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# Measure of diversity of a metagame

Discussion in 'Pokémetrics' started by X-Act, Dec 15, 2008.

1. ### X-Actnp: Biffy Clyro - Shock Shock

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As you might know, I was researching into a method of finding the centralisation of a metagame as a number. Refer to these two threads.

At the conclusion of the former thread, I finally found a good measurement, though not for centralisation, but for diversity, which can be understood as being inversely proportional to centralisation. The method of measurement is repeated here.

First, we convert each Pokemon usage U_i in descending order to the probability of it being in a team. This is done by multiplying it by 6 and dividing it by the sum of all usages U. This number, incidentally, is related to how much that Pokemon is centralising in that metagame; the higher it is, the more centralising is that Pokemon. In fact, we can define it as being the centralisation of that particular Pokemon in that metagame.

Then start summing up all these numbers until the answer first exceeds 5, and count all the numbers summed up. This would be the diversity D of the metagame.

From the definition above, it is clear that the lowest value D can have is 6 (if all teams are the same), and the highest value it can be is one more than 5/6 of the number of Pokemon in that metagame (if all usages are the same). Thus, we can find the percentage diversity which gives us the level of diversity of that metagame compared to the number of Pokemon it contains. We basically would like this percentage to be 0% if D=6 and to be 100% if D = 1 + 5P/6, where P is the number of Pokemon in that metagame. This is achieved by:

Code:
`Percentage Diversity PD = 120 x (D-6) / (P-6)`
This assumes that a metagame has more than 6 possible Pokemon to choose from. In a metagame having 6 Pokemon, the measure of diversity would be meaningless, as every team will be forced to contain exactly those 6 Pokemon. Hence the (P-6) in the denominator poses no trouble whatsoever.

The above measurement isn't very important from the actual diversity's point of view; it is only interesting as a way to compare different metagames.

So, to summarise:

Code:
```C_i = 6 x U_i / U

D is the smallest number of C_i's that, when summed up together, the answer is greater than 5.

PD = 120 x (D-6) / (P-6)

where C_i is the centralisation of Pokemon i
U_i is the usage of Pokemon i
U is the sum of all Pokemon usages
D is the diversity of a metagame
PD is the percentage diversity of a metagame
P is the number of Pokemon in the metagame```
So we don't have a formula for centralisation, but we have one for diversity. :)

Here is a table containing the diversity of each metagame from July to November:

Code:
```    Ladder         Diversity
Jul  Aug  Sep  Oct  Nov
------------------------------------
Standard   60   63   67   62   66
UU   52   61   65   58   55
Uber   22   25   27   23   24
Suspect   --   54   52   41   --
CAP   --   --   59   --   51
Little Cup   --   --   27   --   30```
And this is a table containing the percentage diversities, which is less important than the above, but is interesting to find the percentage of diversity with respect to the number of Pokemon in that metagame:

