  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.
