Algebraic solution towards unit and faction balancing?

[xbriannova]

I don't know where to put this up as there's no questions section in the open development forums section so I might as well ask about this right here:

Is there any mathematical formula for use in balancing factions and units? I'm having a really hard time deciding whether my units are balanced or not in my campaign, Guardian Order. I'm just wishing that there's a scientific way of doing it.

If there isn't then I'm working on it. What I'm doing now is that I'm trying to take into account every single aspect of a unit and then expressing them in terms of algebra. I've got a prototype formula right now if you guys want it. It's extremely long and an extreme headache to behold though. I'm not sure if it'd even remotely work. Haven't tested it yet.

[Soliton]

Is there any mathematical formula for use in balancing factions and units?

No.

[tsr]

From previous discussions I recall that there isn´t an algebraic formula for this. If you are doing one I wish you good luck and I really hope you succeed.

I just wonder if you in your extremely complex formula are accounting for the company of the unit, the current location of the unit and the current opposition of the unit. Because without that your formula wont be even near complete.

/tsr

[thespaceinvader]

There really isn't. Balance depends on more than just units. It depends on maps, and it depends on how the units are used, and what units they are facing. Factions are not balanced in isolation, they are balanced in relation to the rest of the MP era or campaign. A faction that is balanced in a campaign is often not balanced in MP. For example, if you took the recruit list from HttT and dropped it into default as an MP faction it would be horribly unbalanced.

[xbriannova]

There really isn't Balance depends on more than just units. It depends on maps, and it depends on how the units are used, and what units they are facing. Factions are not balanced in isolation, they are balanced in relation to the rest of the MP era or campaign. A faction that is balanced in a campaign is often not balanced in MP. For example, if you took the recruit list from HttT and dropped it into default as an MP faction it would be horribly unbalanced.

From previous discussions I recall that there isn´t an algebraic formula for this. If you are doing one I wish you good luck and I really hope you succeed.

I just wonder if you in your extremely complex formula are accounting for the company of the unit, the current location of the unit and the current opposition of the unit. Because without that your formula wont be even near complete.

/tsr

So far my current prototype algebraic formula is only taking into account unit abilities such as movement, healthpoints, attack damage and number of attacks all the way down to movement costs on terrain as well as defense percentage on each type of terrain. I knew that I may need to take into account company and stuff as usually, in strategy games, players would certainly rely on more than one unit and would often rely on a combined arms approach but... I haven't factored all those in yet.

My mathematical abilities are limited. I've excelled in basic math before all the way up to 10th grade and slightly beyond, but my additional/advanced maths score is like 5%-20% . Therefore I may not be able to crack this. Already, my sides are splitting.

Anyway, hopefully there might be a way to accomplish all this without factoring in everything outside of a unit's abilities.

Is there any mathematical formula for use in balancing factions and units?

No.

Oh well that means I'll have to do it myself. It doesn't have to be all that accurate. I'm going to somehow conjure a formula with a 5%-10% cut-off rate. I forgot the term for when exceptions are made mathematically. Anyway, I will attempt to follow the principles of Occam's razor as well. It's already a headache the way it is.

[Velensk]

I'd strongly suggest that you abandon trying to do this mathmatically, and instead do it in a far more reliable way. Which is repeated testing. Since this is a campaign your units do not have to be balanced like a multiplayer faction. All you need to do is make it so that the campaign is as difficult as you want it to be. So test it, tweak it, test it, tweak it until you get something that is the difficulty you desire.

If you were balancing it for multiplayer then you could not balance each unit individually you'd need to balance the faction. If you look at mainline, by any point system you could devise there would be overpowered units; however the factions are pretty well balanced againt one another. Poachers would be utterly useless for the rebels, but on the knalgans they are a usefull unit. Also some combinations of units are very powerful. A berserker can be made much more intimidating by putting it in a faction with a fast reliable slower, or on a unit that is also a skirmisher. There is no way to simply add the abilities/attributes together to get the value of any one unit.

[xbriannova]

I'd strongly suggest that you abandon trying to do this mathmatically, and instead do it in a far more reliable way. Which is repeated testing. Since this is a campaign your units do not have to be balanced like a multiplayer faction. All you need to do is make it so that the campaign is as difficult as you want it to be. So test it, tweak it, test it, tweak it until you get something that is the difficulty you desire.

