paths that the enemy can take to moscow, where an infantry in west russia can't hit.
infantry at 3 and armour at 5 is well balanced on this map.

now move to ww2v3.  infantry and single movements have slightly less advantage because of the increase in scale.  things got bigger and there are more paths to take to moscow/caucus.  But single movement units can still position themselves to block things pretty well.
infantry at 3 and armour at 5 is well balanced on this map.

now move to europe1940.  suddenly infantry's 1 movement just doesn't seem to cut it.  Tanks can move past infantry without the chance of infantry blocking them off in some fashion.  Tanks, with their extra 2 attack, extra defense, and extra movement are such a good buy for 5 pu's, that I would consider buying almost entirely tanks as a good purchase.  So for this bigger map, you have to up the cost by 1.
infantry at 3 and armour at 6 is well balanced on this map.

NWO goes even further.  Things just got a lot bigger, so the price of single movement units just went down to compensate.  
infantry at 2 and armour at 5 is well balanced on this map.


Also something to note:
NWO has 2 major versions, Normal and Small's.
Small's mod of NWO has infantry priced at 3, among other price tweaks.  His map is completely balanced too.  The difference is only in play style.  People still buy tons of infantry, but less so than normal NWO.  This results in more mobile stacks and faster action.  BUT its still balanced, and all units are still purchased for certain situations.



Also u-boat, your calculations above forget to take into account the most important stat of all: # of hitpoints.
Hitpoints is the reason people still buy infantry at all, even when tanks are so much better of a purchase.
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Re: Unit Balance Theory

U-boat
Yeah, I know it doesn't take into account a lot of things and by no means is it accurate.
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Re: Unit Balance Theory

RODTHEGOD
In reply to this post by U-boat
Anyone who has ever made a map, I’m sure has tried thinking of an equation of some sort to calculate the price of their units. I’ve tried many times. Perhaps we should try to make a general equation that map makers can use that makes balanced units for their costs.

Here's a little starting equation. It's not perfect but it's not meant to be, it's just to get ideas flowing.

 Cost=((attack+defence+movement)/2)+1

Here are some examples of this equation in action.
ex1: Infantry: ((1+2+1)/2)+1 = $3
ex2: Armour: ((3+3+2)/2)+1 = $5

Notes
---The 1 in this equation represents a 1 hit point unit, so i'm not sure what it would be for a 2 hit point unit.
---I'm thinking that the movement rate should be a modifier of the attack value rather then a value on its own. It wouldn't really effect the defence value because when you're on the defence, you are not moving.

What do you guys think?
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Re: Unit Balance Theory

U-boat
That only works for very basic units that are at a low cost. I think it's a good start though.
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Re: Unit Balance Theory

RogerCooper
In reply to this post by RODTHEGOD
Your formula is a decent starting point but in practice attack is more important than defense, because unit can threaten attack to multiple areas but can only defend 1.

Moreover, value is not the same as cost. In WW2, Battleships were not cost-effective compared to Carriers, and few were built. Cavalry had much the same problem.

For WW2, I would just use the values in the later versions of the boardgame. They are balanced to the point that almost every type of unit will be build at some point.
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Re: Unit Balance Theory

Veqryn
Administrator
In reply to this post by RODTHEGOD
rod, i use that same formula as a starting point,..

attack is worth more than defense

movement matters, but matters more for units with attack

blitz only matters when units have good movement

artillery only matters when you have units to support

transport cost

marine

size of map

hitpoints

ability to repair (this is expensive, i usually value the ability to repair as at least 2 attack points)

etc.


for example, on Napoleonic Empires, I have the units balanced like this:

Fusiliers = 1 att, 2 def, 1 movement, 4 cost (2 transport cost, isSupportable)

Howitzers = 1 att, 1 def, 1 movement, 4 cost (3 transport cost, isArtillery)

here you can see that howitzers are basically 2 att / 1 def units when paired with fusiliers.  I did not want to make them 2 att / 1 def as that should cost more, as attack is worth more than defense.  By giving them artillery, they can be priced at 4.
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Re: Unit Balance Theory

