Cyclic impartial games with carry-on moves

In an impartial combinatorial game, both players have the same options in the game and all its subpositions. The classical Sprague-Grundy Theory was developed for short impartial games, where players have a finite number of options, there are no special moves, and an infinite run is not possible. Subsequently, many generalizations have been proposed, particularly the Smith-Frankel-Perl Theory devised for games where the infinite run is possible, and the Larsson-Nowakowski-Santos Theory able to deal with entailing moves that disrupt the logic of the disjunctive sum. This work presents a generalization that combines these two theories, suitable for analyzing cyclic impartial games with carry-on moves, which are particular cases of entailing moves where the entailed player has no freedom of choice in their response. This generalization is illustrated with sc green-lime hackenbush, a game inspired by the classic green hackenbush.

💡 Research Summary

The paper presents a unified theoretical framework for impartial combinatorial games that simultaneously exhibit cycles and a special class of entailing moves called “carry‑on moves.” Traditional Sprague‑Grundy Theory (SGT) handles short impartial games without cycles or special rules, assigning each position a Grundy number and using the Nim‑sum (⊕) to compute the value of disjunctive sums. Smith‑Frankel‑Perl Theory (SFPT) extends SGT to cyclic impartial games by introducing the concepts of reversibility and infinite Grundy values (∞) for positions that belong to a cycle, together with a set D of exit Grundy numbers that can be reached from the cycle. Larsson‑Nowakowski‑Santos Theory (LNST) deals with entailing moves, where a player’s move forces the opponent to make a specific response; when the forced response set has at most one element, the move is termed a “carry‑on move.”

The authors model an impartial game as a finite directed graph (G = (V, x)) where vertices are colored white (free) or gray (forced). Moving a piece to a gray vertex obliges the opponent to immediately move the same piece again, effectively collapsing the two moves into a single turn for the original player. This captures the essence of carry‑on moves.

A key contribution is an algorithm (Algorithm 2) that computes Grundy values for such games. The algorithm initializes all non‑terminal vertices with ∞ and iteratively applies the mex rule. If a vertex has an option whose current Grundy value exceeds the mex (m) and every higher‑valued option can be forced back to a position of value (m), the vertex is assigned the value (m). The process repeats until a fixed point is reached. Vertices that remain ∞ after convergence form cyclic zones; for each such zone the algorithm records the set (D) of Grundy numbers of options that exit the cycle. The final Grundy value of a position is either a finite integer or the pair (\infty D), indicating an infinite cyclic component together with its possible exits.

To handle disjunctive sums, the paper defines a graph product (V * U) that mimics the usual sum of impartial games while respecting the gray‑vertex restriction (pairs of gray vertices are excluded because two pieces cannot simultaneously occupy forced positions). The authors then extend the Nim‑sum to operate on the enriched Grundy values:

• For two finite Grundy numbers (a) and (b), the sum is the ordinary XOR (a \oplus b).
• For a finite value (a) and an infinite value (\infty D), the result is (\infty (D \oplus {a})), i.e., each element of (D) is XOR‑combined with (a).
• For two infinite values (\infty D_1) and (\infty D_2), the result is (\infty (D_1 \oplus D_2)), where the set‑wise XOR is taken.

Theorem 2.5 proves that this extended operation correctly yields the Grundy value of the sum, provided that at most one piece lies inside any cyclic zone (otherwise a player could not make more than one move in a turn). This theorem unifies the treatment of cycles and carry‑on moves, something neither SFPT nor LNST could achieve alone.

The framework is illustrated with a new game called “green‑lime hackenbush,” a variant of classic green hackenbush where some edges are colored lime to represent carry‑on moves. In ordinary green hackenbush, Grundy values correspond to the height of a branch. Introducing lime edges creates forced immediate moves and can generate cycles. Applying the authors’ algorithm, each vertex receives either a finite Grundy number or an (\infty D) value. The authors demonstrate how to compute the outcome of arbitrary positions by evaluating the extended Nim‑sum of component values, showing that the new theory predicts winning strategies that were inaccessible to previous methods.

In conclusion, the paper successfully merges two previously independent extensions of Sprague‑Grundy theory, delivering a practical algorithm for a broader class of impartial games. It opens avenues for further research, such as handling carry‑on moves with multiple forced options, extending to infinite game boards, or incorporating probabilistic elements. The work not only advances the theoretical understanding of impartial games with cycles and entailing moves but also provides concrete tools for analyzing and designing new combinatorial games.