If you were balancing it for multiplayer then you could not balance each unit individually you'd need to balance the faction. If you look at mainline, by any point system you could devise there would be overpowered units; however the factions are pretty well balanced againt one another. Poachers would be utterly useless for the rebels, but on the knalgans they are a usefull unit. Also some combinations of units are very powerful. A berserker can be made much more intimidating by putting it in a faction with a fast reliable slower, or on a unit that is also a skirmisher. There is no way to simply add the abilities/attributes together to get the value of any one unit.

You might be right, but there has to be a way. Albert Einstein sure did came up with the relativity theory. What I aim to do is to at least present a rudimentary, if not very primitive, formula that could be relied upon 90%-95% of the time. There must be a way. If only we have a resident mathematician/Physicist around

[Velensk]

Balancing a game isn't like physics. In physics you are trying to describe how things act by getting the numbers you have to match the other numbers you have. Balancing a game isn't about making things equal, it's not about making the two sides match up. It's about making sure that neither side is inherently disadvantaged, and these disadvantages are frequently to subtle for mathematics. It's not about raw numbers, it's about the practical end of the stick which although it is determined by the numbers used, is far more complex in the interaction than the numbers could ever be.

As it is, you are making a campaign. Things should never be "balanced". What you are trying to do is make it so that the player can make up the difference.

EDIT: as it is, I know that many of Wesnoths community are electrical engineers, programmers, and of other professions that tend to be math inclined.

[xbriannova]

Balancing a game isn't like physics. In physics you are trying to describe how things act by getting the numbers you have to match the other numbers you have. Balancing a game isn't about making things equal, it's not about making the two sides match up. It's about making sure that neither side is inherently disadvantaged, and these disadvantages are frequently to subtle for mathematics. It's not about raw numbers, it's about the practical end of the stick which although it is determined by the numbers used, is far more complex in the interaction than the numbers could ever be.

As it is, you are making a campaign. Things should never be "balanced". What you are trying to do is make it so that the player can make up the difference.

EDIT: as it is, I know that many of Wesnoths community are electrical engineers, programmers, and of other professions that tend to be math inclined.

The way I see it, it depends on what campaign you're talking about. Two factions could be equally balanced, but the way difficulty can be tweaked... You can change many factors affecting the chance a player may emerge victorious, other than the strength of the units making up each faction. For example, you could increase or decrease the base gold of each faction, or you could increase or decrease the income of each faction, or you could change the ai of your players' opponents to make things harder.

Ultimately, no one is right. Neither you nor me. It is entirely up to the campaign author's discretion- whether he wants to change the difficulty by altering the units or altering the game parameters. The way I work, I wanna alter the game parameters as it provides a more controllable environment for the author.

I don't think anything can be too subtle for maths. It all depends on how good we are in mathematics, and whether a way to calculate certain things are discovered or not. I wonder if any of those mathematically inclined would want to take up the challenge haha?

[Daxion]

If have to disagree a bit with some here.
Is it possible to come up with a math. formula to balancing? I do not see why not, at least not from a theoretical standpoint. The sheer complexity of the problem if a scenario is balanced with everything taking into account, we can safely say: it not possible in the near future to come up with a practical formula.

However I do feel that the wesnoth community has provided us campaign makers with a useful tool, which feels much like a mathematical formula:
recruitment costs.
This does sound a bit too simple, but think about it:
It reflects the worth of a unit in "average" conditions. Two armies with units worth 100 Gold, should roughly be equal in strenght.

Now what this is really missing is the terrain. There's not going to much help from math here. If it would be easy, then I'd expect that Wesnoth would have a wunderful brilliant AI
So its basically up to you, of how much you feel the terrain is worth in starting cost and income. Yes, I also think it might be possible to a have a quick an dirty general rule of thumb, that does not take into account the exact distribution of Hex tiles (a bit more on that below).

But you have to understand what such a forumla is: It's a tool. Think of balancing as a recursion process. You try if a certain setting is balanced (on the first try it won't). You change something and try again. Repeat until you are satisfied. This can take a long time. If you would have a good starting point, you might be able to cut down that time. However you will still need quite some time playtesting. There is no way around it really. Just a way to have a more sensible start. I belive most people will choose their "startingpoint" much more efficient by experience, than you could with any algebraic solution.


a few words on a quick formula:
If we have dominant terrain feature (e.g. forrest with a few clearings), elves are clearly at home. Orcs are not. So they would probably need to get a bonus in income and starting gold to compensate for that.

