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Let us go back about one year. Our now beautiful strategy forum was poluted beyond all legal restriction with Z>>>>P threads, and we were happy to see more than a handful of protoss in the topp 30 KESPA ranking. The zerg was not dominating though. The top 30 KESPA had as many terrans as zergs, if not more terrans. This seemed a bit strange to me at first: if zerg is the privileged race when it comes to imbalance, how come there are not more zerg players in the top? Having done my fair share of mathematics in my studies, I began to create a model of races in progaming.
I have now developed a mathematical model that explains how a Z>>P imbalance, together with smaller T>Z, P>T imbalances, causes a TERRAN dominance.
I will here explain it to those who are interested in three steps of increasing mathematical difficulty. If you find it too hard, then just skip ahead to the conclusions at the bottom, they are made so that you do not need to have read everything to be understandable.
[I realise that this is a loooong post, but I feel that I have found, and proved to some extent, something that can increase the understanding of progaming.]
Basic principle
First thing is to assume that ZvP is imbalanced in zergs favour by quite a bit, while TvZ favours T slightly, and PvT favours P slightly. If this is true or not is NOT the topic of this thread. instead assume that that is the case, and see what happens. I will later leave the balancing of the three matchups free to choose, see conclusions.
The principle is that zerg make its prey go extinct. As the zergs kick out all protosses from progaming, they will no longer have any easy wins. Instead, there will be loads of zergs which the terran can abuse due to the slight imbalance in terrans favour in the TvZ matchup. Also there will be very few protoss that can keep the terran numbers down. So as the zerg finish of all the protosses, they will find themselves owned by the terran that get free reign by the absence of protoss.
Easy right? let us now make a first model just to formalise.
First Toy Model
+ Show Spoiler +This model is not very realistic, but it will show the effect introduced above. The model builds on these asumptions: 1)There is a large pool of potential progamers of roughly equal skill. 2)To become a progamer you need to have an overall win probability of 50% or more, averaged over all other progamers. 3)The matchups are imbalanced like this: Z beats P in 70% of the games. P beats T in 60% of the games. T beats Z in 60% of the games. These asumptions are not very accurate, but it is enough to show the point. Name the fraction of Terran progamers t, the fraction of zerg progamrs z, and the fraction of protoss progamers p. z,t,p are numbers between 0 and 100%, and they need to add up to 100% together, to account for all progamers. z + t + p = 100% To have a stable situation, each race must win with 50% probability. Otherwise one of the races will be at the loosing end with a win statistics of below 50%, and will be kicked out according to 2). You cant have one race winning more than 50% without haveing another loosing more than 50%.  let us now try to calculate z,p and t. for that we will look at one race at a time. Terran will of course win 50% against other terran and win 60% agianst Z, but only win 40% against P. For that to add up to a total of 50% we need to have as many protoss as zerg progamers. If there are more protoss, then terran will win less than 50%, and if there are more zerg, then terran will win more than 50%. So z=p Now look at protoss. They win 50% against other P, win 60% against terran, but only 30% against zerg. For those to add up to a total of 50% we need to have double as many terran as zerg to compensate for the heavy ZvP imbalance. So we get here t=2z Looking at zerg now. They win 70% against protoss and win 40% against terran. To add up to 50% we need double number of terrans as protoss: t=2p Solving these three equations (four equations with z+t+p=1) we now find: Z = 25% t = 50% p = 25% This means that 50% of the progamers will be terran players! Of the rest 25% will play toss and 25% zerg. So terran dominates completely in this scenario! It is not even more zergs than protosses! This clearly demonstrates th principle that zerg suffers from exterminating their "food" protoss. It even overstimates it. Zerg doesnt do THAT bad in the real starleagues. This concludes the Toy model. let us now move on to the real model and see how it does.
Cascades progaming model
I will here move a bit faster and not explain the details I find too trivial. If you do not follow, ask in a reply, or skip ahead to the conclusions.
The model is based on these asumptions:
1) Progamers have different skill levels s. The number of progamers at a given skill s is proportional to
e^(-a s)
That is the the number of players falls of exponentially as skill increases. "a" is a parameter that decides how quickly the number of gamers fall. A large "a" means that the very best playes in the worlds is not THAT much better that the ones ranked around 100. A small "a" means that the top players completely own lower ranked players, even if they are not very mucher lower ranked.
This distribution can be discussed. Other ideas are welcome.
2) The probability of a player of skill s1 to beat a player of skill s2 is
1/( 1 + e^(s2-s1) )
the winning probability as function of skill difference:
This probability has the properties one could expect from a winning probability (such as the probability for the other playing winning is 1 - your winning prob). I do not need to add a parameter in the exponential as it essencially would be the same parameter as "a" above.
3) the worst progamer of each race must have the same winning probability averaged over all other progamers.
this is very intuitive. To barely make it as a progamer you need to win at least maybe 40% of your games. Any less, and you will be kicked out. I am not saying where that limit is, but the limit is the same for all races. If a terran gets kicked out at 39%, then a zerg will too.
4) Racial imbalance is represented by giving the favoured race a skill bonus.
So for example if a zerg player of skill 4 plays a protoss of skill 5, the zerg will get a bonus, for example 3, and the probability of the zerg winning will be calculated in 2) as if the zerg had a skill level of 4 + 3 = 7. I will not here fix how imbalanced the matchups are, but leave them for later. Also this is a fairly reasonable asumption in my opinion.
Solving the model
To find the race fractions z,t,p we need to do the following:
1) set up an equation describing the average winning probability of the worst player of each race. call these Wz, Wt and Wp. Each of these will be a function of all the z,t and p.
2) Solve Wz(z,t,p)=Wt(z,t,p)=wp(z,t,p) for z,t and p with help of z + t + p = 1.
I have done this.  
With help from the brilliant program Mathematica, but still. to give a hint of how complicated it was, and for scientific credibility I'll show you my code.
+ Show Spoiler +Note that you can see the expression for Wp. it is a three line function involving hypergeometric functions and Poly gammas. I dont even know what a polygamma is, but mathematica handled them.  You can also see that I solved the equations numerically.
To make it more accissible for non-mathematicians I have reformulated the skill bonuses and "a" in paramteers that are more intuitive, like average ZvP winning %.
Conclusions, go here if you do not understand!!
I have now found a way of calculating how many zerg, terran and protoss progamers there is. To do this I use how imbalanced the three matchups are, and how big the skill difference is between top and bottom progamers.
That is:
I have a program on my computer that I've written. You give the program 4 numbers:
1) Average probability to win a ZvP
2) Average probability to win a TvZ
3) Average probability to win a PvT
4) The probability that the 5:th best player in the world beats the 50:th best player in the world.
With these four number, the program will calculate how big % of all progamers will play zerg, terran and protoss.
As an example I will plug in typical value for about a year ago. At least this is what I guess is typical values. The beauty of this is that I can try any values I want. Anyways, using this estimate of imbalance and skill:
ZvP wins 70%.
TvZ wins 60%.
PvT wins 60%.
the 5:th best player beats the 50:th best player with 65% probability.
Then the program calculates that:
35.8% of all progamers will play Zerg.
42.2% of all progamers will play Terran.
22.0% of all progamers will play Protoss.
As these values are close to the reality a year ago, it hints that my model is working! I tried a few other imbalances and skill-differences with the following result:
Note the second last row: Even when ZvP is just SLIGHTLY more imbalanced than the other two matchups, protoss will be suppresed, and terran will dominate.
I invite all of you to give me sets of these four paramters, and I will calculate z,p,t four your parameters. You dream can come true! You can finally get an answer to that "what if". what if ZvP was balanced? what if terran would roll over toss as well? What if top ten would dominate everyone else with 80-90% statistics?
these results could also be of interest for korean mapmakers. Im imagine they are interested in having equally many of each race (Who wants to whatch mirror matchups?). If they need to get rid of the terran dominance, maybe they should make sure to balance the maps in ZvP primarily? Not what they would have guessed.
Thanks for reading.
Lollipops for those that read eveything!
+ Show Spoiler +
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I read through only about a fraction of this and was enjoying myself will finish later nice analysis.
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Very interesting work man. Re-reading it to make sure I undestand your post in its entirety.
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Pretty interesting model. I read everything except for some of the Mathematica code. T_T (school has Maple licensed instead lawl, not like I know that level of statistics anyway)
I'm just not quite sure that the imbalance was on the order of 70-30. The last row in the chart is probably more like it was.
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Dang
really nice work about the matchup balanaces
i got so fing confused aroudn the mathematica part lol D:
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Wow, this is great. I didn't understand the more complex mathematics, of course, but the mere fact that maths could be used to calculate something like this blows me away. I have never been fond of maths, and never saw its applications in real life. Tahnks to people like you, I'm beginning to change my opinions.
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This is quite interesting, good work.
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wow very nice work, i was kinda confused on some points on the first read but im sure ill get it on the second read
this model of the terran dominance only applies to tournament games tho right?
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lol i dunno dude....u cant generalize that pvz is imba z for that much (70% chance of winning) off my own experience i beat zergs more than i beat terrans
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On November 03 2007 11:44 CustomXSpunjah wrote:
lol i dunno dude....u cant generalize that pvz is imba z for that much (70% chance of winning) off my own experience i beat zergs more than i beat terrans
Reread the first few paragraphs and the last paragraph. This thing is not based on 100% true facts. He is making an assumption purely for the basis of his math. He is not saying it IS the case.
