Diminishing Returns on HP/Defense Spreads.

Deck Knight

Deck Knight

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Alright, so I'm not a math genius like X-Act or Dragontamer, but I wanted to figure out at which point in a defensive spread do base stats achieve their maximum defensive value.

To do this I used a fairly simple method using Base stats in increments of 10. For each 10 Base increment in HP, 5 was removed from defense and special defense.

I used 100/100/100 for my initial base.

BHP = Base HP
BDef = Base Defense
BSD = Base SD
AHP = Actual HP
ADef = Actual Defense
ASD = Actual SD
Total = Total Defenses
Difference = Difference from previous number.

Code: 
[B]No EV's Table (0/0/0)[/B]
BHP	BDEF	BSD   Total	AHP  ADef    ASD     Total	Diff.
100	100	100	300	341	236	236	80476	

10	145	145	300	161	326	326	52486	
20	140	140	300	181	316	316	57196	4710
30	135	135	300	201	306	306	61506	4310
40	130	130	300	221	296	296	65416	3910
50	125	125	300	241	286	286	68926	3510
60	120	120	300	261	276	276	72036	3110
70	115	115	300	281	266	266	74746	2710
80	110	110	300	301	256	256	77056	2310
90	105	105	300	321	246	246	78966	1910
100	100	100	300	341	236	236	80476	1510
110	95	95	300	361	226	226	81586	1110
120	90	90	300	381	216	216	82296	710
130	85	85	300	401	206	206	82606	310
[B]140	80	80	300	421	196	196	82516	-90[/B]
150	75	75	300	441	186	186	82026	-490
160	70	70	300	461	176	176	81136	-890
170	65	65	300	481	166	166	79846	-1290
180	60	60	300	501	156	156	78156	-1690
190	55	55	300	521	146	146	76066	-2090
200	50	50	300	541	136	136	73576	-2490
210	45	45	300	561	126	126	70686	-2890
220	40	40	300	581	116	116	67396	-3290
230	35	35	300	601	106	106	63706	-3690
240	30	30	300	621	96	96	59616	-4090
250	25	25	300	641	86	86	55126	-4490

Using 300 Base stats in Defenses, The point where diminishing returns occurs between 130/85/85 and 140/80/80.

There is basically no difference in a split EV's table with 252/124/124. In fact, diminishing returns appears to happen sooner by a small margin.

Code: 
[B]Split EV Table (252/124/124)[/B]
BHP	BDEF	BSD	Total	AHP	Adef	ASD	Total	Diff.
100	100	100	300	404	267	267	107868
	
10	145	145	300	224	357	357	79968	
20	140	140	300	244	347	347	84668	4700
30	135	135	300	264	337	337	88968	4300
40	130	130	300	284	327	327	92868	3900
50	125	125	300	304	317	317	96368	3500
60	120	120	300	324	307	307	99468	3100
70	115	115	300	344	297	297	102168	2700
80	110	110	300	364	287	287	104468	2300
90	105	105	300	384	277	277	106368	1900
100	100	100	300	404	267	267	107868	1500
110	95	95	300	424	257	257	108968	1100
120	90	90	300	444	247	247	109668	700
130	85	85	300	464	237	237	109968	300
[B]140	80	80	300	484	227	227	109868	-100[/B]
150	75	75	300	504	217	217	109368	-500
160	70	70	300	524	207	207	108468	-900
170	65	65	300	544	197	197	107168	-1300
180	60	60	300	564	187	187	105468	-1700
190	55	55	300	584	177	177	103368	-2100
200	50	50	300	604	167	167	100868	-2500
210	45	45	300	624	157	157	97968	-2900
220	40	40	300	644	147	147	94668	-3300
230	35	35	300	664	137	137	90968	-3700
240	30	30	300	684	127	127	86868	-4100
250	25	25	300	704	117	117	82368	-4500

The point of diminishing returns is unchanged. It is still between 130/85/85 and 140/80/80.

Now for a fun experiment. Shuckle has the most base defenses invested of any pokemon, at a staggering 480.

