The balance formula thread

[CarRepairer]

Haha, I wasn't even thinking much when I said that. I wish I would have come up with a better formula. That's great though, now I'm immortalized in springbash alongside the greats like emmanuel.

[FLOZi]

balance = (guesswork + intuition) * playtesting ^ x

as x -> infinity, balance -> perfect balance

[Gota]

Making balance changes every few weaks as part of "intuition and playtesting is ridiculous and will just result in endless balance tweaks which are later forgoten and repeated.
The lack of ability or desire to see and try to understand how a small balance change can later on disbalance several other units creates an endless cycle of balance tweaks that dont ever end...see CA,ba,xta.
So it goes like this:
A unit is tweaked after several weeks or days people start noticing some unit is now very good but they fail to realize it has become this good cause of that small tweak to another unit...
BA was meant to be an unchanging "finished" mod.
If it actually stayed that way perhaps We would all see after a few years of playing an unchanged balanced that it's actually more balanced than we though.

[aegis]

balance = (guesswork + intuition) * playtesting ^ x

as x -> infinity, balance -> perfect balance

awesome, going to incorporate this formula into a PBA release.

[TheFatController]

See also

[SwiftSpear]

Post here your balance formula ideas.








I present you my balance formula:

45.81 = (hitpoints*acceleration*maxvelocity*turnrate*weapondamage*range*areaofeffect) / (reloadtime*buildcostenergy*buildcostmetal*buildtime)

Explanation:

All variables in dividend increase units 'strength' as they increase, and decrease it when they decrease.
All variables in divisor decrease units 'strength' as they increase, and increase it when they decrease.
Quotient (45.81) was calculated by using variables from an arguably balanced unit, the core instigator from BA.

This particular one is pretty abusable.

HP is on a 1 to 1 ratio to weapon damage? any unit with low HP is going to be reasonably overpowered, any unit with high build time is going to be reasonably overpowered.

The best way to test formulas is try to find scenarios in which they are broken.

[Saktoth]

Culture of Balance
You cant just balance to baww though, not even with very good players. Even Starcraft, played at such a high level, keeps evolving as a game as new strats become popular and new counters are adopted (Terran Mech vs Zerg is way more common these days).

Even if you dont change a game, the culture will change over time. Perfect the use of a unit and it will seem massively OP. Perfect the strat to counter that unit, and it may suddenly seem useless. In order to have an interesting game, you must by necessity have an utterly vast possibility space. Even the best players cannot completely explore that whole possibility space.

Tick and Roach in CA have gone from UP to OP and back again several times without ever needing to change their stats. This is because they are very hard to use and while totally raping some units, are useless against others. Where they lie on UP/OP relies primarily on how people are playing.

It is a very bad idea to keep buffing an underused unit until people start using it. This is what leads to weasel rush etc. Quantum used to do this all the time to early CA with some very lol results (roaches that could kill llts and cost less than them). People will get used to not incorporating the unit into their stategy and by the time they get around to using it, you'll have buffed it into OPness. You must wait until it IS being used and actively incorporated into strats before you change it again.

So, if you want a balanced game, you want to ensure that there is a counter for everything. A game of rock paper scissors where a win with Rock gets you 5$, scissors 3$ and paper $1 is balanced. Rock is massively OP, but because scissors conters it, this makes scissors almost a safe bet. But paper, though the weakest, counters that.

Even with massive stat imbalances in your game (and they are unavoidable) as long as every unit has an effective counter, the game remains balanced!

Counter Mechanics
How and why are counter structures unavoidable anyway and doom any math based balance system?

A theoretical unit with 100 range and 1 dps and 1 HP and 10 speed, given the right circumstances, will beat an infinite number of units with 90 range, 100 HP 100 DPS and 9 speed. The most extreme example of this is artillery: artillery will beat an infinite number of static defense, as long as the enemy is stupid enough to keep building them.

Equally, Unit A has 100 reloadtime and 100 damage per shot doing only 1 DPS. Unit B has 1 reloadtime and 10 damage per shot for 10 DPS. Both units have 100 HP. Who wins? The unit with 1 DPS, not the unit with 10 DPS.

On the flipside, Unit A fights Unit C. Unit C costs 1/10th as much and has only 10 HP. With 10x as many targets, Unit A wastes 90 damage and dies to the other 9 C's (connect the dots to find a decent RPS triangle here).

Its about situational appropriateness. In a game with huge amounts of highly effective static defense, artillery owns. In a mod with 0 static defense and heaps of really fast units (even all balanced to the same cost formula as the static defense), artillery is virtually useless.