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but his calculations are off progamer(which tend to have closer skill levels from one another) matchs not public games.. too much skill variation so it becomes unpredictable
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I think you're right, but I suspect the best way to use this would be to invert what you're doing. Create a formula where you can plug the number of T, Z and P progamers (say the tope 30 or 50 progamers) in as variables and see if it accurately predicts the winning percentage of each race in each matchup. This way you can determine racial imbalance without making any generalized assumptions. If you do it this way you can determine, through observance of variance from the mean, to what extent map imbalance and personal skill affects the likely outcomes of games.
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very interesting, nicely done!
it might be fun to add another layer into your model for per-map skill bonuses. say add another layer where the maps would be selected from a discrete uniform distribution. then each map has its own values for skill imbalances.
then you could answer questions that might be of even MORE interest to tournament organizers and mapmakers. e.g. what is the expected effect on racial distribution of having a mercury (z >>> p) or a paradoxxx (p >>> z) in the map pool? or given 4 maps in my map pool, what remaining 5th map should i pick to maximize racial balance?
edit: one more comment, for picking your 'a' parameter, maybe you should look into fitting it to data of the winning percentages of some players (say from TLPD's ELO top players). that would be more convincing, since if you can pick 'a' arbitrarily, it seems relatively easy for you to pick a value that matches last year's player distributions.
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Thanks for the feedback everyone.
On November 03 2007 11:19 Myrmidon wrote:
Pretty interesting model. I read everything except for some of the Mathematica code. T_T (school has Maple licensed instead lawl, not like I know that level of statistics anyway)
I'm just not quite sure that the imbalance was on the order of 70-30. The last row in the chart is probably more like it was.
On November 03 2007 11:44 CustomXSpunjah wrote:
lol i dunno dude....u cant generalize that pvz is imba z for that much (70% chance of winning) off my own experience i beat zergs more than i beat terrans
70 is probably too much as you both say. So give me better suggestions! Im no sc expert, so I will need help to pick realistic values for the 4 percentages.
Also, this model uses a global average for imbalances, so one single players stats doesnt really matter a lot. But if you want it is possible to make P>Z but setting the ZvP paramter below 50%.
On November 03 2007 11:38 kamehameha wrote:
wow very nice work, i was kinda confused on some points on the first read but im sure ill get it on the second read
this model of the terran dominance only applies to tournament games tho right?
i made the model to fit the entire progaming scene. You "loose" when your team kicks you from the team because you dont deliver. But I think it could be applied also to other groups.
On November 03 2007 11:58 kamehameha wrote:
but his calculations are off progamer(which tend to have closer skill levels from one another) matchs not public games.. too much skill variation so it becomes unpredictable
Well, good thing that you can change the closeness as a parameter then. 
For progamers it will in general be fairly close even when top 5 plays top 50. So set the pramter to 65% win or whatever you may think is appropriate.
for the case of a bunch of random people on bnet the difference in top 5 and top 50 will probably be much bigger, so set the winning probability to like 80% instead. The model covers both cases.
On November 03 2007 12:28 Tadzio00 wrote:
I think you're right, but I suspect the best way to use this would be to invert what you're doing. Create a formula where you can plug the number of T, Z and P progamers (say the tope 30 or 50 progamers) in as variables and see if it accurately predicts the winning percentage of each race in each matchup. This way you can determine racial imbalance without making any generalized assumptions. If you do it this way you can determine, through observance of variance from the mean, to what extent map imbalance and personal skill affects the likely outcomes of games.
It is a good thought, but it is not possible. The imbalances are 3 parameters, while the race fraction have only two free parameters: if you know how many % zerg and terran players there are, then you know also how many protoss players there are, since the rest must be protoss. the equation
z+t+p=1
removes one degree of freedom.that means that the mapping (my program) from the 3 imbalances to the race fractions does not have an inverse. many different imbalances will correspond to the same race fractions, so it is impossible to go in the other direction.
Let me also stress that this is not a "flaw" in my model, but is a general property of broodwar. Would there have been only two races, it would have been possible (one free paramter on each side) while 3 or more races is not possible with (n-1)*(n-2) matchups to balance, and only n-1 free racefractions.
On November 03 2007 13:08 Polemarch wrote:very interesting, nicely done! it might be fun to add another layer into your model for per-map skill bonuses. say add another layer where the maps would be selected from a discrete uniform distribution. then each map has its own values for skill imbalances. then you could answer questions that might be of even MORE interest to tournament organizers and mapmakers. e.g. what is the expected effect on racial distribution of having a mercury (z >>> p) or a paradoxxx (p >>> z) in the map pool? or given 4 maps in my map pool, what remaining 5th map should i pick to maximize racial balance? edit: one more comment, for picking your 'a' parameter, maybe you should look into fitting it to data of the winning percentages of some players (say from TLPD's ELO top players). that would be more convincing, since if you can pick 'a' arbitrarily, it seems relatively easy for you to pick a value that matches last year's player distributions.
maps are averaged over in this model. Adding a z>>>p map would push the ZvP percentage a bit higher. All effect of the type of dodging certain maps etc are neglected. ZvP is the percentage that a Z will beat a P over many games on different maps in different situations. So mapmakers can easily control the three imbalances by creating maps favouring the different matchups.
For the ELO thing: It is probably a good idea. I guess it would take some time since i need to check the ELO rating of every guy they played, and preferably their rating at the time of the game. :/ To see the difference of 60% or 65% you would need about 50 samples of games of players around rank 5 against players of rank around 50. Preferably different players in different matchups. Maybe you feel like contributing with that?
Im of to sleep now. Will run any proposed percentages tomorrow.
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On November 03 2007 13:19 Cascade wrote:For the ELO thing: It is probably a good idea. I guess it would take some time since i need to check the ELO rating of every guy they played, and preferably their rating at the time of the game. :/ To see the difference of 60% or 65% you would need about 50 samples of games of players around rank 5 against players of rank around 50. Preferably different players in different matchups. Maybe you feel like contributing with that?
Why not just calculate the chance mathematically? That's the whole point of ELO ratings, after all.
For example, between the current #5 (Anytime) vs. the current #50 (Yellow) there is a difference of 172 points which gives a 73% winning probability for the #5 player. (The exact players/races don't matter of course).
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Also, historically there have been 112 Protoss pros, 131 Terran pros, and 160 Zerg pros (numbers from TLPD). This is influenced by the early trend of Koreans playing Z due to the initial strong imbalance (pre-patch) in Z's favour and continuing to do so for a while afterwards, but it gives %s of about 28% P, 33% T, 40% Z, which from looking at your table seems to be a better match for a 60/40 (or less) all-time ZvP imbalance than a 70/30 one. That's a better match for the actual known results as well.
It would be interesting to have the proportions of races for players who played their first pro game from 2002 or so onwards, too.
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yeah i was suggesting that you look into extending the per-map stuff to your model. only if you're interested in extending it, of course, but i think that could make it much more useful.
e.g. a good player would probably rather have one grossly imbalanced map out of 5 that they just throw away; rather than 5 moderately imbalanced maps. your extended model could quantitfy that sort of stuff in terms of overall racial balance for map pools.
for picking the 'a' parameter... i wouldn't put in THAT much work into it. here's a relatively easy way that's still based on the data. for a given value of 'a', draw the top 50 skill values out of a sample of say 10000, assign them random races, and treat those as your progamers. then have them randomly compete against each other. check the entropy of the winning percentages and see how it compares to the entropy of the winning percentages of the top 50 players in TLPD. (repeat the expeirment a few times to get a better sample). adjust 'a' until they're about equal. you could probably analyze it mathematically, but a simple simulation like that might be easier.
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4492 Posts
Wow, this is an amazing article, thank you very much for that hard work! I think the model could accept some fine-tuning, but the general principle looks fine.
Now let's make this featured!
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Really interesting, but I'll be honest I couldn't understand the math part. I stop math a long time ago but anyways, the intro and conclusions and some parts explained by words rather than math made a lot of sense based on assumptions, you get yourself hooked with someone or (progaming database) that has all the accurate percentages and you can further this for sure.
Nice writing, keep it up 
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You, my friend, love math!
Very interesting theory though, given you agree with the imbalance in the first place
-Granted, I basically had to jump from theory to conclusion, skimming through the actual math, but that is mainly related to my godgiven incompetence in the subject :p
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Good job
But skill plays more of a factor than the imbalance, look at Bisu, and the top 3 kespa rankers arent terran
But good work on finding why there are more T progamers
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First, very nicely done!
On November 03 2007 13:19 Cascade wrote:Show nested quote +On November 03 2007 12:28 Tadzio00 wrote:
I think you're right, but I suspect the best way to use this would be to invert what you're doing. Create a formula where you can plug the number of T, Z and P progamers (say the tope 30 or 50 progamers) in as variables and see if it accurately predicts the winning percentage of each race in each matchup. This way you can determine racial imbalance without making any generalized assumptions. If you do it this way you can determine, through observance of variance from the mean, to what extent map imbalance and personal skill affects the likely outcomes of games. It is a good thought, but it is not possible. The imbalances are 3 parameters, while the race fraction have only two free parameters: if you know how many % zerg and terran players there are, then you know also how many protoss players there are, since the rest must be protoss. the equation z+t+p=1 removes one degree of freedom.that means that the mapping (my program) from the 3 imbalances to the race fractions does not have an inverse. many different imbalances will correspond to the same race fractions, so it is impossible to go in the other direction. Let me also stress that this is not a "flaw" in my model, but is a general property of broodwar. Would there have been only two races, it would have been possible (one free paramter on each side) while 3 or more races is not possible with (n-1)*(n-2) matchups to balance, and only n-1 free racefractions.
