This is so far a very incomplete answer. I suppose 200 items, which we sample independently and uniformly. If there is any prior information on the ordering, one could maybe do better than that. I will represent the items as the integers $1,2,\dotsc, 200$ and suppose (in the simulations shown) that the true ordering (for one person) is the usual ordering of integers. Then I suppose that the response on the pairs given is without error. We will need later to pass by such strong assumptions. So the question is: "how many pairs do we have to sample to get to know the true ordering?"
I will answer a some what different question. We will represent the sampled pairs as a directed graph, calculate the connected components of that graph and find the length of the longest connected component. First I show a plot of a such graph, made with the
This graph is simulated as above with 50 vertices and 50 pair comparisons, its largest component has 42 vertices and there is 8 isolated vertices. Looking closer at the largest component:
From this graph we can see that, for instance, both 17 and 21 is between 6 and 39, but there is no comparison between 17 and 21 themselves. So from such a graph we can calculate some fuzzy ranking, but it should be clear that if you want a complete ranking, the number of pair comparisons required will be very large!
To advance more I guess we will need some probability model on rankings, I will try to come back with that.