We consider secret-key generation between several agents and a base station that observe independent and identically distributed (i.i.d.) realizations of correlated random variables. Each agent wishes to generate the longest possible individual key with the base station by means of public communication. All keys must be jointly kept secret from all external entities. We do not require them to be kept secret among the agents. In this many-to-one secret-key generation setting, it can be shown that the agents can take advantage of a collective protocol to increase the sum-rate of all the generated keys. However, when each agent is only interested in maximizing its own secret-key rate, agents may be unwilling to participate in a collective protocol. Furthermore, when such a collective protocol is employed, how to fairly allocate individual key rates arises as a valid issue. We study this tension between cooperation and self-interest with a game-theoretic treatment. We establish that cooperation is in the best interest of all agents and that there exists individual secret-key rate allocations that incentivize the agents to follow the protocol. Additionally, we propose an explicit and low-complexity coding scheme based on polar codes and hash functions that achieves such allocations.