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Food safety is a pressing issue affecting public health, and strengthening food safety regulation has become a widespread consensus. This study addresses the three-party game involving food production enterprises, third-party testing agencies, and government regulatory authorities. Considering the influence of random disturbances and time delays on the strategy-choosing and behavioral evolution of game players, a three-party stochastic evolutionary game model for food safety regulation with time delays is established. The stability of the model is analyzed by using Lyapunov's method, and the strategy evolution of the game players is investigated while combining numerical simulations. The study shows that the net benefit of the players' expected strategies is key in determining whether the overall game system can reach the optimal equilibrium. The level of penalties imposed by the central government on local government regulators is key to food safety. The main factors influencing the strategy-choosing of game players are the level of penalties imposed by the local government regulators on third-party testing agencies, together with the supervision costs of the local government regulators. Random disturbances have a certain impact on strategy-choosing of the game players, with a greater impact on testing agencies, followed by food production enterprises, and a weaker impact on government regulatory authorities. The time delay factors exhibit a certain delay effect on the evolutionary path of strategy-choosing among the game players, which aligns with the actual situation. The findings of the study have implications for food safety regulation.

Inspired by the idea of game learning, this paper studies the Cournot game model of the multi-oligopoly market and studies the realization of the Nash equilibrium by using adaptive differential evolution algorithm based on relaxation strategies. The realization path of Nash equilibrium is attained through an adaptive differential evolution based on a relaxation strategy procedure applied to an objective function with best response dynamics. Among them, the objective function is derived from the sum of the payoff changes of each oligopoly under best response decisions. And theoretically proved that the Nash equilibrium is the optimal solution of the objective function. [ABSTRACT FROM AUTHOR]

The keys to constructing numerical schemes for nonlinear partial differential equations are accuracy, handling of the nonlinear terms, and physical properties (energy dissipation or conservation). In this paper, we employ the exponential scalar auxiliary variable (E-SAV) method to solve a semi-linear wave equation. By defining two different variables and combining the Crank−Nicolson scheme, two semi-discrete schemes are proposed, both of which are second-order and maintain Hamiltonian conservation. Two numerical experiments are presented to verify the reliability of the theory. [ABSTRACT FROM AUTHOR]

This paper focused on approximating a second-order nonlinear hyperbolic optimal control problem. By introducing a new variable, the hyperbolic equation was converted into two parabolic equations. A second-order fully discrete scheme was obtained by combining the Crank-Nicolson formula with the finite element method. The error estimation for this scheme was derived utilizing the second-order sufficient optimality condition and auxiliary problems. To validate the effectiveness of the fully discrete scheme, a numerical example was presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we first construct a complete metric space Λ consisting of a class of strong vector equilibrium problems (for short, (SVEP)) satisfying some conditions. Under the abstract framework, we introduce a notion of well-posedness for the (SVEP), which unifies its Hadamard and Tikhonov well-posedness. Furthermore, we prove that there exists a dense Gδ set Q of δ such that each (SVEP) in Q is well-posed, that is, the majority (in Baire category sense) of (SVEP) in δ is well-posed. Finally, metric characterizations on the well-posedness for the (SVEP) are given. [ABSTRACT FROM AUTHOR]