[Paper Review] On the Inducibility of Stackelberg Equilibrium for Security Games

Qingyu Guo, Jiarui Gan+4 more|arXiv (Cornell University)|Nov 9, 2018

Infrastructure Resilience and Vulnerability Analysis25 references4 citations

TL;DR

This paper introduces the Inducible Stackelberg Equilibrium (ISE) as a more reliable solution concept for security games, addressing the critical flaw that Strong Stackelberg Equilibrium (SSE) may not be inducible under resource constraints. By formalizing the defender's utility guarantee and proposing ISE as the strategy with the highest such guarantee, the authors show ISE always exists and significantly reduces overoptimism and sub-optimality seen in SSE, while remaining computationally feasible via polynomial reduction to SSE computation.

ABSTRACT

Strong Stackelberg equilibrium (SSE) is the standard solution concept of Stackelberg security games. As opposed to the weak Stackelberg equilibrium (WSE), the SSE assumes that the follower breaks ties in favor of the leader and this is widely acknowledged and justified by the assertion that the defender can often induce the attacker to choose a preferred action by making an infinitesimal adjustment to her strategy. Unfortunately, in security games with resource assignment constraints, the assertion might not be valid; it is possible that the defender cannot induce the desired outcome. As a result, many results claimed in the literature may be overly optimistic. To remedy, we first formally define the utility guarantee of a defender strategy and provide examples to show that the utility of SSE can be higher than its utility guarantee. Second, inspired by the analysis of leader's payoff by Von Stengel and Zamir (2004), we provide the solution concept called the inducible Stackelberg equilibrium (ISE), which owns the highest utility guarantee and always exists. Third, we show the conditions when ISE coincides with SSE and the fact that in general case, SSE can be extremely worse with respect to utility guarantee. Moreover, introducing the ISE does not invalidate existing algorithmic results as the problem of computing an ISE polynomially reduces to that of computing an SSE. We also provide an algorithmic implementation for computing ISE, with which our experiments unveil the empirical advantage of the ISE over the SSE.

Motivation & Objective

• To address the overoptimism in Strong Stackelberg Equilibrium (SSE) when the defender cannot induce the desired follower response due to resource constraints.
• To formalize the concept of utility guarantee for defender strategies, revealing that SSE often overestimates actual achievable utility.
• To introduce and characterize the Inducible Stackelberg Equilibrium (ISE) as the solution concept with the highest possible utility guarantee.
• To demonstrate that ISE is computationally feasible, as computing ISE polynomially reduces to computing SSE.
• To empirically validate that SSE is significantly overoptimistic and suboptimal in practice, while ISE achieves substantially better utility guarantees.

Proposed method

• Formalize the utility guarantee of a defender strategy as the minimum expected utility the defender can achieve under any follower response, even in worst-case tie-breaking.
• Define the Inducible Stackelberg Equilibrium (ISE) as the strategy that maximizes this utility guarantee, ensuring robustness against non-inducible equilibria.
• Prove that ISE always exists and is optimal in terms of utility guarantee, even when SSE fails to be inducible.
• Show that computing ISE reduces in polynomial time to computing SSE, preserving the scalability of existing algorithms.
• Implement an algorithm using column generation with heuristic pruning to solve large-scale ISE problems efficiently.
• Conduct extensive experiments on randomly generated security games with varying numbers of targets, resources, and schedules to evaluate inducibility, scalability, overoptimism, and sub-optimality.

Experimental results

Research questions

• RQ1 Can the defender reliably induce the Strong Stackelberg Equilibrium (SSE) in security games with resource assignment constraints?
• RQ2 To what extent does SSE overestimate the defender’s actual achievable utility due to non-inducibility?
• RQ3 Is there a solution concept that guarantees a higher and more reliable utility than SSE under the same constraints?
• RQ4 Can the new solution concept (ISE) be computed efficiently without sacrificing the scalability of existing SSE algorithms?
• RQ5 How significant is the overoptimism and sub-optimality of SSE in realistic security game settings?

Key findings

• SSE is significantly overoptimistic in practice, with utility guarantees often much lower than claimed, especially in constrained settings like port protection or air marshaling.
• In 15% to 30% of instances, only a modest fraction of targets are inducible, indicating that inducibility is neither trivial nor guaranteed.
• The percentage of overoptimistic and sub-optimal SSE strategies exceeds 50% across multiple parameter settings, confirming the issue is widespread.
• The average utility guarantee of ISE consistently exceeds that of SSE, with statistically significant improvements shown in 95% confidence intervals.
• The computational cost of computing ISE is nearly identical to that of computing SSE, demonstrating that ISE maintains scalability.
• Experiments confirm that ISE is a safer and more reliable alternative to SSE, particularly in real-world domains with scheduling and resource constraints.