  The main result of this note is a parametrized version of the Borsuk-Ulam
theorem. We show that for a continuous family of Borsuk-Ulam situations,
parameterized by points of a compact manifold W, its solution set also depends
continuously on the parameter space W. Continuity here means that the solution
set supports a homology class which maps onto the fundamental class of W. When
W is a subset of Euclidean space, we also show how to construct such a
continuous family starting from a family depending in the same way continuously
on the points of the boundary of W. This solves a problem related to a
conjecture which is relevant for the construction of equilibrium strategies in
repeated two-player games with incomplete information. A new method (of
independent interest) used in this context is a canonical symmetric squaring
construction in Cech homology with coefficients in Z/2Z.
