  The displacement $\lambda$-convexity of a nonstandard entropy with respect to
a nonlocal transportation metric in finite state spaces is shown using a
gradient flow approach. The constant $\lambda$ is computed explicitly in terms
of a priori estimates of the solution to a finite-difference approximation of a
nonlinear Fokker-Planck equation. The key idea is to employ a new mean
function, which defines the Onsager operator in the gradient flow formulation.
