Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Positional s-of-k games

Eric Duchêne, Valentin Gledel, Miloš Stojaković

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.

Positional s-of-k games

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.
Paper Structure (22 sections, 18 theorems, 2 equations, 21 figures, 2 tables)

This paper contains 22 sections, 18 theorems, 2 equations, 21 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Scoring Positional Games
  3. $s$-of-$k$ games
  4. Exposition of the work
  5. General results
  6. Triangular grid, triangle, $k=3$
  7. Triangle with $s=1$
  8. Triangle with $s=2$
  9. Triangle with $s=3$
  10. Square grid, square, $k=4$
  11. Square with $s\in\{1,4\}$
  12. Square with $s=2$
  13. Square with $s=3$
  14. Triangular grid, rhombus (in all three directions), $k=4$
  15. Rhombus with $s=1$
  16. ...and 7 more sections

Key Result

Theorem 3

Let $\mathcal{{\cal G}}=(V,\mathcal{F})$ be a $k$-uniform hypergraph with $n$ hyperedges, we have $\text{SC}({\cal G},k) \le \underset{e \in \mathcal{F}}{\sum} 2^{-|e|} = n2^{-k}$ and $\text{SC}({\cal G},1) \ge n(1-2^{-k})$.

Figures (21)

  • Figure 1: Pairing for Maker, triangle game with $s=1$.
  • Figure 2: The pairing for the triangular grid and $s=2$, and two pairs of matched triangles of the same color.
  • Figure 3: Pairing for Maker, triangle game with $s=3$.
  • Figure 4: Pairing for Maker, square game with $s=2$.
  • Figure 5: Pairing on the square grid with $s=3$. Left: The pairing on one level, where the vertices marked red are not paired. Right: The side squares (yellow) and the corner squares (red).
  • ...and 16 more figures

Theorems & Definitions (36)

  • Theorem 3: Erdős-Selfridge Theorem, scoringPG
  • Theorem 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • Theorem 8
  • ...and 26 more