Energy Games over Totally Ordered Groups

TL;DR

This paper introduces energy conditions over totally ordered groups, refutes a conjecture about union closure of certain winning conditions, and explores their properties in finite arenas.

Contribution

It defines a new class of energy conditions over totally ordered groups and demonstrates their non-closure under union, challenging previous conjectures.

Findings

Refutes conjecture on union closure of prefix-independent half-positional conditions.

Introduces energy conditions over totally ordered groups.

Shows union of two such conditions may not be half-positional.

Abstract

Kopczy\'{n}ski (ICALP 2006) conjectured that prefix-independent half-positional winning conditions are closed under finite unions. We refute this conjecture over finite arenas. For that, we introduce a new class of prefix-independent bi-positional winning conditions called energy conditions over totally ordered groups. We give an example of two such conditions whose union is not half-positional. We also conjecture that every prefix-independent bi-positional winning condition coincides with some energy condition over a totally ordered group on periodic sequences.