Suppose you have $n$ players play a round robin tournament. A win adds 1 to the winning score of a player while a loss adds nothing. After the round robin tournament is finished the players are assigned ranks based on their scores, with ties being broken at random, the top player receiving a rank of $n-1$ and the bottom player receiving a rank of $0$. What is the relationship between the expected value of the $\textbf{score}$ of player $i$ and the expected value of the $\textbf{rank}$ of player $i$?
Results of a sample tournament with 5 players: $scores=(3,3,2,1,1)$ and $ranks=(3,4,2,1,0)$
Addition information: A player $i$ has an attribute $\lambda_i$. The probability that player i beats player $j$ in a game is $\frac{\lambda_i}{\lambda_i+\lambda_j}$.
Numerical simulations suggest that the two ($\textbf{score}$ and $\textbf{rank}$) are perfectly correlated if taken across all the players, but I don't know how to prove this. Any references would be appreciated.
You don't want the expected value of the rank of player $i$. It is what it is. You are claiming that the expected value of the score matches the rank. Look at two players $i$ and $j$ with attributes $\lambda_i$ and $\lambda_j$. The expected score for $i$ is $\sum_{k\neq i}\frac {\lambda_i}{\lambda_i+\lambda_k}$ If $\lambda_i\lt\lambda_j$, the expected score for $i$ is clearly less than that for $j$, term by term.