K/D/A indexes and KDA ratio

When talking about how good or bad we did play in a certain game, we often refer to K/D/A indexes and KDA ratio as one way of judgement. Why are these indexes and what do they actually mean?

1. K/D/A indexes v.s. KDA ratio

First off, for clarity, we have four different indexes of discussion: K, D, A, and KDA. The KDA index or KDA ratio is calculated by:

KDA = (K + A)/D

These indexes convey different meanings which are not simply "the numbers of kills, deaths, assists, and the ratio". Especially when being combined into the index of KDA ratio, they really have insightful meanings, which are very important for us, since they may serve as one of the ways to judge how bad or good we are on playing League from different perspectives.

The first thing we can easily notice is that, if we make no death (mathematically, D = 0) this index of KDA ratio will be infinite. Which meanings does this case convey? Since we cannot eliminate the case, would the KDA ratio and all the indexes convey completely wrong senses?

2. The meanings: single points on a process

Basically for every team game, every goal or achievement must be the work of the whole team, not just of a certain single player. We have ten players for each game, five players each team. If one of our team mates gets a kill, he cannot get the kill alone: if he is at top, for example, he still needs other team mates to do their work at bot and mid and jungle, so that the opponents will be kept there and not go to top. A team cannot miss any player, and so any single kill taken must be considered to be the work of the whole team of five players for most exactly.

So, for the most exactly, one kill must be:

+) counted one kill for the player who did the killing blow

+) counted four assists for the four other team mates

This point, a problem appears: who is the player who did the killing blow? The player who made the last-hit, the player who dealt the most damage on the slain opponent, or even the player who dealt the most damage on all the opponents in total, etc? The other problem may be: why would it count an assist for those who had died before the kill and dealt no damage to the slain opponent?

One other way, not that much exactly in nature but more reasonably, is to consider those counts for just the players who did participate in the fights. In the meaning that, the players must deal damage within the fights, or at least help other players deal damage or protect them within the fights.

Obviously, each way of choosing the indexes has its own reasons and paradoxes. I can safely say that no way is perfect. But we have to choose one way nonetheless. For some reasons, Riot Games chose the second way, which in general is[1]:

The K index (kill) is counted for the only player who did the killing blow of last-hitting.

The A index (assist) is counted for the player(s) who did like any amount of damage on the slain opponent within a pretty short duration right before the skill.

The D index (death) is counted after a player gets slain, of course.

Note that, this choice of counting is of Riot Games' decision, it still has paradoxes[2] like I have just said. For example, the players who dealt the largest amount of damage on all the opponents may not be those who get the most kills, while the players who dealt the least damage on all the opponents may have highest KDA ratio like supporters for example.

Relatively, we will be considered to be a better player with more kills and assists and less deaths (greater K and A indexes with a smaller D index).

+) For just a single game, these three indexes of K and D and A may be enough to judge how good or bad a player did on the particular game. In this case of a single game, we may not need any additional index for more complicated.

+) The complexity is when we consider how good or bad we are on a set of a very large number of games, like hundreds or thousands of games with the same scales for the indexes of K and D and A. How can we say how good or bad a player is from like hundreds or thousands of games, with hundreds or thousands of K and D and A? One of the best ways, which Riot Games did choose, is to consider how effective the player do in one life; this is where the index of KDA ratio enters the League.

KDA ratio represents how effective a player plays one single life (in the meaning of before a single death). With one single life (or before each of his own death), how many times the player participates in taking down an opponent. On this perspective, the K and A indexes (kills and assists) should be treated as the same, since as said before: the work of taking down an opponent should be considered to be the achieve of all the players participating in the fight.

If your KDA ratio is 3:1, for example, that means for one single life (before a single death) you can make three times of participating in taking down an opponent.

Single points on a process: Basically, we can consider all the games we have played as a process of playing League, and each single game as a point on the process (which together consists of the process as a whole). When we say "how good or bad we are on playing League" (or some other games as well), we often make the reference to the process rather than a certain single match of the game.

The KDA ratio can be described as the characteristic of a process, while the K/D/A indexes should be described as the characteristics of a certain single match. These two kinds of indexes (KDA ratio and K/D/A indexes) are in fact very different in nature, due to the reason.

