Analysis of the Pareto equilibrium in multi-objective games using semi-tensor product

Fanyueyang ZHANG; Jun'e FENG

doi:10.1631/FITEE.2400945

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Analysis of the Pareto equilibrium in multi-objective games using semi-tensor product

Regular Papers | Updated：2025-07-28

- Analysis of the Pareto equilibrium in multi-objective games using semi-tensor product
  Enhanced Publication
- 利用半张量积分析多目标博弈中的Pareto均衡
- Frontiers of Information Technology & Electronic Engineering Vol. 26, Issue 7, Pages: 1222-1236(2025)
- Affiliations：
  
  School of Mathematics, Shandong University, Jinan 250100, China
- Author bio：
  
  E-mail: zhangfanyueyang@mail.sdu.edu.cn
  ‡ Corresponding author
- Funds：
- DOI：10.1631/FITEE.2400945
  CLC： O225
- Received：24 October 2024，
  
  Revised：2025-01-24，
  
  Published：2025-07
- Accepted：
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Fanyueyang ZHANG, Jun'e FENG. Analysis of the Pareto equilibrium in multi-objective games using semi-tensor product[J]. Frontiers of information technology & electronic engineering, 2025, 26(7): 1222-1236.
DOI：

Fanyueyang ZHANG, Jun'e FENG. Analysis of the Pareto equilibrium in multi-objective games using semi-tensor product[J]. Frontiers of information technology & electronic engineering, 2025, 26(7): 1222-1236. DOI： 10.1631/FITEE.2400945.

摘要

近年来受到广泛关注的多目标博弈（MOG）是一类具有向量收益的博弈。本文基于半张量积（STP），从静态和动态两个角度讨论该MOG模型的Pareto均衡存在性、有限步可达性和有限步可控性。首先，利用多层图提出MOG的概念，并利用STP将收益函数转化为代数形式。然后，从静态角度分别提出验证所有玩家是否都能满足其期望和验证局势是否为Pareto均衡的两个充分必要条件。进而，从动态角度设计策略更新规则来研究演化MOG的有限步可达性。最后，通过添加伪玩家分析演化MOG的有限步可控性，并给出一种寻找最短演化过程和控制序列的反向搜索算法。

Abstract

Multi-objective games (MOGs) have received much attention in recent years as a class of games with vector payoffs. Based on the semi-tensor product (STP)

this paper discusses the MOG

including the existence

finite-step reachability

and finite-step controllability of Pareto equilibrium of this model

from both static and dynamic perspectives. First

the MOG concept is presented using multi-layer graphs

and STP is used to convert the payoff function into its algebraic form. Then

from the static perspective

two necessary and sufficient conditions are proposed to verify whether all players can meet their expectations and whether the strategy profile is a Pareto equilibrium

separately. Furthermore

from the dynamic perspective

a strategy updating rule is designed to investigate the finite-step reachability of the evolutionary MOG. Finally

the finite-step controllability of the evolutionary MOG is analyzed by adding pseudo-players

and a backward search algorithm is provided to find the shortest evolutionary process and control sequence.

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Keywords

references

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