Dont get me wrong, even i use a basic DPS/cost HP/cost comparison to get a baseline of balance on units. But for designing the WHOLE game, you have to think 'what is good against what', and the network of relationships within which the units fit.

[Llamadeus]

A game of rock paper scissors where a win with Rock gets you 5$, scissors 3$ and paper $1 is balanced. Rock is massively OP, but because scissors conters it, this makes scissors almost a safe bet. But paper, though the weakest, counters that.

Man, no wonder CA balance is so backwards.

[SpliFF]

So, if you want a balanced game, you want to ensure that there is a counter for everything. A game of rock paper scissors where a win with Rock gets you 5$, scissors 3$ and paper $1 is balanced. Rock is massively OP, but because scissors conters it, this makes scissors almost a safe bet. But paper, though the weakest, counters that.

Err.. nope. The power in RPS is perfectly balanced, paper is not the weakest because it can cover rock. In fact in your game paper is the STRONGEST because you can buy 5 papers for every enemy rock, despite them being equal in overall strength.

Of course if you had an army of only paper then you'd lose to a single scissor unit, but my point is your example is not remotely balanced.

PS. If you added a lawyer command unit they could win the battle using only paper, however you could defeat them with 1 well-aimed rock or strategically placed scissors.

[Llamadeus]

Err.. nope. The power in RPS is perfectly balanced, paper is not the weakest because it can cover rock. In fact in your game paper is the STRONGEST because you can buy 5 papers for every enemy rock, despite them being equal in overall strength.

Of course if you had an army of only paper then you'd lose to a single scissor unit, but my point is your example is not remotely balanced.

I don't see how it's any less balanced than regular rock paper scissors. You have to use all three to win, and you have to use them at a particular ratio. It just so happens that the ratio is 5:3:1 and not 1:1:1.

[Evil4Zerggin]

http://en.wikipedia.org/wiki/Nash_equilibrium

[SpliFF]

If you put it that way you're right. However i was referring to balancing the units against each other 1on1. Using your logic any game is balanced since all players have the same unit choices and can therefore take advantage of the same exploits and OP units. If you have a $1 nuke that destroys rock paper and scissors (but has no weakness other than other nukes) the game is still fair as long as everyone always takes nukes.

[Llamadeus]

But the 5:3:1 game fulfills another criterion of balance: that no unit is obsolete, which most people seem to agree is a necessary component of game balance.

[Google_Frog]

It's not that rock, paper or scissors costs 5/2/1$ each. A win with them gives you that much money. This would be more interesting than RPS because there are now 3 separate choices to double, triple etc... think your opponent with.

[Llamadeus]

The optimal solution is about as trivial as regular rock-paper-scissors though. 5/9 paper, 3/9 rock, 1/9 scissors.

[Google_Frog]

Every unit having a counter doesn't mean it's balanced. Jack-of-all-trades units aren't balanced even if they have a counter because if they beat most situations then they're the safest bet. The risk/benefit cost of every unit should average out. The strengths of a unit should be balanced with it's weaknesses. So if a unit is useless in most situations it should be really good at the situation it's good in. Jack-of-all-trades shouldn't be really good at anything.

[Llamadeus]

Thinking about it, the rock-paper-scissors example kind of shows how every unit is balanced by having a counter. The more expectation (or perceived expectation) the unit has over the field, the more ubiquitous its counter becomes.

[Gota]

But statistically,after all the mind games(assuming the oped unit has a very clear counter that when built renders the unit useless but is weak against other units),you'd still be building X more than others.

An important factor of spring is also the fact counter movements actions are not instantaneous like say in a fighting game,because units need to get into firing range to be able to spread their "purpose".

That's why i think That in rts games,especially in spring a strong rps system is inferior cause it asks X to guess what unit Y will use when the units that X is making right now in the lab will reach their destination.

This makes the players either just spam mixes of units hoping they get the ratio right or just hope that the unit they are spamming is so good at a perticuler situation that it might,when it gets to the front line,out perform its counters.

To make an rps system work well in a mod with the diversity of BA,you'd have to slow the game down to
#1 allow players to attempt to properly scout their enemies.
#2 have time to think and build the appropriate counters.
#3 when unit assessment was wrong and you'v built the wrong unit and you see the enemy has built counters for it you will still have time to counter react and build units to counter the now known enemy unit types.

If not it just becomes a guessing game without any strategy but instead just an attempt by players to salvage a random situation into victory.
A situation that accured by chance and pure luck.