3. Some important conclusions

In case of a single game, the three indexes of K/D/A are enough to judge how good or bad we did play the game. To mention KDA ratio of a single game is not always a good idea, since this index may be very different for different games, and may get the infinite value in case a player makes no death (which is difficult and unnecessary to explain).

On the other hand, in case of a process (a set of hundreds or thousands of games), we must use KDA ratio to judge how good or bad we were in the process (at playing the set of games). To mention the separate indexes of K/D/A in this case is not a good idea, since it should makes nonsense to compare the sets of three numbers of hundreds or thousands of different players, as well as to talk about the meanings they convey in comparison.

Also for that reason, the number of deaths would never be zero for a set of hundreds or thousands of games. Or more strictly, we can say that the KDA ratio should only apply to the case of many games, many enough for the D index to get a non-zero value.

Different roles may have pretty different K and A indexes (the same for the D index). Support and assassin for example: support role would spend more time on warding and observing map, so may less participate in fighting relatively in comparison with AD assassin (whose main duty is to find a victim to take down).

Although players always favor few specific roles and always play very many games as their favorite roles, KDA ratio still serves as the relative indication of a player's skill level. This is especially true when it is used to compare those players favoring the same roles, and play the same numbers of games on their favorite roles.

Because the difficulty increases for higher Divisions and Tiers, KDA ratios of a player would be different and decrease for higher Divisions and Tiers. It would be hard to say which value of KDA ratio would be considered to be excellent, so the scale below is just for reference:

KDA = 3:1 is fair

KDA = 4:1 is good

KDA = 5:1 is excellent

Finally, due to the differences between the champions, the KDA ratios of a player would be different on different champions as well. A player may be excellent on playing Lux, meanwhile may be very bad on playing Jax, for example.

This post was completed; there would be just minor adjustments if any on this post

Last update: March 10, 2016

The Silencekeeper

References

[1] Note that, this discussion may have a gap in comparison with what Riot Games really do and did choose to do. Like they may choose whether or not an assist is counted for those players who didn't deal any damage on the slain opponent, but did help protect the one who made the killing blow. The nature of the issues doesn't change, nonetheless.
[2] Also note that, paradoxes are not always bad, and not always wrong

Go to the full discussion: “K/D/A indexes and KDA ratio”

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Initial Elo and ‘fresh account’, provisional matches and ‘soft reset’

DISCLAIMER: Due to the hidden policy of Riot Games on these issues (MMR related issues[1]), it is impossible for a third party to claim any fact regarding the issues. This post, on the same spirit, is not to claim: “this is what Riot has been doing or used to do”. The purpose of this post is to provide a probe on how the system works, by providing the strict reasoning – which in my opinion is the optimizational method for the system to work properly.

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In the post regarding the issues of MMR/ Elo and LP, we discussed the principle of the methods used to rate League players (to establish the table of standings). To understand the principles is one problem, to interpret the table of standings obtained is an other problem. The later (how to interpret the table of standings) provides very important interests of us, the summoners, like:

+) The initial Elo and fresh account (who never play a certain queue, ranked queue for example): How is this initial value assigned?

+) Soft reset: How is Elo reset for a new season?

+) Provisional matches, and the like.

As the final record of players' skill levels on playing League (all players as a whole community), the table of standings would provide us critical insights of ourselves in relation to other players as a whole, if we can interpret it the right way. This work (interpreting the table of standings), however, may invoke many complicated concepts, and in this post I will try to provide some view on these issues – in the clearest and easiest way as I can do, and use fewest statistical concepts possible.

These concepts (soft reset, provisional matches, fresh account, initial Elo,...), on one hand, are the "League concepts"; on the other hand, they are also the concepts of statistics; on yet an other hand[2], they also are the concepts of daily life (how to rate people on their work of doing something). In other words, there are three different angles of view to look at these concepts; each way of view gives us different information about the issues. To be able to look at the concepts from the different points of view would give us a complete and full understanding of them.

1. Elo spectrum (the distribution of Elo indexes): the bell curve

First off and very importantly: the Elo spectrum has the bell shape.

In a perfect case, the bell curve describes a distribution in which almost of the values (like 70%) are somewhat at the center of the bell: the average value or the median value (M). The remaining portion (like 25%) distributes equally on the both sides, decreases when the tails advance towards the vanishing point(s) of zero. This kind of distributions plays a vital role in the natural as well as social world, including Elo spectrum of League players.