Code: 
[B]Shuckle No EVs Table (0/0/0)[/B]
BHP	BDEF	BSD	Total	AHP	Adef	ASD	Total	Diff.
20	230	230	480	181	496	496	89776
	
10	235	235	480	161	506	506	81466	
20	230	230	480	181	496	496	89776	8310
30	225	225	480	201	486	486	97686	7910
40	220	220	480	221	476	476	105196	7510
50	215	215	480	241	466	466	112306	7110
60	210	210	480	261	456	456	119016	6710
70	205	205	480	281	446	446	125326	6310
80	200	200	480	301	436	436	131236	5910
90	195	195	480	321	426	426	136746	5510
100	190	190	480	341	416	416	141856	5110
110	185	185	480	361	406	406	146566	4710
120	180	180	480	381	396	396	150876	4310
130	175	175	480	401	386	386	154786	3910
140	170	170	480	421	376	376	158296	3510
150	165	165	480	441	366	366	161406	3110
160	160	160	480	461	356	356	164116	2710
170	155	155	480	481	346	346	166426	2310
180	150	150	480	501	336	336	168336	1910
190	145	145	480	521	326	326	169846	1510
200	140	140	480	541	316	316	170956	1110
210	135	135	480	561	306	306	171666	710
220	130	130	480	581	296	296	171976	310
[B]230	125	125	480	601	286	286	171886	-90[/B]
240	120	120	480	621	276	276	171396	-490
250	115	115	480	641	266	266	170506	-890

Shuckle doesn't get diminishing returns on defenses until it hits 230 Base HP, with no EV investment.

Code: 
[B]Shuckle Split EVs Table (252/124/124)[/B]
BHP	BDEF	BSD	Total	AHP	Adef	ASD	Total	Diff.
20	230	230	480	244	527	527	128588
	
10	235	235	480	224	537	537	120288	
20	230	230	480	244	527	527	128588	8300
30	225	225	480	264	517	517	136488	7900
40	220	220	480	284	507	507	143988	7500
50	215	215	480	304	497	497	151088	7100
60	210	210	480	324	487	487	157788	6700
70	205	205	480	344	477	477	164088	6300
80	200	200	480	364	467	467	169988	5900
90	195	195	480	384	457	457	175488	5500
100	190	190	480	404	447	447	180588	5100
110	185	185	480	424	437	437	185288	4700
120	180	180	480	444	427	427	189588	4300
130	175	175	480	464	417	417	193488	3900
140	170	170	480	484	407	407	196988	3500
150	165	165	480	504	397	397	200088	3100
160	160	160	480	524	387	387	202788	2700
170	155	155	480	544	377	377	205088	2300
180	150	150	480	564	367	367	206988	1900
190	145	145	480	584	357	357	208488	1500
200	140	140	480	604	347	347	209588	1100
210	135	135	480	624	337	337	210288	700
220	130	130	480	644	327	327	210588	300
[B]230	125	125	480	664	317	317	210488	-100[/B]
240	120	120	480	684	307	307	209988	-500
250	115	115	480	704	297	297	209088	-900

It's the same number here, but notice the drastic difference in defense numbers between the two tables.

To get a more concrete number, I'll use the two from my experiments in relation to where there defenses are.

These are the diminishing returns numbers for actual stats:

196/421 = 0.608

286/601 = 0.467

These are the numbers that do not yet have diminishing returns:

206/401 = 0.514

296/581 = 0.509

My Hypothesis would be: Whenever you are creating or evaluating a defensive spread with a limited amount of points available for defenses, maximum durability is achieved when actual (not base) defenses are roughly 50% of actual (not base) HP.

In other words, the best overall defensive spread when you have 300 preset points to alocate is (roughly) 140/80/80.

If you wanted the bulkiest possible pokemon with only 200 points to allocate, start with the total Base Stats Vs. Base HP stats provided here.

220/480 = 0.458

140/300 = 0.466

So start your Base HP around 46% of the 200 total, then go up and down the 200 stat total.

200 * 0.46 = 92. Your remaining points are 108, for 54 each.

