The use of time-delay gravitational lenses to examine the cosmological expansion introduces a new standard ruler with which to test theoretical models. The sample suitable for this kind of work now includes 12 lens systems, which have thus far been used solely for optimizing the parameters of $Lambda$CDM. In this paper, we broaden the base of support for this new, important cosmic probe by using these observations to carry out a one-on-one comparison between {it competing} models. The currently available sample indicates a likelihood of $sim 70-80%$ that the $R_{rm h}=ct$ Universe is the correct cosmology versus $sim 20-30%$ for the standard model. This possibly interesting result reinforces the need to greatly expand the sample of time-delay lenses, e.g., with the successful implementation of the Dark Energy Survey, the VST ATLAS survey, and the Large Synoptic Survey Telescope. In anticipation of a greatly expanded catalog of time-delay lenses identified with these surveys, we have produced synthetic samples to estimate how large they would have to be in order to rule out either model at a $sim 99.7%$ confidence level. We find that if the real cosmology is $Lambda$CDM, a sample of $sim 150$ time-delay lenses would be sufficient to rule out $R_{rm h}=ct$ at this level of accuracy, while $sim 1,000$ time-delay lenses would be required to rule out $Lambda$CDM if the real Universe is instead $R_{rm h}=ct$. This difference in required sample size reflects the greater number of free parameters available to fit the data with $Lambda$CDM.
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