
100. Re: Mathematical PROOF that K/D Ratio Does NOT Equal Skill
mdub Mar 17, 2009 11:52 AM (in response to Clan of One Man)
in the amount of time it took you to come up with this equation you could have been getting laid. think about!!!real life, cod5 k/d ratio arguement. i hope you get my point.Clan of One Man wrote:
I will begin by working through a specific example with numbers (Part 1). This is the easiest way for most people to understand the underlying logic behind the mathematical arguments. After that, I will solve the equations for the completely general case (Part 2). This will demonstrate that the arguments are valid, regardless of the specific numbers involved.
 
OVERVIEW
Edit: Since people seem to be getting upset, let me explain the jist of the idea first. Your spread is really what is important when considering how effective you are as a player. You would rather have a player who is +10 than a player who is +5 regardless of the K/D. As an example, would you rather have a teammate who goes 6/1 (+5, 6:1 K/D) or one who goes 20/10 (+10, 2:1 K/D). I think the answer to that question is pretty obvious. I have derived a formula to express your average spread as a function of Kills per Minute and Kills/Deaths. It turns out that a player with a lower K/D can have a better average spread if his KpM is higher. If all other factors are EXACTLY equal, the player with the better average spread will win the game. This means that average spread is a better indicator of player skill than K/D. Don't get me wrong, it is obvious that K/D and player skill are highly correlated. I am just saying that K/D is a poor predictor of performance unless it is qualified by the amount of time in which that ratio is achieved. K/D is not the end all be all predictor of how good you are as a player that many people think it is. Like all statistics, this will have to be calculated individually for each game type to be meaningful. You can't compare apples to oranges.
 
DEFINITIONS
Spread : This is the quantity (Kills  Deaths) for a particular player in a particular match
KpM : This is the average kills per minute over a players entire gaming career. It is their total Kills divided by their play time in Minutes
K/D : This is the average Kill to Death ratio over a players entire gaming career. It is their total Kills divided by their total Deaths.
X : For the sake of this argument, X will represent the elapsed time for a particular match in minutes
 
PART 1
Let us consider two players. We will call them player A and player B. Player A has an outstanding 2.0:1 K/D ratio and averages 1.5 KpM. Player B has a very respectable 1.6:1 K/D ratio and averages 2.1 KpM. Both players are on the 10th prestige. They each have thousands of games played and more than 16 days of play time. As a result, their respective K/D ratios are fixed and have not changed in several hundred games.
One day, player A decides to get a group of friends together and challenge player B to a friendly game of Team Death Match. Player B accepts the challenge and gets his own group of friends together. Everyone checks the enemy teams stats while waiting in the pregame lobby. As it turns out, both teams have exactly identical stats except for player A and player B. The match begins with player A's team feeling sure of their victory since player A is "better" than player B.
Now we will assume that we are living in an ideal mathematical world. This means that everyone will always perform exactly according to their statistics. The match progresses and finally draws to a close with player B's team being the victor. Player A thinks "that was just a coincidence" and challenges player B's team to a rematch. They end up playing 100 games and player B's team wins every time. Player A is finally forced to accept that player B has the better team but cannot understand how this is possible. Player B then explains.
Remember that we are living in an ideal mathematical world. Since both teams have identical statistics (except for player A and player B) both teams will always put out identical performances. Therefore, the outcome of the match is solely determined by the performance of player A and player B.
Now we must compare the performance of player A and player B to determine the winner. Please note that not every match will end in the score limit being reached. I am sure that everyone has played in a match where the time limit was reached before the score limit was reached. As a result, we need more information than just K/D to determine the winner. The other piece of information that we need is the average Kills per Minute (KpM).
In examining player performance, we must take into account both the player's kills and the player's deaths. The player with the better spread will win the game. We will now define Spread:
Spread = Kills  Deaths
Both the amount of Kills and Deaths are dependent upon the length of the match. We can determine the kills as follows:
Kills = KpM*X  where X is the length of the match in minutes
Similarly, the number of deaths can be expressed by the total Kills divided by the K/D ratio. A quick dimensional analysis verifies this expression (K/1)/(K/D) = (K*D)/K = D. We then have:
Deaths = (KpM*X)/(K/D)
The expression for the player's spread at any arbitrary time X is then given by:
Spread(X) = KpM*X  (KpM*X)/(K/D)
For the sake of example, lets assume that the score limit was not reached. The match had a 10 minute time limit.
