Team's seasonal win probabilities

We consider a competition involving $r$ teams, where each individual game involves two teams, and where each game between teams $i$ and $j$ is won by $i$ with probability $P_{i,j} = 1 - P_{j,i}$ . We suppose that $i$ and $j$ are scheduled to play $n(i,j)$ games and say that the team that wins the most games is the winner of the competition. We show that the conditional probability that $i$ is the winner, given that $i$ wins $k$ games, is increasing in $k$ . We bound the tail probability of the number of wins of the winning team. We consider the special case where $P_{i,j} = {v_i}/{(v_i + v_j)}$ , and obtain structural results on the probability that team $i$ is the winner. We give efficient simulation approaches for computing the probability that team $i$ is the winner, and the conditional probability given the number of wins of $i$ .