This claim can be seen as the origin of all my discussion on this post, and can be seen as the origin of all the issues relating to the concepts of initial Elo, fresh account, provisional matches,...

To claim this is easy, to prove it is very difficult (without using complicated concepts of statistics, which is unnecessary for us, the summoners). So, I will illustrate this via two exemplary models which have the same nature with the Elo spectrum; it would be easy to accept that due to the similarity of these models, the Elo spectrum must fall into the same category of distributions: the bell curve.

The acceptance of the truth is very important. However, no one can accept what is so-called the truth if it is not proved strictly without even one single flaw. So at the end of this section, I will provide some reasoning which can be used to prove it, for people who want to go until the final limits of the truth. All of these, although is necessary for people to accept the truth, is not necessary for us (the summoners); so if you don't want to check the reasoning, simply accept it and ignore the spoiler part and go to the next sections.

Exemplary model 1: distribution of grades of students

This is one kind of distributions which falls into the category of bell curve. More exact the teaching and training as well as the rating processes are, more bell-like curve will be obtained for the spectrum of grades when the number of students increases. This proven theory is used very popularly, in education and pedagogy for rating learners for example.

In the ideal scene, if the grades are rated on the 5 scales (A, B, C, D, and E) for example, a large number of students would fall into the range from D to B, while a very small groups would get A and E.

Exemplary model 2: distribution of IQ indexes (Intelligent Quotient[2])

Although many people may argue or doubt or disagree with many aspects of this index for rating intelligence, almost agree that the spectrum of IQ distribution must have the bell curve shape to exactly describe the phenomenon.

In many standardized IQ tests, the average value or median value (M) is normalized to be 100 IQ.

Elo spectrum of League players

Reasonably, we don't have to say much more to see the Elo index, just like the two examples of grades and IQ, must fall into this category of distributions: if it exactly describes the phenomenon and really reflects the skill levels of players on playing League.

Some bell curve distributions. (1) When we organize the values into smaller ranges, the curve will be come the bars, like in the two upper images. (2) Many distributions, due to different reasons, don't have the exact shape of a bell, but somewhat similar, like the two bottom images.

Both of these cases (regarding the bar-style organization and the ugly bell shapes) may be applied for League: Divisions and Tiers may be organized into the bars, and the bell shape may be changed accordingly, when the bars are changed by changing some parameters in the algorithms for example.

The scientific theory

Scientifically, we can use theories of probability and statistics to prove the truth: Elo index obeys the distribution of bell curve. I will point out the major characteristics of the Elo index which will scientifically prove the fact. To do that, we will consider the Elo values of League players as discrete random variables, and the set of all the players' Elo(s) as the set of discrete random variables.

+) These variables are independent. Although each game consists of ten different players, each player must in fact determine the result themself, especially when the number of games they play is large enough. For more clarity, after hundreds or thousands games, for example, a player's skill level is what defines the player's Elo, independent from all the other players they played with in the games. Elo value represents a player's skill level; consequentially, it is independent as well.

+) Each of the value of the variables is in fact a sum (or average) of the independent variables. Since the Elo of a player is obtained through many games, it is subtracted and added many times to have the final value. The final Elo of a player, so, is in fact a sum of many Elo values.

+) The variables are identically distributed, since they have the same probability distribution (or the same probability mass function): their values are always found within the range from the minimum Elo to the maximum Elo with the same corresponding probability for every Elo within the range.

+) The variance is finite: the average value of the squared deviations of the Elo values and the arithmetic mean of the Elo values. All of these numbers can be calculated, so the final value or the variance is able to calculate: it is finite.

These four characteristics, according to the central limit theorem[3], ensure that the distribution approaches the normal distribution when the number of variables grows, or the Elo spectrum has the bell shape when the number of players is large enough (the number of few thousands would be considered to be "large enough" in this case, let alone the number of millions).

2. Initial Elo and fresh account, provisional matches

Initial Elo

No matter which methods and algorithms are used to calculate the Elo indexes, the calculations must have a number to begin: we cannot begin from nowhere. When we start playing a certain queue (ranked queue for example), we must be assigned an initial Elo number. How is this initial Elo assigned?

One of the best initial values is the median value or the average value obtained from the bell curve of the Elo spectrum[4]. Why? Since this is the average Elo of all players, it would be fair for all players to have the same value as the initial one for the process of being rated.