Player A: KpM = 1.5, K/D = 2.0
Spread(10) = (1.5*10)  (1.5*10)/(2) = +7.5
Player B: KpM = 2.1, K/D = 1.6
Spread(10) = (2.1*10)  (2.1*10)/(1.6) = +7.875
Remember that the performance of both teams are identical except for player A and player B. We can clearly see that player B's team wins by the slimmest of margins, despite player B having a much worse K/D than player A! Note that our argument remains self consistent. Player A still had a K/D of 2.0 for the match and player B still had a K/D of 1.6. Both players performed exactly as predicted by their statistics yet player B was the victor. As I will discuss in the next section, player B's team will always win, regardless of the length of the match.
 
Part 2
We can easily examine the spread of each player as a function of any arbitrary time X. Simply grab a graphing calculator and enter in the two equations for spread as follows:
Player A:
Y1 = (1.5*X)(1.5*X)/2
Player B:
Y2 = (2.1*X)(2.1*X)/1.6
The time is plotted along the X axis. The player's spread is plotted along the y axis. Note the form of the equation for spread. Both terms in the equation are directly proportional to X. This means that when time = 0, spread = 0. This is what you would expect since you can't kill anyone before the match has started. Note that the plot of spread is linear with intersection at X = 0. This means that at any time X>0 the equation with the greater slope will have the greater spread. In other words, player B's team will always win, regardless of when the match ends.
From the above arguments, it is clear that what really determines the winner of the game is the slope of the player's spread as a function of time. In the most general case:
Spread(X) = KpM*X(KpM*X)/(K/D)
The slope is then given by the first order time derivative of the spread, ds/dx
ds/dx = KpM  KpM/(K/D)
We will call this new quantity (ds/dx) the "effective impact" that a player will have on the game. The effective impact is a far better predictor of player performance than either K/D or KpM alone. Like all of our current statistics for WaW, this quantity should be calculated independently for each game type and then once overall. The accuracy of this quantity could be improved by some relatively simple statistical techniques. Any proper matchmaking system should rate players with a complex formula (similar to the way the BCS works in college football) that utilizes the effective impact as one of its cornerstones. The teams could then be divided up according to the player ratings to provide the closest possible match of skill.
Now that we have finished our analysis, lets take a moment to go back and qualify our assumptions. We have assumed that we are living in a perfect mathematical world when clearly we are not. Player performance is highly variable from game to game. The results of any given game are questionable at best. However, we can be reasonably confident in the accuracy of the K/D ratio and KpM once enough data has been acquired. If we assume these numbers are close to their "true" values, then the results of many matches between player A and player B will be accurately predicted by their effective impact on the game.
In conclusion:
Let us define player "skill" as the ability to win the game. We can clearly see that player skill is correlated to K/D ratio. However, K/D ratio alone is an extremely poor predictor of player performance and must be qualified by the time in which that ratio was achieved. 
101. Re: Mathematical PROOF that K/D Ratio Does NOT Equal Skill
Vindexer Mar 17, 2009 12:00 PM (in response to Clan of One Man)
in the amount of time it took you to come up with this equation you could have been getting laid. think about!!!real life, cod5 k/d ratio arguement. i hope you get my point.[/quote]mdub wrote:
in the time it took you to post this, you could have tried to get your kpm even close to mine, so that you wouldn't have to waste 7 days still not having as many kills as i do. 
102. Re: Mathematical PROOF that K/D Ratio Does NOT Equal Skill
UnbeltedMedusa Mar 17, 2009 12:15 PM (in response to Clan of One Man)this thread is funny, i mean:
How is it possible that someone is wasting his time for something i could not care less about, very very funny. Is here anyone who actually has read all this stuff? XD 
103. Re: Mathematical PROOF that K/D Ratio Does NOT Equal Skill
manbearpig894 Mar 17, 2009 12:17 PM (in response to Clan of One Man)Clan of One Man wrote:
I will begin by working through a specific example with numbers (Part 1). This is the easiest way for most people to understand the underlying logic behind the mathematical arguments. After that, I will solve the equations for the completely general case (Part 2). This will demonstrate that the arguments are valid, regardless of the specific numbers involved.
