Massively Winning Configurations in the Convex Grabbing Game on the Plane

TL;DR

This paper analyzes the convex grabbing game on the plane with binary weights, constructing large point sets where Bob can secure nearly 3/4 or all of the total weight, answering a prior open question.

Contribution

It constructs large point sets demonstrating Bob's ability to win nearly 3/4 or all the weight, addressing an open problem in the convex grabbing game.

Findings

Bob can obtain almost 3/4 of total weight in certain large sets.

Bob can secure the entire weight in even-sized point sets.

The paper discusses conjectures on optimal moves for both players.

Abstract

The convex grabbing game is a game where two players, Alice and Bob, alternate taking extremal points from the convex hull of a point set on the plane. Rational weights are given to the points. The goal of each player is to maximize the total weight over all points that they obtain. We restrict the setting to the case of binary weights. We show a construction of an arbitrarily large odd-sized point set that allows Bob to obtain almost 3/4 of the total weight. This construction answers a question asked by Matsumoto, Nakamigawa, and Sakuma in [Graphs and Combinatorics, 36/1 (2020)]. We also present an arbitrarily large even-sized point set where Bob can obtain the entirety of the total weight. Finally, we discuss conjectures about optimum moves in the convex grabbing game for both players in general.