For the problem of efficiency-robust estimation of location, approximate versions are developed of optimally robust Pitman-type estimators. These are shown to have full asymptotic efficiency for a prototype family of distributions, used to define the estimators. The asymptotic efficiency under other distributions of interest is also characterized. For sample sizesn= 20, 30, 40, 50, and 100 the efficiencies were estimated by Monte Carlo methods under the following distributions: normal, logistic, double-exponential, and contaminated normal (one percent, five percent, ten percent). Over all these shapes the efficiencies obtained are approximately 88 percent or more for alln; they rise to approximately 91 percent or more forn= 100. Some theoretical and numerical comparisons with other estimators are given.