In contradiction to @dmckee, I'm not sure that this would easily meet the requirement of an objective primary winning criterion. The common compression formats are really rules for how to uncompress, and there aren't canonical levels of CPU effort to expend in compression. So even GNU zip with the same options may give slightly different results for the same input in different versions. In fact, some zip/gzip implementations are non-deterministic and may give sizes differing in a few bytes according to the contents of your entropy pool.
If you try specifying versions as "GNU zip 3.0.2 with options -9" then it's possible that a lot of people would have to find the source and compile it themselves. Technically that doesn't make it subjective, but it does make it a lot harder to check.
Also, bearing in mind that most archive formats include the original filename, you're going to have to either specify single-character names or penalise people whose languages use long filename extensions.
If you require people to submit the compressed versions of the file then it is objective, but there are some serious disadvantages. All the submissions will be base-64 encoded and opaque, and it gives an advantage to people who are able to run non-deterministic compressors on lots of high-powered computers.