Wouldn't it be possible then, if we have a scenario where there's a general consensus on the imbalance between say T and P, and a situation with the ZvP as was the case 1 year ago, to use this (modified a bit, of course) work and settle the dispute between "whiners" and "man-up's" based on an up-to date Kespa rankings (6 months average or something) ?
On November 03 2007 11:02 Cascade wrote:
As these values are close to the reality a year ago, it hints that my model is working!
If we look at the situation with brutal honesty (lol) we would have roughly speaking the following:
model (assumptions) -> realty.
(here we look at the model as a program and the assumptions as arguments)
So we have two parameters on the left, namely the model and the set of assumptions, and having produced the correct reality it could be that:
a) the model is good and the assumptions are right,
or
b) the model is not good (the assumptions are arbitrary here).
I don't think b) is the case though, the model seems well thought of (even though to be honest i don't know anything about mathematica, or mathematical modelling for that matter), so I expect you can use it to predict the kespa racial distribution for the next several months (you'll need correct assumptions for this, but that's what teamliquid is here for ).
Two technical remarks are:
a) a typo
On November 03 2007 11:02 Cascade wrote:
while TvP favours T slightly, and PvT favours P slightly.
,and
b) the question why is the winning probablity not exactly 0,5 when the skill difference is exactly 0?
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Belgium6766 Posts
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You are insane man
gj anyways!
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This is incredible. Nice work man, you should get payed for this. Or at least have your name on some of the future seasons OSL/MSL/PL maps...
to bad you don't speak Korean (or do you?)
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This is kinda like what happened in the Yellowstone national park.
People noticed that there are much less trees than like 20 yrs ago.
..
(long investigation)
..
.
They figured out that 20 yrs ago was when the wolves were extinct from there > plant eaters grow in population > plants cant grow big, they get eaten.
So they populated the wolves again > plants start growin again.
Nature is a fascinating thing, such beauty relying on shaky feet of perfect balance.
The fact that Starcraft resembles natural balance show us just how great this game is.
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On November 03 2007 14:54 gravity wrote:Show nested quote +On November 03 2007 13:19 Cascade wrote:For the ELO thing: It is probably a good idea. I guess it would take some time since i need to check the ELO rating of every guy they played, and preferably their rating at the time of the game. :/ To see the difference of 60% or 65% you would need about 50 samples of games of players around rank 5 against players of rank around 50. Preferably different players in different matchups. Maybe you feel like contributing with that? Why not just calculate the chance mathematically? That's the whole point of ELO ratings, after all. For example, between the current #5 (Anytime) vs. the current #50 (Yellow) there is a difference of 172 points which gives a 73% winning probability for the #5 player. (The exact players/races don't matter of course).
Oh, didnt know you could do that! Thanks a lot.
ok, so assuming that the ELO ranking is accurate to set the 5-50 parameter to 73%, there are only the three imbalances left to tune. I plugged in some values and here are the results:
ZvP___TvP___PvT____z____t____p
60____55____55_____37___35__28
65____55____55_____40___37__23
60____58____53_____33___40__28
60____53____58_____41___30__29
In general a higher 5-50 winning % will even out the three race ratios as skill will outweight imbalance. Also please help me finding realistic values for the three imbalances, I'm really not feeling very confident in my guesses.
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Some more replys here:
On November 03 2007 16:37 Polemarch wrote:
yeah i was suggesting that you look into extending the per-map stuff to your model. only if you're interested in extending it, of course, but i think that could make it much more useful.
e.g. a good player would probably rather have one grossly imbalanced map out of 5 that they just throw away; rather than 5 moderately imbalanced maps. your extended model could quantitfy that sort of stuff in terms of overall racial balance for map pools.
for picking the 'a' parameter... i wouldn't put in THAT much work into it. here's a relatively easy way that's still based on the data. for a given value of 'a', draw the top 50 skill values out of a sample of say 10000, assign them random races, and treat those as your progamers. then have them randomly compete against each other. check the entropy of the winning percentages and see how it compares to the entropy of the winning percentages of the top 50 players in TLPD. (repeat the expeirment a few times to get a better sample). adjust 'a' until they're about equal. you could probably analyze it mathematically, but a simple simulation like that might be easier.
Ok, I think I see what you mean. Right now I can't come up with an easy way to include that in the model though. If I would not use the "average over everything" approach and look more in detail on separate maps it would be an entirely new model, and probably much more complicated. :/
it could probably be done, but I think you would have to use a different method to solve the model then. I guess you could just keep insertnig new random progamers and simulate everything. it would require more computing time, but probably it could be done.
I will not do it though, as it would basically be starting over from the beggining. using that kind of simulation would however make it a lot easier to include new effects, so it has potential. I'll leave it for someone else.
for the a: Sounds reasonable. Maybe I'll try that later. Actually I can directly calculate average winning percentage for a player of given skill and race, but it is that function with hypergeometric functions.
oz:
Yes, if you fix the 5-50 parameter (from ELO or whatever) and one of the imbalances, then the other two can be fixed from race ratios. You would of course have the whiners settle on which imbalance they could agree on first.
On the correctness of the model: of course we can not be sure at all. As it is right now I do not have any accurate figures for the imbalances or the race ratios, so it is hard to say. it would be interesting to somehow get the "correct" imbalance statistics and race ratios of today (last 6 months or so) and see if the model accurately predicts the ratios from the imbalances. I find it a bit strange that it is so hard to find a total statistic for all progaming TvP,PvZ,ZvT played last 6 months. Or to find out how many currently active progamers of each race there are. I would help A LOT if someone could get those numbers! With those numbers a REAL test of the model could be made, instead of all this guessing about.
I'll fix the typo. The probabily is 50%. look at the plot again. maybe you missread 0.6 in the y-scale as 0.5? Those numbers doesnt show very well. Also 1/(1+e^0) = 1/(1+1)=50%.
On November 03 2007 20:23 minus_human wrote:This is incredible. Nice work man, you should get payed for this. Or at least have your name on some of the future seasons OSL/MSL/PL maps... to bad you don't speak Korean (or do you?)
I can say Han-bang-OH, teugeling-eh, YYYEEEEEHH!!!! and one gate-OH!! I don't think I could explain my model very well using those 4 words only.
I cant see anyone wanting to translate that essay, so they will have to learn english.
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Absolutely amazing work, just genius for coming up with the idea of modeling the progaming scene like this. My maths knowledge isn't super advanced, so correct me if I'm wrong.
On November 03 2007 11:02 Cascade wrote:
Looking at zerg now. They win 70% against protoss and win 40% against terran. To add up to 50% we need double number of terrans as protoss:
You don't even need to look at the zerg, you've already got your answer from what you already have. Moreover because you already have z+t+p=100%, and then one additional equation for looking at each race, that's altogether 4 equations in 3 unknowns and unless they're linearly dependent there's no solution. In this case they are, perhaps because two of the matchups have equal percentages. So unless I'm missing something here, the 'toy model' just fails completely.
Didn't consider the more advanced model in too much detail, but it's really interesting and I'll look over it when I have the time. But surely the matchup %ages aren't that bad, if I recall PGT stats correctly from when it was still up, even on supposedly pvz imba maps like lost temple, pvz was 45:55, and the matchups were never more than 60% for any map that had at least a hundred games played on it. What distribution do you get when you have 55% zvp, 52% pvt, and 52% tvz?
Edit: Actually thinking about it a bit more, the toy model is consistent but it gives bad results, such as the "best" race, with the most and least balanced of the imbalanced matchups for the matchups that they lose and win respectively, is not the most common race.
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Russian Federation18 Posts
What a great job! My respect Cascade!
I agree that precise TLPD statistics would be of invaluable help in testing your model. What's more, I guess you could actually perform multivariate optimization to get those 4 parameters from empirical data. You could minimize say least square deviation between theoretical and TLPD percentages vectors. I'm sure Mathematica has some robust routines for that.
By the way, are you familiar with this model:
Lotka-Voterra equation?
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This is very interesting indeed.You did great work in figuring out something like this. Hope you post more mathematical models concerning starcraft.
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NOTE: I have skimmed other people's responses, and I may be repeating some of their statements here, but that is only because they are wholly my opinion and I'm using it to complete my argument. If you had answered a part of my argument previously, just skip over it - chances are I have read it already.
I think that your work is thorough and very well done, but I have to disagree on a few certain points. Though your data is accurate for the past, as you have shown, whether or not it will be accurate for the future may have nothing at all to do with this model. The depth to which you can extrapolate this is shallow indeed, because even now we are seeing Protoss rising over Zergs and Terrans falling to Zergs consistently. The individual sways of progamers over the data are too great - one player's fall from the limelight and one player's rise could completely denounce your model, and not just on an outlier-only scale. Already, having 2 Protosses at ranks of 1 and 2 of KESPA is saying something about the accuracy of the model. Sure, you may argue that "overall" (and like I said, even this may be a stretch) your model will stand true, but what good does it do if it can't work for the top 10 of KESPA?
