Surely this sort of problem must have a name. Because it does not have a well defined answer, I am asking for (1) the name of this type of problem and (2) for general approaches to this question that make various kinds of assumptions.

For a clear example of why this problem does not have a well defined answer, take the following example. Let there be three teams, A, B and C (associated with rows/cols 1,2,3 respectively) with the true probability of winning matrix of row team beating col team as $$ \begin{bmatrix} - & 0.5 & 0.5 \\ 0.5 & - & 0.9 \\ 0.5 & 0.1 & - \end{bmatrix}$$

If at the current point in a season each team has played the other an equal number of times, Team A's expected win percentage is 50%, Team B's expected win percentage is 70%, and Team C's expected win percentage is 30%.

Now suppose Team C faces Team A. Team C's win percentage is 30% and Team A's win percentage is 50%. A naive approach is to assume the probability of C winning is 40%. However, the true probability is higher at 50%.

In essence, by summarizing a teams performance with a win percentage statistic, a lot of information is lost about individual games. And also, a lot can be manipulated to non-intuitive effect by varying the values in the matrix.

However, it seems to me that the values in the matrix are not arbitrary and must be based on some idea of the comparative skills of a team. My question is what sorts of assumptions can we place on these values so that given the win percentages of each team (assuming equal number of games against all teams) we can estimate accurately the probability of one team beating another in a specific matchup?

(My only thought at present is to minimize the variance of the entries in the matrix along rows and columns. But surely this problem has been studied well.)


One way of doing so is through the theory of ELO scores. For team $i$ playing against team $j$, we model the probability of team $i$ beating $j$ as: $P_{ij}=\sigma(c(R_i-R_j))$, where $\sigma(x)=\frac{1}{1+e^{-x}}$.

Here $R_i,R_j$ are the ratings of team $i,j$ respectively. Thus you can train a logistic regression model to estimate $c,R_i,R_j$, since you know the given $P_{ij}$s. This will allow you to compare any pair of teams.


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