Let \mu be a probability measure on a group. A function satisfying the mean-value property with respect to \mu is called \mu-harmonic. The space of bounded harmonic functions plays a central role in understanding both the algebraic and geometric properties of groups, as well as the long-term behavior of the associated random walk. A measure \mu is called Liouville if every bounded mu-harmonic function is constant (equivalently, if the Poisson boundary is trivial). In this talk, I will show that the set of non-degenerate Liouville measures on a countable amenable group with the infinite conjugacy class property is not closed under convex combinations. This talk is based on joint work with Joshua Frisch from the University of California in San Diego.