However, since this initial Elo is the same for every player to begin; on one hand it would be fair; on the other hand it may least exactly reflect players' skill differences: different players must have different skill levels, more or less. Those whose real skill levels are higher than the median will gradually climb towards the top, while those whose real skill is lower will finally drop down.

The initial Elo, consequentially, is not the most suitable value for every player to begin: it doesn't reflect the differences in skill levels of different players, as well as the direction a player would go for the coming games.

Provisional matches

One of the best ways to have a more exact Elo value for each player is to adjust the initial Elo via a certain set of "initial games"; the number of these initial games must be enough to see more information about the player: the difference in skill level in comparison with the initial Elo and other players, the trend the player would go for the coming games: advance or fall down. These games are called "provisional matches" in League and include ten games in total.

Initial Elo value and provisional matches together will provide the beginning Elo of a player to be rated in playing League. This "beginning Elo" is the adjusted value of the initial Elo by the ten provisional matches: it is different for different players, so more exact reflects the differences in skill levels of different players, and somewhat reflects the trend each player will go for the coming games: up to the top or downwards.

For the best of the purpose (to see the most difference possible of each player's skill level as well as the trend for the coming games), each provisional match should weight much more than a regular match. In the meaning that a winning or losing game should give a huge amount of Elo exchanged (added to or subtracted from the player's Elo). No one but Riot Games knows how much weight these provisional games are, but it must be like five times greater than a regular game (and the ten provisional games might be equivalent to like fifty regular games, for example).

Fresh accounts and soft reset

The players who (or more exactly, the accounts which) have never played a certain queue (ranked queue for example) are called "fresh accounts". The process of assigning the initial Elo and obtain the beginning Elo is applied for all fresh accounts who have just started to play a certain queue (ranked queue for example).

For the accounts which are played many games (more than the ten provisional games), obviously, their Elo much more exactly reflects their skill levels. So, it should be best to use their Elo of the previous season (instead of the initial Elo as fresh accounts) to obtain their beginning Elo for the upcoming season.

The process of obtaining the beginning Elo for the upcoming season based on the Elo of the previous season are called "soft-reset" in League. If the process uses the initial Elo of fresh accounts for all players (including players who did play many games in the previous season), the process is called "hard-reset" in League: in hard-reset, all players' initial Elo will be reset to be equal and to be that of fresh accounts.

Just like all other competitive games (tennis, football, etc.), League's Elo indexes and table of standings must be reset for a new season. This process (soft reset) is to compress the Elo spectrum towards the center point of the spectrum (the average value or the median value of the bell curve). So that all the players will start the new season in a reasonable basis.

3. Some special conclusions

The average Elo value or the median Elo value may change slightly for different seasons, and the trend of changing would be increasing since all the players may get better over time. This means, if the median Elo for the first season is 1200 (for example, which may be equivalent to silver V of the first season), the median Elo for the sixth season may increase to 1300 (silver IV of the sixth season, for example).

The soft reset should more exactly reflect players' skill levels in comparison with hard reset, since it is based on the Elo the players got the whole previous season. The hard reset, which puts all players at the same initial Elo, would bring topmost players into the games with lowest players; this is obviously not a good idea. That should be why Riot Games always prefers a soft reset for a new season.

Due to the compressing nature of the process, basically after the soft reset, very low-rated players at the bottom of the spectrum would receive a higher Elo, while very high-rated players at the top of the table of standings would receive a lower Elo (in comparison with their Elo of the previous season). And the players somewhat at the center of the spectrum would see least or even no change in their positions.

Finally, by changing parameters of MMR rating algorithms, Riot Games can vastly change the shape of the curve of the distribution. No matter how enthusiastically and curiously people want to know about the true shape, they have their own reasons for the hidden policy:

+) If the shape (of the computational MMR values) is really the bell curve (as that of the theoretical Elo indexes), that means their methods and especially their MMR system's theories and algorithms are very exact in scientific meanings.

+) If the shape is not like the bell curve, that means their methods and theories and algorithms have critical flaws.

In both ends, on business perspective, they are definitely right to hide the MMR index: no matter whether it is to hide their business secrets and achievements or their flaws. They design the system, so they possess the rights on how to use the system.

