Abstract
This aim of this paper is to provide the immune particle swarm optimization (IPSO) algorithm for solving the single-leader–multi-follower game (SLMFG). Through cooperating with the particle swarm optimization (PSO) algorithm and an immune memory mechanism, the IPSO algorithm is designed. Furthermore, we define the efficient Nash equilibrium from the perspective of mathematical economics, which maximizes social welfare and further refines the number of Nash equilibria. In the end, numerical experiments show that the IPSO algorithm has fast convergence speed and high effectiveness.
Highlights
In 1950, Nash equilibrium was formulated based on noncooperative games formed among all players, and the existence of an equilibrium point was proven [1,2]
This paper considers a single-leader–multi-follower game with a bilevel hierarchical structure
We define the efficient Nash equilibrium by refining of the traditional Nash equilibrium with efficiency; this efficient Nash equilibrium is beneficial to all followers and greatly reduces the number of Nash equilibria, which means social welfare maximization
Summary
In 1950, Nash equilibrium was formulated based on noncooperative games formed among all players, and the existence of an equilibrium point was proven [1,2]. A single-leader–multi-follower game (SLMFG) is a special form of the leader–follower game, called the bilevel programming problem. Yu [7] introduced the Nash equilibrium point existence theorem for the SLMFG and multi-leader–follower game. With the development of biological evolution and heuristic algorithms, swarm intelligent algorithms have displayed the potential for possibly solving nonlinear bilevel programming problems. Many scholars have tried to solve the Nash equilibrium of the SLMFG by using swarm intelligence algorithms, including a dynamic particle swarm optimization algorithm [19], genetic algorithms [20,21], and a nested evolutionary algorithm [22]. We present the model of the single-leader–multi-follower game, the efficient Nash equilibrium of the SLMFG, and some assumptions of the SLMFG. Several numerical experiments showed that the IPSO algorithm is practicable: the efficient Nash equilibrium is solved and the number of.
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