Chart:

Code: 
[B]Hypothesis Test No EVs[/B]
BHP	BDEF	BSD	Total	AHP	Adef	ASD	Total	Diff.
92	54	54	200	325	144	144	46800
								
100	50	50	200	341	136	136	46376	
98	51	51	200	337	138	138	46506	130
96	52	52	200	333	140	140	46620	114
94	53	53	200	329	142	142	46718	98
92	54	54	200	325	144	144	46800	82
90	55	55	200	321	146	146	46866	66
88	56	56	200	317	148	148	46916	50
86	57	57	200	313	150	150	46950	34
84	58	58	200	309	152	152	46968	18
[B]82	59	59	200	305	154	154	46970	2[/B]
80	60	60	200	301	156	156	46956	-14
78	61	61	200	297	158	158	46926	-30

Our estimate wasn't perfect, but it got it pretty close (HP and defensive point total correlate only roughly). If you had 200 points to allocate on a spread and you wanted the bulkiest pokemon possible, you would use 82/59/59 defenses.

154/305 = 0.504

Including this with our two other examples:

220/480 = 0.458

140/300 = 0.466

82/200 = 0.41

By the way, I know this was tl;dr, but I thought it might be useful. If someone can help me clean it up I'll gladly put it in Contributions + Corrections. Basically all you'd need is an easy way to check Actual HP vs. Actual Defenses with a preset amount of stats to allocate.

Eg:
Total Defensive Base stats to allocate
Find: Actual stats where HP = Defense*2.
Convert into Base stats.
 
I actually had calculated this using calculus 2 days ago... lol.

My result was the following:

If you have n base points to allocate, then the best base stats totalling n with the best defenses is (roughly):

HP: n/2 - 18
Def: n/4 + 9
SpD: n/4 + 9

So, for example, for 300, the best spread would be:

HP: 300/2 - 18 = 132
Def: 300/4 + 9 = 84
SpD: 300/4 + 9 = 84

So 132/84/84 is the best spread.

For 200, the best spread would be:

HP: 200/2 - 18 = 82
Def: 200/4 + 9 = 59
SpD: 200/4 + 9 = 59

So 82/59/59 is the best spread, matching your analysis.

For your Shuckle example (480):

HP: 480 / 2 - 18 = 222
Def: 480 / 4 + 9 = 129
SpD: 480 / 4 + 9 = 129

Sp 222/129/129 would be the best spread.

This is actually a rough estimate, since the best spread actually depends on the EVs you decide to allocate, but it would be the best spread if you allocated no EVs.

I used my normalised base stats to get to my answer by the way.
 
X-Act said:
I actually had calculated this using calculus 2 days ago... lol.
Click to expand...

Darn, I'm still using an abacus, deductive reasoning, and tables LOL.

My result was the following:

If you have n base points to allocate, then the best base stats totalling n with the best defenses is (roughly):

HP: n/2 - 18
Def: n/4 + 9
SpD: n/4 + 9

So, for example, for 300, the best spread would be:

HP: 300/2 - 18 = 132
Def: 300/4 + 9 = 84
SpD: 300/4 + 9 = 84

So 132/84/84 is the best spread.

For 200, the best spread would be:

HP: 200/2 - 18 = 82
Def: 200/4 + 9 = 59
SpD: 200/4 + 9 = 59

So 82/59/59 is the best spread, matching your analysis.

For your Shuckle example (480):

HP: 480 / 2 - 18 = 222
Def: 480 / 4 + 9 = 129
SpD: 480 / 4 + 9 = 129

Sp 222/129/129 would be the best spread.

This is actually a rough estimate, since the best spread actually depends on the EVs you decide to allocate, but it would be the best spread if you allocated no EVs.

I used my normalised base stats to get to my answer by the way.
Click to expand...

Very cool, that. Much better than guestimating and manually dragging down numbers in Excel.
 
Hmm... This is an interesting timing. X-Act did it two days ago, you did it now, and I had just started doing this exact calculation (my method was also going to be calculus-based rather than Excel-based :toast: )
 
Yeah I used this also for the Create-a-Pokemon thread, to show that stat total is irrelevant (I showed that you can create a 515 stat total Pokemon having better defenses than a 520 stat total one).

Also, I seriously doubt that Gamefreak goes into such details when assigning base stats to Pokemon.
 
X-Act said:
Also, I seriously doubt that Gamefreak goes into such details when assigning base stats to Pokemon.
Click to expand...
Which is a damn shame.

Its all very interesting, I love reading analysis like this. Things like these make me want to try and fully optimize my pokemons EV spreads and movesets. Its what draws me back to pokemon every time.
 
Hah, I was just going to do this.
I mentioned in the C-A-P thread that you get a greater return on BST by going higher HP than defenses and was going to find the correct proportion and here it is.
 