Also, you did not include anything on mirror matches - there is obviously always the chance of Zergs eliminating their own kind, and thus leaving less players up for the supposed Terran killing. This of course cannot be controlled as it is almost entirely random and unpredicted, but it still makes a model such as yours inaccurate on the grand scale because the model you have put forth, as far as I can tell, simply ignores mirror matches all together.
Assumptions you made such as "First thing is to assume that ZvP is imbalanced in zergs favour by quite a bit, while TvZ favours T slightly, and PvT favours P slightly," and the winning percentages were not well-proven in my opinion; in fact I disagree with both statements. I believe that the percentage of any MU is wholly dependent on the skill level of the players playing them, and occassionally the map if it is conceretely imbalanced. So let's say back in February the percentage was 70% Zerg wins over Protoss, as I believe you had said. But since then, what has happened? Bisu beat Savior, and the PvZ became a MU where the P could say "it's just another zerg." Of course not every P player is Bisu, but also not every Z player is Savior. On the whole, it is proper to say shifts are occuring.
Like I said in my first paragraph - I admire what you did here, and your findings are pretty interesting based on concrete data and your assumptions. But I think like others have said, there is too much variety, chaos, and unpredictability in this game of skill to create such a model.
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You are correct about the toy model Wonders. The last equation is not needed to find a solution, and I write it down only as a consistency check.
I didn't want to go into this a bit technical detail, but as you ask:
Those three equations will always be linearly dependent. Proof:
+ Show Spoiler +
Call the imbalances pt, tz and zp. So for example a terran has probability tz to beat a zerg and 1-pt to beat a protoss. the equation for T total 50% will be
(1-pt)*p + tz*z + 0.5*t = 0.5
use p+t+z=1
(1-pt)*p + tz*z + 0.5*(1-p-z) = 0.5
Move terms with p to LHS, terms with z to RHS. constant terms cancel.
(0.5-pt)*p=(0.5-tz)z
In the same way for the other races we will get the equations
(0.5-zp)*z=(0.5-pt)*t
(0.5-tz)*t=(0.5-zp)*p
now define
(0.5-pt)=a
(0.5-tz)=b
(0.5-zp)=c
and we can rewrite the three equations as:
a*p = b*z
c*z = a*t
b*t = c*p
These are linearly depend since
c*(first equation) - b*(second eq) = a*(third equation).
This completes the proof.
So the toy model is self consistent and will always have solutions. i think yuo understand why I didnt want to include that discussion part in the op. The only reason the toy model is there is for those who find the serious model too complicated, so adding a long part about why it is self consistent wouldn't make sense.
oh, more imbalance suggestions!!
55,52,52 with the ELO inspired 73% 5-50 parameter gives:
z = 36.1
t = 33.8
p = 30.1
using 65% for 5-50 parameter (and same imbalances) gives
z = 37.4
t = 34.5
p = 28.2
With these smaller imbalances protoss doesn't get THAT suppressed, so terran does not feel a very big effect. t is fairly close to the standard 33.3% in both cases.
We also see (again) that larger skill gaps (higher 5-50 parameter) pushes the race fractions towards 33.3% since skill then is more important than imbalance.
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Why is this a featured thread?? As long as a post contains pretty pictures and mentions Starcraft theories, is it put into Featured? Even if it's useless shit that nobody understands?
I stopped reading as soon as I got to:
I have now developed a mathematical model that explains how a Z>>P imbalance, together with smaller T>Z, P>T imbalances, causes a TERRAN dominance.
This part makes sense. End of story. We don't need math to prove this. Math won't really prove it. We see that Z eliminates all the P, so terran can statistically beat the remaining Z without having to worry about playing against protoss.
What's the point of all the weird math that makes no sense at all?
1) Progamers have different skill levels s. The number of progamers at a given skill s is proportional to
e^(-a s)
That is the the number of players falls of exponentially as skill increases. "a" is a parameter that decides how quickly the number of gamers fall. A large "a" means that the very best playes in the worlds is not THAT much better that the ones ranked around 100. A small "a" means that the top players completely own lower ranked players, even if they are not very mucher lower ranked.
This distribution can be discussed. Other ideas are welcome.
2) The probability of a player of skill s1 to beat a player of skill s2 is
1/( 1 + e^(s2-s1) )
e^(-a s) is not proportional to s (skill). It is not even exponentially proportional to s.
Also, 1/( 1 + e^(s2-s1) ) = P(player of skill s1 to beat player of skill s2).
Example 1: What if they have the same skill level? Probability = 1/1 = 100%
Example 2: What if one has no skills at all? Probability = 1/(1+e^x) < 100%
What the results SHOULD have been, if one person didn't have skills, the probability to win was 100%. And when the skill level was the same, the probability should be 50%.
Why are people buying your bullshit mathematics?
BECAUSE THEY DON'T READ IT.
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Didn't read the entire thing, but i would imagine i could sum up what your going to say in about a paragraph.
Z kncoks out all protoss out of leagues, which is the anti terran, and terran dominates Zerg, so without any Protoss to stand in a Terrans way he wins. And the fact that ZvP was considered the most imbalanced, meaning there would be so few protosses, and the few there were would get knocked out by a Z and then the Z inevitably falling to a Terran.
Then we would have a lot of terrans who don't have protosses KOing them, and zergs getting beaten by terrans and then there being no protosses to balance the cycle due to it being more imbalanced then say ZvT.
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Probability(s1 beats s2) = (s1-s2+1)/2 is a better equation.
(1) s1 = 1, s2 = 0: P = 100%
(2) s1 = 0, s2 = 1: P = 0%
(2) s1 = 0.5, s2 = 0.5: P = 50%
If s1 is slightly better than s2, probability is slightly higher than 50%.
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Russian Federation18 Posts
On November 04 2007 00:56 WhatisProtoss wrote:
Why are people buying your bullshit mathematics?
BECAUSE THEY DON'T READ IT.
Well, YOU ARE A FAG!
If s1 = s2 then
1/(1 + exp(s2 - s1)) = 1/(1 + exp(0)) = 1/(1 + 1) = 1/2 = 50%
And noone said s >= 0.
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On November 04 2007 00:56 WhatisProtoss wrote:Why is this a featured thread?? I stopped reading as soon as I got to: Show nested quote +I have now developed a mathematical model that explains how a Z>>P imbalance, together with smaller T>Z, P>T imbalances, causes a TERRAN dominance. What's the point of all the weird math that makes no sense at all? Show nested quote +1) Progamers have different skill levels s. The number of progamers at a given skill s is proportional to
e^(-a s)
That is the the number of players falls of exponentially as skill increases. "a" is a parameter that decides how quickly the number of gamers fall. A large "a" means that the very best playes in the worlds is not THAT much better that the ones ranked around 100. A small "a" means that the top players completely own lower ranked players, even if they are not very mucher lower ranked.
This distribution can be discussed. Other ideas are welcome.
2) The probability of a player of skill s1 to beat a player of skill s2 is
1/( 1 + e^(s2-s1) ) e^(-a s) is not proportional to s (skill). It is not even exponentially proportional to s. Also, 1/( 1 + e^(s2-s1) ) = P(player of skill s1 to beat player of skill s2). Example 1: What if they have the same skill level? Probability = 1/1 = 100% Example 2: What if one has no skills at all? Probability = 1/(1+e^x) < 100% What the results SHOULD have been, if one person didn't have skills, the probability to win was 100%. And when the skill level was the same, the probability should be 50%. Why are people buying your bullshit mathematics? BECAUSE THEY DON'T READ IT.
e^(-a.s) gives the distribution of players at the skill level. So if the skill difference between S class and lower class is say s, and there's b times more S class than lower class, then the number of players s skill lower than the lower class is b^2*[number of S class]. I don't know what you mean by exponentially proportional.
The win% formula is the actual formula used in the ELO system. e^0 is 1 not 0.
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fanatacist:
Im not sure if you are refering to the toy model or the real model. In any case I think you should reread the op. Both models account for mirror matchups. I promise. If you reread and still do not understand, then tell me which model you are looking at and i will explain.
the assumption Z>>P>T>Z i made in the toy model was just an example to illustrate a principle. When you reread the conclusions you will notice that my real model can take any balance % you want and make a prediction.
this model also is not conserned with individual players in the limelight, but rather global properties of a large group of players.
Whatisprotoss:
ok, so confusion on the winning probability. I'll explain a bit more.
first of, the exponential function.
e^0 = 1.
e^(-infinity) = 0
e^(infinity) = infinity
so if you have to players of same skill the winning probability is
1/(1+e^(0)) = 1/(1 + 1) = 1/2 = 50%
as it should.
Also note that skill = 0 is not the lowest possible skill level. Skill goes on down to minus infinity and up to + infinity. Probability to beat someone of skill = -infinity is
1/(1+e^(-infty)) = 1/(1+0) = 1 = 100%
as it should. To beat someone of skill + infinity you get
1/(1+e^infinity) = 1/(infinity) = 0%
again as it should. Look at the plot in the op.
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On November 04 2007 01:13 cmr.Pent wrote:Show nested quote +On November 04 2007 00:56 WhatisProtoss wrote:
Why are people buying your bullshit mathematics?
