I have a number of categories and a bunch of data about pairwise comparisons between those two categories; how do I learn information about overall rankings/ratings/etc. of all the categories?
Let me provide more detail. I'm trying to learn what the general population is most worried about, by surveying them. I have about 20 categories of potential concerns, call them $A, B, ..., T$. I have picked a random sample of people. For each person in the sample I have done the following. I picked a pair of concerns, e.g., $E$ and $Q$. Then I asked the respondent: which are you more worried about, $E$ or $Q$? They could select one of three possible answers: $E$, $Q$, or "about the same for both". I have about 300 responses. Each respondent got a different, independently random selection of a pair of concerns.
How should I analyze this data, to extract as much information as I can? For instance, can I learn anything about what concerns people are generally most worried about, or least worried about? Can I compute some approximate rankings or ratings or some comparisons? And, of course, I am concerned about computing statistical significance for any conclusions drawn. Any advice?
P.S. Feel free to choose any reasonable model for how people will respond. For instance, you could assume transitivity for each individual (each person has an internal ranking of concerns, and answers the question based on that ranking; so if they would answer that they're more worried about $P$ than $Q$ if asked about $P$ vs $Q$, and would say they're more worried about $Q$ than $R$ if asked about $Q$ vs $R$, then if asked about $P$ vs $R$ they would say they're more worried about $P$ than $R$).