We give a short proof of a recent result of Claesson, Dukes, Franklín and Stefánsson, connecting the number $S_n$ of score sequences and the Erdős-Ginzburg-Ziv numbers $N_n$ from additive number theory. Our proof utilizes the lattice path representation of score sequences by Erdős and Moser, and remarks by Kleitman added to an article of Moser regarding cyclic shifts of such paths. The connection between $S_n$ and $N_n$ is an instance of the Lévy-Khintchine formula from probability theory. We highlight the utility of such formulas, by giving a short proof of Moser's conjecture that $S_n\sim C4^n/n^{5/2}$, where $C$ is described in terms of $N_n$.