A Novel Method to Determine the Best Player in a Team

This method can be used in teams games where the first to a fixed number of points wins and where players gain points that are the result of a specific individual on the opposite team playing poorly. This type of analysis would work for a game like Halo or any first person shooter – by killing a specific enemy player you gain a point. The basic idea is that if you die a lot you’re bad because the enemy gains points from you dying (because they killed you) and killing a lot is good.

I’ve been in countless arguments about what the best method is to determine the best player using only the information of kills and deaths in the game. Some players argue that the kill death ratio (kills divided by deaths) is the best measure of skill. I used to take the stance that kills minus deaths is the best measure. But I’ve created a novel measure that takes account of all available information to determine the bets player.

Hypothetical scenario: you’re going to play in a game of first 50 kills in a 2v2 against the strongest opponents in the world. You know that in any given encounter the probability you kill is 0.3 and the probability you die is 0.7. You may choose either of the following team mates.

xXx1337sNiPeRXxX who is guaranteed to have 47 kills 48 deaths.
DogrA_aRMA who is guaranteed to have 2 kills and 1 death.

Using either the kill death ration or kills minus deaths would lead you to believe that DogrA_aRMA was the best player to choose. But in fact your chances of winning are dramatically higher with xXx1337sNiPeRXxX.

With xXx1337sNiPeRXxX you would need 3 kills to win the game. The enemy needs 2 kills. There can be at most 3 + 2 – 1 = 4 encounters, this is because the closest the kills at the end can be is 50 to 49. It can therefore be assumed that there will be exactly 4 encounters and that you need to have 3 or more in these encounters to win. It doesn’t matter that some of these kills would result in games where one team has over 50 kills – it still is equivalent to enumerating the permutations of wining when there are no kills beyond the limit.

The probability of winning takes a binomial distribution. The chances of you winning with xXx1337sNiPeRXxX is

With DogrA_aRMA there are at most 97 events. You need at least 48 of these to be kills. The probability of winning is therefore

As you can see, this is dramatically lower than with the other player, even though our measures of his kill death ration and kills minus deaths was higher. Strange indeed. It suggests that these statistics are flawed.  In the kill death ratio the information of the number of encounters is completely lost. With kills minus deaths the number of encounters is also lost but you can determine a minimum number of encounters from it. But then how should we figure out who the best player is?A new method is needed. One in which no information is lost… a method which will put and to the debate. One statistic to rule them all. It works as follows.

The basic idea to evaluate the strength of a player is to figure out the team’s probability of winning without player by replacing him with a player who has the skill level of the average of the rest of the team. This value is then subtracted from the probability of the team winning a rematch with no replacements. This gives us the score.

Not listed is the probability of Team 1 and 2 winning which is 0.159661361932217 and 0.840338638067789.

The formula for column H is

This one minus the probability that you lose and don’t get the kills required to win. It’s done like this because of how the BIDOMDIST function in spreedsheets works. 119 is the maximum number of encounters and 59 is the least number of deaths you need to lose.

The higher the score, the more that player did towards making his team win the game. Here, it shows player K made the biggest contribution to the game, player L came in second and D in third. The scores do not represent skills when comparing between teams. But within the teams they do. So we know that within team 2 the best to worst is K, L, J, G, H and for team 1 is D, E, F, B, A, D. It would be wrong to say that player D is better than player G due to his score being higher.

Download the open source spreedsheet here if you’re interested.


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