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Published May 24, 2022 | Submitted
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Aggregation of Models, Choices, Beliefs, and Preferences


A natural notion of rationality/consistency for aggregating models is that, for all (possibly aggregated) models A and B, if the output of model A is f(A) and if the output model B is f(B), then the output of the model obtained by aggregating A and B must be a weighted average of f(A) and f(B). Similarly, a natural notion of rationality for aggregating preferences of ensembles of experts is that, for all (possibly aggregated) experts A and B, and all possible choices x and y, if both A and B prefer x over y, then the expert obtained by aggregating A and B must also prefer x over y. Rational aggregation is an important element of uncertainty quantification, and it lies behind many seemingly different results in economic theory: spanning social choice, belief formation, and individual decision making. Three examples of rational aggregation rules are as follows. (1) Give each individual model (expert) a weight (a score) and use weighted averaging to aggregate individual or finite ensembles of models (experts). (2) Order/rank individual model (expert) and let the aggregation of a finite ensemble of individual models (experts) be the highest-ranked individual model (expert) in that ensemble. (3) Give each individual model (expert) a weight, introduce a weak order/ranking over the set of models/experts, aggregate A and B as the weighted average of the highest-ranked models (experts) in A or B. Note that (1) and (2) are particular cases of (3). In this paper, we show that all rational aggregation rules are of the form (3). This result unifies aggregation procedures across different economic environments. Following the main representation, we show applications and extensions of our representation in various separated economics topics such as belief formation, choice theory, and social welfare economics.

Additional Information

The authors gratefully acknowledge support from Beyond Limits (Learning Optimal Models) through CAST (The Caltech Center for Autonomous Systems and Technologies) and partial support from the Air Force Office of Scientific Research under awards number FA9550-18-1-0271 (Games for Computation and Learning) and FA9550-20-1-0358 (Machine Learning and Physics-Based Modeling and Simulation). The first version of the paper was written during the first author's Ph.D. studies with many helpful comments from Federico Echenique and Kota Saito. The first author thanks his Ph.D. advisors Jaksa Cvitanic, Federico Echenique, Kota Saito, and Robert Sherman. For helpful discussions, the first author thanks Itai Ashlagi, Kim Border, Martin Cripps, David Dillenberger, Drew Fudenberg, Simone Galperti, Michihiro Kandori, Igor Kopylov, Jay Lu, Fabio Maccheroni, Thomas Palfrey, Charles Plott, Luciano Pomatto, Antonio Rangel, Pablo Schenone, Omer Tamuz, and Leeat Yariv.

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August 20, 2023
October 24, 2023