Geometric Give and Take

2026-03-09

AI Summary

This study addresses a geometric special case of Spencer’s balancing game under a planar straight-line arrangement: Alice must prevent Bob from emptying any cell by sequentially selecting lines, using the minimum possible initial number of stones. Focusing on the cell decomposition induced by $n$ lines in general position, the work integrates tools from computational geometry, combinatorial game theory, and algorithmic complexity. By leveraging dual graphs and a carefully designed potential function, the authors construct an effective strategy for Alice. The main contribution is the first proof that the minimum initial number of stones required is $\Theta(n^3)$, accompanied by a polynomial-time algorithm to compute this value, thereby fully characterizing the complexity of this geometric balancing game.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider a special, geometric case of a balancing game introduced by Spencer in 1977. Consider any arrangement $\mathcal{L}$ of $n$ lines in the plane, and assume that each cell of the arrangement contains a box. Alice initially places pebbles in each box. In each subsequent step, Bob picks a line, and Alice must choose a side of that line, remove one pebble from each box on that side, and add one pebble to each box on the other side. Bob wins if any box ever becomes empty. We determine the minimum number $f(\mathcal L)$ of pebbles, computable in polynomial time, for which Alice can prevent Bob from ever winning, and we show that $f(\mathcal L)=\Theta(n^3)$ for any arrangement $\mathcal{L}$ of $n$ lines in general position.

Problem

• balancing game
• geometric arrangement
• pebbling
• line arrangement
• combinatorial game

Innovation

• geometric balancing game
• line arrangement
• pebbling strategy
• polynomial-time computability
• combinatorial geometry