Is there a known algorithm to solve this "playoff ranking" problem?

I would like to compute the probabilities of "playoff rankings" given
a set of n teams playing a fixed set of pairwise games (with
probabilities of each team winning at a certain percentage), some
pairs occur more than once and some pairs not at all.

Now, it seems like I could formulate this as a graph problem with
teams as vertexes (or verticies if you prefer) and games as edges,
labelling the edges with the probabailities, perhaps encoding it all
as an adjacency matrix. Then, I could compute some sort of
transititive closure to get the final probabilities of each possible
outcome.

However, it also seems I could do something with a search method (like
minimax), and cut-off evaluating some games once I know they won't
effect the outcomes I'm interested in.

More importantly, this seems like the kind of problem that someone
somewhere has already studied and there are some off-the-shelf
approaches I could simply program.

The program I am comparing to, does the analysis by Monte Carlo
simulation (see www.nfl-forecast.com), but it seems to me that there
must be a more closed-form way of calculating the results.

So, please enlighten me on some well-known algorithms that seem
relevant to my problem (and pointers to papers that explain them,
hopefully relatively simply, like one might read in an undergraduate
class, as I'm more than 25 years past my last schooling).

Thanks,
-Chris
.

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