Incidence, a Scoring Positional Game on Graphs
Guillaume Bagan, Quentin Deschamps, Eric Duch\^ene, Bastien Durain,, Brice Effantin, Valentin Gledel, Nacim Oijid, Aline Parreau

TL;DR
This paper introduces scoring positional games on graphs, focusing on the Incidence game, and analyzes its computational complexity, revealing PSPACE-completeness in one version and polynomial-time solutions in another.
Contribution
It defines scoring positional games on graphs, analyzes the Incidence game, and establishes complexity results including PSPACE-completeness and polynomial-time computability.
Findings
Computing the score in Maker-Breaker Incidence is PSPACE-complete.
In Maker-Maker Incidence, the score can be computed in polynomial time.
Score formulas for paths and cycles are derived using Milnor's universe.
Abstract
Positional games have been introduced by Hales and Jewett in 1963 and have been extensively investigated in the literature since then. These games are played on a hypergraph where two players alternately select an unclaimed vertex of it. In the Maker-Breaker convention, if Maker manages to fully take a hyperedge, she wins, otherwise, Breaker is the winner. In the Maker-Maker convention, the first player to take a hyperedge wins. In both cases, the game stops as soon as Maker has taken a hyperedge. By definition, this family of games does not handle scores and cannot represent games in which players want to maximize a quantity. In this work, we introduce scoring positional games, that consist in playing on a hypergraph until all the vertices are claimed, and by defining the score as the number of hyperedges a player has fully taken. We focus here on Incidence, a scoring positional game…
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Taxonomy
TopicsArtificial Intelligence in Games · Game Theory and Applications · Sports Analytics and Performance
