Solved – Winning probability in a game with multiple players

gamesprobability

I do not come from a mathematical background, and hope you can answer this (probably very basic?) question.

I got a game group with friends where we play board games, normally 3-6 players each night. I have created an application which keeps track of game history, and keeps track of a Elo rating for each player. These Elo ratings can be used to determine the winning probability between two players.

However, how can I use these winning probabilities to get the probability of a player winning a multi-way game?

Let's say we have a three way game, two players with a Elo rating of 1200 and one with 1400. The two with 1200 have 50% chance of winning against each other and the 1400-player has a 76% chance of winning in a heads up match against a 1200-player. How can I find the probabilities that each player wins the three-way game?

Best Answer

If you assume that the weaker players won't gang up on the stronger player (a very strong assumption!), then a reasonable model would be the following. (I'm following the notation of the "theory" section of the Wikipedia article on ELO.)

  • let $R_A, R_B, R_C$ be the ratings of the three players.
  • let $Q_A = 10^{R_A/400}$; define $Q_B, Q_C$ similarly.
  • the probability of $A$ winning is $Q_A/(Q_A + Q_B + Q_C)$ and similarly for the other two players.

With the numbers you gave, $C$ has a probability of about $0.613$ of winning, and $A, B$ each have probability $0.194$ of winning.

This seems like the "obvious" generalization of the Elo math. The most obvious problem, to me, is that I wouldn't know how to update these ratings after a multi-player game is played.

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