Equilibrium analysis for multi-defender Stackelberg security games under a logit tie-breaking rule

AbstractWe introduce a general multi-defender Stackelberg security game where multiple independent defenders jointly protect a same set of targets from being attacked by a common attacker. In the game, Strong Stackelberg Equilibrium is fundamentally problematic, because the notion of ‘breaking ties in defender's favour’ is no longer well defined, as we must specify which defender will receive the favour. To address this issue, we define a new equilibrium concept under a newly defined tie-breaking rule. We characterise Logit Stackelberg Multi-Defender Equilibrium, corresponding to a logit tie-breaking rule, as well as an equivalent Nash Equilibrium among defenders, and exhibit algorithms for computing the equilibrium solutions. We find that Logit Stackelberg Multi-Defender Equilibrium and its' equivalent Nash Equilibrium may not exist, which motivates us to find an approximate equilibrium. We design a revised exclusion algorithm to find the approximate ε-Nash Equilibrium in which no defender gains more than ε by deviating.KEYWORDS: Multi-defender security gametie-breaking rulelogit Stackelberg multi-defender equilibriumNash equilibriumexclusion method AcknowledgmentsThe authors thank the editor and the anonymous referees for their valuable comments and suggestions, which greatly helped improve the content and presentation of this article.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThe paper is supported by the National Natural Science Foundation of China [grant numbers 61572095, 61877007] and Department of Science and Technology of Shanxi Province [grant numbers 20210302124303, 202203021222251].Notes on contributorsLing ChenLing Chen received the B.S. degree in statistics from Shandong University, Weihai and the Ph.D. degree in mathematics from Dalian University of Technology. Starting from 2021, she is working as a lecturer at Taiyuan Normal University. Her current research includes security game, optimisation and decision-making.Mingchu LiMingchu Li received the B.S. degree in mathematics from Jiangxi Normal University, in 1983, the M.S. degree in applied science from the University of Science and Technology Beijing, in 1989, and the Ph.D. degree in mathematics from the University of Toronto, in 1997. He was an Associate Professor at the University of Science and Technology Beijing, from 1989 to 1994. He was engaged in the research and development of information security at Long View Solution Inc. and Compute Ware Inc., from 1997 to 2002. Since 2002, he has been a Full Professor with the School of Software, Tianjin University. He has been at the School of Software Technology, Dalian University of Technology, as a Full Professor, a Ph.D. Supervisor, and the Vice Dean. His main research interests include theoretical computer science and cryptography. His other research interests include graph theory, network security, and game theory.