Hopefully you don't mind the massive influx of double and triple posting from me, but I was going through the old forums that I was part of (link in my sig), and a lot of these guides were written by some of the members during Serebii's "Golden Competitive Age" (circa 2009). Jesusfreak and a couple of other people may still remember some of these members (Salavoir, Bulldogs, Calum, zapper22011, jrcxyz, me, etc.),but they had an elitist separation from Serebii and had a treasure trove of fckin good info on competitive battling. Considering we're approaching a Golden Age in the Serebii Competitive forums once again (thanks to Saph and MMS), I thought now would be a good time to bust these articles out. and fyi, you can't find these anymore on smogon, so drink them up while you can children.
**NOTE**: THESE EXAMPLES ARE WRITTEN WITH THE GEN. IV METAGAME IN MIND, SO IF A MOD COULD HELP ME IRON OUT SOME KINKS TO BETTER FIT THE GEN VI META I'D GREATLY APPRECIATE IT.
Potential: Defensive and Offensive proofs
Warning! Defensive potential does not work the way we think it does! The reason my proof is fine was because I had forgotten about the necessary rounding in the damage formula. This rounding throws everything off, and so defensive potential does not work as well as the proof suggests but has errors! Do NOT use the defensive potential for a precision spread!
A more precise statement of Potential
Determining defensive potential (e.g. determining whether one combination of HP and SDef is better than another combination of HP and SDef on the special side) is quite simple. Multiply your Pokemon's HP stat by one of its Defense stats to get a number. Try a different combination of HP and that same defensive stat then multiply them again. Compare the two numbers: (the larger number x 100 / the smaller number) - 100 = combination that created the larger number is w% better than the combination that created the smaller number.
Or you can use: the smaller number x 100 / the larger number = combination that created the smaller number takes hits w% as well as the combination that created the larger number. Offensive potential (e.g. determining whether a given SAtk stat and move is better than the same SAtk and another move (or a different SAtk and same move) on the special side) is similar and is compared with the exact same formulas. The numbers you use are found by multiplying an attack stat by the base power of the appropriate move. In both cases, the larger the number you get from multiplication, the better the defensive or offensive potential is.
More precisely still, let H0 = an arbitrary stat value of HP and D0 = an arbitrary stat value of the defensive stat fixed, where "defensive stat" can be either SDef or Def and "defensive stat fixed" can be either SDef or Def but only one for the rest of this discussion, that is "defensive stat fixed" can be only one of SDef or Def but must remained fixed as SDef or Def for the rest of the discussion (e.g., if the "defensive stat fixed" is SDef, it must remain SDef for the remainder of this discussion). That is, let H = {H0, H1, ..., Hn} and D = {D0, D1, ..., Dn}. Let H0D0 = k0, where k0 is an element of T = {k0, k1, ..., kn}. Then, choose another arbitrary stat value of HP, say H1, such that H1 may or may not = Ho, and choose another arbitrary stat value of the defensive stat fixed, say D1, such that D1 may or may not = Do. Let H1D1 = k1.
If k1 > k0, then having the two stat values H1 and D1 is better than having the two stat values H0 and D0. If k0 > k1, then having the two stat values H0 and D0 is better than having the two stat values H1 and D1. k1 = k0 if and only if H0D0 = H1D1 (one case happening when H0 = H1 and D0 = D1), which means it is neither better nor worse having either pair of stat values. Let Clarge = the larger of k1 and k0 and let Csmall = the smaller of k1 and k0. Then let w% = 100(Clarge/Csmall) - 100. Letting Clarge = HlargeDlarge while Csmall = HsmallDsmall, we say that having the two stats Hlarge and Dlarge is w% better than having the two stats Hsmall and Dsmall (the first pair takes w% less damage than the second pair). Letting z% = (100Csmall)/Clarge, we say that having the two stats Hsmall and Dsmall allows our Pokemon to take hits z% as well as having the two stats Hlarge and Dlarge.
We call k0 "the defensive potential resulting from having the two stat values H0 and D0," k1 "the defensive potential resulting from having the two stat values H1 and D1," ..., kn "the defensive potential resulting from having the two stat values Hn and Dn." Or for short, we call the numbers k0, k1, ..., kn "the defensive potential of a Pokemon" or "the defensive potential" or simply "defensive potential.” That is, defensive potential is an element of the set T. The larger the defensive potential, the "better" or "greater" the defensive potential is.
