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How to balance (calculate strength of) units in any strategy game

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Date Editor Before After
7/5/2016 11:41:11 PMDErankBrackman before revert after revert
7/5/2016 11:37:48 PMDErankBrackman before revert after revert
7/5/2016 11:37:36 PMDErankBrackman before revert after revert
7/5/2016 11:37:23 PMDErankBrackman before revert after revert
7/5/2016 11:23:25 PMDErankBrackman before revert after revert
Before After
1 This kind of ideal balance you are referring to cannot and should not be true for every pair of units, specifically not if they are of different types as you said correctly. This first equation is only used as a motivation for the following integral equation, which says that this "ideal balance" should be true for a unit only on average. For example a raider will be more effective vs skirmishers and less effective vs riots. The integral calculates an average cost for that. 1 This kind of ideal balance you are referring to cannot and should not be true for every pair of units, specifically not if they are of different types as you said correctly. This first equation is only used as a motivation for the following integral equation, which says that this "ideal balance" should be true for a unit only on average. For example a raider will be more effective vs skirmishers and less effective vs riots. The integral calculates an average cost for that.
2 \n 2 \n
3 The problem are infinite effectivities as described in my first post. But there are ways to solve that, for example only comparing same unit types or considering unit groups. 3 The problem are infinite effectivities as described in my first post. But there are ways to solve that, for example only comparing same unit types or considering unit groups.
4 \n 4 \n
5 chi and ψ are multisets which means functions that assign a number to every unit in the game to describe unit groups. For example a group of 2 Glaives and 1 Rocko would be described as ψ( Glaive) =2, ψ( Rocko) =1, ψ( everything else) =0, where units are unit stat vectors. ϕ depends on those functions and we integrate over those whole function spaces. [spoiler]I added a description of ϕ as the unit group generalization of the superiority function f. [/spoiler] 5 chi and ψ are multisets which means functions that assign a number to every unit in the game to describe unit groups. For example a group of 2 Glaives and 1 Rocko would be described as ψ( Glaive) =2, ψ( Rocko) =1, ψ( everything else) =0, where units are unit stat vectors. ϕ depends on those functions and we integrate over those whole function spaces. [spoiler]I added a description of ϕ as the unit group generalization of the superiority function s. [/spoiler]
6 Maybe it's better to remove the terms 1/chi(X) and integral_(Z in Omega\{X})chi(Z)f(Z)delta(Z)dZ in the last formula in "Generalization on unit groups".. Then it would be also much easier to calculate. 6 Maybe it's better to remove the terms 1/chi(X) and integral_(Z in Omega\{X})chi(Z)f(Z)delta(Z)dZ in the last formula in "Generalization on unit groups".. Then it would be also much easier to calculate.