Things the formula would need to take into account:

• dominante type of terrain
  how dominant that type actually is
  size of the map (I'm thinking this might be important since this is sort of giving the magnitude of advantage disadvante. Elves just move faster in forrests, hence it's easier to control large areas

[Ken_Oh]

I'm sure it's "possible" to come up with something like this, but it would be an colossal task. Is there a term that sounds bigger than colossal? If so, use that one instead.

It's far easier to use intuition and testing for this. Our brains are really powerful, but not much of that power goes into conscious thinking. This would be like trying to figure out the physics of throwing a ball before learning to throw.

Just throw the damn baseball, and if it doesn't quite move the way you want it to, experiment in throwing it a little differently.

[xbriannova]

If have to disagree a bit with some here.
Is it possible to come up with a math. formula to balancing? I do not see why not, at least not from a theoretical standpoint. The sheer complexity of the problem if a scenario is balanced with everything taking into account, we can safely say: it not possible in the near future to come up with a practical formula.

However I do feel that the wesnoth community has provided us campaign makers with a useful tool, which feels much like a mathematical formula:
recruitment costs.
This does sound a bit too simple, but think about it:
It reflects the worth of a unit in "average" conditions. Two armies with units worth 100 Gold, should roughly be equal in strenght.

Now what this is really missing is the terrain. There's not going to much help from math here. If it would be easy, then I'd expect that Wesnoth would have a wunderful brilliant AI
So its basically up to you, of how much you feel the terrain is worth in starting cost and income. Yes, I also think it might be possible to a have a quick an dirty general rule of thumb, that does not take into account the exact distribution of Hex tiles (a bit more on that below).

But you have to understand what such a forumla is: It's a tool. Think of balancing as a recursion process. You try if a certain setting is balanced (on the first try it won't). You change something and try again. Repeat until you are satisfied. This can take a long time. If you would have a good starting point, you might be able to cut down that time. However you will still need quite some time playtesting. There is no way around it really. Just a way to have a more sensible start. I belive most people will choose their "startingpoint" much more efficient by experience, than you could with any algebraic solution.


a few words on a quick formula:
If we have dominant terrain feature (e.g. forrest with a few clearings), elves are clearly at home. Orcs are not. So they would probably need to get a bonus in income and starting gold to compensate for that.

Things the formula would need to take into account:

• dominante type of terrain
  how dominant that type actually is
  size of the map (I'm thinking this might be important since this is sort of giving the magnitude of advantage disadvante. Elves just move faster in forrests, hence it's easier to control large areas

Okay, what I have is that I'm taking into account the innate abilities of a unit, say health and attack damage, no. of attacks, and I'm taking into account how the terrain would affect him- in terms of movement and defence.

I'm sure it's "possible" to come up with something like this, but it would be an colossal task. Is there a term that sounds bigger than colossal? If so, use that one instead.

It's far easier to use intuition and testing for this. Our brains are really powerful, but not much of that power goes into conscious thinking. This would be like trying to figure out the physics of throwing a ball before learning to throw.

Just throw the damn baseball, and if it doesn't quite move the way you want it to, experiment in throwing it a little differently.

Well, I tried throwing the baseball, and I'm not happy with what I have. There's also something to do with curiosity- a feature that humankind has. I have that curiosity- I wanna explore this option a little. Who knows? A person who did might just invent something for the rest of the community, the way Isaac Newton was able to invent the lightbulb after thousands of failures. Just light the damn candle! Everyone kept telling him haha

Reply to everyone: Okay, I've got some good news guys, I think my formula is taking shape. What the formula does is that it calculates how much a single gold coin is worth in Wesnoth, or rather how powerful is a 1-gold unit (How powerful Should be the opposite). From here, all you need to do is the input the number of gold a unit costs.

It doesn't end there. Each factor, say hp=healthpoint and M=movement are highly malleable entities. An M may be worth a certain amount of hp which means the individual factors could be converted around. It's very complex really. I'll try to put in the formula now... Please hold while I edit.