[KingRaptor]

An unbelievably long essay I wrote about the fallacy of the traditional Tired model of mathematical balancing. Yes, the author information at the end is a bad joke. Ignore it and concentrate on the points.

tl;drs provided at various points for those of you who have ADD.



The Myth of Mathematical Balancing

Abstract
The issue of mathematical balancing is a hotly contested one in the Spring community, with its adherents fiercely insisting that it is the best way, indeed the only way to achieve true balance. This essay will study the failings of the traditional Tired method of using math balancing on games as a whole, and note the places where a less ambitious formula can be used to supplement, not supplant, conventional balance methods.

Background
The concept of mathematical balancing was originally introduced to the Spring community by Tired. The topic was mentioned in a number of places (here for instance), and an actual formula was posted here. Tired also constructed a spreadsheet containing all the unit stats and their resultant costs (downloadable sample), which was used in Tired Annihilation. This system will hereafter be referred as the Tired model.

Other people have also experimented with mathematical balance. It has been used in the Supreme Annihilation and Supreme Legacy mods, and dabbled with elsewhere; Evil4Zerggin wrote about mathematical balancing on the CA wiki, and I attempted to construct a new spreadsheet design which attempted to overcome weaknesses in the Tired model, but ultimately proved to be too difficult to create, and was abandoned (hereafter referred to as the KR model) ├óÔé¼ÔÇ£ a sample can be downloaded here.

The Linear Error
Despite its proponents praising the depth and complexity of the Tired method, it is in reality exceedingly simple mathematics-wise: a number of unit variables (easily less than a third of the ones that matter) and some constants are multiplied together, and the output is used to price the unit. The given equation in the Basic Mod Balance thread is:

Speed%*((Weapon Value% + Health%)/2)*LoS%*Special Value%, where Weapon Value = DPS*Range*RoundedAOE.

The patent nonsense implied by such a formula is immediately obvious to anyone who spends a few moments thinking about it. In particular, range and speed are two attributes whose effects can hardly be described as linear.

Take the CA Janus, for example. It fires twin high-damage missiles with a ten second reload time, outranging most ├óÔé¼┼ôT1├óÔé¼┬Ø units but can be hit by LLTs as it approaches them. It has low health and relies mainly on its mobility to survive. Now double its range (enough to easily outrange HLTs and almost all ├óÔé¼┼ôT2├óÔé¼┬Ø units), and halve its HP to compensate. Such a unit would be nothing short of massively overpowered ├óÔé¼ÔÇ£ yet under the above equation, it would be balanced with only a 25% cost increase!

Nor is this restricted to any particular unit or stat. For instance, would a Flash in BA be balanced if it had 50% less DPS but 50% more speed? Could the KP Byte be balanced by halving its range in order to double its HP? How about giving Gundam├óÔé¼Ôäós Char double range at half DPS? At the other extreme, increasing a unit├óÔé¼Ôäós range may not increase its performance significantly ├óÔé¼ÔÇ£ for instance, an *A nuclear silo can already cover the entirety of all common maps regardless of where it├óÔé¼Ôäós placed, and a BA/CA Big Bertha can similarly reach the opponent├óÔé¼Ôäós base from the player├óÔé¼Ôäós own in most cases. Increasing their range would not impact their performance in a proportional manner.

The KR model attempts to resolve the problem by using exponential terms where appropriate; however, this increases the difficulty of finding the correct formula. In just the sample units tested, for instance, the Stumpy, Janus and Gator are severely undervalued, while the Glaive (Peewee) apparently costs much less than it should. Tweaking value weightage could solve these issues, but at the cost of ruining the valid outputs of other units. This, incidentally, reflects another fallacy of the Tired model ├óÔé¼ÔÇ£ rather than retaining cohesive balance by tweaking all units at once, the spreadsheet method cripples it by ├óÔé¼┼ôfixing├óÔé¼┬Ø that which ain├óÔé¼Ôäót broke.