This post was completed; there would be just minor adjustments if any on this post
Last update: February 08, 2016

The Silencekeeper

References

[1] For the purpose of this post, the term "Elo" will be used in almost of the discussion (instead of the term "MMR"), in the meaning that it is the theoretical value of Elo: which, in theory, truly represents players' skill levels. MatchMaking Rating system is designed to be expected to be able to calculate this theoretical Elo, and "MMR" is the computational value the system actually obtains.
[2] Of course, we only have two hands, ikr! xD

[3] For more details: https://en.wikipedia.org/wiki/Intelligence_quotient (section Current Tests)
[4] For more details:
     https://en.wikipedia.org/wiki/Central_limit_theorem
     https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables

[5] This value is estimated to be around 1200 in League (which is equivalent to the Divisions of Silver V or IV or III); however, this cannot be verified due to the publisher's policy.

Go to the full discussion: “Initial Elo and ‘fresh account’, provisional matches and ‘soft reset’”

Issues of MMR/ Elo and LP (League Point)

You love League! No doubt.

That is why you have decided immediately to read this post once your eyes catch the title: a kind of the most fundamental, basic, exciting, secret, ambiguous, and... complicated – but also a kind of by far  the most familiar, regular, boring, popular, unclear, and... simplest issues on playing League. On one hand, we have to deal with them every day of playing League; on the other hand, we almost have no certain understanding of them: nothing about them is ever published or confirmed by the Publisher – Riot Game(s)[1].

How should we deal with the paradoxes? How can we know what we want to know, master what we want to master, improve what we want to improve,... if we have no certain understanding of them? Anyways, “every problem always has more than one solution”[2], at least in my opinion. And in this post, I want to try to cast some light on these issues – my way, my own solution. So, let's start our journey!

Introduction

Like other competitive games (chess, tennis, basketball,...), one of the most important interest for us is to have an exact table of standings, to be able to see how good or bad we – or other players or teams we love or “hate” – are in comparison with others. While the problem of establishing such standings for single competitor versus single competitor (two-player) games (like chess) is not so easy to do (and the situation for establishing such standings for teams in team games is somewhat similar to that of two-player games, when we consider a team just as a single player of the game), the problem of establishing such standings for individuals of team games like basketball and football would be by far the most complicated issue of the field.

For the nature of the issue, say, you can kinda easily agree that at a certain time, Roger Federer or Novak Djokovic is the number one in tennis, and quite the same for Barcelona or Manchester United in football; however, to say "Messi or Ronaldo is the number one in football" is a completely different issue. Unfortunately, the situation is applied for us (the summoners): we can say a certain team is the number one in a certain region at a certain period of time, but to say "some body is the number one (in a certain region at a certain period of time)" is a different problem.

So, the question is how we are rated on playing League? How we gain and lose LP (League Point) and how we are put in Divisions and Tiers? How the standings for individual summoners can be established? Prior to the second season, League of Legends used an Elo system for rating players[3], and we will start with the famous rating system: Elo rating system, to answer the questions.

1. Elo rating system

Elo is a rating system named after the author Arpad Elo[4], a chess master and a Hungarian-born American professor of physics. Elo was originally created for rating chess players, but then gradually adopted to use for many games. The term “Elo” is now very popular in many games to indicate the rating of players in comparison. The most fundamental principles of Elo rating system are[5]:

(1) Each player who is rated is represented by a number (the Elo); the Elo number then will change based on the outcomes of games with other rated players.

(2) For each game, the winner takes points from the loser.

(3) The difference between the ratings of the two players determines how many points will be exchanged:

+) For a particular game, the higher-rated player is expected to win. So, if the higher-rated player does win, he only takes less points from the lower-rated player. If the lower-rated player makes an upset win, however, the player takes many points from the higher-rated player – much more than those he would be taken by the higher-rated player if he lost.

+) Particular in chess, the lower-rated player also takes few points from the higher-rated player for a draw (a tie).

At this point, I think those are enough for us to have an idea of what is a rating system and what is the Elo rating system in particular. More details and deeper analyses on this matter may be not necessary and may have to invoke much more complicated issues which may be uncertain for me myself.

2. MMR (MatchMaking rating system)

Obviously, using an Elo rating system designed for two-player games like chess for a team game like League of Legends is not that much effective, for many reasons (which we may discuss at an other time). That would be why Riot Games must design their own rating system for League of Legends, and the rating system is named “Matchmaking rating” system (MMR).