1G=2.25hp+0.625m+(0Da+3.125M+3.125A+0Du+0Fw+0Sw)+2.875xp+(0.625fas+0.25fns+0fsp)+(0.5sas+0.0125sns+0ssp)
+(0.25ar+0br+0cr+0fr+0ir+0pr)+(0.25fd+0.75cd+0.375sd+0.75vd+0.5ffd+0.625hd+0.75md+0.25ud+0.25srd+0.5ccd+0.25dd
+0.25swd+0.375crd+0.5id+0.625fffd+0.625mgd)+(0.375fm+0.0125cm+0.25sm+0.0125vm+0.0125ffm+0.25hm+0.375mm+0um
+0.375ssm+25ccm+0dwm+0.375swm+0.25crm+0im+0.25fffm+0.25mgm)

G=Gold
hp=Healthpoints
m=movement points
Da=Dawn advantage
M=Morning advantage
A=Afternoon advantage
Du=Dusk advantage
Fw=First Watch advantage
Sw=Second Watch advantage
xp=experience points
fas=first attack strike damage
fns=first attack number of strikes
fsp=first attack special
sas=second attack strike damage
sns=second attack number of strikes
ssp=second attack special
ar=arcane resistance
br=blade resistance
cr=cold resistance
fr=fire resistance
ir=impact resistance
pr=pierce resistance
fd=frozen defence
cd=castle defence
sd=sand defence
vd=village defence
ffd=flat defence
hd=hills defence
md=mountain defence
ud=unwalkable defence
srd=swamp defence
ccd=cave defence
dd=deep water defence
swd=shallow water defence
crd=coastal reef defence
id=impassable defence
fffd=forest defence
mgd=mushroom grove defence
fm=frozen movement cost
cm=castle movement cost
sm=sand movement cost
vm=village movement cost
ffm=flat movement cost
hm=hills movement cost
mm=mountain movement cost
um=unwalkable movement cost
ssm=swamp movement cost
ccm=cave movement cost
dwm=deepwater movement cost
swm=shallow water movement cost
crm=coastal reef movement cost
im=impassable movement cost
fffm=forest movement cost
mgm=mushroom grove movement cost

[Caphriel]

So far my current prototype algebraic formula is only taking into account unit abilities such as movement, healthpoints, attack damage and number of attacks all the way down to movement costs on terrain as well as defense percentage on each type of terrain.
...
Anyway, hopefully there might be a way to accomplish all this without factoring in everything outside of a unit's abilities.

Emphasis added. Movement cost and defense on terrain are balanced against the frequency and position of the terrain on the maps. You can't ignore the map composition. A unit that got 70% defense on flat would be not be overpowered on a map that was almost entirely swamp and sand. Movement balance depends on the distances involved. Movetype balance depends on the terrain used, and how the paths are laid out.

The value of abilities like healing and leadership varies based on how many units the player has, how spread out they are, what levels they are, in addition to all the fun stuff like average damage for a turn. And most of those factors are going to depend on how far into the campaign the player is, how much gold the campaign designer is providing, stuff like that.

Even, hypothetically, if you developed a formula for balance, how would you know that it's right? By playtesting, of course. Furthermore, there will certainly be effectively infinite solutions to an equation with that many variables. Most of those solutions probably won't be fun, even if they are perfectly balanced. Skip the headache on trying to develop a formula, and just go right to playtesting. Make the campaign fun; don't worry if the factions achieve some balance.

Besides which, even if they are balanced in the hands of humans, the AI players differently.

On an side note, document your work. If you succeed, you'll be able to write a doctoral thesis.

On a second side note, Edison invented the lightbulb.

tl;dr : Game design - ur doin it wrong. I know that sounds harsh, but I can think of a couple dozen examples of balanced factions and units that would not be fun to play with. You can't include a variable in your equation to maximize fun, so trying to mathematically achieve balance runs the risk of ruining the campaign.

[xbriannova]

So far my current prototype algebraic formula is only taking into account unit abilities such as movement, healthpoints, attack damage and number of attacks all the way down to movement costs on terrain as well as defense percentage on each type of terrain.
...
Anyway, hopefully there might be a way to accomplish all this without factoring in everything outside of a unit's abilities.

Emphasis added. Movement cost and defense on terrain are balanced against the frequency and position of the terrain on the maps. You can't ignore the map composition. A unit that got 70% defense on flat would be not be overpowered on a map that was almost entirely swamp and sand. Movement balance depends on the distances involved. Movetype balance depends on the terrain used, and how the paths are laid out.

The value of abilities like healing and leadership varies based on how many units the player has, how spread out they are, what levels they are, in addition to all the fun stuff like average damage for a turn. And most of those factors are going to depend on how far into the campaign the player is, how much gold the campaign designer is providing, stuff like that.

Even, hypothetically, if you developed a formula for balance, how would you know that it's right? By playtesting, of course. Furthermore, there will certainly be effectively infinite solutions to an equation with that many variables. Most of those solutions probably won't be fun, even if they are perfectly balanced. Skip the headache on trying to develop a formula, and just go right to playtesting. Make the campaign fun; don't worry if the factions achieve some balance.