Offensive potential is defined in a similar manner. I would go through the whole thing again, but this is supposed to be informal! So I will leave it at letting A0 = an arbitrary value of the offensive stat fixed ("offensive stat" and "offensive stat" fixed are defined in the same way as "defensive stat" and "defensive stat fixed" substituting Atk for Def and SAtk for SDef in the preceding discussion) and B0 = an arbitrary value of the base power of an arbitrary move of the same parity of the offensive stat fixed ("parity" defined here means being either "special or physical" instead of the usual "odd or even" e.g., if the offensive stat fixed is special, the move from which the base power comes must be special).
Let A = {A0, A1, ..., An} and B = {B0, B1, ..., Bn} having elements which may or may not be equal to each other and let A0B0 = j0, A1B1 = j1, ..., AnBn = jn while letting Elarge = the larger of two elements of J = {j0, j1, ..., jn} = AlargeBlarge while letting Esmall = the smaller of the two elements chosen to determine Elarge = AsmallBsmall. J has elements which may or may not be equal to each other. Then substituting j0, j1, ..., jn for the corresponding k0, k1, ..., kn, A0, A1, ..., An for the corresponding H0, H1, ..., Hn, B0, B1, ..., Bn for the corresponding D0, D1, ..., Dn, Elarge for Clarge, Esmall for Csmall, "offensive" for "defensive," and "give hits" for "take hits" in the discussion on defensive potential shows how "offensive potential" is to be defined. This shows that offensive potential is an element of the set J.
Proof of Offensive Potential
We must prove that the properties of the offensive potential can be deduced from known mechanics of the Pokemon game. We begin with the damage formula.
By the work of the effective base power equations, the damage formula gives ((Basepower * Mod1 * K) + 2) * Mod2 * Mod3 * Type1 * Type2 * N * STAB, where K = (L * A * (1/50))/D, A = [Sp]Atk, D = [Sp]Def, L = (Level × 2 ÷ 5) + 2, and N = R * 0.01 [R is the random number]. For a given defending Pokemon Mod2 * Mod3 * Type1 * Type2 * STAB is a constant, say Mod2 * Mod3 * Type1 * Type2 * STAB = M, and K/A [K/A = L *(1/(50D))] is a constant too. Although N isn’t really a constant, it is a variable which can be brought outside of the scope of the damage formula by multiplying it by everything after the rest of the calculations (as shown later when it cancels out of our equation). Mod1 is also constant for a fixed defending Pokemon.
Let Mod1(K/A) = W. Then the damage formula becomes ((BasepowerAW) + 2)MN = ((Basepower * A)W) + 2)MN = (BasepowerA)WMN + 2MN = V(BasepowerA) + F, where V = WMN (a constant) and F = 2MN (another constant). Thus, the only two variables which affect the equation are the base power of the move used and the offensive stat of the attacking Pokemon. The greater BasepowerA is, the greater the damage will be, and so BasepowerA can be used as a relative measure of how much damage you will do to a fixed, arbitrary Pokemon provided Basepower and A have the same parity.
Now say that the Basepower is an arbitrary Basepower, say B0, by our earlier definitions, and say that A is an arbitrary value of the offensive stat fixed, say A0, by our earlier definitions. Then choose another arbitrary Basepower, say B1, and another arbitrary value of the offensive stat fixed, say A1, ..., then choose another arbitrary Basepower, say Bn, and another arbitrary value of the offensive stat fixed, say An, where all elements of B and A are of the same parity. Then we get V(B0A0) + F = V(j0) + F, V(B1A1) + F = V(j1) + F, ..., V(BnAn) + F = V(jn) + F. So we see that the greater the offensive potential is, the better it is, and that offensive potential can be used as a relative measure of how much damage you will do to a fixed, arbitrary Pokemon. That is, if j1 > j0 then the equation containing j1 does more damage while the situation is reversed for j0 > j1. If j1 = j0, then they do the exact same amount of damage, provided the elements of B and A have the same parity, which means it is neither better nor worse having the Basepower and offensive stat fixed that forms either of them.
Now choose two arbitrary values for Basepower, say Bi and Bk, and choose two arbitrary values for A, say Ai and Ak (where BiAi = ji and BkAk = jk are elements of J). Then we get VBiAi + F = V(ji) + F and VBjAj + F = V(jk) + F for the amount of damage done to a fixed, arbitrary Pokemon. Since the larger damage done corresponds to the larger of ji and jk and since both are elements of J, these equations become VElarge + F and VEsmall + F, the larger damage done corresponding to VElarge + F, of course. Comparing these shows (VEsmall + F)/(VElarge + F) = (WEsmall + 2)/(WElarge + 2) [the MN’s cancel out].
