Graph games and the pizza problem

TL;DR

This paper introduces a class of two-player graph-based games inspired by a pizza-cutting problem, providing a complete solution for certain game subclasses and algorithms for optimal play and partitioning sequences.

Contribution

It offers a complete solution and linear time algorithm for a simplified class of pizza-like graph games, and develops a novel partitioning algorithm for sequences.

Findings

A linear time algorithm for a simplified class of pizza games.

A quadratic time algorithm to determine game value and optimal moves.

Development of a new sequence partitioning method with potential broader applications.

Abstract

We propose a class of two person perfect information games based on weighted graphs. One of these games can be described in terms of a round pizza which is cut radially into pieces of varying size. The two players alternately take pieces subject to the following rule: Once the first piece has been chosen, all subsequent selections must be adjacent to the hole left by the previously taken pieces. Each player tries to get as much pizza as possible. The original pizza problem was to settle the conjecture that Player One can always get at least half of the pizza. The conjecture turned out to be false. Our main result is a complete solution of a somewhat simpler class of games, concatenations of stacks and two-ended stacks, and we provide a linear time algorithm for this. The algorithm and its output can be described without reference to games. It produces a certain kind of partition of a given finite sequence of real numbers. The conditions on the partition involve alternating sums of various segments of the given sequence. We do not know whether these partitions have applications outside of game theory. The algorithm leads to a quadratic time algorithm which gives the value and an optimal initial move for pizza games. We also provide some general theory concerning the semigroup of equivalence classes of graph games.