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# Measure of centralisation in a metagame

Discussion in 'Pokémetrics' started by X-Act, Dec 10, 2008.

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

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Here is a simple way to have a clue as to how centralised is a particular metagame.

As you noticed from my recent stats, I wasn't just providing the list of Pokemon that were OU, but also a list of Pokemon that were 'frequently used'. Both of these are used to check for the centralisation of a metagame.

To remind you, a Pokemon is OU if its usage is greater than the sum of all usages divided by 176.1407. A Pokemon is frequently used if its usage is greater than the sum of all usages divided by 37.7113.

Count the number of Pokemon that are OU; this is equal to O. Then count the number of Pokemon that are frequently used; this is equal to F.

The measure of centralisation is then this very simple formula:

Code:
Central = floor(500 / (O - F))
Here is the centralisation of various metagames from September onwards:

It looks like the OU metagame and the UU metagame have low centralisation, with UU having slightly less centralisation than OU. Uber has a lot of centralisation. But this is something that you all know about. :)

What is interesting is the Suspect ladder metagame centralisation. For September, it was only slightly more centralised than for OU, while for October it was much more centralised. The reasons for this are not entirely clear to me.

I thought to share this piece of information with all of you. :)
2. ### Matt_Lane

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Thanks - yet again - X-Act!
3. ### cimhappiness is such hard work

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I think that, although useful, we don't know how to apply the word "centralized" to an Uber argument. Most people use "how much impact does it have on my team" as "centralization", and I don't know if this numerical definition or the subjective one should have more weight.

This is part of why I wanted Rayquaza test, so we can see these numbers in an obviously "one Uber" metagame.
4. ### Ancien Régimecapitalism delenda est

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Ubers isn't supposed to be balanced so "Centralization" is wholly irrelevant.
5. ### Pombo

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Thanks again, but... tell me something: whats the university you made? You got some talent/good teachers/both.
6. ### X-Actnp: Biffy Clyro - Shock Shock

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Well, if a metagame containing "OU + Rayquaza" would have more centralisation than "OU only", then this formula should show that. Whether or not a centralising Pokemon condemns it to uber or not is a matter of debate, though.

Of course, but knowing that the uber metagame is very centralised is a reason why it's a good test case to show that the formula works.
7. ### Caelumqibz official stalker

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Where did the 500 come from?
8. ### cimhappiness is such hard work

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I'm talking about Suspects, not the Uber tier.
9. ### AeroGP

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493 + 5 formes of Rotom + Shaymin-S + Giratina Origin = 500.
10. ### devoninja

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+ Deoxys forms =/= 500

More likely it's an arbitrary number which only affects how big the final numbers are without having any statistical meaning itself.

If you had 5000 instead all the results would be 10x bigger but would mean the same thing.

For clarification, in this instance 'OU' does not mean "banned from UU but not from Ubers" rather it refers to pokemon which are Overused within their tier, such that there would be a number of 'OU' pokemon for the UU tier, correct?
11. ### Evil Hamster

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It if was that, 500 would only apply to the uber tier, it would have to be reduced for the other tiers to accomodate all the Pokemon banned in the tier. It's probably just an arbitrary number to make the statistics more sensible numbers as opposed to tiny fractions.
12. ### CardsOfTheHeart

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The actual number I think we're looking for is 505, but we've already established that it's not important. The focus is on the relative centralization between the tiers--particularly between Uber and the other tiers.

EDIT: 505 = 493 + 3 alternate Deoxys forms + 2 alternate Wormadam forms + 5 alternate Rotom forms + 1 Giratina-O + 1 Shaymin-S
13. ### X-Actnp: Biffy Clyro - Shock Shock

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To answer how the 500 came to be, I need to explain how the whole formula came to be. Feel free to skip reading this post if you're not interesting in how the formula was created, as it is rather long.

For me, it seemed obvious enough that a centralised metagame should have a high F and a low O, while a metagame that is not very centralised should have a low F and a high O. (Remember: O is the number of OU Pokemon while F is the number of frequently used Pokemon.) For example, in one case, a particular Standard metagame had F=6 and O=50, while a particular uber metagame had F=14 and O=28. As you can see, the Standard metagame has a low F and a high O, while the uber metagame has a high F and a low O. However, intuition isn't always right, so I kept my intuition to myself until I could prove that it is correct.

To test my intuition, I decided to find the number of Pokemon that are used in 1 out of every T = 1, 2, 3, 4, etc. teams, up to 20. (Remember: a Pokemon is frequently used if it is in one out of every 4 teams, while a Pokemon is OU if it is in one out of every 20 teams.) Then I plotted these numbers against T and compared different graphs obtained from different metagames.

I noticed that the graphs for the standard and UU metagames were very similar except that the UU graph was slightly steeper, while the uber metagame graphs were much less steep. Since 'steepness' is determined by gradient, I thought "so centralisation is related to the gradient of this graph; the higher the gradient, the less centralised the metagame is".

