Winning Criteria for Open Games: A Game-Theoretic Approach to Prefix Codes

TL;DR

The paper analyzes two-player Gale–Stewart games on infinite regular trees with open winning sets, aiming to determine which player has a winning strategy. It establishes a simple Kraft-type bound that yields a sufficient condition for Player 2 to win and a necessary condition for Player 1 to win, depending on the structure of the open winning set. A central contribution is the explicit equivalence between open winning sets that guarantee Player 1's win and maximal prefix codes, extending to finite alphabets, which provides a concrete, coding-theoretic criterion for determinacy. By bringing in Nielsen–Schreier theory and Schreier/covering graphs, the paper derives algebraic criteria: infinite index of generated subgroups implies Player 2 win, and it develops Kraft-like inequalities via coverings. Overall, the work builds a bridge between combinatorial game theory, prefix codes, and group-theoretic methods, yielding computable conditions to assess who can win open-Gale–Stewart games on trees.

Abstract

We study two-player games with alternating moves played on infinite trees. Our main focus is on the case where the trees are full (regular) and the winning set is open (with respect to the product topology on the tree). Gale and Stewart showed that in this setting one of the players always has a winning strategy, though it is not known in advance which player. We present simple necessary conditions for the first player to have a winning strategy, and establish an equivalence between winning sets that guarantee a win for the first player and maximal prefix codes. Using this equivalence, we derive a necessary algebraic condition for winning, and exhibit a family of games for which this algebraic condition is in fact equivalent to winning. We introduce the concept of coverings, and show that by covering the graph with an infinite labeled tree corresponding to the free group, we can derive a simple trait of maximal prefix codes.

Figures (5)

• Figure 1: Illustration of the Schreier graph $\mathrm{Sch}(\langle\,\rangle)$, its core $\mathrm{Core}(\langle\,\rangle)$, the Schreier graph $\mathrm{Sch}(H)$, and its core $\mathrm{Core}(H)$ for the free group $F = F(\{a,b\})$ and the subgroup $H = \langle b,\, aba,\, ab^{-1}a \rangle$. The vertices of each Schreier graph are in bijection with the right cosets in $H \backslash F(\mathcal{A})$. In this example, the coset set decomposes as $\{H,\, Ha,\, Hab\} \;\cup\; \{Ha^{2}w,\; Ha^{-2}w' \mid w,w' \in F(\mathcal{A}) \text{ with no cancellation}\}$.
• Figure 2: A Stallings folding of two edges, labeled $a$ with the same starting vertex and direction
• Figure 3: A diagram of a bouquet graph and the bouquet graph of the set from previous example $\{ aba, ab^{-1}a ,b\}$
• Figure 4: For example, let $x = \langle 1,0 \rangle$ and $c = \langle 0,1 \rangle$, which together determine the position $\langle 1,0,0,1 \rangle$ in the full binary tree $\{0,1\}^{\leq \mathbb{N}}$. See the diagram on the right, where this position is highlighted. The corresponding positions in the covering Schreier graph associated with $x = \langle 1,0 \rangle$ and $c = \langle 0,1 \rangle$ are $\langle 1,0,0,1 \rangle$, $\langle 1,0,0,1^{-1} \rangle$, $\langle 1,0^{-1},0^{-1},1 \rangle$, $\langle 1,0^{-1},0^{-1},1^{-1} \rangle$. See the diagram on the left.
• Figure 5: For example, let $x = \langle 1,0 \rangle$ and $c = \langle 0,0 \rangle$, which together determine the position $\langle 1,0,0,0 \rangle$ in the full binary tree $\{0,1\}^{\leq \mathbb{N}}$. See the diagram on the right, where this position is highlighted. The corresponding positions in the covering Schreier graph associated with $x = \langle 1,0 \rangle$ and $c = \langle 0,0 \rangle$ are $\langle 1,0,0,0 \rangle$, $\langle 1,0^{-1},0^{-1},0^{-1} \rangle$. See the diagram on the left.

Theorems & Definitions (44)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Lemma 2.6
• Lemma 2.7
• Theorem 2.8
• Theorem 2.9
• Lemma 2.10