How much freedom does an effectiveness metric really have?

It is tempting to assume that because effectiveness metrics have free choice to assign scores to search engine result pages (SERPs) there must thus be a similar degree of freedom as to the relative order that SERP pairs can be put into. In fact that second freedom is, to a considerable degree, illusory. That is because if one SERP in a pair has been given a certain score by a metric, fundamental ordering constraints in many cases then dictate that the score for the second SERP must be either not less than, or not greater than, the score assigned to the first SERP. We refer to these fixed relationships as innate pairwise SERP orderings. Our first goal in this work is to describe and defend those pairwise SERP relationship constraints, and tabulate their relative occurrence via both exhaustive and empirical experimentation. We then consider how to employ such innate pairwise relationships in IR experiments, leading to a proposal for a new measurement paradigm. Specifically, we argue that tables of results in which many different metrics are listed for champion versus challenger system comparisons should be avoided; and that instead a single metric be argued for in principled terms, with any relationships identified by that metric then reinforced via an assessment of the innate relationship as to whether other metrics are likely to yield the same system-versus-system outcome.