  In the typical analysis of a data set, a single method is selected for
statistical reporting even when equally applicable methods yield very different
results. Examples of equally applicable methods can correspond to those of
different ancillary statistics in frequentist inference and of different prior
distributions in Bayesian inference. More broadly, choices are made between
parametric and nonparametric methods and between frequentist and Bayesian
methods.
  Rather than choosing a single method, it can be safer, in a game-theoretic
sense, to combine those that are equally appropriate in light of the available
information. Since methods of combining subjectively assessed probability
distributions are not objective enough for that purpose, this paper introduces
a method of distribution combination that does not require any assignment of
distribution weights. It does so by formalizing a hedging strategy in terms of
a game between three players: nature, a statistician combining distributions,
and a statistician refusing to combine distributions. The optimal move of the
first statistician reduces to the solution of a simpler problem of selecting an
estimating distribution that minimizes the Kullback-Leibler loss maximized over
the plausible distributions to be combined. The resulting combined distribution
is a linear combination of the most extreme of the distributions to be combined
that are scientifically plausible. The optimal weights are close enough to each
other that no extreme distribution dominates the others.
  The new methodology is illustrated by combining conflicting empirical Bayes
methodologies in the context of gene expression data analysis.