Conversely, even the most subtle variations in a unit├óÔé¼Ôäós stats can completely unbalance it. At one point, CA ran a standardization script that allowed most units to see further than they could shoot. This allowed unsupported Rockos to spot LLTs (and other targets) on their own, and destroy them without fear of damage. It was considered so OP that the unit├óÔé¼Ôäós range had to be nerfed (but this had other knock-on effects, refer to Balancing in a Vacuum below). Or suppose that an Arm LLT had 50 more range than a Core LLT ├óÔé¼ÔÇ£ the extra range is pretty much meaningless against most units, but it means that you could easily kill a Core LLT by placing an Arm one just outside its range. How much more should this cost? As the CA balance guidelines state: ├óÔé¼┼ôThe use of a unit is not always determined by good stats, but often by the right combination and values. There is a huge difference between just outranging an LLT, and having the same range as an LLT. Try to think more in terms of unit relationships than pure numbers.├óÔé¼┬Ø

tl;dr: Treating nonlinear values as being linear, and ignoring how subtle changes can completely change a unit, is an invitation to fail.


Game Design and Regression to the Mediocre

So, we├óÔé¼Ôäóve established that the Tired model falls apart with unique units and overlooks the small changes with big effects. How does Tired resolve the problem? Simple ├óÔé¼ÔÇ£ dodge it. A not invalid complaint about TiA (and its successor, SA) is the blandness of the unit selection; readily apparent by such changes as the dulling of unit strengths (for instance, the Peewee is now just a short ranged fighter rather than a fast raiding unit) and across-the-board speed reductions, creating a more homogenous unit selection that brings to mind Francis Galton├óÔé¼Ôäós famous phrase (albeit used in a different context): ├óÔé¼┼ôregression to mediocrity├óÔé¼┬Ø.

Tired was even willing to sacrifice his own design principles in order to conform to the formula. The original reasoning behind the massive price increase of the Jeffy and Weasel in Tired Annihilation over BA (>3x) was that the Jeffy is ├óÔé¼┼ôfree information├óÔé¼┬Ø ├óÔé¼ÔÇ£ i.e. it allows the player to gain intel on his opponent for minimal cost. Yet, the same Tired reduced the spybot├óÔé¼Ôäós cost to less than a quarter of what it was (in TiA 6.0 and SA 1.0 it has an adjusted cost of 59, less than two-fifths of a Flash).

In fact, Tired went so far as to declare (admit, rather), that any unit is balanced if it fits the formula regardless of what the formula actually is. Consider the following quote:

For a balanced equation, I could say that speed was worth 10x as much as Health. As such, it would take many times - perhaps 5x - as much metal invested in Flash Tanks as invested in a Sumo to kill a Sumo. In BA, that would equate to something like 10,000 metal in Flash Tanks. This would be balanced.

I could say that range was worth only 2% as much as dps, and redesign a Flea that could fire from one corner of a 16x16 map to an opposing corner and cost 200 metal without changing any other basic unit stats. This would be balanced.

I could say that airplanes should cost 57x as much as comparable ground units, put wings on a Stumpy, and send 11,400 metal PteroStumps to their doom against normally priced Slashers. Would this PteroStump be effective? No. Would it be balanced? Yes.

Thus we see how more balance can actually lead to a less fun game.

tl;dr: Tired circumvented problems with his formula by changing the game, not the formula. And not for the better.


Balancing in a Vacuum

Tired tells us that any stat can have a single arbitrary value assigned to it ├óÔé¼ÔÇ£ but can it?

Even the best spreadsheet will ultimately be balancing in a vacuum, outputting sterile values without reflecting actual unit vs. unit situations which are likely to be encountered in the field. In an actual game, in addition to all the unit stats which are or are not considered by the equation, we have the one completely uncontrollable variable on top of it all ├óÔé¼ÔÇ£ human interaction.

Unit attributes combine to form the unit in a way that far exceeds the mere sum of its stats. In BA, the Flash doesn├óÔé¼Ôäót look so hot compared to the Peewee on paper ├óÔé¼ÔÇ£ it has slightly less HP/cost and half the DPS/cost, easily loses in terms of turn rate and acceleration, and can├óÔé¼Ôäót go up hills like its Kbot counterpart can ├óÔé¼ÔÇ£ yet in an actual combat situation, the Flash is simply superior in every way due to the way it works ├óÔé¼ÔÇ£ its higher per-unit HP allows it to survive long enough to dish out damage, fewer units means Flashes are less prone to crowding each other out than Peewees are, the greater speed means the Flash can close in on traditional enemies such as LLTs faster, and the Flash doesn├óÔé¼Ôäót chain explode with shrapnel like the Peewee does.

As hinted above, different units value different stats; while DPS is generally useful for all combat units, range is much more valuable for artillery and standoff units than HP is, while rushers require more HP to survive. Range is especially important on a building, much more than a mobile unit (as it can├óÔé¼Ôäót move to be able to hit its target), and so forth.

