We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for $u_t = u \Delta u + u \int_\Omega |\nabla u|^2$ in bounded domains $\Omega\subset\mathbb{R}^n$ which arises in game theory. We prove that solutions converge to 0 if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. In particular, it is shown that in this case the blow-up set coincides with $\overline{\Omega}$; i.e., the finite-time blow-up is global