Federated learning (FL) is an emerging collaborative machine learning method. In FL processing, the data quality shared by users directly affects the accuracy of the federated learning model, and how to encourage more data owners to share data is crucial. In other words, how to design a good incentive mechanism is the key problem in FL. In this paper, we propose an incentive mechanism based on the enhanced Shapley value method for FL. In the proposed mechanism, the enhanced Shapley value method is proposed to measure income distribution, which takes multiple influence factors as weights. The analytic hierarchy process (AHP) is used to find the corresponding weight value of the influence factors. Finally, the numerical experiments are carried to verify the performance of the proposed incentive mechanism. The results show that compared with the Shapley value method considering the single factor, the income distribution of all participants can better reflect multiple factor contribution when using the enhanced Shapley value method.

Federated learning (FL) is a distributed machine learning framework that can effectively help multiple players to use data to train federated models while complying with their privacy, data security, and government regulations. Due to federated model training, an accurate model should be trained, and all federated players should actively participate. Therefore, it is crucial to design an incentive mechanism; however, there is a conflict between fairness and Pareto efficiency in the incentive mechanism. In this paper, we propose an incentive mechanism via the combination of the Shapley value and Pareto efficiency optimization, in which a third party is introduced to supervise the federated payoff allocation. If the payoff can reach Pareto optimality, the federated payoff is allocated by the Shapley value method; otherwise, the relevant federated players are punished. Numerical and simulation experiments show that the mechanism can achieve fair payoff allocation and Pareto optimality payoff allocation. The Nash equilibrium of this mechanism is formed when Pareto optimality payoff allocation is achieved.

In this article, we consider the following new Kirchhoff-type problem: $\begin{array}{rl}-\left(a-b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{\Delta }u& =|u{|}^{p-2},\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u& =0,\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$ where a and b are positive constants, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with $C^1$ boundary $\mathrm{\partial }\mathrm{\Omega }$, $p\in [2,2^{\ast })$ with $2^{\ast }=2N/\left(N-2\right)$ if $N\ge 3$, and $2^{\ast }=+\mathrm{\infty }$ if N = 1, 2. We show that the problem possesses infinitely many sign-changing solutions by using combination of invariant sets of descent flow and the Ljusternik–Schnirelman type minimax method. And an example for p = 2 is illustrated our results.

The distributed training of federated machine learning, referred to as federated learning (FL), is discussed in models by multiple participants using local data without compromising data privacy and violating laws. In this paper, we consider the training of federated machine models with uncertain participation attitudes and uncertain benefits of each federated participant, and to encourage all participants to train the desired FL models, we design a fuzzy Shapley value incentive mechanism with supervision. In this incentive mechanism, if the supervision of the supervised mechanism detects that the payoffs of a federated participant reach a value that satisfies the Pareto optimality condition, the federated participant receives a distribution of federated payoffs. The results of numerical experiments demonstrate that the mechanism successfully achieves a fair and Pareto optimal distribution of payoffs. The contradiction between fairness and Pareto-efficient optimization is solved by introducing a supervised mechanism.