Code:
```    Ladder              Percentage  Diversity
Jul      Aug      Sep      Oct       Nov
---------------------------------------------------------
Standard   13.76%   14.52%   15.38%   14.12%    15.16%
UU   20.60%   24.63%   26.42%   23.28%    21.94%
Uber    3.91%    4.64%    5.06%    4.10%     4.34%
Suspect     ---    12.28%   11.62%    8.82%      ---
CAP     ---      ---    13.25%     ---     11.25%
Little Cup     ---      ---    15.85%     ---     18.11%```
2. ### Archer

Joined:
Dec 22, 2007
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It's interesting to see that the Metagame has become slightly more diverse during November. Of course the Skymin ban should increase it for December.

I can't see a way to calculate centralisation when it takes logic to see what is in the metagame as a result of another pokemon. You'll probably find a way, though. This is interesting, though.
3. ### X-Actnp: Biffy Clyro - Shock Shock

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Well, isn't a metagame with a lot of diversity one with not much centralisation and vice-versa?
4. ### SHUCKLE MAN

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Would we be able to use something like this to determine when we should stop banning Pokemon when we test UU?

Could we say, that we ban Pokemon to BL until we get a metagame that has a diversity of 60 or higher, or something?

I think that would be good, to stop the community banning 50 Pokemon. People need to get in the mindset that we're banning as little as possible, if we're very lucky, we won't have to ban anything at all!
5. ### Yashichi

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Mar 6, 2007
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Not necessarily. Diversity describes the number of Pokemon commonly used, whereas centralization describes the degree to which other Pokemon are excluded from use. A metagame with 100000 Pokemon allowed where 500 are commonly used is very diverse, but also very centralized. Similarly, a metagame where ten Pokemon are allowed and nine are commonly used is not very diverse, but not very centralized.
6. ### X-Actnp: Biffy Clyro - Shock Shock

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Your definition of centralisation is inversely proportional to my definition of 'percentage diversity'.

Look at the original post. You'll see that even though UU is less diverse than OU (due to having less Pokemon to choose from), its percentage diversity is higher, since 55 out of 274 Pokemon is higher than 66 out of 481 Pokemon.

To SHUCKLE MAN, I know what you're talking about, but finding a cut-off point where we say that a Pokemon would be banned is difficult.
7. ### gorm

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Oct 4, 2006
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hah, a statistical definition of uber, the holy grail of pokemon x__x

we're getting there but the problem is still "too noisy" lol.

and by "we" i mean "mostly x-act". this is amazing work man, not much to say. i read through the formula briefly and it makes a good deal of sense. seems like all that's left is "mapping" and "defining an "eccentricity" that would be considered above the "uber threshold". finding that can probably done with a set of "obvious tests" with ubers in standard.

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Odd how October had the Diversity decreased and such, though recalling off the top of my head this was when Platinum stuff was released and Deoxys-S was still around.

UU is really scattered around. >_>
9. ### gorm

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Oct 4, 2006
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yeah it's interesting how uu vs ou is a clear indicator of "relatively balanced metagames are stabler with more people" just as predicted by the formula =)
10. ### SHUCKLE MAN

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Apr 26, 2006
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Yeah, that's true I guess. Hopefully we'll find a way to stop ourselves going overboard with the bannings (I have no idea how we'll do this though).

The percentage diversities are quite interesting. The one for Ubers makes me 'lol'.

And the diversities overlapped for OU and UU which surprised me. And I don't know what made September so diverse in UU.

Anyway, thanks for the stats X-Act!
11. ### Yashichi

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Mar 6, 2007
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I apologize, my last post was of the "Yes, but technically no" variety. What I meant to say is that you're mistaken about the importance of the D value. A D-value alone cannot give any information about centralization. The PD-value is much more useful to determine centralization.

Say we have some metagame M in which there are 12 Pokemon (P), each of which has equal usage. So the D-value of this metagame is equal to 11. This metagame is obviously not centralized, since all of the allowed Pokemon are equally used. Now say we add 988 Pokemon to M, but none of them are used, for whatever reason. D remains the same, but M is obviously more centralized since more Pokemon are being excluded. D is an absolute measure, independent of the number of allowed Pokemon, but centralization (C) is not. The PD-value also depends on the number of allowed Pokemon, so it can be treated as the inverse of C. For some constant diversity, as the number of Pokemon increases, the percentage diversity will decrease and the centralization will increase.

The problem comes down to this: do we want to minimize centralization (and thus maximize percentage diversity), or do we want to maximize diversity? Many users seem to use these interchangeably, but they are subtly different. Since maximizing percentage diversity requires banning a large number of Pokemon (with the very percentage-diverse UU as my example), I take it that we are attempting to maximize diversity, which is consistent with banning the most centralizing Pokemon. This renders the concept of centralization rather moot, and we should use diversity instead.
12. ### wildfire393

Joined:
Oct 3, 2008
