A Little Aggression Goes a Long Way

TL;DR

This paper analyzes a two-player troop placement and attack game on graphs, proving its computational complexity is NP-complete even under restricted conditions, and explores specific graph structures like matchings and cycles.

Contribution

It establishes the NP-completeness of determining optimal moves in the game and examines the game's behavior on matchings and cycles.

Findings

NP-complete to find optimal moves even with known second-player strategy

Game analysis on matchings and cycles reveals specific strategic insights

Complexity results inform game design and AI development

Abstract

Aggression is a two-player game of troop placement and attack played on a map (modeled as a graph). Players take turns deploying troops on a territory (a vertex on the graph) until they run out. Once all troops are placed, players take turns attacking enemy territories. A territory can be attacked if it has $k$ troops and there are more than $k$ enemy troops on adjacent territories. At the end of the game, the player who controls the most territories wins. In the case of a tie, the player with more surviving troops wins. The first player to exhaust their troops in the placement phase leads the attack phase. We study the complexity of the game when the graph along with an assignment of troops and the sequence of attacks planned by the second player. Even in this restrained setting, we show that the problem of determining an optimal sequence of first player moves is NP-complete. We then analyze the game for when the input graph is a matching or a cycle.