  The main topic of this paper are algorithms for computing Nash equilibria. We
cast our particular methods as instances of a general algorithmic abstraction,
namely, a method we call {\em algorithmic boosting}, which is also relevant to
other fixed-point computation problems. Algorithmic boosting is the principle
of computing fixed points by taking (long-run) averages of iterated maps and it
is a generalization of exponentiation. We first define our method in the
setting of nonlinear maps. Secondly, we restrict attention to convergent linear
maps (for computing dominant eigenvectors, for example, in the PageRank
algorithm) and show that our algorithmic boosting method can set in motion {\em
exponential speedups in the convergence rate}. Thirdly, we show that
algorithmic boosting can convert a (weak) non-convergent iterator to a (strong)
convergent one. We also consider a {\em variational approach} to algorithmic
boosting providing tools to convert a non-convergent continuous flow to a
convergent one. Then, by embedding the construction of averages in the design
of the iterated map, we constructively prove the existence of Nash equilibria
(and, therefore, Brouwer fixed points). We then discuss implementations of
averaging and exponentiation, an important matter even for the scalar case. We
finally discuss a relationship between dominant (PageRank) eigenvectors and
Nash equilibria.
