This is an investigation of stochastic choice when only the average of the choices is observable. For example when one observes aggregate sales numbers from a store, but not the frequency with which each item was purchased. The focus of our work is on the Luce model, also known as the Logit model. We show that a simple path independence property of average choice uniquely characterizes the Luce model. We also characterize the linear Luce mode, using similar tools. A linear version of the Luce model is used most frequently in empirical work by applied economists.
Our characterization is based on the property of path independence, which runs counter to early impossibility results on path independent choice. From an empirical perspective, our results provide a small-sample advantage over the tests of Luce's model that rely on estimating choice frequences.