BECAUSE THEY DON'T READ IT. Well, you are a fag!If s1 = s2 then 1/(1 + exp(s2 - s1)) = 1/(1 + exp(0)) = 1/(1 + 1) = 1/2 = 50%
Okay. Then explain his first point.
Why does he have the number of terran to be twice as much as zerg or protoss, then feign surprise when the number of terran is 50%, zerg, 25%, and protoss 25%??
Idiot.
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On November 04 2007 01:18 WhatisProtoss wrote:Show nested quote +On November 04 2007 01:13 cmr.Pent wrote:On November 04 2007 00:56 WhatisProtoss wrote:
Why are people buying your bullshit mathematics?
BECAUSE THEY DON'T READ IT. Well, you are a fag!If s1 = s2 then 1/(1 + exp(s2 - s1)) = 1/(1 + exp(0)) = 1/(1 + 1) = 1/2 = 50% Okay. Then explain his first point. Why does he have the number of terran to be twice as much as zerg or protoss, then feign surprise when the number of terran is 50%, zerg, 25%, and protoss 25%?? Idiot.
He has terran twice as much as zerg and protoss based on the matchup percentages. He 'feigns surprise' (more like exclamation), presumably because he has successfully demonstrated how such a model of progaming might work, and how it agrees with the current figures.
Learn some maths before you go calling the maths bullshit.
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Thanks Wonders. And thanks for feedback everyone.
Let us focus on the main model! The toy model is only an illustration.
On November 04 2007 00:14 cmr.Pent wrote:What a great job! My respect Cascade! I agree that precise TLPD statistics would be of invaluable help in testing your model. What's more, I guess you could actually perform multivariate optimization to get those 4 parameters from empirical data. You could minimize say least square deviation between theoretical and TLPD percentages vectors. I'm sure Mathematica has some robust routines for that. By the way, are you familiar with this model: Lotka-Voterra equation?
I guess I could yes. Seems like a lot of work though. Im still hoping that someone will be able to find statistics for this without having to add up loads of things in TLPD.
I've not heard of that model under that name, but I've done similiar models in some math course long ago. You also have similar things in chemistry with different molecules that brakes or accelaretes other processes iirc.
On November 04 2007 01:05 MoNKeYSpanKeR wrote:
Didn't read the entire thing, but i would imagine i could sum up what your going to say in about a paragraph.
Z kncoks out all protoss out of leagues, which is the anti terran, and terran dominates Zerg, so without any Protoss to stand in a Terrans way he wins. And the fact that ZvP was considered the most imbalanced, meaning there would be so few protosses, and the few there were would get knocked out by a Z and then the Z inevitably falling to a Terran.
Then we would have a lot of terrans who don't have protosses KOing them, and zergs getting beaten by terrans and then there being no protosses to balance the cycle due to it being more imbalanced then say ZvT.
yes, that's a good summary.
What I continue to do then is:
1) Prove that it actually is so in my real modelas well.
2) Quantify if by making predictions of exactly what the race fractions will be.
I'm not sure what you meant with the last paragraph. It will not loop around infinitely, but rather reach a stable state. Even if there are more terrans than toss, the zerg will still be ok since they really beat the crap out of the few P that are left, but only have a smaller disadvantage from the more numerous T.
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MaTRiX[SiN]
Sweden1282 Posts
"many different imbalances will correspond to the same race fractions, so it is impossible to go in the other direction."
wouldnt this mean that even thougt your result is close to reality it still doesnt have to be correct?
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On November 04 2007 00:56 WhatisProtoss wrote:Why is this a featured thread?? As long as a post contains pretty pictures and mentions Starcraft theories, is it put into Featured? Even if it's useless shit that nobody understands? I stopped reading as soon as I got to: Show nested quote +I have now developed a mathematical model that explains how a Z>>P imbalance, together with smaller T>Z, P>T imbalances, causes a TERRAN dominance. This part makes sense. End of story. We don't need math to prove this. Math won't really prove it. We see that Z eliminates all the P, so terran can statistically beat the remaining Z without having to worry about playing against protoss. What's the point of all the weird math that makes no sense at all? Show nested quote +1) Progamers have different skill levels s. The number of progamers at a given skill s is proportional to
e^(-a s)
That is the the number of players falls of exponentially as skill increases. "a" is a parameter that decides how quickly the number of gamers fall. A large "a" means that the very best playes in the worlds is not THAT much better that the ones ranked around 100. A small "a" means that the top players completely own lower ranked players, even if they are not very mucher lower ranked.
This distribution can be discussed. Other ideas are welcome.
2) The probability of a player of skill s1 to beat a player of skill s2 is
1/( 1 + e^(s2-s1) ) e^(-a s) is not proportional to s (skill). It is not even exponentially proportional to s. Also, 1/( 1 + e^(s2-s1) ) = P(player of skill s1 to beat player of skill s2). Example 1: What if they have the same skill level? Probability = 1/1 = 100% Example 2: What if one has no skills at all? Probability = 1/(1+e^x) < 100% What the results SHOULD have been, if one person didn't have skills, the probability to win was 100%. And when the skill level was the same, the probability should be 50%. Why are people buying your bullshit mathematics? BECAUSE THEY DON'T READ IT.
haha basically, my first reply sounded very much like this, but i thought i wouldnt put the guy down, he obviously put a lot of effort into it. But i agree tho, the idea is all thats worth mentioning here. Zerg overkills the terran killers which are then safe. Thats it.
The rest of the post is messing around with made-up numbers put into basic equations.
I guess people are surprised that math sometimes can actually be used for something, even if the whole story is obvious without it.
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Russian Federation18 Posts
On November 04 2007 02:45 niteReloaded wrote:Show nested quote +On November 04 2007 00:56 WhatisProtoss wrote:Why is this a featured thread?? As long as a post contains pretty pictures and mentions Starcraft theories, is it put into Featured? Even if it's useless shit that nobody understands? I stopped reading as soon as I got to: I have now developed a mathematical model that explains how a Z>>P imbalance, together with smaller T>Z, P>T imbalances, causes a TERRAN dominance. This part makes sense. End of story. We don't need math to prove this. Math won't really prove it. We see that Z eliminates all the P, so terran can statistically beat the remaining Z without having to worry about playing against protoss. What's the point of all the weird math that makes no sense at all? 1) Progamers have different skill levels s. The number of progamers at a given skill s is proportional to
e^(-a s)
That is the the number of players falls of exponentially as skill increases. "a" is a parameter that decides how quickly the number of gamers fall. A large "a" means that the very best playes in the worlds is not THAT much better that the ones ranked around 100. A small "a" means that the top players completely own lower ranked players, even if they are not very mucher lower ranked.
This distribution can be discussed. Other ideas are welcome.
2) The probability of a player of skill s1 to beat a player of skill s2 is
1/( 1 + e^(s2-s1) ) e^(-a s) is not proportional to s (skill). It is not even exponentially proportional to s. Also, 1/( 1 + e^(s2-s1) ) = P(player of skill s1 to beat player of skill s2). Example 1: What if they have the same skill level? Probability = 1/1 = 100% Example 2: What if one has no skills at all? Probability = 1/(1+e^x) < 100% What the results SHOULD have been, if one person didn't have skills, the probability to win was 100%. And when the skill level was the same, the probability should be 50%. Why are people buying your bullshit mathematics? BECAUSE THEY DON'T READ IT. haha basically, my first reply sounded very much like this, but i thought i wouldnt put the guy down, he obviously put a lot of effort into it. But i agree tho, the idea is all thats worth mentioning here. Zerg overkills the terran killers which are then safe. Thats it. The rest of the post is messing around with made-up numbers put into basic equations. I guess people are surprised that math sometimes can actually be used for something, even if the whole story is obvious without it.
Yes it is obvious when the op has explained it in detail. I don't think it was really obvious for everyone that Z>>P leads to Z's disadvantage.
The most important thing though is that Cascade managed to describe the effect quantitatively.
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On November 04 2007 02:07 MaTRiX[SiN] wrote:
"many different imbalances will correspond to the same race fractions, so it is impossible to go in the other direction."
wouldnt this mean that even thougt your result is close to reality it still doesnt have to be correct?
The result is correct (if my model is correct at least). What it says it that IF the imbalances are X, THEN the race fractions will be Y.
The fact that many different X will predict the same Y says that we CANNOT go the other way. That is, we cannot say "look, the race fractions are Y! That means that the imbalances must be X!".
So the prediction goes in only one way.
On November 04 2007 02:45 niteReloaded wrote:Show nested quote +On November 04 2007 00:56 WhatisProtoss wrote:Why is this a featured thread?? As long as a post contains pretty pictures and mentions Starcraft theories, is it put into Featured? Even if it's useless shit that nobody understands? I stopped reading as soon as I got to: I have now developed a mathematical model that explains how a Z>>P imbalance, together with smaller T>Z, P>T imbalances, causes a TERRAN dominance. This part makes sense. End of story. We don't need math to prove this. Math won't really prove it. We see that Z eliminates all the P, so terran can statistically beat the remaining Z without having to worry about playing against protoss. What's the point of all the weird math that makes no sense at all? 1) Progamers have different skill levels s. The number of progamers at a given skill s is proportional to
e^(-a s)
That is the the number of players falls of exponentially as skill increases. "a" is a parameter that decides how quickly the number of gamers fall. A large "a" means that the very best playes in the worlds is not THAT much better that the ones ranked around 100. A small "a" means that the top players completely own lower ranked players, even if they are not very mucher lower ranked.