So, first of all, MMR stands for MatchMaking System – the rating system Riot designs and use for League. Consequently, in League we must use the term MMR for more exactly, but due to the popularity of the term “Elo”, the two words are always used in the same meaning in League: to talk about the particular numbers representing the skill levels of players in playing League, measured by the MMR system.

Now, we should probe the main characteristics of this MMR system for our main purpose: to have an idea of how our skill on playing League is measured. According to Riot Games[6]:

+) The goal of the MMR system is to create games which is as fair as possible, so that the chance of winning for the both teams would be equal (50%).

+) The process is described in three steps: (1) determine strength of players, (2) determine eligible opponents, and finally (3) build a match. With adjustments for whether the players are solo or pre-made and some protection for new players.

+) Players have a separate MMR for each queue, developing independently from each other.

+) Initially, the system only matches players of similar MMR. The MMR difference then will be slowly increased to some degree.

3. League Point (LP), Division, and Tier: the table of standings

With all the basic knowledge of the rating systems, it now should be easy for us to have a clear idea on the issues we interest most: Division and Tier, LP.

+) Based on MMR, a player will gain or lose a certain amount of points for a winning or losing game; these points are called League Points (LP). When a player reaches a little bit more than 100 LP, they will reach the higher level and their LP will be reset. These levels which are equivalent to a little bit more than 100 LP are called Divisions.

+) When a player overcomes the fifth division, they will be put in the next higher stage (and their division and LP will be reset). These stages which are equivalent to 5 divisions (about 500 LP) are called Tiers. There are 7 Tiers at the moment: Bronze, Silver, Gold, Platium, Diamon, Master, and Challenger.

We can say that, Riot Games uses these concepts (LP and Division and Tier) to organize the table of standings of League players rated by the MMR system. This is “the table of standings” of League players on the bottom line.

Some important conclusions

The Elo, or more exactly, MMR index of League players (which is rated by the MMR system) is hidden by Riot's policy; until now, it is not officially publicized. Consequentially, all other sources which provide what is so-called Elo or MMR of League players are not official, and definitely cannot use the rating system Riot Games designs – the MatchMaking Rating system (MMR system) due to copyright issues. They can, however, use an other and somewhat similar to the MMR system developed outside of Riot Games. In other words, the "Elo" index other sources provide cannot be called "MMR".

+) The MMR for each queue develops independently, but this doesn't mean that the initial MMR for each queue is treated in the same way, especially for fresh accounts which have never played a certain queue yet. This is unclear.

+) Gradually, the MMR difference would be increased; this means more many games we play, more different players we may meet: may be very higher-skill players as well as very lower-skill players in comparison with ourselves.

The LP gained and lost should be calculated in somewhat the same way with Elo and MMR. This would be the reason why we may gain or lose very different amount of LP for different games. In addition, if we win many consecutive games, the system should match us with very higher skill players; consequentially, the LP gained for each winning game would be very large, while the LP lost for each losing game would be very small: since the opponents are supposed to be rated much higher than us, on the Elo perspective. The opposite dimension would be, in case we lose many consecutive games.

+) It would be a big problem for smurf accounts: it should be very hard to determine a new account is a smurf or not on the internet world of anonymity. The same issue if very high skill players play accounts of low skill players for certain purposes.

Finally, although LP and Divisions and Tiers reflect the standings of us on playing League, the most fundamental index – which would exactly describe how good or bad we are on playing League – is not LP or Division or Tier. That is MMR index, which is hidden by Riot Games' policy. Consequentially, it should be best for us to deal with the MMR index rather than focus on LP or Divisions or Tiers: on the perspective of rating systems, these concepts (LP and Division and Tier) don't convey any meaning. But this is a different problem, I guess!

This post was completed

There would be just minor adjustments if any on this post

The Silencekeeper

References

[1] “Riot Games” (plural form) but they have published only one game so far, right?! xD
[2] For more morals from League: Morals from League
[3][5] According to wikipedia.org: https://en.wikipedia.org/wiki/Elo_rating_system
[4] For more details: https://en.wikipedia.org/wiki/Arpad_Elo
[6] From the official guide at: https://support.riotgames.com/hc/en-us/articles/201752954

Go to the full discussion: “Issues of MMR/ Elo and LP (League Point)”

Issues of AR/ MR reduction/ penetration

DISCLAIMER: due to the ambiguous nature of the issues which have never been officially confirmed by the publisher – Riot Games, all the references herein are just to back the reasoning and assumption, they can not be considered to be unquestionable proofs to trust. Furthermore, also due to the ambiguity, the degree of certainty of the discussion should be judged by the readers themselves.

