Second-order induction in prediction problems

Significance How do people generate beliefs about economic, political, and social events? A natural formula for the predicted value of a variable is its weighted average value in the past, where past values are given a higher weight if they were observed in circumstances more similar to the current ones. Agents learn from the data the best way to assess the similarity of past cases to the present one. The basic formula and this learning process appeared both in statistics and in psychology and they thus make sense for modeling economic agents. We study this process and identify circumstances under which agents are likely to agree on predictions and conditions under which disagreement over predictions may be reasonably expected. 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 “empirically optimal similarity function” (EOSF) 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, nonuniqueness is the rule, and finding the EOSF 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.

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