RODTHEGOD
Ya I've been trying to think on how the map size would have an effect on movement(which in turn has an effect on attack). I started thinking that the average distance between capitals is the number that I would have to work with but now I'm thinking that it would be the number of territories accessible to that unit (for example: for land units, you look at the number of land territories, for sea units, you look at the number of sea territories and for air units, you look at the number of all territories). As well seeing as there are different rules for the 3 types of territories, there would be different modifications to that total number (an example would be for air units you would take the total number of territories and divide by 2 because air units have to land on a land territory or a carrier).

The following equation I was still working with the idea of the average distance between capitals. I realise now that it is quite flawed but I'll put it here anyways:

Cost = (a*m/D+d)+h

a=attack
m=movement
D=average distance between capitals (in this case I was using the revised version of triplea which has an average distance of 5.8)
d=defence
h=hit points

and here are some examples of the equation in action

ex1: Infantry (1a,2d,1m) 1hp = $3.172...
ex2: Armour (3a,3d,2m) 1hp = $5.034...

and I was thinking those numbers would be rounded of course.
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Re: Unit Balance Theory

Pulicat
Instead of average distance between capitals, I would look at average adjacent accessible territories. For example, say on a particular map, each land territory had an average of 4.2 accessible adjacent land territories and 3.9 accessible adjacent sea territories, etc...

I'm thinking it would be useful to value the movement aspect exponentially, for example: A tank with movement 2 on a map with average 4.2 accessible adjacent land territories means that it has (4.2)^2 possible paths to take.

puli
how now brown cow?
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Re: Unit Balance Theory

U-boat
That makes sense
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Re: Unit Balance Theory

RODTHEGOD
In reply to this post by Pulicat
Thats an interesting idea. Would you include impassible territories in the average count?
As well I'm wondering if that number should be in relation to the total territory count (like using your 4.2^2 as an example, you would then divide by the total number of land territories or something like that) because when you get to bigger and bigger maps, there is bound to be a maximum territory connection average at some point(my prediction would be 6 as a maximum). As well, how should I consider land-sea connections? Should they have a different value compared to land-land connections or sea-sea connections?
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Re: Unit Balance Theory

grizzly
That equation actually minimizes attack and movement impact and makes defense the most important.
if you look tanks get 1.something cost for their attack and movement values, and 3 of their cost from their defense

your value territory distance value is too high, and it causes the offensive values be tiny.
Just my 2 cents...
psi
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Re: Unit Balance Theory

psi
In reply to this post by U-boat
It's more complicated.

If one unit has 1 defense and 1 offence and 1 movement and another has double those values, pricing the latter at double the cost of the former actually makes the latter useless.  The reason is that the lower price of the former even though it is exactly proportional, itself gives the former an advantage by making instances of it to have twice the numbers versus the former.

For ex.  20 of the first unit versus 10 of the second unit results in an overwhelming victory for the former.

Discounting movement, there is a way to be mathematically precise in balancing unit prices but as I said it is more complicated. (or just use the battle calculator) With movement, it is even more complicated.
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Re: Unit Balance Theory

RODTHEGOD
In reply to this post by RODTHEGOD
Alright after being busy for a while I've come back to continue this equation thing.

Working with the revised map I found the land(going to a land territory) and sea(going to a sea territory) connections of all the territories.

After doing that, I then made 3 equations that I think work for the various unit types (land, air or sea). this number would be the average # of connections for the territories that  unit can travel through.

