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In a recent discussion with friends we were discussing win-rates in a video game. One person suggested it made sense that every champion could have a less than 50% win rate, while another suggested this was impossible because every loss suffered for a champion would result in a win for another.

In practice it seems highly unlikely to ever have a <50% win rate for every character in a team based game like a MOBA, however we weren't able to come to the an answer and none of us are good enough at statistics for a definitive answer.

Some constraints and details to consider given the ruleset of a MOBA like League of Legends:

  • Win rate is defined as the percentage of games won for a specific character.
  • Every game has an equal number of winning and losing players (5 on each side). I would prefer to disregard scenarios in which a player leaves in which case there would be 4 on one side and 5 on the other.
  • Each player will select the character they play before the game begins from a list of characters. This list is larger than 10 so their will be characters whose win rates are not affected if they do not appear in a game.
  • Players are unable to switch characters mid game.
  • Each character may appear once on each team, which means their are a minimum of 5 characters and a maximum of 10 appearing in each game.
  • If a character is on both teams in a game the result would count as both a loss and a win for that character, otherwise if the character is "unmirrored" it would only be a win or a loss depending on the result.
  • Draws are not allowed, or at least do not affect win rates.
BenHaef
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    For those of us not familiar with MOBAs, could you explain how a character's win rate is defined? – Karl Oct 10 '23 at 00:55
  • Win rate in this case is defined as a percentage of games won out of games played for a specific character across all players. There is a subcategory known as "unmirrored winrate" which would be win rate when the enemy team does not feature the same character, but this question is primarily concerned purely with normal win rate. – BenHaef Oct 10 '23 at 00:58
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    Does each game have the same number of winners and losers? – Karl Oct 10 '23 at 00:59
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    Yes, in a Moba there is 5 winning players and 5 losing players. Team sizes are always the same. However, each player can choose different characters, which means the characters specific win rate may or may not be affected based on if they are selected by any of the players for a specific game. Additionally each character is only select-able once per team. They are unable to change once selected. – BenHaef Oct 10 '23 at 01:01
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    @BenHaef Might be worth noting that even if you did consider unmirrored winrate, you still have the same number of wins as losses recorded for each game (e.g., when there are two mirrors, then $3$ wins and $3$ losses will be recorded), so Karl's answer still applies. – M W Oct 10 '23 at 02:36
  • You need to specify further constraints, such as "games have, at maximum, 1 instance of any given character" and "every game ends in a win for one team and loss for the other." This would be much easier when you specify the particular game and game mode. DotA doesn't allow same characters in normal game modes, LoL does (on different teams though.) DotA also has (fun) game modes with more than 2 teams. – knallfrosch Oct 10 '23 at 10:04
  • "this was impossible because every loss suffered for a champion would result in a win for another" - that is correct, and a good, simple way to think about it (putting edge cases aside). – NotThatGuy Oct 10 '23 at 11:07
  • Please edit necessary detail into the question. – Carsten S Oct 10 '23 at 11:31
  • Is MOBA an acronym? – Teepeemm Oct 10 '23 at 14:33
  • "this [win rate < 0.5] was impossible because every loss suffered for a champion would result in a win for another" - does this game allow draws? – Trang Oul Oct 10 '23 at 14:41
  • @Teepeemm I assume Multiplayer Online Battle Arena. The second entry from Google is Metro Omaha Builders Association, so probably not that. – Michael Richardson Oct 10 '23 at 14:45
  • MOBA in this case is referring to Multiplayer Online Battle Arena's like League of Legends, Dota, and Smite. I've detailed some constraints and clarifications in the question. Let me know if there is anything else that would help. – BenHaef Oct 10 '23 at 16:49

4 Answers4

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Let $W_i$ be the number of games won by character $i$ and $L_i$ the number of games lost by character $i$. Since every game has the same number of winners and losers, the total number of player wins is equal to the total number of player losses:

$$\sum_iW_i=\sum_iL_i.$$

But if every character's win rate is less than 50%, we have $W_i<L_i$ for each $i$, which implies

$$\sum_iW_i<\sum_iL_i.$$

Karl
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The tiniest of side notes that I couldn't put in a comment because I don't have the reputation yet: You can be on the winning team and still get a loss by having abandoned and then rejoined the game, or by having your team win 4v5. I'm not sure how Riot counts these edge cases specifically when combining data, but considering the player receives a loss for all intents and purposes on their end, one could say that the average win rate is in fact below 50% considering no game can have above 5 winners, but technically one (or more) people on the winning team can "lose".

This is similar to why average K/D scores in shooter games is below 1 unless there are revival mechanics, you have people killing themselves every so often and you get a death in the game without a kill.

So technically if every champ hovered at almost exactly 50% win rate, you might overall get 49.9% win rates across the board.

Another note is that the datasets any collection site for LoL uses is incomplete. There have been occasions where they are more keen on picking up data from players actively using those sites, and in those cases you're usually looking at more "hardcore" gamers, leading to statistics showing an overall average win rate of over 50%. Last time I saw this was several years ago though.

Tony
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  • Another potential edge case is if someone leaves before picking a character (if such matches continue and stats are recorded for them). – NotThatGuy Oct 10 '23 at 11:13
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none of us are good enough at statistics for a definitive answer.

The only thing you really need to know about probabilities and statistics to solve this is that expectation is linear, which is a fancy phrase to say that the expected total number of winners must be equal to the weighted average of the participating characters' winrates.

Since you've confirmed in comments that the number of winners is always 50%, then the characters' winrates must average to 50%.

An average is always bounded between the min and the max; if the average winrate is 50% then there must exist at least one character with a winrate below or equal to 50% and a character with a winrate above or equal to 50%; and if there exists a character with a winrate strictly less than 50%, then there must also exist a character with a winrate strictly greater than 50%.

Stef
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    Note that the characters' winrates will average to 50% if either (a) they all play the same number of games, or (b) you do a weighted average, weighted by the number of games played. Otherwise you can get a situation where (for example) player A wins one game against players B through Z, with no other games played, leaving one person with a win rate of 100% and 25 players with a win rate of 0%. A straight average over players would yield something like a 4% average win rate. – Michael Seifert Oct 10 '23 at 17:06
  • @MichaelSeifert Exactly. That's why I say "weighted average" in the first paragraph. – Stef Oct 10 '23 at 17:51
  • ::: smacks self on head ::: How did I miss that? – Michael Seifert Oct 10 '23 at 17:59
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The answer is "yes", but it's because of a detail you didn't consider - it is possible for games to involve more players on one side than the other. The game wouldn't be balanced, but it would still be a game.

Exploiting this loophole means the desired <50% win rate is possible. For simplicity, consider a 3v3 MOBA and there are 6 champions: A, B, C, D, E, F.

  • Game 1: A + B + C vs. D, D wins.
  • Game 2: A + D + F vs. E, E wins.
  • Game 3: A + B + E vs. F, F wins.
  • Game 4: D + E + F vs. A, A wins.

Now every champion has <50% win rate.

If you impose the condition that every game must have six players, and ignore other loopholes like leavers causing an extra "loser", then it's not possible for every character to have <50% win rate, per Karl's answer.

Allure
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