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When we play defensive-leaning champions like tanks, one of the most important issues for us to consider is AR and MR: how we should build and how much AR and MR we should have for the best effectiveness. Meanwhile when we play very offensive champions like ADC, mid mage, assassin,... we may have to concern how to most effectively damage the opponents, especially the tanks with very high indexes of AR and/or MR.

Very interestingly and in fact, these issues are very closely related, and together very close to health point (HP). Basically, the defensive players want to increase the indexes of their own champions to lose lowest HP possible, while offensive players want to ignore the indexes of the opponents to take highest HP possible in fighting.

At this point, the concepts converge on the central issue: how the defensive indexes of AR and MR reinforce or diminish the index of health point (HP). Understanding this will help us improve the playing of both kinds of defensive- and offensive-leaning champions. Due to the equivalence between the couple of armor and physical damage with the couple of magic resistance and magic damage, we will only mention the former (the later of magic resistance and magic damage will be considered completely the same).

There are two types of index to increase physical damage: armor penetration and armor reduction; for each type, there are two different forms: flat and percent. Finally, all we have are four forms: flat AR reduction, percent AR reduction, flat AR penetration, and percent AR penetration. The sources for these forms include: champions' abilities, items, runes, and masteries; example:

+) Flat AR reduction: Rammus' E-PuncturingTaunt reduces the target's AR by 10/ 15/ 20/ 25/ 30 for 1.25/ 1.5/ 1.75/ 2/ 2.25 seconds
+) Percent AR reduction: The Black Cleaver reduces targets' AR up to 30% for 6 seconds

+) Flat AR penetration: Youmuu's Ghoshblade gives +20 AR penetration
+) Percent AR penetration: Last Whisper gives +30% bonus AR penetration

1. Issues with questionable degree of certainty[1]

+) The order in which the forms are applied in case the attackers have more than one form is[2]

(1) Flat AR reduction

(2) Percentage AR reduction

(3) Flat AR penetration

(4) Percentage AR penetration

+) Example: one tank with 200 AR taken damage from an ADC with 10 AR reduction, 30% AR reduction, 20 AR penetration, and 45% AR penetration (according to the rule):

(1) Firstly, 200 AR will be reduced to 190 AR, by 10 AR reduction

(2) Next, 190 AR will be reduced to 133 AR, by 30% AR reduction

This value of 133 AR will become the AR index of the target as shown on the table of stats instead of the original of 200 AR (always in a short duration of a few seconds). Consequentially, the damage from other team mates on the tank will be treated with the new value of 133 AR instead of 200 AR (within the duration).

(3) Then, 133 AR will be considered as 113 AR for the damage from the ADC, by 20 AR penetration

(4) Finally, 113 AR will be considered as 62.15 AR for the damage from the ADC, by 45% AR penetration

For the particular ADC, the damage dealt on the particular tank will treat with the final value of 62.15 AR, not 133 AR as damage from other team mates nor 200 AR as original.

+) The damage dealt in fighting by attackers always differs from the damage taken by the targets (always gets reduced by defensive indexes of the targets). One of the relations for the damage reduction by AR index is supposed to be:[3]

For the example above, the damage the tank takes from the (damage dealt by the) ADC, according to the relation, is:

This taken damage will be directly subtracted from the total health (HP) of the tank until the tank gets slain. In this case, we can say: "the AR index makes the tank take only 62% damage from the ADC" or "the AR index helps the tank ignore 38% damage from the ADC".

2. Issues with high degree of certainty

No matter how exact the equations are, we can still get something good from the nature of the relations: more AR we have, the HP will be harder to lose; in other words: the AR increases the effectiveness of the HP.

In the example above, if a certain AR index makes the tank reduce 50% the damage from the ADC (we don't need it to be exactly 200 or so for the uncertainty of the equation), this means the ADC will need double damage to take away the same HP from the tank. In other words, the AR index doubles the HP of the tank in fighting against the particular ADC. Here, we have just come to two important conclusions:

(1) AR-HP equivalence: AR and HP are just two equivalent forms of the same index and can be transformed into each other by using some mathematical relations.

