# Perimeter-defense Game on Arbitrary Convex Shapes

arXiv: https://arxiv.org/abs/1909.03989 · HTML: https://mlsanity.com/paper/1909.03989/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2019.

## Built on by (most cited first, 2 of 2)

- [Embedded Mean Field Reinforcement Learning for Perimeter-defense Game](https://mlsanity.com/paper/2505.14209/) (2025)
- [Game-Theory-Assisted Reinforcement Learning for Border Defense: Early Termination based on Analytical Solutions](https://mlsanity.com/paper/2603.15907/) (2026)

## Abstract (arXiv)

This paper studies a variant of multi-player reach-avoid game played between intruders and defenders. The intruder team tries to score by sending as many intruders as possible to the target area, while the defender team tries to minimize this score by intercepting them. Specifically, we consider the case where the defenders are constrained to move on the perimeter of the target area. Since it is challenging to directly solve the multi-player game due to the high dimensionality of the joint state space, we leverage the solutions to smaller scale problems. First, we solve the one vs. one game, for which existing works either rely on numerical approaches or make simplifying assumptions (e.g., circular perimeter, or equal speed). This paper accommodates target areas with any arbitrary convex shapes and provides analytical solution which lends itself to a useful geometric interpretation. We also provide a detailed discussion on the optimality of the derived strategies. Secondly, we solve the two vs. one game to introduce a cooperative pincer maneuver, where a pair of defenders team up to capture an intruder that cannot be captured by either one of the defender individually. Finally, we introduce how the aforementioned building blocks are used in three different assignment-based defense strategies.