I've been trying to show that increasing $N$ (where $N\in\{2,3,...\}$) shifts the following function rightward:
$$P(x;N,k):= \frac{k(N-1)e^{-kx}}{\left(1+(N-1)e^{-kx}\right)^2}, \ (k>0).$$
I made a graph in Desmos just in case it's helpful: https://www.desmos.com/calculator/fi698ohyds.
Note/Fun(?) Fact: For those who are interested, this relates to the "Contest Success Function (CSF) in differences" discussed in Hirschleifer (1989); these are used to model the probability of winning a winner-takes-all "contest" (e.g. a conflict). The function above corresponds to $\frac{\partial}{\partial C_1} p_1(\mathbf{C})\bigg|_{\mathbf{C} = (1,0,...,0)}$, where $\mathbf{C}=(C_1,...,C_N)$ and $p_1$ is as in equation (5) of that paper. Anyway, it's a neat paper, I think :)
Using properties of the exponential function we obtain $$ (N-1) e^{-kx} = e^{\ln(N-1)} e^{-kx} = e^{\ln(N-1)-kx} = e^{-k \left( x - \ln(N-1) / k \right) } $$ Therefore $$ P \left( x;\ N,\ k \right) = P \left( \left( x - \tfrac{\ln(N-1)}{k} \right)\!;\ 2,\ k \right) $$