How to guage a units strength

The PA team would almost certainly have their own (possibly bullshit) formulas for unit balance. Almost every designer, no matter how amateur, plugs their unit stats into an excel spreadsheet at some point and tries to discern some formula for stat tradeoffs.

This reminds me a bit of the work that Evil4Zergin did a while ago, it was very good and expressed strictly mathematically. This was back when a bunch of people tried to mathematically model BA and failed badly. Evil4Zergin's analysis was the best I've seen, but it was on the old CA site which no longer exists.

As I recall he first put HP and DPS together as one stat, which was sort of a 'Damage over lifetime': IE how much damage a unit will do when taking x DPS (Which is linear for x regardless of how you trade off DPS vs HP).

Equally you can put speed and range together, to calculate the 'bonus' damage from a unit closing range based on the range difference and the speed difference, which becomes infinite if one unit has both more range and more speed, and can also potentially be infinite if the bonus damage exceeds the targets HP (It dies before getting in range).

AoE can be considered alongside lancasters square law. It allows a unit to exceed the max 50% weight advantage. Given an equal metal cost in armies, as the enemies comparative weight decreases, the advantage of AoE increases. It is hard to apply this model exactly though because the amount of units you hit depends on positioning, but the bonus does essentially relate to comparative weight advantage for a given metal cost. It can also be applied to lancasters square law in that it can force the opponent to split their army, potentially having less weapons in range at a time as would happen with an enfilade. If PA has supcom-like armies that walk in big grids, you can probably calculate optimal spacing for said grids and from that determine how many units can engage an enemy at one time without taking AoE damage (and add the speed divided by the distance between units to the 'bonus damage' accrued from entering the enemies range one at a time rather than as a group). You could also model the units and their spacing as an arc, which is the optimal formation in many RTS's, and figure out the number of units that can fit in a 180 degree arc. From there you can calculate the number of units in the arc at which damage from AoE makes them stop making cost vs a given unit. These spacing mechanics also relate to footprint as much as AoE, and the spacing along the arc can also model shot blocking.

Of course these are all idealized, abstracted situations, but that's mathematics for you.

It'd be great if you could include a mention of us, even as a counter-example where your model breaks down, in your thesis! Wikipedia will never be able to get rid of us then! :P

quote: It'd be great if you could include a mention of us, even as a counter-example where your model breaks down, in your thesis! Wikipedia will never be able to get rid of us then! :P

I'll try to think of a way. As Zero-K have a much more refined balance and many more unit variables than PA it is hard to just step in with mathematical analysis actually evaluate the balance.

Like both players in Zero-K pretty much have the same options and the initial factory choices really changes with maps so it is hard to say if something is overpowered or not or if it just accepted to be the meta on different maps. Like how would you mathematically determine if airplanes versus shieldbots is balanced or not?

Like my work would only provide some guidelines which would make it easier to evaluate the balance and I intend to use the metrics without evaluating the map balance.

Well sort of true. Once bombers commit to an attack they should go in and it isn't so much about fighting for positions like raiders do. Although the players still have very different strategic goals like if the bomber player should bomb mexes, raiders, AA, cons or suicide to take out the commander. Then you also have stochastic elements where the land player have to guess when and if the air players switches to land and the airplayer have to guess where the enemy AA is and how much defenses he requires to not get bumrushed and if the enemy made so much AA that he have to switch to land production.

Yeah, but the positioning of the AA have to be considered as continuous rather than discrete since there are so many possible locations where you can place your AA.
Then there is also pre-game choices, especially for the air player as he have to decide if he should make Avengers and scout the enemy or not. Scouting the enemy means that he knows if the enemy went air or not and he also have a chance to spot incoming raiders and kill them with Avengers but it also shows that he went air first.
Avengers are also good scouts later in the game as they can usually boost past defences to scout the enemy without dying.
I've been reading up on game theory lately and usually moves are either simultaneous(blind) or games with stages(taking turns).
Like the choice of factory is a simultaneous blind choice so the airplayer might have to consider that the enemy also started air. Then there is taking turns like the air player decides to scout with avengers and the the ground player responds to this by making AA to defend, make lots of Raiders to rush down the airplayer, make lots of cons so he can expand and outeco the airplayer or a mix of those reactions where the airplayer have different ability to assess what the ground actually did.
Just making a build order planner in Starcraft is pretty challanging: http://www.aaai.org/ocs/index.php/AIIDE/AIIDE11/paper/viewFile/4078/4407
Anyway we/you could try :D

google sheets file
//slasher is anomaly since it does not account for stop to fire behaviour
//scorcher damage is set as 15 instead of 30 due to heatray effects
//for raiders, skirmishers, and assults (i have not made an arty or strider category
//special unit classes such as suicide units, air, sea not included due to having special targeting categories unusual attack modes


power^ : combat utility (fairly accurate to achivable ingame results)
(asumes unit survives battle to repair, finite income, conservative playstyle)

utility^ : "spamability" if you have "infinite" income, how usefull is the unit to simply spam till doomsday

power || utility (no "^") : these values are true if the unit can not kite the enemy target "effectively"

/cost and /sqrt(cost) : scales with weight due to potential to survive and repair

||range*speed : skirming power (allways /cost does not scale with weight)
||range*speed*DPS : skirming power+damage capacity while skirm
||hp*dps : lifetime damage
||hp*dps/sqrt(cost) : lifetime damage, weight adjusted, a mesure of the units survivability assuming the unit attempts to retreat when damaged




//does not count special damage classes, shields, AOE effects
//although many results are true to ingame, not absolute accuracy
//"^" and non-"^" are not normalized, they are NOT comparable to eachother
//stats taken from zero-k.info/Static/UnitGuide, are not curent
//if anyone can make a crawler to take the stats directly from the code, please do

//definitions of " " terms
//spamability and infinite : if loosing a unit is acceptable(or unavoidable) and/or, there is a preferance to kill the enemy base cause economic damage with attack patterns that will lead to attrition
(also true for literal cases of infinite income games where buildpower is the limiting factor)

//effectively : the capacity to do damage to the target on a consistent basis while avoiding damage yourself



edit beter table, defined what the colums actualy show, seperators, utility