  We introduce the Colonel Blotto game with favoritism, an extension of the
famous Colonel Blotto game where the winner-determination rule is generalized
to include pre-allocations and asymmetry of the players' resources
effectiveness on each battlefield. Such favoritism is found in many classical
applications of the Colonel Blotto game. We focus on the Nash equilibrium.
First, we consider the closely related model of all-pay auctions with
favoritism and completely characterize its equilibrium. Based on this result,
we prove the existence of a set of optimal univariate distributions -- which
serve as candidate marginals for an equilibrium -- of the Colonel Blotto game
with favoritism and show an explicit construction thereof. In several
particular cases, this directly leads to an equilibrium of the Colonel Blotto
game with favoritism. In other cases, we use these optimal univariate
distributions to derive an approximate equilibrium with well-controlled
approximation error. Finally, we propose an algorithm -- based on the notion of
winding number in parametric curves -- to efficiently compute an approximation
of the proposed optimal univariate distributions with arbitrarily small error.