Besides which, even if they are balanced in the hands of humans, the AI players differently.

On an side note, document your work. If you succeed, you'll be able to write a doctoral thesis.

On a second side note, Edison invented the lightbulb.

tl;dr : Game design - ur doin it wrong. I know that sounds harsh, but I can think of a couple dozen examples of balanced factions and units that would not be fun to play with. You can't include a variable in your equation to maximize fun, so trying to mathematically achieve balance runs the risk of ruining the campaign.

Okay first of all. WAH HAHAHA! Sorry about the Isaac Newton thing! My mistake I was typing too fast while thinking at the same time.

And yes, I'm documenting it here.

Oh and about the frequency of the types of terrain present/map composition, that's part of the game environment I'm talking about- To affect the difficulty of the game, you can change that. I wouldn't want my campaign to appear monotonous either with average difficulty for every scenario. The value of abilities can be determined mathematically I believe. I know a way.

Okay, how do I test the formula? By factoring in a unit and see if it all makes sense I'm seeing some consistency when I did that in my formula. What happened is that I put in the Spearman into my formula and everything looks good, but the movement is at 8.75. So what I conclude is that each factor, or a unit's statistic, could be exchanged for one another like how you do it in an RPG game. It is algebra after all- the actual numerical value of each expression is never actually determined. I believe, by exchanging one factor with another, I will eventually end up with the actual spearman's statistics and this is how we balance units- by giving it disadvantages to bestow advantages to it.

My next step is to determine the value of each algebraic expression/factor then I might be closer to achieving my goals.

[xbriannova]

1G=2.25hp+0.625m+(0Da+3.125M+3.125A+0Du+0Fw+0Sw)+2.875xp+(0.625fas+0.25fns+0fsp)+(0.5sas+0.0125sns+0ssp)
+(0.25ar+0br+0cr+0fr+0ir+0pr)+(0.25fd+0.75cd+0.375sd+0.75vd+0.5ffd+0.625hd+0.75md+0.25ud+0.25srd+0.5ccd+0.25dd
+0.25swd+0.375crd+0.5id+0.625fffd+0.625mgd)+(0.375fm+0.0125cm+0.25sm+0.0125vm+0.0125ffm+0.25hm+0.375mm+0um
+0.375ssm+25ccm+0dwm+0.375swm+0.25crm+0im+0.25fffm+0.25mgm)

G=Gold
hp=Healthpoints
m=movement points
Da=Dawn advantage
M=Morning advantage
A=Afternoon advantage
Du=Dusk advantage
Fw=First Watch advantage
Sw=Second Watch advantage
xp=experience points
fas=first attack strike damage
fns=first attack number of strikes
fsp=first attack special
sas=second attack strike damage
sns=second attack number of strikes
ssp=second attack special
ar=arcane resistance
br=blade resistance
cr=cold resistance
fr=fire resistance
ir=impact resistance
pr=pierce resistance
fd=frozen defence
cd=castle defence
sd=sand defence
vd=village defence
ffd=flat defence
hd=hills defence
md=mountain defence
ud=unwalkable defence
srd=swamp defence
ccd=cave defence
dd=deep water defence
swd=shallow water defence
crd=coastal reef defence
id=impassable defence
fffd=forest defence
mgd=mushroom grove defence
fm=frozen movement cost
cm=castle movement cost
sm=sand movement cost
vm=village movement cost
ffm=flat movement cost
hm=hills movement cost
mm=mountain movement cost
um=unwalkable movement cost
ssm=swamp movement cost
ccm=cave movement cost
dwm=deepwater movement cost
swm=shallow water movement cost
crm=coastal reef movement cost
im=impassable movement cost
fffm=forest movement cost
mgm=mushroom grove movement cost

Here it is. Be aware that some factors are grouped together due to their similarity, which means they might have the same value but its just that they bestow different advantages. Some factors may not be important, such as the impassable movement cost, unless someone decides to make a unit that can pass through the impassable which is actually possible by the way. Be aware that some factors could have negative values.

From here, my next step is to determine the value of each factor as I mentioned earlier and for now, it looks relatively easy. movement cost factors could easily be considered to be of the same value for example. Another example is defence. Such factors could be exchanged for one another. Therefore, it is possible to come up with an algebraic expression to represent these groups of raw factors. For example, I could just add up all the movement costs and call it mc=movement cost.