Energy Games over Totally Ordered Groups
Alexander Kozachinskiy

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.
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Taxonomy
TopicsGame Theory and Applications · Opinion Dynamics and Social Influence
