Multi-Defender Single-Attacker Perimeter Defense Game on a Cylinder: Special Case in which the Attacker Starts at the Boundary

TL;DR

The paper analyzes a multi-defender single-attacker perimeter defense game on a cylinder, focusing on the nontrivial case where the attacker starts near a defended boundary and moves faster than the defenders. By assuming homogeneous defenders and full knowledge of all positions, it derives closed-form conditions for when the attacker can win, encapsulated in a maximum defendable circumference $C_{\max}$ that depends on the number of defenders $n$, defender speed $v_{\mathrm{agent}}$, and defense-region length $d_{\mathrm{agent}}$. The key result shows that the attacker wins when $\frac{C}{d_{\mathrm{agent}}} > n + (n-1)\gamma$ with $\gamma = \frac{2 v_{\mathrm{agent}}}{v_a - v_{\mathrm{agent}}}$, or equivalently when $\frac{v_a}{v_{\mathrm{agent}}} > 1 + \frac{2(n-1)}{(C / d_{\mathrm{agent}}) - n}$. This framework clarifies how circumference, speed differentials, and defender count jointly constrain perimeter security and offers insights transferable to other 1D boundary scenarios.

Abstract

We describe a multi-agent perimeter defense game played on a cylinder. A team of n slow-moving defenders must prevent a single fast-moving attacker from crossing the boundary of a defensive perimeter. We describe the conditions necessary for the attacker to win in the special case that the intruder starts close to the boundary and in a region that is currently defended.

Figures (3)

• Figure 1: In this work we consider a special case of a multi-agent perimeter defense game in which the defense boundary exists in a topological cylinder and an attacker starts close to the perimeter boundary. The attacker moves more quickly than the defenders --- maximum speed of the attacker and defenders is indicated by the red and blue arrow lengths. Note that the cylinder wraps around such that the left and right sides of the environment are connected (as illustrated by the depiction of Defender 3).
• Figure 2: Top: The cylindrical game we consider can be used as an approximation to a defense game with a circular boundary whenever the difference in radii between inside and outside the circular boundary are negligible. Bottom: Circles are homomorphic to non-self-intersecting cycles of other shapes. Defense of a circular perimeter has previously been used as an approximation to defense of other boundary shapes (this requires that the effects of turning and curvature are negligible, for example, with respect to how the agents' velocities and defensive radii interact with the boundary shape).
• Figure 3: Two starting scenarios for a well balanced cylinder patrol game. The attacker is red, $n$ defenders patrol a perimeter (black). The distance sensed by each defender is gray. The attacker moves with a quicker speed $v_{a}$ (red arrow) than the defenders' speeds $v_{i}$ (blue arrows). In the homogeneous case, all defenders move at the same speed $v_{\mathrm{agent}} = v_{i}$ for all $i$ and all defenders are able to sense the same distance $d_{\mathrm{agent}} = d_{i}$ for all $i$. The length of the gap between the regions sensed by defenders $i$ and $j$ is denoted $d_{i,j}$.