The land equation = (all land and sea connections of all land territories)/(2*#of land territories)

-I found that the number of land and sea connections for land territories(excluding impassible territories) was 223.
-I found that the number of land territories was 63
-In my calculations I included 0 connection values (for land and sea) as islands are important in this value. so I originally had the bottem value be equal to the number of land and sea connections of a territory but i realised that it was just the # of land territories multiplied by 2.
-so after plugging the numbers in, we find that the average # of connections for land units is 1.77

LCA = 1.77

The air equation = (all land + sea connections of all territories)/(# of all territories)

-I found that the number of land and sea connections for all territories(excluding impassible territories) was 580.
-I found that the # of all territories was 127
-After plugging the numbers in, we find that the average # of connections for air units is 4.30

ACA = 4.30

The sea equation = (all sea connections of all sea territories)/(# of sea territories)

-I found that the number of sea connections for all sea territories was 274
-I also found that the number of sea territories was 64
-After plugging the numbers in, we find that the average # of connections for sea units is 4.28

SCA = 4.28



-notice the air and sea values are actually quite similar.
-Anyways, I'm not sure yet how these numbers should apply to the units cost yet but i'm working on it.
aka
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Re: Unit Balance Theory

aka
In reply to this post by U-boat
Hi ppl,


this is what I have found about unit cost and value.:


1. If income per terretory is high and prices of units is high too, its easyer to find a fitting price for a kind of unit. Lets say all costs and prices are tripled.
An inf cost 9 and an Ari is 12 by all income tripled too, thats no prolem - same game. But than u have more tolerance in pricing - Its possible to scale it up a little like put Inf to 10 now or make 11 for Ari. Or maybe you like to see just a little more research. You can make it 14 instead of 15. Thats allso useful in finding a fitting price for land-income.
   
2. A unit that can do not walk, attack or defence can at least one thing.  -   It can die ! Thats why low cost units I think are too chaep. In your formula a "non-do nothing but die unit" cost just "1". Some to cheap for me. I say that all formulas get worth the lower the unit is. Or the lower the land value is.

3. For all AI skriptors it would be heaven to find out a value of one specific unit on the map.
When u see a top player sacrifice a bomber for 1 inf to delay the block on suezchanel one round.
Or how sweet some single UK-inf block some 8 tanks blitzing moskwa.
I understand that the value of one unit could be priceless.
The same unit type could be nearly useless same round just other place.

4. because of that its not possible to find a formula for finding a fittin price. The price is never fitting and the price will be an approximation for ever.

5. Be careful in priceing "double hit units" . If good played they are able to take hits every round - just bring that to your mind. Armys of "double hit units" can change the ballance of the game. Make them way epansive.

6. 1point speed has more value than 1attack or 1defence.





Sorry I did not read the entire post. I just feel fine to add on some I found.


bang the rocks together, guys.





solong


aka
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Re: Unit Balance Theory

RODTHEGOD
In reply to this post by RODTHEGOD
Well I thought I made some progress on this equation but I was wrong. I'll show you what I did.

The first thing I did was remove the /2 in the LCA equation because i didn't think it worked. (again for the revised map only)

so I now have these three values:

Land connection average (LCA): 3.54
Air connection average (ACA): 4.30
Sea connection average (SCA): 4.28

I also have these numbers

Percentage of land territories = 63/127
Percentage of sea territories = 64/127

From this and the unit stats, I came up with these two equations:

Equation # 1: is looking at the attack value of a unit
AP = attack product
A = attack value
CA = connection average (based on whether its a land, air or sea unit)
M = movement value
P = percentage of accessable territories (based on whether its a land, air or sea unit)

AP = (A*(CA^M)*P)/(A*M)


Equation # 2: is looking at the defence value of a unit
DP = defence product
D = defence value
CA = connection average (based on whether its a land, air or sea unit)
aMa = average applicable movement average (average movement rate of all the units that can attack it)
P = percentage of accessable territories (based on whether its a land, air or sea unit)

DP = (D*CA*P)/(aMa)


Equation # 3: is determining the cost of the unit
AP = attack product
DP = defence product
$ = cost of unit

$ = AP + DP


Example: I then started to experiment.

