What's the best way to convert pair-wise ranking to a ranked list? I have a set of ordered items A > B > C ... > F. For each element of the set I have a feature vector. Using these features I trained a neural network to predict the probability that A > B for any pair of items A and B. The neural network predictions are noisy. The output from the network may not be perfectly consistent with the true ranking. 
My question is how do I go from having these pairwise ranking probabilities to a best-guess for the total ordering of the items in the set?
 A: If you have sufficiently few items to do an exhaustive search of all orderings, decide on a criterion for what a "best guess" means for the overall ordering, and report the ordering that maximizes your criterion. For example, a simple "best-guess" criterion could be the number of A:B pairs that were called in the correct order (A>B vs B>A) by the neural network, regardless of the probability values. (The proportion of correct pairwise rankings is the basis of the area under the curve, AUC, criterion for evaluating receiver-operating-characteristic curves.)  Or you might want to add a criterion that more strongly weights the neural-network calls made with high probability. But you would have to choose such weights, depending on what you mean by "best guess."
A: Assuming that pair-wise rank means a series of binary values - either 0 or 1 depending on whether the team won or lost (is better or worse) against another team and there are a multiple such binary outcomes for different teams, then a overall ranking can be built using the Bradley Terry model which will convert these series of binary scores to a real number which can then be compared against one another. The team with the higher value is ranked higher.
The advantage of these type of models is that there need not be n(n-1)/2 comparisons. As long as all teams are sampled enough times, a tiny subset is enough to create a reasonable ranking list. In your case, since the results are noisy (and also automatically generated you may want to consider more combinations. Alternatively strive for higher accuracy in your model (maybe even manually classifying a few samples) and then even very few pair-wise results will help generate a reasonable scores
