Attack Level and Defense Level Demystified

The latest entry in the popular Demystified series: Attack Level and Defense Level

Please check this PWI WIKI article for a comprehensive guide to Attack/Defense levels and all other game mechanics.

TL; DR Formula
If Attack Level > Defense Level:
damage = damage dealt * (1 + (A - D)/100)
else
damage = damage dealt / (1 + 1.2 * (D - A)/100)

Background:

Alright so there was some confusion about the effects of combining attack level and defense level. There were two theories...

1) damage = damage dealt * (1 + (A - D)/100)
2) damage = damage dealt * (1.01)^A * (0.99)^D

where A is attacker attack level and D is defender defense level

Test:
Feel free to skip this section. I tested on a target, Linnkotsu, in fb19 with a combination of lvl 10 Frenzy and the Dominance Blessing. My normal test dummy, Quilue, was unavailable. This was a duel using only punches which deal a fixed physical damage.

Raw damage in a duel using punches 
with dominance blessing and frenzy.

          ATT+00 ATT+10  ATT+20  ATT+30
         _______________________________

+00DEF |   141    155     169     183
+10DEF |   126    141     155     169

Dividing out by the base damage 141 and keeping 3 significant figures:

Normalized Damage
          ATT+00 ATT+10  ATT+20  ATT+30
         _______________________________

DEF+00 |   1.00   1.10    1.20    1.30
DEF+10 |   0.90   1.00    1.10    1.20

Values Predicted by (1.01^A * 0.99^D)
          ATT+00 ATT+10  ATT+20  ATT+30
         _______________________________

DEF+00 |   1.000  1.105   1.220   1.348
DEF+10 |   0.904  0.999   1.104   1.219

This exactly matches: (1 + (A - D)/100) and disproves 1.01^A * 0.99^D.

Looking at Higher Defense Levels:
After my initial results I was told that the formula doesnt hold for higher defense levels and collected data to test this. For this I used unarmed attacks against a psychic with varying defense levels.

ATT	DEF	DAM	DEC% 	ATT-DEF
0	0	1227	0	0
0	41.8	822	-33.00	-41.8
0	36.3	856	-30.23	-36.3
20	41.8	980	-20.13	-21.8
20	36.3	1029	-16.13	-16.3
0	5.5	1157	-5.70	-5.5
20	5	1411	14.99	15
20	0	1472	19.96	20

(1+(A-D)/100) matches exactly for positive values of (A-D) but not negative negative.
azukaya was able to find a matching formula in this post:
http://pwi-forum.perfectworld.com/showpost.php?p=9805402&postcount=110

For D > A: 1/(1 + 1.2(D - A)/100)

Effect of Full Sharding:
Let's see the effect on the effective enemy HP with full 24 stone sharding in addition to +8 attendance blessings.

Effective HP is how much damage an enemy can absorb before dying:

effective HP = base HP / defense reduction

          ATT+00 ATT+24 ATT+32 
         ____________________

DEF+00 |   1.000  0.806  0.758
DEF+24 |   1.288  1.000  0.926
DEF+48 |   1.576  1.288  1.192
DEF+56 |   1.672  1.384  1.288

If you have a ATT+32 advantage they are 24.2% easier to kill.
If enemy has a DEF+24 advantage they are 28.8% harder to kill.
If enemy has a DEF+48 advantage they are 57.6% harder to kill.
If enemy has a DEF+56 advantage they are 67.2% harder to kill.


Here's a chart showing the effectiveness of defense level sharding versus various attack levels

http://imgur.com/w81Eh.png

The blue line is for an attacker with attack level = 0 and corresponds to PVE pretty much.
The red line (attack level = 15) is for someone wearing the anniversary blessing.
The yellow (attack level for someone wearing Jones blessing.
The green is someone DoT Sharded with attack level = 45

You can see that defense sharding actually becomes more effective after you get more defense level then their attack level.


Conclusion:
The correct attack level formula is:

If Attack Level > Defense Level:
damage = damage dealt * (1 + (A - D)/100)
else
damage = damage dealt / (1 + 1.2 * (D - A)/100)

Attack level exactly cancels out defense level but attack level has diminishing returns while defense level has constant returns. The full sharding with DEF+1 stones isn't really more effective than Citrine gems or Vit stones for most classes unless they have high base HP. DEF+2 sharding is nice but when comparing them to Vit Gems keep in mind that Vit Gems add to both HP and defense.