Secret Sharing Scheme, with joining players, no central entity
This is a classic problem that can be solved with secure multiparty computation. When a new player joins, the $K$ parties who approve them joining run a secure computation protocol to generate their share. Concretely, if Shamir sharing is used, then the function computed is "reconstruct the polynomial $p$ from the $K$ shares input, and output the share $p(\alpha_{N+1})$, where $\alpha_1,\ldots,\alpha_{N+1}$ are field elements and $\alpha_i$ is associated with party $P_i$.