 
OVERVIEW
Edit: Since people seem to be getting upset, let me explain the jist of the idea first. Your spread is really what is important when considering how effective you are as a player. You would rather have a player who is +10 than a player who is +5 regardless of the K/D. As an example, would you rather have a teammate who goes 6/1 (+5, 6:1 K/D) or one who goes 20/10 (+10, 2:1 K/D). I think the answer to that question is pretty obvious. I have derived a formula to express your average spread as a function of Kills per Minute and Kills/Deaths. It turns out that a player with a lower K/D can have a better average spread if his KpM is higher. If all other factors are EXACTLY equal, the player with the better average spread will win the game. This means that average spread is a better indicator of player skill than K/D. Don't get me wrong, it is obvious that K/D and player skill are highly correlated. I am just saying that K/D is a poor predictor of performance unless it is qualified by the amount of time in which that ratio is achieved. K/D is not the end all be all predictor of how good you are as a player that many people think it is. Like all statistics, this will have to be calculated individually for each game type to be meaningful. You can't compare apples to oranges.
 
DEFINITIONS
Spread : This is the quantity (Kills  Deaths) for a particular player in a particular match
KpM : This is the average kills per minute over a players entire gaming career. It is their total Kills divided by their play time in Minutes
K/D : This is the average Kill to Death ratio over a players entire gaming career. It is their total Kills divided by their total Deaths.
X : For the sake of this argument, X will represent the elapsed time for a particular match in minutes
 
PART 1
Let us consider two players. We will call them player A and player B. Player A has an outstanding 2.0:1 K/D ratio and averages 1.5 KpM. Player B has a very respectable 1.6:1 K/D ratio and averages 2.1 KpM. Both players are on the 10th prestige. They each have thousands of games played and more than 16 days of play time. As a result, their respective K/D ratios are fixed and have not changed in several hundred games.
One day, player A decides to get a group of friends together and challenge player B to a friendly game of Team Death Match. Player B accepts the challenge and gets his own group of friends together. Everyone checks the enemy teams stats while waiting in the pregame lobby. As it turns out, both teams have exactly identical stats except for player A and player B. The match begins with player A's team feeling sure of their victory since player A is "better" than player B.
Now we will assume that we are living in an ideal mathematical world. This means that everyone will always perform exactly according to their statistics. The match progresses and finally draws to a close with player B's team being the victor. Player A thinks "that was just a coincidence" and challenges player B's team to a rematch. They end up playing 100 games and player B's team wins every time. Player A is finally forced to accept that player B has the better team but cannot understand how this is possible. Player B then explains.
Remember that we are living in an ideal mathematical world. Since both teams have identical statistics (except for player A and player B) both teams will always put out identical performances. Therefore, the outcome of the match is solely determined by the performance of player A and player B.
Now we must compare the performance of player A and player B to determine the winner. Please note that not every match will end in the score limit being reached. I am sure that everyone has played in a match where the time limit was reached before the score limit was reached. As a result, we need more information than just K/D to determine the winner. The other piece of information that we need is the average Kills per Minute (KpM).
In examining player performance, we must take into account both the player's kills and the player's deaths. The player with the better spread will win the game. We will now define Spread:
Spread = Kills  Deaths
Both the amount of Kills and Deaths are dependent upon the length of the match. We can determine the kills as follows:
Kills = KpM*X  where X is the length of the match in minutes
Similarly, the number of deaths can be expressed by the total Kills divided by the K/D ratio. A quick dimensional analysis verifies this expression (K/1)/(K/D) = (K*D)/K = D. We then have:
Deaths = (KpM*X)/(K/D)
The expression for the player's spread at any arbitrary time X is then given by:
Spread(X) = KpM*X  (KpM*X)/(K/D)
For the sake of example, lets assume that the score limit was not reached. The match had a 10 minute time limit.
Player A: KpM = 1.5, K/D = 2.0
Spread(10) = (1.5*10)  (1.5*10)/(2) = +7.5
Player B: KpM = 2.1, K/D = 1.6
Spread(10) = (2.1*10)  (2.1*10)/(1.6) = +7.875
Remember that the performance of both teams are identical except for player A and player B. We can clearly see that player B's team wins by the slimmest of margins, despite player B having a much worse K/D than player A! Note that our argument remains self consistent. Player A still had a K/D of 2.0 for the match and player B still had a K/D of 1.6. Both players performed exactly as predicted by their statistics yet player B was the victor. As I will discuss in the next section, player B's team will always win, regardless of the length of the match.