Ex. 1: Infantry

Equation # 1
AP = attack product
A = 1
CA = 3.54
M = 1
P = 63/127

AP = (1*(3.54^1)*63/127)/(1*1)
AP = 1.76


Equation # 2
DP = defence product
D = 2
CA = 3.54
aMa = 14/5
P = 63/127

DP = (2*3.54*63/127)/(14/5)
DP = 1.25


Equation # 3
AP = 1.76
DP = 1.25
$ = cost of unit

$ = 1.76 + 1.25
$ = 3.01


I thought 3.01 was pretty darn close to 3 so I decided to try out this equation again with armor

Ex. 2: Armor

Equation # 1
AP = attack product
A = 3
CA = 3.54
M = 2
P = 63/127

AP = (3*(3.54^2)*63/127)/(3*2)
AP = 3.11


Equation # 2
DP = defence product
D = 3
CA = 3.54
aMa = 14/5
P = 63/127

DP = (3*3.54*63/127)/(14/5)
DP = 1.88


Equation # 3
AP = 3.11
DP = 1.88
$ = cost of unit

$ = 3.11 + 1.88
$ = 4.99


Again I thought that 4.99 was pretty darn close to 5 so I decided to try it again except on artillery but this time i discovered a flaw in my equation.

Ex. 3: Artillery

Equation # 1
AP = attack product
A = 2 (I tried both 2 and 3 here as infantry is such common unit that in most cases the artillery does provide support)
CA = 3.54
M = 1
P = 63/127

AP = (2*(3.54^1)*63/127)/(3*2)
AP = 1.76


Equation # 2
DP = defence product
D = 2
CA = 3.54
aMa = 14/5
P = 63/127

DP = (2*3.54*63/127)/(14/5)
DP = 1.25


Equation # 3
AP = 1.76
DP = 1.25
$ = cost of unit

$ = 1.76 + 1.25
$ = 3.01


Hardly the answer I wanted, and I quickly discovered why: In the Attack Product equation, I had what is basically A/A which is equal to 1 so the attack value had no effect on the Attack product.

Alright well unfortionately I need to get to bed, so I'll leave it at that.
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Re: Unit Balance Theory

RODTHEGOD
Alright I had an epiphany.
Basically I did an experiment where I compared the victory probabilities of Armour vs infantry (in revised) at various levels of investment. I found that the value of a hitpoint on a unit is not a linear relationship (I haven't found the relationship yet) but rather is dependent on the number of that unit you have.

To sum that up I discovered that 3 armour vs 5 infantry is not the same as 30 armour vs 50 infantry (attack or defence) despite having the same cost and ratio of units which is kind of obvious when you think about it.

I came to the conclusion that the production values on the map influence the cost of units. Essentially as you spend more and more money on certian types of units, their hitpoint value goes down. So far I havent found the appropriate relationship.

So far I've tried multiplying the attack and defence products from above by the hit point product below.

 HP = 1/(2.63492^.2)

In this case the one is the number of hits a unit has.
The 2.63492 is the total amount of land production divided by the number of land territories
And the .2 is the percentage of the average starting owned territories.
it comes out to .82385

So the big equation is

$ = ((.82 x 1/6 x 3.54 x 63/127) + (.82 x 2/6 x 3.54 x 63/127))

It didn't work unfortionately
Another thing I tried was to find the average starting production which turned out to be 33.2 and tried putting the attack and defence products (added up) to the power of 1/33.2.
That also did not work.

So anyone have any ideas?



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Re: Unit Balance Theory

Zim Xero
ATT + DEF + 1/2 Mov + 2 gives an approximate value for unit comparison
adding about .5 for each special ability
+3 for double hit without repairability

If you want to simply compare for an attack, use ATT total + #total attackers / DEF total +#total defenders.

Not adding for number of units in an equation leads to an imbalance as is seen in the original A&A, where low cost units are simply the best buy... in that case infantry.

There is also a floating variable on all of this relating to maps and prices themselves, but it is not normally a huge differentiator.
'thats the way it is' makes it neither desireable nor inevitable
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