The KR model does attempt to counteract such problems by:

• Using unit categories (though the only one shown in my sample spreadsheet is the active/passive defender distinction) to allow different stat weights for different units;
• Attaching an exponent to the final unit cost (to reflect the advantages of a higher weight class), but this is an imperfect solution;
• Including ├óÔé¼┼ômagic numbers├óÔé¼┬Ø that can account for such differences, but as Saktoth has pointed out in the past, we may as well not have a formula if we resort to such kludges.

Even the number of units coming to an engagement can change the outcome completely. Two AKs can beat one Rocko in BA, simply by running up to it, dodging its rockets, and pew-pewing it to death. Four AKs can likewise beat two Rockos. Six AKs beat three Rockos. But what about 10 AKs vs. five Rockos, or 20 vs. 10? The massive target density simply means the large AK clump cannot hope to dodge the rocket salvo, and many units will be hit and killed (filling the survivors with shrapnel that hurts them ├óÔé¼ÔÇ£ see above Flash-Peewee comparison). The stragglers can easily be picked off by the Rockos. This is something that no formula can account for.

Changing a unit, or a specific class of units, can also have knock-on effects that make other units overpowered or underpowered even though those other units haven├óÔé¼Ôäót been changed themselves. For instance, when CA reduced its Rocko range, riot units such as the Outlaw that were previously countered by it became too powerful against it, and had to have their own range nerfed as well. However, that in turn meant they couldn├óÔé¼Ôäót fight raiders as effectively, and those too had to be rebalanced.

Even if you could design a formula that compares not only the units to a benchmark, but to every other unit, adding unit mixing to the equation only screws it up further. When a Slasher line is attacking your LLTs, you can run your Peewees/Glaives out and whack them. But wait, what if they├óÔé¼Ôäóre screened by Levelers? Your raiders would simply get slaughtered, and you have to come up with something else. What if instead of Slashers, it├óÔé¼Ôäós the nearly identical Samson? Arm doesn├óÔé¼Ôäót have Levelers. How do you balance the Peewee to fit this problem?

You may now be asking ├óÔé¼┼ôBut if a computer can├óÔé¼Ôäót perform such complex comparisons, what can a human do?├óÔé¼┬Ø It is a fallacy to assume that computers are superior in dealing with every possible mathematical problem; otherwise AIs would be routinely beating humans at Go just like they do in chess. Among other things, computers cannot properly handle integration (antidifferentiation), and where humans can use analytical methods to arrive at the exact value for an integral, computers are forced to use more primitive numerical methods to arrive at the solution (See: one two). Similarly, while a human may not be able to intricately compare hundreds of thousands of unit combinations, he/she can work out the big picture and actually get a better result, by not missing the forest for the trees.

If you told a biologist that you could use such simplistic mathematical models to simulate a real-life ecosystem, he would laugh in your face. Yet that├óÔé¼Ôäós exactly what advocates of the Tired formula claim to be able to do.

tl;dr: Units do not exist in a vacuum and cannot be balanced as such.


Understanding our Limitations

So, if the spreadsheet approach doesn├óÔé¼Ôäót work, does that mean that mathematical balance is useless? Not quite. Every time we compare two unit├óÔé¼Ôäós stats, we are performing balance comparisons using ad hoc mathematics.

There are simple formulae which, by virtue of not overestimating their capability, can give better results than the universal spreadsheet. Evil4Zerggin├óÔé¼Ôäós Basic Efficiency Models are a good example ├óÔé¼ÔÇ£ such methods can be use to evaluate a small number of units in a specific area without overextending the results to other aspects. Rather than pretending to factor in every single unit attribute, we take a few, acknowledge the contribution of the ones we don├óÔé¼Ôäót, and then run tests to determine the extent of their effect.

tl;dr: Simpler math can work better, by doing it right.


Conclusion

Despite claims to the contrary, the Tired model is nowhere near as complex as it wants to be, or needs to be. Any attempt to create a more complicated formula runs into the difficulty of finding appropriate values that can be applied uniformly to all units. Nevertheless, simpler models applied in a more specific manner can act as a useful tool to identify potential imbalances; combined with testing, this remains the best way to balance a Spring game.

And finally, to quote Tired once more:

Balance out of context means nothing.

Final tl;dr: There are some things math can balance; for everything else, there├óÔé¼Ôäós playtesting.


KingRaptor is a senior Complete Annihilation developer and an expert on Spring game design. He is the author of such bestselling books as The Peet Delusion and Smoth vs. Argh: A Tale of Forum Drama.

[Peet]

The Peet Delusion