 
Part 2
We can easily examine the spread of each player as a function of any arbitrary time X. Simply grab a graphing calculator and enter in the two equations for spread as follows:
Player A:
Y1 = (1.5*X)(1.5*X)/2
Player B:
Y2 = (2.1*X)(2.1*X)/1.6
The time is plotted along the X axis. The player's spread is plotted along the y axis. Note the form of the equation for spread. Both terms in the equation are directly proportional to X. This means that when time = 0, spread = 0. This is what you would expect since you can't kill anyone before the match has started. Note that the plot of spread is linear with intersection at X = 0. This means that at any time X>0 the equation with the greater slope will have the greater spread. In other words, player B's team will always win, regardless of when the match ends.
From the above arguments, it is clear that what really determines the winner of the game is the slope of the player's spread as a function of time. In the most general case:
Spread(X) = KpM*X(KpM*X)/(K/D)
The slope is then given by the first order time derivative of the spread, ds/dx
ds/dx = KpM  KpM/(K/D)
We will call this new quantity (ds/dx) the "effective impact" that a player will have on the game. The effective impact is a far better predictor of player performance than either K/D or KpM alone. Like all of our current statistics for WaW, this quantity should be calculated independently for each game type and then once overall. The accuracy of this quantity could be improved by some relatively simple statistical techniques. Any proper matchmaking system should rate players with a complex formula (similar to the way the BCS works in college football) that utilizes the effective impact as one of its cornerstones. The teams could then be divided up according to the player ratings to provide the closest possible match of skill.
Now that we have finished our analysis, lets take a moment to go back and qualify our assumptions. We have assumed that we are living in a perfect mathematical world when clearly we are not. Player performance is highly variable from game to game. The results of any given game are questionable at best. However, we can be reasonably confident in the accuracy of the K/D ratio and KpM once enough data has been acquired. If we assume these numbers are close to their "true" values, then the results of many matches between player A and player B will be accurately predicted by their effective impact on the game.
In conclusion:
Let us define player "skill" as the ability to win the game. We can clearly see that player skill is correlated to K/D ratio. However, K/D ratio alone is an extremely poor predictor of player performance and must be qualified by the time in which that ratio was achieved.
Math???????Ugh.....this is why I play Call of Duty....to get away from this stupid crap... 
104. Re: Mathematical PROOF that K/D Ratio Does NOT Equal Skill
Nemesis JP94 Mar 17, 2009 12:19 PM (in response to Clan of One Man)dude i do agree with some of them that the post was pretty geeky and myself i did not read it but while i was scanning it you made some pretty hard calculations 
105. Re: Mathematical PROOF that K/D Ratio Does NOT Equal Skill
skeptile Mar 17, 2009 12:49 PM (in response to Clan of One Man)Good catch!! It was in the wee hours of the early morning when your original post got my wheels turning. I will edit my original post. Looks like I screwed up the number of player B's deaths somehow. I think I accidentally figured his deaths by his K/D ratio x 10. Great post!! 
106. Re: Mathematical PROOF that K/D Ratio Does NOT Equal Skill
mjc0603 Mar 17, 2009 12:52 PM (in response to Clan of One Man)Pretty long winded  but i get the gist of it.
Well presented.
A+ for effort
CHEERS! 
107. Re: Mathematical PROOF that K/D Ratio Does NOT Equal Skill
Clan of One Man Mar 17, 2009 12:53 PM (in response to Clan of One Man)Vindexer wrote:
to push the example to the extreme though, having a team, all of whom have a higher k/d ratio than another team, whom all have a higher kpm. will lead to the team with the higher k/d winning. it is certainly important... i like this analysis... i will have to look into it. i've been bored in advanced econometrics... shame treyarch probably won't ever let me touch large quantities of player data.
Actually as it turns out, this just isn't the case. It is very easy to make that mistake. What really happens is very counter intuitive. If you look at my original post, consider a team of 6 player A's vs a team of 6 player B's.
player A: K/D = 2.0, KpM = 1.5
player B: K/D = 1.6, KpM = 2.1
Now consider a match that lasts 5 minutes.