If ji = jk, then this becomes 1, showing that the damage done by having Esmall does 100% as much damage as having Elarge (or equivalently, the damage done by having Elarge is 0% more damage than the damage done by having Esmall, as seen by using the reciprocal of the fraction as a comparison which involves {[WElarge + 2]/[WElarge + 2] – 1}100 = [1 – 1]100 = 0). More generally, the expression depends entirely on Esmall and Elarge. Ignoring the 2 for simplicity shows that Esmall/Elarge [the W’s cancel if the 2 is ignored] is a measure of the relative damage done to a fixed, arbitrary Pokemon. Ignoring the 2 is fine since in practice Esmall and Elarge are quite a bit larger than 2, and so the error involved in ignoring the 2 is quite small. Even for level 1s and 5s the error is small considering the lowest base power of a move is 10 (try and see for yourself!).
Letting z = Esmall/Elarge [which means z% = (100Esmall)/Elarge] and w = Elarge/Esmall [which means w% = ((100Elarge)/Esmall) - 100] shows that the damage resulting from Elarge is w% more than the damage resulting from Esmall, or equivalently, the damage resulting from Esmall is z% as much as the damage resulting from Elarge. But according to our definitions, Esmall and Elarge describe offensive potential and so hold true for ji and jk. Since those are arbitrary elements of J, they hold true for any two elements of J. Thus, for any two offensive potentials, one offensive potential is w% better (does w% more damage) than the other and the other is z% as good (does z% as much damage) as the first offensive potential.
Therefore, all properties of the offensive potential can be deduced from known Pokemon game mechanics, Q.E.D.
Proof of Defensive Potential
We must prove that the properties of the defensive potential can be deduced from known mechanics of the Pokemon game. The proof is similar to that of the offensive potential, but it is done afterwards because it is harder to accomplish, especially because of the backwards thinking involved and extra variable. We begin with the damage formula as before.
By the work of the effective base power equations, the damage formula gives ((Basepower * Mod1 * K) + 2) * Mod2 * Mod3 * Type1 * Type2 * N * STAB, where K = (L * A * (1/50))/D, A = [Sp]Atk, D = [Sp]Def, L = (Level × 2 ÷ 5) + 2, and N = R * 0.01 [R is the random number]. For a given attacking Pokemon Mod2 * Mod3 * Type1 * Type2 * STAB is a constant, say Mod2 * Mod3 * Type1 * Type2 * STAB = M, and KD [KD = L * A * (1/50)] is a constant too. As before, Mod1 is also constant for a fixed, attacking Pokemon, but so is Basepower for a fixed, arbitrary move from a fixed, attacking Pokemon. Letting Basepower * Mod1 * KD = U and G = UMN changes the formula to ((U/D) + 2)MN = ((UMN)/D) + 2MN = (G/D) + F [remember that F = 2MN], which shows that the damage done by a Pokemon depends on D. The greater D is, the smaller the damage is.
However, percents are what matter in Pokemon. The percent damage done by a fixed, arbitrary attacker is ((100G/D) + 100F)/HP = ((100G)/(HP*D)) + (100F)/(HP) = ((100G)/(HP*D)) + (100FD)/(HP*D) = (100G + 100FD)/(HP*D). Now, here we have an interesting formula, since the percent damage is affect by both (HP*D) and D [since HP and D are the only variables], which just goes to show how much more complicated defensive potential is. However, as you can see, increasing (HP*D) decreases the percent damage, so the greater (HP*D) is, the better that combination of HP and D is. Now say that HP is an arbitrary HP, say H0, by our earlier definitions, and say that D is an arbitrary value of the defensive stat fixed, say D0, by our earlier definitions. Then choose another arbitrary value for HP, say H1, and another arbitrary value of the defensive stat fixed, say D1, ..., then choose another arbitrary value of HP, say Hn, and another arbitrary value of the defensive stat fixed, say Dn. All arbitrary values of HP and the defensive stat fixed are elements of H and D, respectively.
