As someone who has past lives both as a litigation attorney and as electronic discovery director hired by the Cadwaladers of the world (or their clients directly) to structure and execute large document reviews like this, I can tell you the document searching is done as part of the document collection process. Search terms are heavily negotiated between the parties to the action, then an outside company runs the search terms and then pulls all documents with a hit to a database, which is then indexed and integrated with coding forms. There is virtually no human interaction with the documents at this stage, and there are an incredible number of documents that get pulled to the database and that must be reviewed by human eyes that contain search term hits that are not truly responsive to the document request-- imagine every piece of spam or personal email you get, then imagine how many might have words like basketball, attendance, paper, class, roy, williams, ugly, shade, blue . . . you get the idea. The law firm then hires a team of 10 - 100+ outside attorneys to review the documents to see if they're truly responsive to the inquiry, with broad issue checkboxes to help further sort the issues of the case. All documents deemed unresponsive are put aside and in the vast majority of cases will never see the light of day again. All responsive documents are then subject to secondary review by law firm associates.
The initial collection and indexing of data is a separate and substantial cost-- must admit I never paid one of these bills, so I can't estimate cost per document, but could approach or even surpass the cost of the first pass review by contract attorneys. Cost for the first pass team of attorneys is somewhere between .45 and .55 per document, so 200,000 documents would cost around $100,000 for the first pass. The second pass will be between 5% and 20% of the original document count depending on how pointed the search terms were and how responsive the documents were, with the big firm's junior associates charging $300+/hr. and a couple senior associates overseeing at $500+/hr. and conferring with the $1K+/hr. partner on the case as needed. It gets expensive quickly.