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

Abstract

We 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.