(2) Relativity nature: the effectiveness of a certain AR index is different for different attackers due to the different AR reduction and AR penetration indexes the attackers possess. In other words, we are talking about the relativity of AR index in different reference to different attackers.

It's so funny but true to consider these properties just like (1) a kind of the famous relation of mass-energy equivalence and (2) a kind of the relativity concepts by Albert Einstein:

+) For the nature of armor penetration: physical damage simply ignores the armor on effect, while the armor index of the target is not reduced in reality and still remains as original (as shown on the table of stats). Damage from other team mates will NOT benefit from the armor penetration indexes.

+) Armor reduction, on the contrary, reduces the armor index of the target in reality (as shown on the table of stats, and always in a short duration of a few seconds); damage from other team mates will also benefit from the reduction (increase, within the duration).

In other words: armor reduction on affected targets is applied for damage from all sources (including damage from other team mates), but is limited in time by a short duration: the armor indexes of the affected targets will return as originals off the limited duration; while armor penetration of a champion is perpetual, but is limited to apply only for the damage from the holder champion.

+) In case the target's armor has been ignored to zero already, flat armor reduction is an exception among the four forms: it can reduce the target's armor further below zero to deal more damage.  The three remaining forms (flat and percent armor penetration as well as percent armor reduction) CANNOT ignore target's armor further below the value of zero.

Pragmatically, these characteristics are reasonable: any percentage of zero is zero, so percent penetration and reduction cannot ignore target's armor further below zero; for flat armor penetration, you need something to "penetrate", so nothing to penetrate if the armor gets zero already. In the same way, "penetration" should be considered as "personal characteristics of the attackers", so it should not reduce the armor index of the targets in reality, and damage from other team mates should not benefit from it; while "reduction" should be made to be "general power of the attackers", so it should reduce the armor of the targets in reality and increase damage from other team mates on the affected targets as well.

Conclusion

Due to the equivalent nature of AR and HP, for defensive champions (tanky), we have at least two-plus-one ways to increase the defense capability: increase AR or increase HP, or increase both at the same time. Which and how to increase for the best should depend on the particular games, players, champions, items and plans, etc. For example, Nautilus benefits much more from HP[4] in comparison with AR, so we should build HP-focus for the best of playing Nautilus; Rammus, on the other hand, benefits much more from AR[5] in comparison with HP, and we should build AR-focus for the best of playing Rammus.

In the same way, for offensive champions (ADC, mage, assassin,...), we also have at least two-plus-one ways to increase the damage capability: increase AD index or increase AR reduction and AR penetration, or increase both at the same time (we can also increase critical damage, critical chance, and AS as well of course). Which and how to increase for the most effectiveness, just like the defensive indexes, should depend on the particular games, players, champions, items and plans, etc.

This post was completed; there would be just minor adjustments if any on this post
Last update: January 26, 2016

The Silencekeeper

References
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First off, I would remind the DISCLAIMER again: due to the ambiguous nature of the issues which have never been officially confirmed by the publisher – Riot Games, all the references herein are just to back the reasoning and assumption, they can not be considered to be unquestionable proofs to trust.
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[1] [2] [3] According to leagueoflegends.wikia.com, reddit.com, forums.na.leagueoflegends.com:

http://leagueoflegends.wikia.com/wiki/Armor
http://leagueoflegends.wikia.com/wiki/Armor_penetration
http://forums.na.leagueoflegends.com/board/showthread.php?t=2538500&page=3
http://forums.na.leagueoflegends.com/board/showthread.php?t=2929417&page=4#post32707861
https://www.reddit.com/r/summonerschool/comments/424rio/how_do_black_cleaver_and_the_new_last_whisper/
https://www.reddit.com/r/summonerschool/comments/424rio/how_do_black_cleaver_and_the_new_last_whisper/cz7m8oo

http://leagueoflegends.wikia.com/wiki/Magic_resistance
http://leagueoflegends.wikia.com/wiki/Magic_penetration

[4] For more details, look at the following post: Nautling: how to Nautilus? Be the Titan of the Dephs!
[5] For more details, look at the following post: P3 The 'null phase' of a game: bans and picks. How? (Section 3. Some typical examples of my own favorite counter-picks, subsection Strategic counter-picks)

Go to the full discussion: “Issues of AR/ MR reduction/ penetration”