This paper considers secret-key generation between several agents and a base station that observe independent and identically distributed 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. 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 their 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. This paper studies this tension between cooperation and self-interest with a game-theoretic treatment. The work establishes that cooperation is in the best interest of all individualistic agents and that there exists individual secret-key rate allocations that incentivize the agents to follow the protocol. Additionally, an explicit coding scheme that achieves such allocations is proposed.