# 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?

• Do you wan't build NN classifier which return probability that the input vector classified to A and B? For example 85% that input vector is A and 15% that B May 13, 2015 at 8:45
• No. I have a vector for A and B and I have also a NN that tells me probability that A > B. I want to go from this to a global ranking. May 13, 2015 at 14:27
• But why you didn't use simple comparison between vectors and get 100% result? May 13, 2015 at 14:34
• What do you mean simple comparison between vectors? There is no simple way from looking at the feature vectors to tell if item A should be ranked higher or lower than item B. That's why I need a NN to predict it. The question I'm asking is how to go from having these pair-wise predictions to a total ordering across a larger set of items. May 13, 2015 at 17:12
• Are you sure that there is transitivity in your rankings? Or might this be like the "rock, paper, scissors" game where A > B and B > C but also C > A? In that case, there is no simple ranking of all items.
– EdM
May 13, 2015 at 17:37