This distribution can be discussed. Other ideas are welcome.
2) The probability of a player of skill s1 to beat a player of skill s2 is
1/( 1 + e^(s2-s1) ) e^(-a s) is not proportional to s (skill). It is not even exponentially proportional to s. Also, 1/( 1 + e^(s2-s1) ) = P(player of skill s1 to beat player of skill s2). Example 1: What if they have the same skill level? Probability = 1/1 = 100% Example 2: What if one has no skills at all? Probability = 1/(1+e^x) < 100% What the results SHOULD have been, if one person didn't have skills, the probability to win was 100%. And when the skill level was the same, the probability should be 50%. Why are people buying your bullshit mathematics? BECAUSE THEY DON'T READ IT. haha basically, my first reply sounded very much like this, but i thought i wouldnt put the guy down, he obviously put a lot of effort into it. But i agree tho, the idea is all thats worth mentioning here. Zerg overkills the terran killers which are then safe. Thats it. The rest of the post is messing around with made-up numbers put into basic equations. I guess people are surprised that math sometimes can actually be used for something, even if the whole story is obvious without it.
Haha, that's sweet of you to not put me down at first! A habbit of a good poster.
I see your point though. I think it is a matter of which level you want to do it at. I certainly agree that it is far from vital for the common teamliquidan to know exactly what race fractions comes out of a certain imbalance setting. But I'm working with research and have a habbit of doing this sort of things properly. Hopefully it was of interest for some.
I do not think however, that this effect was common knowledge. Maybe you didn't say that? I'm not sure exactly what you refer to with "the whole story", so just disregard if I've misinterpreted you.
I also think that even though the exact percentages are not of great interest for most, it IS of interest that they CAN BE FOUND. This is my opinion as a physicist, and I completely understand if you do not agree on that point. Anyway, it would be kinda stupid to say that I COULD find exact values without explaining how I did it and presenting examples.
EDIT: Just saw "made up numbers in basic equations". Hehe, ok, slightly unfair imo. The 4 numbers that needed to be "made up" I left for us all to find from statistics. And I don't know how to say this without sounding like a jerk so: I do no think very many on this board could have solved that model. I came of as an elitist now right? D'oh!
again tnx for feedback.
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So, could you work this backwards? Take the KeSPA top 30, look at race distributions (and gains/losses in position), and determine how unbalanced that season's maps were and in what areas.
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MaTRiX[SiN]
Sweden1282 Posts
On November 04 2007 04:57 Cascade wrote:Show nested quote +On November 04 2007 02:07 MaTRiX[SiN] wrote:
"many different imbalances will correspond to the same race fractions, so it is impossible to go in the other direction."
wouldnt this mean that even thougt your result is close to reality it still doesnt have to be correct? The result is correct (if my model is correct at least). What it says it that IF the imbalances are X, THEN the race fractions will be Y. The fact that many different X will predict the same Y says that we CANNOT go the other way. That is, we cannot say "look, the race fractions are Y! That means that the imbalances must be X!". So the prediction goes in only one way.
yes but I mean, with the imbalances you assumed you got race fractions that were close to reality, but since more than one imbalance could create that race fraction this doesnt really mean anything? we allready knew what the race fractions were, what's interesting is what the imbalances (if any) are...
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I agree with WhatisProtoss.
This thread contains one or two nice ideas, but mathematical models for explenation of T dominance? Come ON, if you really knew enough about maths to formulate complex models on the basis a lots of variables, you simply would know that this hypothesis isn't viable at all. Even given that all the assumptions that are made prior to the equations were right (which I really doubt), there are lots of historical and random data that simply can't be managed in such a short abstract. And everyone who doesn't understand that is just a n00b at math...
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It's funny that this thread comes along during the dominance of a Protoss player whose PvZ is better than his PvT.
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On November 04 2007 05:12 MaTRiX[SiN] wrote:Show nested quote +On November 04 2007 04:57 Cascade wrote:On November 04 2007 02:07 MaTRiX[SiN] wrote:
"many different imbalances will correspond to the same race fractions, so it is impossible to go in the other direction."
wouldnt this mean that even thougt your result is close to reality it still doesnt have to be correct? The result is correct (if my model is correct at least). What it says it that IF the imbalances are X, THEN the race fractions will be Y. The fact that many different X will predict the same Y says that we CANNOT go the other way. That is, we cannot say "look, the race fractions are Y! That means that the imbalances must be X!". So the prediction goes in only one way. yes but I mean, with the imbalances you assumed you got race fractions that were close to reality, but since more than one imbalance could create that race fraction this doesnt really mean anything? we allready knew what the race fractions were, what's interesting is what the imbalances (if any) are...
We can see both the imbalances and race fractions easily from just looking at the statistics on all the maps, and by simply counting numer of players of each race. What this model does it that it explains how they fit togheter, and explains how they affect each other. It does not generate any previously unknown statistics about present och past progaming.
A possible use is the one I adressed in the op:
If there, for example, are too many terran for the organisers likeng. Too many mirrors maybe makes the view rates drop. they cannot directly change the race ratios with less that forbidding terran players to play, which probably would not help their ratings. They could however change the balance of the maps to indeirect effect the ratios of the races. my model helps them to understand how changes in map balance effects the race ratios.
Did that answer your consern?
On November 04 2007 05:26 Pinselstrich wrote:
I agree with WhatisProtoss.
This thread contains one or two nice ideas, but mathematical models for explenation of T dominance? Come ON, if you really knew enough about maths to formulate complex models on the basis a lots of variables, you simply would know that this hypothesis isn't viable at all. Even given that all the assumptions that are made prior to the equations were right (which I really doubt), there are lots of historical and random data that simply can't be managed in such a short abstract. And everyone who doesn't understand that is just a n00b at math...
Welcome to Teamliquid.
You need to be a bit careful with attacking people like that. I know my maths. And if I were not capable of "formulating complex models on the basis of a lot of variables", then they would have kicked me from my position as PhD in theoretical physics.
That said: if you have opinions on my asumptions I invite you to express them and we can discuss them. As you can read in the op I am not myself convinced by all assumptions, so I'd be happy if yuo have some ideas of what should be changed. but please dont just tell me that I am wrong without specifying better than that. :/
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Sry, but I don't have the nerve to discuss a matter to death that's very clear IMO. But I'll give it a try.
First, if you want to be anything but unscientific, state a clear hypothesis, all variables involved (including those possibly involved and the odds that they are/are not). Basically failing in that regard is what allready screws your whole model. You sure u got that PhD?... At university they would kick me out for such sloppy work.
Your point 2:
Winning probability as a function of skill difference?
You can't be serious and if you are you don't know very much about e-sports or sports in general obviously. Especially if you consider that most progaming leagues are played in tournaments which of course screws any distribution, because maybe better players got kicked out earlier in the tournament. (fe Stork and Hwasin this MSL)
btw in inferential statistics the expected frequencies can easily differ from observed fre's. That's what we have our lovely tests of significance for.
the worst progamer of each race must have the same winning probability averaged over all other progamers.
As you say yourself, this is very intuitive. I don't believe that players are kicked out simply based on their winning %, because in Korea SC is big business, it's about marketability, sometimes second chances, sometimes not, personality, teamplay and so on. Lots and lots of undefined factors. Also I believe that it is easier to kick terrans out, because they are easier to play and therefor there is a much larger pool of good T's that can take your place than with Z or P. But that's just my opinion... problem is, you have to proove the opposite or modify your hypothesis before you can go on, not just ignore it.
Also, you have to realize that the less Protoss or Zerg players there are the more the Terran players will only train for TvT. Which reduces their skills level considering TvP and TvZ.
At the end, I'd like to point out that through the history of SC there were different patches that very much influenced the statistical percentages of wins between the races. I can't understand how someone that calls himself serious mathematician can overlook this and the impossibility to find the REAL percentages of the imba's, because of this, especially if you look upon this matter historically.
...
That's it for now, because as I scroll further through your abstract I realize that I'd loose very much time in specifying everything I don't like. And asking myself where'd you get that PhD from of course. xD
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On November 04 2007 05:26 Pinselstrich wrote:
I agree with WhatisProtoss.
This thread contains one or two nice ideas, but mathematical models for explenation of T dominance? Come ON, if you really knew enough about maths to formulate complex models on the basis a lots of variables, you simply would know that this hypothesis isn't viable at all. Even given that all the assumptions that are made prior to the equations were right (which I really doubt), there are lots of historical and random data that simply can't be managed in such a short abstract. And everyone who doesn't understand that is just a n00b at math...
Um... that's why they're called models. Is economics also useless, just because it uses models and assumptions to represent real-life situations even when there are 100s of variables out there that can turn things around and each individual corporation has its own determinants for success? Not at all, because when you look at things on a large scale (like he is) these trends can actually be studied and can be applicable.
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On November 04 2007 07:09 teamsolid wrote:Um... that's why they're called models. Is economics also useless, just because it uses models and assumptions to represent real-life situations even when there are 100s of variables out there that can turn things around and each individual corporation has its own determinants for success? Not at all, because when you look at things on a large scale (like he is) these trends can actually be studied and can be applicable.
