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For a 5-a-side football match, 2 teams (A and B) are created by randomly picking players from a pool of 100.

The problem is to predict the probability of team A winning.

All these players have played with and against every other player from the pool many times before. Thus the win % of every player when paired with a particular player is known. Similarly, win % of every player when playing against a particular player is known.

There is a strong interaction between players, i.e. some players complement each others strengths when playing together, while some players are weak when playing against particular players.

How do we then calculate the probability of Team A winning while taking these interactions into account?

Aditya Gadgil
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    This can't be answered if you can't precisely specify how players complementing each other affect the percentages. – Henrik supports the community Apr 06 '17 at 20:36
  • We know that when player 1 and player 2 play together the average win % of their team is x. When player 1 plays against player 2 the average win % of player 1's team is y. We know these values for every player (1-100) with and against every player. Is it then possible to calculate the win probability of a team of random players against another random team? – Aditya Gadgil Apr 06 '17 at 21:01

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This is a complex question. Any answer will depend on lots of (implicit) assumptions. Here's one possible way.

For team A with 5 players there are 10 pairs of players. You know the 10 win probabilities for each pair playing together on the team. Average those to get the team "strength" independent of the opposition.

Now do the same for team B.

For each of the 25 pairs of players (one from each team) you know the win percentages (for A over B - some of those will be less than 0.5). Average those.

Decide how you you want to combine (weight) these three summary averages.

You might want to try out this algorithm for teams of 2 players chosen from a small pool (not 100), using made up data with some extreme values, to check that it produces sensible answers. You should be able to predict the answers if there's one superstar who always wins, or one person who always loses. But what would happen if they were on the same team?

Ethan Bolker
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  • Thanks for you reply Ethan. I was wondering if there was a mathematically sound way to combine these probabilities rather than trial and error. – Aditya Gadgil Apr 06 '17 at 21:04