  We study competing first passage percolation on graphs generated by the
configuration model. At time 0, vertex 1 and vertex 2 are infected with the
type 1 and the type 2 infection, respectively, and an uninfected vertex then
becomes type 1 (2) infected at rate $\lambda_1$ ($\lambda_2$) times the number
of edges connecting it to a type 1 (2) infected neighbor. Our main result is
that, if the degree distribution is a power-law with exponent $\tau\in(2,3)$,
then, as the number of vertices tends to infinity and with high probability,
one of the infection types will occupy all but a finite number of vertices.
Furthermore, which one of the infections wins is random and both infections
have a positive probability of winning regardless of the values of $\lambda_1$
and $\lambda_2$. The picture is similar with multiple starting points for the
infections.