Player A Stats:
1.5KpM*5M = 7.5 Kills
8Kills/(2.0Kills/Death) = 3.75 Deaths
Player B Stats:
2.1KpM*5M = 10.5 Kills
10.5Kills/(1.6Kills/Death) = 6.5625 Deaths
Now we must stop for a moment. It is very difficult to compare teams A and B directly. This is because the numbers will never work out right. Lets examine why. The formula for spread is given by:
Spread = KpM*X  KpM*X/(K/D)
From this formula you can see that a positive kill to death ratio (K/D > 1) always implies a positive spread. It is therefore impossible for both teams to perform according to their average statistics in the same match. Someone must be negative. To compare the two teams we will have to conduct a "Thought Experiment." Assume that we have created a team of perfect Artificial Intelligences. These AI, will always respond to the same situation, exactly the same way, every time. They have been programmed to handle ALL possible situations in CoD WaW. The AI team has a "neutral" skill. This means that if the results of playing many matches against all possible combinations of all possible players in CoD WaW were taken, the AI K/D ratio is 1.0 and their KpM is 1.0. Note that this does not mean that the AI team will always perform this way. The averages just prove that the team has "neutral skill" and can be used as an unbiased reference point.
We can now play against this team of AI to compare team performances. For statistical purposes, we can think of this AI team as a control group. We will have to make an assumption here. Player A and player B must have different playing styles. This means that when they play the team of AI, the game will progress differently. The assumption we will make is that the KpM and K/D of player A and player B are close to their "True" values. If these stats are calculated from the results of a very large number of games, then this assumption is reasonable. All this really means, is that player A and player B's ability to react to a wide variety of situations is built in to their KpM and K/D statistics. It is therefore "fair" to have Team A and Team B play the AI team separately and compare the results. Assuming that the KpM and K/D are close to their "True" values, team A and team B will perform according to their statistics.
Now we will have each team play our AI team and note the results. Remember we have two teams of 6 identical players that will perform according to their statistics.
Team A:
Kills = 7.5*6 = 45 Kills
Deaths = 3.75*6 = 22.5 Deaths  Note that the team kill to death ratio is 2.0
Team B:
Kills = 10.5*6 = 63 Kills
Deaths = 6.5625*6 = 39.375 Deaths  Note that the team kill to death ratio is 1.6
Interpretation of results:
Both teams won their matches. The spread for team A is 4522.5 = +22.5. The spread for Team B is 6339.375 = +23.625. You can see that team B won their game by MORE than team A. This would tend to imply that team B is the better team. This is because our AI team represents the neutral skill in ALL of CoD.
Some final comments on the above points. The only real way to compare the skill of team A and team B is to have them play each other. However, since we do not have a physical team A and team B, we cannot do this. It is impossible to create a mathematical analysis of the results of a match between team A and team B without a "thought experiment" like the one I have posted. Note that we only conduct this experiment so that we can handle a formal mathematical analysis for a general case, independent of the specific teams involved. If you solve these formulas for the general case, you will find that team B will always do better than team A, regardless of the length of the match.
It is interesting to note the results of this thought experiment. The difference in respective spreads of team A and team B is 23.62522.5 = 1.125. Dividing the difference in spreads by 6 yields the result 1.125/6 = 0.1875. Multiplying this number by 2 we get 0.375. If you been avidly following this topic, you will remember that 0.375 was the difference in spreads from player A and player B in the original post. In the original post, the length of the match was 10 minutes, not 5 minutes. This is why we had to multiply by 2. Essentially, we have just scaled the original argument by a proportionality constant of 6. This makes sense because all we really did was create a team of 6 identical player A's and 6 identical player B's.
In conclusion: We have taken the original argument to the extreme of an entire team of players with a higher K/D ratio than the other team. As in the original argument, the team with the lower K/D ratio still "won". The thought experiment confirms the validity of the original argument. The equation will predict player performance independent of the length of match. Furthermore, the equation remains valid, regardless of the number of players we take into consideration. 
108. Re: Mathematical PROOF that K/D Ratio Does NOT Equal Skill
knightmare699 Mar 17, 2009 12:55 PM (in response to Clan of One Man)you have way to much time on your hand 
109. Re: Mathematical PROOF that K/D Ratio Does NOT Equal Skill
jcrocx Mar 17, 2009 1:14 PM (in response to Clan of One Man)Slow and steady wins the race. Or match I should say. A high KPM just means you run around like an idiot. I would rather move slower and have a higher KDR than run around like an idiot with a lower one. Patient people in the game perform better. For sure their KPM is lower but who cares? A straight up 2:1 ratio group against 1.5:1 group the 2:1 would statistically win no matter what the KPM was.
My theory: Great stats, though they indicate a good player, are not conclusive.