Revision as of 08:36, 23 May 2010

This is my talk page, where I will very rarely dump little bits of math that I am working on despite my inability to comprehend their significance. Also, I might just dump my strats here since I won't be posting my province and KD #s anywhere.

Weighted Average Cost of Army per Acre

A possibly useful formula for attackers, this seeks to demonstrate the economic viability of running a predetermined OPA and a predetermined DPA through the most population-efficient means. It does not take in to consideration variables beyond troop stats and cost. Also, it doesn't account for DPA w/ 50% or 75% of your elites out, which could hurt your ability to hybridize.

Detailed Explanation

In order to evaluate the economic viability of race's armies, we'll need a goal for them to reach. For attackers, this is obviously a goal Offense per Acre and Defense per Acre. They will be noted in our formula as gOPA and gDPA. Let's get specific so it is easier to discuss and go with 90 OPA and 70 DPA - fair raw numbers for a heavy attacker. Let's assume we're Avians too so that we can give concrete examples alongside the arcane symbols.

In the interest of population - the most important resource of all - we'll be using as many elites as possible. Since no race has a primarily defensive elite, we'll be getting all of our offense from elites and adding defensive specialists to bolster what's left. This means we can already get our first derived variable - Elites per acre (ePA)!

ePA = gOPA / eO

ePA = 90 / 8 = 11.25

(eO is Elite Offense, the offensive strength of the race's elites)

Now, we can do the same for our gDPA. However, we have to account for the defense provided by the number of elites we've just determined we'll need. So, we'll subtract out the elite defense to find the gap between our current and our goal defense and use specialists to fill it in. In terms of Defensive Specialists per Acre (dSPA), it will go like this:

dSPA = [gDPA - (eD * ePA) ] / dSD

dSPA = [70 - (3 * 11.25)] / 5 = 7.25

(eD is Elite Defense, the defensive strength of the race's elites. dSD is Defensive Specialist Defense, the defensive strength of the race's defensive specialists because remember Humans have a +1 to this)

Now that we know how many Elites and Defensive Specialists we'll need to train per acre, it's simple addition to figure out our Army size per Acre (APA). For our Avian, this is 18.5. Furthermore, we know how much of our army is divided between elites and specialists - simply divide elites per acre by army per acre and likewise for specialists. Combine this with the fact that the costs for these things are preset by the game, and we have all the components for a Weighted Average Cost per Troop!

For the math illiterate: let's say you're throwing a party and need to buy beer. You buy a case of one beer for $22 and a case of another for $25. You'd get the average cost per case by simply adding them together and dividing by the number of cases, right? So it's $23.50 per case. Let's say you bought 3 cases of the cheaper beer. What's the average now? If you simply add them together and divide like you normally would, you'll be wrong because the two items cost different amounts. 3 out of 4 (75%, or .75) of the cases were $22 while only the 1 out of 4 (.25) was $25. So, you'd do the math as .75*22 + .25*25 which equals $22.75 per case of beer. The same logic applies with what we're doing now.

So, to find the Weighted Average Cost Per Troop, we'd do this:

[(ePA / APA) * cE] + [(dSPA / APA) * cdS]

(cE is the cost to train an elite while cdS is the cost to train a defensive specialist)

But wait! We know that APA is Army per Acre, or (ePA + dSPA), so we could expand it to this:

[(ePA / [ePA + dSPA]) * cE] + [(dSPA / [ePA + dSPA]) * cdS]

For our Avian that would be

[(11.25 / [11.25 + 7.25]) * 800] + [(7.25 / [11.25 + 7.25]) * 350]

[(11.25/18.5)*800] + [(7.25/18.5)*350]

(.6081*800) + (.3919*350)

486.48 + 137.17

624

So, now that we know how much is costs us on average to train a troop, we can compare the costs of maintaining attacker-level OPA and DPAs between races, right? Wait! Different races will need a different APA, so we'll need to account for that by multiplying that WACT by APA. For our Avian, this is 11,537gc.

The Formula

Above, we realized that we could break APA down into known terms. Say you wanted to start this formula from scratch with no intermediate steps? It would look like this (abbreviations are explained in the above section):

[([(gOPA / eO) / ([gOPA / eO] + [(gDPA - [eD * ePA] ) / dSD] )] * cE) + ([([(gDPA - [eD * ePA] ) / dSD]) / ([gOPA / eO] + [[(gDPA - [eD * ePA] ) / dSD]])] * cdS)] * [(gOPA / eO) + ([(gDPA - [eD * ePA] ) / dSD])]

It's absolutely mad, isn't it? If you rerun these numbers in a future age, I recommend including the intermediate steps in a spreadsheet program so you don't have tons of errors. For now, I've done the math for you below.

Age 47 (Justice) Numbers

For Age 47 with 90 OPA and 60 DPA (note the 10 less DPA from above), the following information applies. I have included in parenthesis things that may affect the overall economic viability of attacker strats with this race, whether directly through an income bonus or reduction in other costs, or indirectly by allowing you more acres than otherwise that you could use for Banks, Libraries, etc. I am not including things such as Townwatch that have a largely detrimental effect to match their benefit nor Spell success bonuses which may allow you to marginalize your already marginal number of Guilds and Towers.

Avian:      10,838gc   16.5 APA    (Attack time redux)
Dark Elf:    9,750gc   15 APA      (No rune cost, Invisibility, Tree of Gold)
Dwarf:      10,886gc   14.57 APA   (Free construction, +30% BE)
Elf:        10,886gc   14.57 APA   (Clear Sight, Fountain of Knowledge)
Gnome:       5,100gc   18 APA      (Halved thief cost)
Halfling:    7,200gc   15 APA      (Sabotage gains, Stealth recovery)
Human:      10,531gc   15.63 APA   (Income bonus)
Orc:        12,000gc   20 APA      (Elite credit training)

Mistake to be Corrected

If you allow dSPA to fall below 0 in your calculations, you'll end up with skewed numbers. I may at some point take the time to clarify this in the above formulas, but I'm a very busy (Re: lazy) man.