  The weighted average is by far the most popular approach to combining
multiple forecasts of some future outcome. This paper shows that both for
probability or real-valued forecasts, a non-trivial weighted average of
different forecasts is always sub-optimal. More specifically, it is not
consistent with any set of information about the future outcome even if the
individual forecasts are. Furthermore, weighted averaging does not behave as if
it collects information from the forecasters and hence needs to be extremized,
that is, systematically transformed away from the marginal mean. This paper
proposes a linear extremization technique for improving the weighted average of
real-valued forecasts. The resulting more extreme version of the weighted
average exhibits many properties of optimal aggregation. Both this and the
sub-optimality of the weighted average are illustrated with simple examples
involving synthetic and real-world data.