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To statistically define "uber", we'd have to look at a couple things:

What percentage of the "centralization" is provided by the Pokemon in question and it's specific counters
How centralized/diverse is the metagame with the Pokemon in question compared to the metagame without the Pokemon in question.
13. ### X-Actnp: Biffy Clyro - Shock Shock

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To be honest, I'm a bit divided on which measurement is the more important.

Some people, like eric the espeon, Obi and Hipmonlee stress that diversity is more important than percentage diversity. Their point is that having 50 Pokemon out of 50 Pokemon being used is the same as having 50 Pokemon out of 400 being used. Their point is that the diversity is the same.

Others, like you, argue that you need to take the number of Pokemon into account, since the former scenario has 100% diversity while the other has much lower diversity.

In view of this, I decided to work out both and let the community sort this dilemma out. :)
14. ### X-Actnp: Biffy Clyro - Shock Shock

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The diversity definition can be extended as follows to provide a better idea of how the teams are diversified.

Instead of summing up until we reach 5, we sum up until we reach 1, then 2, then 3, then 4 and then 5.

This gives us the number of Pokemon that are typically among 2, 3, 4, 5 and 6 of the six in a team respectively.

For example, the diversity of the November Standard metagame can be written as 5+7+10+15+29 (=66). This means that a typical team has two of the top 5 Pokemon in the list, has three of the top 12 (5+7) Pokemon, has four of the top 22 (5+7+10) Pokemon, has five of the top 37 (5+7+10+15) Pokemon and has six of the top 66 (5+7+10+15+29) Pokemon.

For Uber, the diversity in November is 3+2+3+5+11 (=24). A typical Uber team has two of the top 3 Pokemon, three of the top 5, four of the top 8, five of the top 13 and six of the top 24.

For UU, the diversity in November is 4+6+10+14+21 (=55). A typical UU team has two of the top 4 Pokemon, three of the top 10, four of the top 20, five of the top 34 and six of the top 55.

This shows the diversity in a fraction of a team.

The Pokemon can be listed in between square brackets, where the ones that are nested the most inside square brackets are the ones that are more likely to be in a team.

For example. for Standard in November, we have:

Code:
`[[[[[Scizor Heatran Salamence Tyranitar Zapdos] Blissey Gyarados Lucario Infernape Gengar Metagross Swampert] Bronzong Azelf Celebi Vaporeon Shaymin-s Starmie Skarmory Suicune Kingdra Machamp] Jirachi Gliscor Togekiss Rotom-h Snorlax Weavile Forretress Electivire Mamoswine Breloom Hippowdon Tentacruel Heracross Magnezone Aerodactyl] Rhyperior Flygon Dusknoir Cresselia Dragonite Jolteon Porygonz Dugtrio Rotom-w Ninjask Alakazam Donphan Empoleon Smeargle Gallade Milotic Umbreon Yanmega Porygon2 Roserade Ludicolo Spiritomb Ambipom Abomasnow Crobat Hitmontop Charizard Slowbro Weezing]`
For Ubers in November:

Code:
`[[[[[Kyogre Rayquaza Groudon] Darkrai Dialga] Mewtwo Palkia Lugia] Scizor Deoxys-S Deoxys-A Giratina Blissey] Latias Mew Metagross Garchomp Bronzong Tyranitar Giratina-O Forretress Ho-oh Manaphy Shedinja]`
For UU in November:

Code:
`[[[[[Hitmontop Clefable Claydol Rotom] Steelix Ninetales Froslass Venusaur Absol Weezing] Drifblim Lanturn Altaria Toxicroak Mantine Hitmonlee Nidoking Swellow Blastoise Hypno] Leafeon Glaceon Drapion Hitmonchan Sharpedo Scyther Primeape Miltank Kabutops Lapras Poliwrath Manectric Jumpluff Probopass] Nidoqueen Camerupt Lopunny Electrode Articuno Kangaskhan Persian Meganium Venomoth Linoone Jynx Grumpig Golduck Omastar Shedinja Kingler Gorebyss Banette Relicanth Flareon Quagsire]`
15. ### Luxormaniac

Joined:
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Can someone pm me a link or the actual usage stats for standard in July-November? I'd like to calculate centralization using my old idea (Sum of top ~10 usage percentages). It'd be cool to post it here once its done, but i could make my own thread.

Another idea: Measure the Individual-centralization.

????

There should be a way to measure how much one pokemon centralizes the metagame.
To do this would involve adding usage percentages up for this pokemon and its most common counters. For example, garchomp when it was unbanned would have a high individual centralization because garchomp+its counters formed a considerable percentage of the metagame.
Although this might be too close to the overall centralization to matter.
If this is true, though, perhaps it could be applied to less commonly used pokemon.

Alternative centralization quantification:
Take 10 experienced panelists, who you know to be effective, and have them all rate the metagame 1-10 and take the average.
Simplest is always best!
16. ### X-Actnp: Biffy Clyro - Shock Shock

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This would be a decent measure of centralisation - in fact, the diversity measurement mentioned in the original post uses the same idea!
17. ### Luxormaniac

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I've read all these threads, so I was probably there. Or maybe i posted it, I don't remember.

So... do you have that data? Or are you going to do it without giving anyone that link at all so you can get the credit?

jk X-act do it if you want. Recognition is nice, but laziness is its own reward.
18. ### X-Actnp: Biffy Clyro - Shock Shock

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I did it already, together with a ton of other ideas that came to my mind. I'm not using it because I wasn't satisfied with how the measurement turned out. It is a decent approximation of centralisation, but that's all.

I'm standing by my (extended) definition of diversity. It makes sense mathematically and makes sense Pokemon-wise.
19. ### Luxormaniac

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Dec 1, 2008
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isnt a decent approximation of centralization all we're going for? anyway, your the expert. May the Force be with you.