Pareto-efficiency is a very important property, but in some cases it cannot be attained (for example, when agents play a game such as the Prisonner's Dilemma). In such cases, it can be useful to measure how far the outcome is from being Pareto-efficient.
A simple measure of approximate Pareto-efficiency (which I found e.g. here and here) uses a multiplicative factor: an outcome is $r$-approximately-efficient if there is no alternate outcome in which each agent’s utility increases by a factor of at least $r$. However, this measure assumes that agents have cardinal utility functions. In economics, it is usually assumed that agents' utility functions are just representations of preference relations. In particular, applying a monotone transformation to a utility function leads to a different function that represents the same preferences. If we apply a monotone transformation to a utility function (even a simple one such as adding or subtracting 1), the multiplicative approximation ratio changes.
MY QUESTION: what is a notion of approximate Pareto-efficiency, which is meaningful even when agents have ordinal preferences?