However, this is only when we focus on defensive capabilities. If, for example, we wanted to assign 252 EVs on an attacking stat (or a mix of Speed and an attacking stat), the importance of the HP base stat diminishes greatly. So, yeah, if you want a pure mixed wall, this is the way, but if you want something bulky that can also dish out good damage, a lower HP stat and higher defenses (relatively, HP should still be more important) would be the way to go.
 
Time Mage said:
However, this is only when we focus on defensive capabilities. If, for example, we wanted to assign 252 EVs on an attacking stat (or a mix of Speed and an attacking stat), the importance of the HP base stat diminishes greatly. So, yeah, if you want a pure mixed wall, this is the way, but if you want something bulky that can also dish out good damage, a lower HP stat and higher defenses (relatively, HP should still be more important) would be the way to go.
Click to expand...

The only advantage the table has over X-Acts Calculus(if any) is that you can do the same thing with a 252 HP assumption.

So using the same 200 base point spread:

Code: 
Hypothesis Test 252 HP (252/0/0)								
BHP	BDEF	BSD	Total	AHP	Adef	ASD	Total	Diff.
92	54	54	200	388	144	144	55872	
								
100	50	50	200	404	136	136	54944	
98	51	51	200	400	138	138	55200	256
96	52	52	200	396	140	140	55440	240
94	53	53	200	392	142	142	55664	224
92	54	54	200	388	144	144	55872	208
90	55	55	200	384	146	146	56064	192
88	56	56	200	380	148	148	56240	176
86	57	57	200	376	150	150	56400	160
84	58	58	200	372	152	152	56544	144
82	59	59	200	368	154	154	56672	128
80	60	60	200	364	156	156	56784	112
78	61	61	200	360	158	158	56880	96
76	62	62	200	356	160	160	56960	80
74	63	63	200	352	162	162	57024	64
72	64	64	200	348	164	164	57072	48
70	65	65	200	344	166	166	57104	32
[B]68	66	66	200	340	168	168	57120	16
66	67	67	200	336	170	170	57120	0[/B]
64	68	68	200	332	172	172	57104	-16
62	69	69	200	328	174	174	57072	-32

So either of these spreads works with max HP.

Note the difference between total defenses here and total defenses for no EVs:

Code: 
No EVs
82	59	59	200	305	154	154	[U]46970[/U]	2

[B]252 HP EVs[/B]
68	66	66	200	340	168	168	[U]57120[/U]	16
66	67	67	200	336	170	170	[U]57120[/U]	0

So basically if you really want to max HP, go right ahead, but it only leaves you 252 EVs to alocate elsewhere.


X-Acts no EV formula works if you want to find out what the most efficent defenses are for natural 101 HP Subs.

n/2 - 18 = 134; n/2 = 152 n = 304
304/4 + 9 = 76
304/4 + 9 = 76
 
I thought the pokemon community had been aware of: "Whenever you are creating or evaluating a defensive spread with a limited amount of points available for defenses, maximum durability is achieved when actual (not base) defenses are roughly 50% of actual (not base) HP." for a while. Which is essentially the only part of this that relates to competitive battling.
 
It's not my calculus LOL. It is Newton and Leibnitz's.

husk, this is important, if anything, to compare different defensive stats and see whichever is better.
 
It helps a lot if you're creating the perfect defensive theorymon with a limited BST. Outside of that, it's not very helpful. EVs can already be calculated in a number of ways, namely breakpoints to avoid metagame 2HKOs.
 
But these calculations would only apply to neutral on def and sdef builds wouldnt they?

EDIT: Nature-wise, i mean.
 
X-Act said:
It's not my calculus LOL. It is Newton and Leibnitz's.

husk, this is important, if anything, to compare different defensive stats and see whichever is better.
Click to expand...

I was referring to it more affectionately as your equations, I know you didn't come up with the theories behind calculus or the proofs.

Unless of course you managed to break Einstein's theory of relativity, thus enabling you to travel back in time and thereby supplanting all previous mathematical theorists, doing potentially untold damage to the original timestream...

Err, yeah, Pokemon, right.
 
Its kind of funny how pokemon is so complicated that you can emerge mathematical representations out of it but it is still being generally viewed as child play.

Pokemon>>Chess.
 
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