Loopy games

Summary

Abstract. We strengthen the usual notion of simplest form for stoppers and show that under the stronger definition, equivalence coincides with graphisomorphism. We then show that the game graph of a canonical stopper contains no 2- or 3-cycles, but may contain n-cycles for all n ≥ 4.

We also introduce several new methods for simplifying games γ whose graphs contain alternating cycles. These include a generalization of dominated and reversible moves.

Introduction

A loopy game is a combinatorial game in which repetition is permitted. The history and basic theory of loopy games are discussed in [Siegel 2009]. In this article we focus on two fundamental problems left unresolved by Winning Ways.

Long irreducible cycles. The first problem concerns the cycles that appear in the game graph of a stopper. Conway showed that every stopper s admits a simplest form [Conway 1978], so one would expect that certain cycles are intrinsic to the play of s. All canonical stoppers discussed in Winning Ways are plumtrees: their graphs contain only 1-cycles. It is therefore natural to ask whether longer canonical cycles are possible, and to attempt to characterize the structure of such cycles.

Conway defined the simplest form of s to be a representation with no dominated or reversible moves. This is not quite strong enough for our purposes, as illustrated by the example t shown in Figure 1. Certainly t has no dominated or reversible options, but it is easy to check that t = on.