Second-order induction : uniqueness and complexity

Agents make predictions based on similar past cases, while also learning the relative importance of various attributes in judging similarity. We ask whether the resulting "empirical similarity" is unique, and how easy it is to find it. We show that with many observations and few relevant variables, uniqueness holds. By contrast, when there are many variables relative to observations, non-uniqueness is the rule, and finding the best similarity function is computationally hard. The results are interpreted as providing conditions under which rational agents who have access to the same observations are likely to converge on the same predictions, and conditions under which they may entertain different probabilistic beliefs