Let $[n]$=$\{1,\dots,n\}$ be a set of players in a round-robin tournament. Each player $i$ has an associated skill parameter, $\lambda_{i}$, and the probability that player $i$ defeats player $j$ is $\frac{\lambda_{i}}{\lambda_{i}+\lambda_{j}}$ (ordinary Bradley-Terry comparison). Once a tournament has concluded the players are ranked by their score, with ties being broken at random, so if a score vector was $(3,2,3,0,2)$ then a valid rank vector would be $(4,2,5,1,3)$.

The expected score for player $i$ is $\mathbb{E}[\lambda_{i}]=\sum\limits_{j\not=i}{\frac{\lambda_{i}}{\lambda_{i}+\lambda_{j}}}$. Is there a closed form expression for the expected rank?

estimatingthe skill parameters (by maximum likelihood) based on a full round robin, the ranks of the estimated parameters will match the ranks of the winning percentages of the players. – cardinal Dec 31 '13 at 16:06