  In federated learning (FL), clients cooperatively train a global model
without revealing their raw data but gradients or parameters, while the local
information can still be disclosed from local outputs transmitted to the
parameter server. With such privacy concerns, a client may overly add
artificial noise to his local updates to compromise the global model training,
and we prove the selfish noise adding leads to an infinite price of anarchy
(PoA). This paper proposes a novel pricing mechanism to regulate
privacy-sensitive clients without verifying their parameter updates, unlike
existing privacy mechanisms that assume the server's full knowledge of added
noise. Without knowing the ground truth, our mechanism reaches the social
optimum to best balance the global training error and privacy loss, according
to the difference between a client's updated parameter and all clients' average
parameter. We also improve the FL convergence bound by refining the aggregation
rule at the server to account for different clients' noise variances. Moreover,
we extend our pricing scheme to fit incomplete information of clients' privacy
sensitivities, ensuring their truthful type reporting and the system's ex-ante
budget balance. Simulations show that our pricing scheme greatly improves the
system performance especially when clients have diverse privacy sensitivities.
