# Probability of win between two agents with known win rates

Probability theory and statistics

### Probability of win between two agents with known win rates

Hey there,

Assuming that we have 2 agents playing the same game in the same environment of other players, with one of them having win rate of e.g. 56% and the other of e.g. 52%, can we actually say what the expected win chance would be in the matchup between them, without any further assumptions (e.g. without assumption of Elo rating distribution of "game skill" or anything like that)?

From common sense in the above scenario probability of the first player winning should be some p higher than 0.5 (as his win rate is higher than his opponent, suggesting him being better) and lower than 0.56 (as his enemy is better than average, with win rate >50%). Moreover, naturally, completely symmetrical but reverse matchup should give the second player winrate 1-p with the same equation.

Is there a closed form equation for the exact value of p?
kristinabui8

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### Re: Probability of win between two agents with known win rat

kristinabui8 wrote:Hey there,

Assuming that we have 2 agents playing the same game in the same environment of other players, with one of them having win rate of e.g. 56% and the other of e.g. 52%, can we actually say what the expected win chance would be in the matchup between them, without any further assumptions (e.g. without assumption of Elo rating distribution of "game skill" or anything like that)?Tweakbox Appvalley

From common sense in the above scenario probability of the first player winning should be some p higher than 0.5 (as his win rate is higher than his opponent, suggesting him being better) and lower than 0.56 (as his enemy is better than average, with win rate >50%). Moreover, naturally, completely symmetrical but reverse matchup should give the second player winrate 1-p with the same equation.

Is there a closed form equation for the exact value of p?

issue got solved

kristinabui8

Posts: 4
Joined: Thu Jul 30, 2020 4:54 am
Reputation: 1