Multiplayer ELO Rating & Matchmaking System

Design of a multiplayer rating & matchmaking system, which could be applied to various card/board games and esports

Game developers rely on these algorithms to make each game as fair as possible for the user. Fair matches incentive players to keep returning as it provides greater satisfaction for users and limits the feeling of being cheated someone may have when they are matched against people with skill levels far superior to their own.

Extending Elo to multiplayer games

Expected scores

Elo only knows how to compare two players at a time. Let’s reframe our multiplayer game as a combination of pairwise matchups. For example, a game with three players has three distinct matchups (A vs. B, A vs. C, and B vs. C). More generally, a game with N players has N(N-1)/2 distinct pairwise matchups.

Let’s use standard Elo to compute all pairwise expected scores using the formula from the previous section. Then we’ll sum each player’s individual expected scores to get a total expected score for each player. Lastly, we’ll scale the scores so that they sum to 1 across all players (allowing us to interpret the scores as probabilities, like we do in standard Elo). In mathematical terms, our expected score for player A looks like this:

Equation 4: Expected score calculation for multiplayer Elo. I’ll leave it up to the reader to validate that the sum of all players’ expected scores is 1, and that this equation converges to the standard Elo equation when N = 2.

The equation looks a little unwieldy, but it’s really not so different from the standard Elo process (compare to Equation 1 above). Step one, done.

Poker fans: one thing to note is that Elo rating is for tournament style games including MTT and Ladder/SNG poker game, so it’s always coming with same entry position, clear result of game ranking, and so on. This type of rating system couldn’t be applied to cash games as it is not purely skill-based with same stake from every player.