Using data from an online learning platform Valentin Wagner, Isabell Zipperle and I estimated the rank response function. It is downloadable at RePEc. Here is the abstract and the main figure:
We estimate rank response functions after receiving rank-order feedback in an online learning platform. We find that the shapes of the rank response functions depend on the outcome measure under consideration. For our effort measure, i.e., whether learners continue to interact with the platform, we can reject neither a linear nor a U-shaped rank response function. For our performance measure, i.e., correctly solved exercises, we find no clear pattern overall but suggestive evidence for a linearly decreasing rank response function for individuals in the lower half of the ability distribution, i.e., the lower the rank the lower the performance.