Technology Category
Game Theory and Economic Paradigms: Cooperative Game TheoryMultiagent Systems: Mechanism DesignConstraint Satisfaction and Optimization: Satisfiability Modulo Theories
Positional s-of-k games
🤖 AI Summary
This study addresses the quantification of scores obtained by players through partial occupation of winning sets in positional games, proposing the **s-of-k game** framework: a player scores by occupying at least *s* elements within a winning set of size *k*. This framework formally introduces a scoring mechanism for partial occupation, unifying the objective functions of various theoretical and practical board games. By modeling game boards as combinatorial structures—specifically triangular, square, rhombus, and hexagonal grids—the work establishes upper and lower bounds on the Maker’s score (denoted SC and SC₂) under both unrestricted play and paired-strategy constraints. These results precisely characterize how strategic restrictions impact the Maker’s scoring capability.
Technology Category
Application Category
📝 Abstract
We introduce a general framework for positional games in which players score points by claiming a prescribed portion of each winning set, extending the notion of scoring Maker-Breaker games. In the scoring variant, Maker gains a point by fully claiming a winning set, while Breaker aims to minimize Maker's total score. In this paper, we generalize these models for all k-uniform positional games by fixing an integer threshold s in {1,2,..., k} so that a player scores a point whenever she claims at least s elements of a winning set of size k. We refer to this class as s-of-k games. Such formulation allows for a flexible description of scoring objectives that appear in both theoretical models and real-life board games. We further investigate the impact of strategy restrictions on the achievable score. In particular, we analyze s-of-k games both under optimal play, where the score is denoted by SC, and under the additional constraint that Maker is restricted to a pairing strategy. The corresponding score in this setting is denoted by SC_2. While the unrestricted score captures the standard notion of optimal play in scoring positional games, the pairing-restricted score allows us to observe Maker's loss incurred by limiting her to these standard strategies. We comprehensively study s-of-k games played on regular grids, which provide a natural and uniform setting for illustrating the general framework. After developing several general tools for the analysis of both scores, we complement them by a number of ad-hoc strategies tailored for particular cases of these games, to obtain both upper and lower bounds for the two scores on triangular, square, rhombus and hexagonal grids.
Problem
Research questions and friction points this paper is trying to address.
positional games
scoring games
s-of-k games
pairing strategy
regular grids
Innovation
Methods, ideas, or system contributions that make the work stand out.
positional games
s-of-k games
scoring games
pairing strategy
combinatorial game theory
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