So what I did at first is to plot a trendline of these points and find the gradient of this trendline. However, this whole process seemed to be way too complicated for the average Smogoner to understand, and the teacher in me decided that this process, although sound, needs to be simplified. :)

Then I remembered that I had already mentioned that frequently-used Pokemon are those used in 1 out of every 4 teams (i.e. the point where T=4). So I connected a line between this point and the 'OU' point (i.e. the point where T=20) and found that the gradient of this line is very near that of the trendline. Thus I found a much simpler way of finding a very good approximation of the trendline gradient.

There was, however, a problem: the higher the gradient, the LESS is the centralisation, and vice-versa... which means that the gradient is inversely proportional to the centralisation. This problem was solved easily by taking the reciprocal of the gradient as being the measure I wanted. Alternatively, you could argue that I found the difference in x divided by the difference in y instead of the other way round.

Okay, so I had a point at (4,F) and another one at (20,O). The difference in x is 16, while difference in y is (O-F). Thus, my initial measure of centralisation was 16 / (O-F). I soon realised that this number is too small for practical purposes, however, so I just multiplied the number by a constant. After testing a few numbers, I settled for a constant equal to around 30 to 32, and since 30x16 = 480 and 32x16 = 512, I decided to use 500 as the constant. Finally, I just removed the numbers after the decimal point so that the measure is a neater number, and the formula floor(500 / (O-F)) resulted.

The fact that (O-F) is in the final formula confirms that a centralised metagame has a low O and a high F, while a non-centralised metagame has a high O and a low F. If O is low and F is high, then O-F is a small number, and hence 500 / (O-F) is high. If O is high and F is low, the O-F is a large number, and hence 500 / (O-F) is small.
14. ### Caelumqibz official stalker

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Thanks X-Act. I initially assumed you did a method similar to the one you described (the form of the equation was a bit of a tip off) but I just couldn't figure out how you'd get 500, didn't think about multiplying by a constant :nerd:. Nice work as always.

(btw; for American high school students here; Gradient = Slope).
15. ### Chou ToshioOver9000

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Nice Job X-Act. If you have a measure of centralization, I forgot that if you track it month by month we can see how stable things are/aren't and that goes a long way towards making an opinion about the centralization. Awesome Job. This is greatly appreciated.

We can see here that despite all the weird tests we do, OU is pretty damn solid. UU varies in centralization levels more! Then again there are fewer players (less data) so that makes sense.
16. ### Bourbon

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Can you do this for pre september, like for all the months when Garchomp was unbanned, so we can see what sort of effect he had compared to now?
17. ### cimhappiness is such hard work

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Less than you'll think, Skiddle. He didn't really limit the pool of usable POkémon, just changed it. If I recall correctly.
18. ### wildfire393

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It would be interesting to see some variations on that formula to check for few different things. For example, testing for weighting on O and testing for weighting on F. Like 500/(2O-F) and 500/(F-2O). This could show us whether the standard metagame has been centralized "the same amount" due to having a near identical number of Overused and Frequently Use Pokemon in the metagames or some other factor.
19. ### jamespicone

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I'd like to see the metagame where all frequently used pokemon are OU - it's infinitely centralised!
20. ### Pisti

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It would be 507 in that case, the 2 alternate Burmy forms.

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Actually, one should expect to have more centralization in UU with fewer players, just like Suspect, no?
22. ### CardsOfTheHeart

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Wrong. The only difference between the Burmy forms is sprites; there's nothing functionally different about them in battle, thus they are still considered to be one Pokemon. Besides, if we counted them, then we would also have to count Shellos's and Gastrodon's East Sea form.

*end off-topic chatter*

I just realized I haven't complemented you for this. Great work as always, X-Act!
23. ### X-Actnp: Biffy Clyro - Shock Shock

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No. The amount of players in a metagame does not affect the centralisation of that metagame in any way.

What affects the centralisation of the metagame is what the players use. If there are 1 million players using exactly the same team, then obviously that metagame is ultra-centralised (F=O=6 in that case, so Central would be infinity, lol). Conversely, if there are 100 players using teams having very little in common, then the metagame is very non-centralised (F would be close to 0 and O would be a large number, hence 500 / (O-F) would be very small).

Suspect is inherently more centralised than Standard because there are many players who try new things on the Standard ladder, which decentralises the ladder. In Suspect, people have much less time to do this, as they're vying for a 1655/65 rating, so they tend to use their tried-and-tested teams instead of innovative teams.
24. ### jamespicone

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Extremely small numbers of people could influence centralisation, though - if only two people play UU, and don't change their team mid-month, only 12 pokemon at max have been used in the ladder - and all 12 of them are OU and frequently used, because they're in half of the teams.

Basically, the less players there are, the less scope there is for different pokemon to be used - but that effect, I imagine, only exists for very small playerbases.
25. ### eric the espeonmaybe I just misunderstood

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That effect can be seen quite clearly in Septembers LC stats which had 11 Frequently used Pokemon but only 32 OUs. This was probably at least partly due to the fact that only 321 battles took place in that time.