Player 1 and Player 2, henceforth called P1, and P2 respectively are in competion with each other. The winner of the competion is the player who wins the most number of games (not the player with the cloestest ratio of wins:loses being 1:1). P1 and P2 play the games on independent devices from each other, start at the same time, and stop when they played all the games they can within the time limit.
P1 choses to play the games as fast as possible, meaning he can play as many games as P2, plus some number $A$. Therefore, the number of games P1 can play is $N+A$. The probability of him winning a game however is only $ℙ$ $(0 < ℙ < 1)$.
P2 chooses to play slower, meaning the probability that he wins is just the same as P1, plus some number $B$. Therefore, probability of P2 winning a game is $ℙ+B$ $(0 < ℙ+B < 1)$ & $(ℙ < ℙ+B)$. However, the number of games P2 can play is only $N$.
What is the probabilities for each player winning the competition?