Exactly my point. Not all parts of course, but very large parts of economics ARE useless. Talking about large scale, if you look at the assumptions of the economic and following neoliberal agenda, namely: "If we have flexibly prices, wages and rates of interest there will exist an equilibrium at the labor market and everyone will have work." The hell everyone will.
Models encourage ppl to make absolutistic statements, but how can you make such a statement if your model is based on probabilistic maths and assumptions?
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On November 04 2007 05:26 Pinselstrich wrote:
I agree with WhatisProtoss.
This thread contains one or two nice ideas, but mathematical models for explenation of T dominance? Come ON, if you really knew enough about maths to formulate complex models on the basis a lots of variables, you simply would know that this hypothesis isn't viable at all. Even given that all the assumptions that are made prior to the equations were right (which I really doubt), there are lots of historical and random data that simply can't be managed in such a short abstract. And everyone who doesn't understand that is just a n00b at math...
WhatisProtoss is an ass, who doesn't deserve to be spoken to directly.
Who knows what and to what extent is entirely his own business and vastly off-topic.
In response to your contribution (and others'), this is how you do mathematical models of things. If you do a model of an ecosystem there are a lot of things you don't consider at first - disease, change of climate effects, etc. and still your model roughly works (gives you the general idea). If you want to be more precise you take more and more things into consideration and the model gets more and more accurate.
To people who say the conclusion is obvious, sure it is, but if blizzard adds a fourth race, say wankers (W) and you have an imbalance which looks something like T : W 46% T : Z 56% T : P 41% P : Z 52% Z : W 53% P : W 44% and I ask you will there be more wankers than terrans in progaming after three months, it might just not come off the top of your head.
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Very interesting model. I read all of it, and i think you're just a bit too genius for me haha. Your models seems pretty accurate to last years statistics. Maybe zergs need to throw out a few games not to lose their supply hahahaha
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thanks Oz.
Pinselstrich:
OK, I'll have a go as well then. if you get bored you don't need to reply, i wont hold it against you. it IS hard to discuss mattes like this on by just writing.
May I ask which univerity you are at, and what you are studying?
1) Ok, first the op is not a paper sent for publication, nor an assignment at uni, but an opening post on a forum. It shouldn't be as formal, or noone would read it. Being scientific is not my first priority. Nevertheless I have in the beggining a paragraph on it's own saying "I have now developed a mathematical model that explains how a Z>>P imbalance, together with smaller T>Z, P>T imbalances, causes a TERRAN dominance." i can't see how that is not scientific enough for this forum.
2) you are listing a lot of things left out of the model altogether. This is interesting, and also important for the model. I have given each progamer only the property "skill" (apart from race), and measured their success only in winning ratio.
You do not like that I set up a formula to calculate winning probability as a function of skill, but do not say why I shouldn't. First I do not have any more information about the players. In one specific match it will of course depend on a lot of other things like style, preparations, current confidence etc. I average over all those other effect. Point is that those effect will, in the long run, effect each race equally much, so they will not change the average outcome. And even if for example zergs (by some reason) tends to be extra nervous and play worse in ZvP, that will then be included in the imbalance.
Same goes for bad luck in tournaments, being kicked out/allowed to stay due to behaviour, marketing etc. All these are factors that, as far as I know, effect all races equally, so they can also be ignored. T being replaced by other T easier than a zerg gets replaced by another zerg doesnt matter at all since I am looking only at race distribution. For all I care, they can change every terran on their team three times a day.
And I cannot, need not prove it. These are assumptions that I make in my model. If i could prove it, they would not be assumptions. I've tried to argue why it is reasonable to assume them, i cannot do more.
the point with terran training more for TvT is a very good one. Same thing as lefthanded tennis/badbinton/etc players are favoured. That actually would dampen the race ratios towards 33.3%.
The imbalances I speak of are not the ones yuo call "real imbalances", but the actual statistics you would get if you take the 100 best players from each race and let them play each other (infinitely) many times on the current seasons maps.
They cannot of course be found to arbitrary precison, but are still defenitely measurable.
And you are so rude for insulting me about being bad at my job! FYI I was picked from quite many applicants.
Ok, peace out. gl with your studies.
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Hi again,
@0z: LoL, ty, I'll keep that in mind. xD
I know what you mean, but if you make such a model you have to explain every single thing you used in it and try to explain why you didn't account for others. As I am used to see most models inapplicable for reality because of their lack of explenation, I still have my doubts about this one. And I am just one person with a single brain, imagine about 30 guys like me bringing up new and new factors that should be considered. What I'm trying to say is that making a model alone and without qualified help is like Sisyphos-work.
@Cascade:
Of course you can, I do statistics and sociology specialising on economy in Vienna. Maybe that's the reason why I don't like models, most of which I've encountered were not applicable for RL situations. Math is only kind of my hobby, so don't expect too much. But from experience I know, it's one thing to analyse the past and present, but a completely different story to forsee the future with a model.
1) Yeah, you're right. But normally we take the 0-hypothesis and try to proove the opposite. If you can't you approove your alternative hypothesis (which would be what your topics about). But since you don't seem to like testing theories, instead calling it "model", I can overlook that. It is indeed a topic in an internet forum.
2) I noticed that, so the thing you name "skill" is each players winning ratio at the end of the day?
You see, I don't think that every short run effect will affect each race equally in the long run. Since there obviously are differences in gameplay and difficulty of gameplay between the races (some maps tend to favor some races, Monty Hall for instance). So you'd have to connect the racial ratios with the maps on which the games are played, for the obvious imba's there are sometimes. So there are not only factor influencing the short run, but also the long run IMO.
You're argumenting with something we call "central limit theorem" and you're right with it. I apologize, it really doesn't matter what race you are in that regard.
Well, then argue. xD
Yeah, and also that P's and Z's will train more PvT and ZvT if T dominates. And the fewer Ps and Zs there are, the more these few will be raping the T's and could end up dominating. IMO that's the reason why it's impossible to have one race dominating the others as long as the game itself is balances (what I believe that SC is, finally).
That's the problem. In statistics we have things we call expected frequencies and observed frequencies. The EF are the thing you mean with letting the best players play infinitely, the OF are reality. And if you have both you can first use measures of compound to see if there's a relation between your expectation and reality. If you're lucky and there is you have to make some tests of significance to be sure that youre experiment isn't just a methodological artifact. As long as you don't do that, from statistical perspective your tables are obsolete and have no viability at all.
About youre PhD I was just kidding, don't take anything personally when it's 2am + beer here. xD
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Ok, Im in the same timezone as you, so I'm about to go to sleep as well.
We seem to be ok on 1). regarding 2: Um, skill is not the same as winning probability. But the winning probability can be calculated if you know both players skill. That is explained in the op. This probability is however modified by which races are playing by giving the favoured race a boost in skill. Also explained in op.
I think the issue here is this: I am averaging over everything.
And yes, the imbalances ARE effected by which maps progamers a playing on. If they suddenly change to maps where Z>>T (I wish...) then I would have to use a new TvZ imbalance parameter in my model.
And yes, since this model is based on averages and statistics, this will only be accurate if many games are played. So as you say, when comparing to actual statitics (if we can find some eventually...) we need to take errors into account (with the normal sqrt(n) and see how many sigmas of I am and yadayada). The values that pop out of my model doesn't have errors though, since I'm doing the calculation for infinitely many games.
Yeah, I though you were fooling around on the PhD, but you never know on t3h Int4rW3b!!
If you're not an experienced forum-poster, then take an advice and be careful: While that kind of fooling around is very funny and enjoyable IRL, the joke part does not always transfer very well into the written words, and you risk ending up in stupid arguments.
remember to drink plenty of water befor you go to bed to avoid the hangover tomorrow.
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HonestTea
5007 Posts
The Korean SC community has been experimenting with mathematical models for years already.
Mathematical models are not meant to tell us whether Bisu will beat Savior. They do not predict single game results, or even champions. (Well, except for Etter's math).
Mathematical models are useful for judging the overall big picture, for treating each player and each result as a piece of data. We can find some interesting correlations and maybe reach some conclusions. As long as everybody understands that the models are not to be the end all, they will open a new area of discussion. In particular, they help us understand macro trends.
Cascade's OP has already led to good discussion.
What I'm trying to say is this:
1) Cascade, thanks for an awesome post.
2) WhatIsProtoss, why do you feel entitled to post your bullshit?
BECAUSE IT'S NOT YOU WHO TOOK THE TIME TO ACTUALLY COME UP WITH THIS SHIT
if you're going to be a critic, at least be a good one.
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Haha, still up. xD
Well, that's what I meant with "at the end of the day." Of course you do some calculating in between, but basically you say "skill + race" modified by your opponents "skill + race" = your winning probability? My problem with this is, as I said, not the definition by itself (that is one of the ideas I actually like) but the definition of the parameters and especially their loading, which could easily end up beeing arbitrary.