Then we get (100G + 100FD0)/(H0*D0) = (100G + 100FD0)/k0, (100G + 100FD1)/(H1*D1) = (100G + 100FD1)/k1, ..., (100G + 100FDn)/(Hn*Dn) = (100G + 100FDn)/kn. So we see that the greater the defensive potential is, the better it is because you take less percent damage, and that defensive potential can be used as a relative measure of how much damage you will take from a fixed, arbitrary Pokemon provided the move used by that Pokemon and the defensive stat fixed are have the same parity. That is, if k1 > k0 then the equation containing k1 takes less damage while the situation is reversed for k0 > k1. If k1 = k0, then they take the exact same amount of damage, provided that the move and defensive stat fixed have the same parity, which means it is neither better nor worse having the HP and defensive stat fixed that forms either of them.
Now choose two arbitrary values for HP, say Hi and Hj, and choose two arbitrary values for D, say Di and Dj (where HiDi = ki and HjDj = kj are elements of T). Then we get (100G + 100FDi)/(Hi*Di) = (100G + 100FDi)/ki and (100G + 100FDj)/(Hj*Dj) = (100G + 100FDj)kj for the amount of damage taken from a fixed, arbitrary Pokemon. Since the larger damage done corresponds to the larger of ki and kj and since both are elements of T, these equations become (100G + 100FDlarge)/Clarge and (100G + 100FDsmall)/Csmall, the smaller percent damage taken corresponding to (100G + 100FDlarge)/Clarge, of course, where Dlarge and Dsmall correspond to Clarge and Csmall, respectively (remember their definitions!). Comparing these shows ((100G + 100FDlarge)/Clarge)/((100G + 100FDsmall)/Csmall) = ((U + 2Dlarge)/Clarge)/((U + 2Dsmall)/Csmall) [the 100MN’s cancel out] = ((U + 2Dlarge)Csmall)/((U + 2Dsmall)Clarge).
If ki = kj and Di = Dj, then this becomes 1, showing that the damage taken having Csmall is 100% as much damage taken by having Clarge (or equivalently, the damage taken by having Clarge is 0% more damage than the damage taken by having Csmall, as seen by using the reciprocal of the fraction as a comparison which involves {[[U + 2Dsmall]Clarge]/[[U + 2Dsmall]Clarge] – 1}100 = [1 – 1]100 = 0). More generally, the expression depends entirely on Dsmall, Dlarge, Csmall, and Clarge, which is a bit more complicated than what we had for offensive potential. Ignoring the 2Dsmall and 2Dlarge for simplicity shows that Csmall/Clarge [the U’s cancel if the 2Dsmall and 2Dlarge are ignored] is a measure of the relative damage taken from a fixed, arbitrary Pokemon.
Ignoring the 2Dsmall and 2Dlarge is fine since in practice U is quite a bit larger than 2Dsmall and 2Dlarge, and so the error involved in ignoring the 2Dsmall and 2Dlarge is quite small. However, with lower level Pokemon, the error becomes more significant. Fortunately, for level 100s U will be much larger than either 2Dsmall or 2Dlarge and so the error is negligible. Unlike for offensive potential though, the error is not negligible for lower level Pokemon Note that 2Dsmall and 2Dlarge disappear without error when Dsmall = Dlarge, but this cannot be one of the possibilities because we know that if we keep the defense fixed and add HP, you’ll take less percent damage and so defensive potential is not useful in that case. Rather, it is more useful when the sum of the EVs creating D and HP is a constant. This means Dsmall =/= Dlarge for our purposes, so there will be an error.
Letting z = Csmall/Clarge [which means z% = (100Csmall)/Clarge, which can be done because Csmall/Clarge results from the smaller damage/larger damage and so “inherits” the damage’s comparing property] and w = Clarge/Csmall [which means w% = ((100Clarge)/Csmall) – 100, which can be done because Clarge/Csmaller results from the larger damage/smaller damage and so “inherits” the damage’s comparing property] shows that the damage resulting from Clarge is w% more than the damage resulting from Csmall, or equivalently, the damage resulting from Csmall is z% as much as the damage resulting from Clarge. But according to our definitions, Csmall and Clarge describe defensive potential and so hold true for ki and kj. Since those are arbitrary elements of T, they hold true for any two elements of T. Thus, for any two defensive potentials, one defensive potential is w% better (takes w% less damage) than the other and the other is z% as good (takes hits z% as well) as the first defensive potential.
Therefore, all properties of the defensive potential would be deduced from known Pokemon game mechanics, after which would follow a "Q.E.D." For reasons noted above though, this is not a valid proof. It remains because I think it provides some interesting insights.....not to mention that it took a whole lot of time to type up.