Efficient algorithm for comparing one-on-one tournaments

I was thinking of doing a simple program, which basically has a list of some items - such as foods, for an example - and which pairs two foods against each other for user to choose his or her favourite out of those two options. Ultimately, a list of ranked candidates would emerge. Behind the scenes, I thought two really simple "algorithms" to work by.

1. Working through each pairing possible to score the list perfectly. Seems really tedious for the user when there are many items in the list.

2. Simple elimination. Two foods enter, one leaves for the next round. But I can't really then get a rank, other than just general estimates of categories of performance.

So I was wondering, what other ranking systems there would be for one-on-one "tournaments" like this?