I'm trying to point out the difficulty of getting viable readings as long as you don't mass lots and lots of data, which you can't, because there have been many changes regarding maps, gameplay and game structure (-> patches), that cannot be accounted for in a model. Not mentioning that this doesn't have to be the end of it at all, Starcraft still seems to be evolving. And evolution -as we all know- is a random process. (Hehe, now I'm being rethorical)
Next problem, "when comparing with actual statistics (if we can find some eventually...)": from which years do we take them. Does it make sense to take anything before that last patch, is old school SC comparable to today's SC and so on. I don't want to be pessimistic again, but it could end up in not having enough material, because one has to rule out different things for various reasons.
Haha, till now I posted on far to many forums, just ask my ex-girlfriends ( xD ), but I really don't care who thinks what of me, as long as the person I'm writing to understands.
Cheers.
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HonestTea
5007 Posts
No harm done by you, Pinselstrich. Your contributions were good discussion.
Good night.
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You have bad hand writing... ... ... =D
jk, nice write up!
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MaTRiX[SiN]
Sweden1282 Posts
thougt some more about this, if the race distributions was decided by imbalance wouldnt the terran numbers continue to grow as we go further into tournaments?
in this thread: http://www.teamliquid.net/forum/viewmessage.php?currentpage=1&topic_id=52660
you can see how the race fractions are roughly the same at all stages off progaming, which would imply that there being more terrans than protosses and zergs has to do with something else.
edit: didnt remember the thread as well as I thougt :p the race distributions seems to change at the offline qualifier stage but not after that...
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Have to say, good work Cascade. I used to think like some of the poster that modeling and economics in general were pretty much fluff compared to like biology or other hard sciences, but I've really come to appreciate how much of it influences our policymaking and government.
To those of you who claim that quanitfying the imbalance has no significance, you should keep in mind that a lot of our lives is significantly shaped by people who made models of how we would act. For example, amount of money you're being charged for your credit card bill (i.e. monthly minimum percentage) is highly tied to mathematical models of how much and how quickly you'll return it.
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On November 04 2007 12:23 Pinselstrich wrote:Haha, still up. xD Well, that's what I meant with "at the end of the day." Of course you do some calculating in between, but basically you say "skill + race" modified by your opponents "skill + race" = your winning probability? My problem with this is, as I said, not the definition by itself (that is one of the ideas I actually like) but the definition of the parameters and especially their loading, which could easily end up beeing arbitrary. I'm trying to point out the difficulty of getting viable readings as long as you don't mass lots and lots of data, which you can't, because there have been many changes regarding maps, gameplay and game structure (-> patches), that cannot be accounted for in a model. Not mentioning that this doesn't have to be the end of it at all, Starcraft still seems to be evolving. And evolution -as we all know- is a random process. (Hehe, now I'm being rethorical) Next problem, "when comparing with actual statistics (if we can find some eventually...)": from which years do we take them. Does it make sense to take anything before that last patch, is old school SC comparable to today's SC and so on. I don't want to be pessimistic again, but it could end up in not having enough material, because one has to rule out different things for various reasons. Haha, till now I posted on far to many forums, just ask my ex-girlfriends ( xD ), but I really don't care who thinks what of me, as long as the person I'm writing to understands. Cheers.
Ok, so we seem to be down to basically one issue: how the f do we choose the imbalances??
You are making two points and I agree on both of them.
1) Making up percentages from what we personally believe are the correct imbalances is not very scientific. This would be remedied if it would be possible to find statistics of games on all maps for, say, the last 6 months, and statistics on the number of active progamers of each race. I've been asking for these number during 3 pages now, but they refuse to appear. Is the best foreign page on progaming really not capable of delivering such basic statistics?
2) even if we get statistics, how long back should we go? This is probably my major concern with the application of the model. While not an issue with the model itself, it still makes the application a lot more complicated.
a) If we go to far back in time, imbalances will no longer be the same, due to strategical evolution and maps (and patches).
b) If we do not go far back enough, we will get few games and low (=bad) statistics.
i've been suggesting 6 months as compromise between these two issues, but I'm not sure.
We should play some games sometime.
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On November 04 2007 19:55 MaTRiX[SiN] wrote:thougt some more about this, if the race distributions was decided by imbalance wouldnt the terran numbers continue to grow as we go further into tournaments? in this thread: http://www.teamliquid.net/forum/viewmessage.php?currentpage=1&topic_id=52660you can see how the race fractions are roughly the same at all stages off progaming, which would imply that there being more terrans than protosses and zergs has to do with something else. edit: didnt remember the thread as well as I thougt :p the race distributions seems to change at the offline qualifier stage but not after that...
I'm not 100% sure about this:
due to the exponential distribution in skill, the race disrtibution predicted in my model will be the same for any number of gamers.
I am though sure on this:
If you are looking at really low number, like 10 or less, stastical fluctuations will be to big, so you cannot really use an statistical aproach like mine. I Think top 30 kespa is about the smallest number of progamers you could look at with this method, and even that is really pushing it and will be subject to big errors.
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On November 04 2007 04:57 Cascade wrote:Show nested quote +On November 04 2007 02:07 MaTRiX[SiN] wrote:
"many different imbalances will correspond to the same race fractions, so it is impossible to go in the other direction."
wouldnt this mean that even thougt your result is close to reality it still doesnt have to be correct? The result is correct (if my model is correct at least). What it says it that IF the imbalances are X, THEN the race fractions will be Y. The fact that many different X will predict the same Y says that we CANNOT go the other way. That is, we cannot say "look, the race fractions are Y! That means that the imbalances must be X!". So the prediction goes in only one way. Show nested quote +On November 04 2007 02:45 niteReloaded wrote:On November 04 2007 00:56 WhatisProtoss wrote:Why is this a featured thread?? As long as a post contains pretty pictures and mentions Starcraft theories, is it put into Featured? Even if it's useless shit that nobody understands? I stopped reading as soon as I got to: I have now developed a mathematical model that explains how a Z>>P imbalance, together with smaller T>Z, P>T imbalances, causes a TERRAN dominance. This part makes sense. End of story. We don't need math to prove this. Math won't really prove it. We see that Z eliminates all the P, so terran can statistically beat the remaining Z without having to worry about playing against protoss. What's the point of all the weird math that makes no sense at all? 1) Progamers have different skill levels s. The number of progamers at a given skill s is proportional to
e^(-a s)
That is the the number of players falls of exponentially as skill increases. "a" is a parameter that decides how quickly the number of gamers fall. A large "a" means that the very best playes in the worlds is not THAT much better that the ones ranked around 100. A small "a" means that the top players completely own lower ranked players, even if they are not very mucher lower ranked.
This distribution can be discussed. Other ideas are welcome.
2) The probability of a player of skill s1 to beat a player of skill s2 is
1/( 1 + e^(s2-s1) ) e^(-a s) is not proportional to s (skill). It is not even exponentially proportional to s. Also, 1/( 1 + e^(s2-s1) ) = P(player of skill s1 to beat player of skill s2). Example 1: What if they have the same skill level? Probability = 1/1 = 100% Example 2: What if one has no skills at all? Probability = 1/(1+e^x) < 100% What the results SHOULD have been, if one person didn't have skills, the probability to win was 100%. And when the skill level was the same, the probability should be 50%. Why are people buying your bullshit mathematics? BECAUSE THEY DON'T READ IT. haha basically, my first reply sounded very much like this, but i thought i wouldnt put the guy down, he obviously put a lot of effort into it. But i agree tho, the idea is all thats worth mentioning here. Zerg overkills the terran killers which are then safe. Thats it. The rest of the post is messing around with made-up numbers put into basic equations. I guess people are surprised that math sometimes can actually be used for something, even if the whole story is obvious without it. Haha, that's sweet of you to not put me down at first! A habbit of a good poster. I see your point though. I think it is a matter of which level you want to do it at. I certainly agree that it is far from vital for the common teamliquidan to know exactly what race fractions comes out of a certain imbalance setting. But I'm working with research and have a habbit of doing this sort of things properly. Hopefully it was of interest for some. [#1]I do not think however, that this effect was common knowledge.) Maybe you didn't say that? I'm not sure exactly what you refer to with "the whole story", so just disregard if I've misinterpreted you. I also think that even though the exact percentages are not of great interest for most, it IS of interest that they CAN BE FOUND. This is my opinion as a physicist, and I completely understand if you do not agree on that point. Anyway, it would be kinda stupid to say that I COULD find exact values without explaining how I did it and presenting examples. EDIT: Just saw "made up numbers in basic equations". Hehe, ok, slightly unfair imo. The 4 numbers that needed to be "made up" I left for us all to find from statistics. And I don't know how to say this without sounding like a jerk so: I do no think very many on this board could have solved that model. I came of as an elitist now right? D'oh! again tnx for feedback.
#1. Ok maybe its not common knowledge, but to me, it seems very obvious that z>>p leads to T dominance. Almost as obvious to comment on it with a "duh".
as for the rest of the post, i still appreciate your work, i myself probly wouldnt know how to make it in mathematica and all that. My point was that usually, in projects like this, the advanced tools of math are used to make a discovery or tell us that something seemingly far fetched can actually work. In this case, you took an obvious(is it?) concept, put some numbers in equations and got some results. The solution didn astonish me personally, its on the same 'level' as the basic idea.
I hope i dont sound rude or anything, and im trying not to ^ ^. This is a good thing you did, certainly worth reading.
P.S. you dont sound like an elitist, you sound like an enthusiastic young man who's willing to use his probly newly-gained knowledge to try to explain some things not related to university. its a good thing.
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