Non-Stopper-Sided Loopy Games

• Non-stopper-sided loopy games are infinite cyclic combinatorial games where play may revisit positions and neither player can force termination.
• They are formulated via a coalgebraic extension of Conway's theory, which highlights unique strategic and topological challenges.
• The framework reveals complex play structures and draw outcomes, complicating canonical classification and the search for a generalized Grundy theory.

A non-stopper-sided loopy game is a combinatorial game, possibly infinite, for which play may revisit previous positions (i.e., the associated digraph contains cycles), and in which neither player can unilaterally force termination under infinite play when playing as "Left" or "Right." This distinguishes such games from "stoppers," where alternation of moves must always yield a finite play, and from "stopper-sided" games, where at least one player possesses a forced-ending strategy. Non-stopper-sided loopy games thus exhibit fundamentally different strategic properties, raise questions of determinacy and contextual equivalence, and pose significant challenges for topological classification and canonical analysis within combinatorial game theory (Honsell et al., 2011, Burke et al., 15 Jan 2026).

1. Fundamental Definitions and Coalgebraic Framework

Non-stopper-sided loopy games are formulated within the coalgebraic extension of Conway's theory of combinatorial games. Let $F : \mathrm{Class}^* \to \mathrm{Class}^*$ be the functor $F(X) = \mathcal{P}(X) \times \mathcal{P}(X)$. A hypergame is an element $G$ of the final $F$-coalgebra $(\mathcal{H}, \text{id})$, i.e., $G \in \mathcal{H}$ with $G = (G^L, G^R)$ for sets of Left and Right options $G^L, G^R \subseteq \mathcal{H}$. This coalgebraic formulation admits non-wellfounded games: $G$ may appear in its own option set, enabling loops and allowing for infinite plays (Honsell et al., 2011).

A loopy game is any $G$ where its play-graph contains a cycle; non-stopper-sided refers specifically to loopy games in which neither player has a forced-ending (stopper) role under standard alternation (Burke et al., 15 Jan 2026). In combinatorial-games folklore, stopper-sidedness refers to the ability of one player to force game termination. When neither Left nor Right can achieve this, the game is non-stopper-sided.

2. Play Structure, Draws, and Strategic Characterization

Players in hypergames alternate moves, forming plays $\pi \in \mathrm{Play}_x$ as finite or infinite sequences $x_0^{K_0}\ x_1^{K_1}\ \ldots$, with $K_i \in${L, R}$$and legal transitions. A finite play is winning for Left (resp., Right) if the final position has no Right (resp., Left) options; an infinite play results in a draw. The sets of plays are partitioned as follows:</p> <ul> <li>$$W\mathrm{Play}_x^L$: Finite, Left-winning plays</li> <li>$W\mathrm{Play}_x^R$: Finite, Right-winning plays</li> <li>$D\mathrm{Play}_x$$: Infinite (draw) plays</li> <li>Non-losing for a player: winning or drawing (<a href="/papers/1107.1351" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Honsell et al., 2011</a>)</li> </ul> <p>A non-losing strategy prescribes moves so that, against any counter-strategy, play remains in the non-losing set$$N\mathrm{Play}_x^L = W\mathrm{Play}_x^L \cup D\mathrm{Play}_x$$(and similarly for Right). Characterization of such strategies is formalized coalgebraically: Left has a non-losing strategy on$$x$iff$x\mathbin{\$}0$$under a greatest fixpoint relation, and symmetrically for$$0\mathbin{\$}x$$(Right).</p> <p>Non-stopper-sided loopy games necessarily feature infinite plays that neither player can prevent, leading to strategic focus on non-losing (as opposed to strictly winning) behavior.</p> <h2 class='paper-heading' id='determinacy-equivalence-and-draw-phenomena'>3. Determinacy, Equivalence, and Draw Phenomena</h2> <p>In hypergames, determinacy is adapted: every position$$x$$admits a non-losing strategy for at least one of the four roles—Left or Right, acting as Player I or II. If one role possesses a winning strategy, no other role can even guarantee non-losing. The coinductive proof establishes mutual exclusivity and ensures every hypergame is non-losing-determined for at least one assignment of sides and roles (<a href="/papers/1107.1351" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Honsell et al., 2011</a>).</p> <p>Equivalences are structured as follows:</p> <div class='overflow-x-auto max-w-full my-4'><table class='table border-collapse w-full' style='table-layout: fixed'><thead><tr> <th>Notion</th> <th>Symbol</th> <th>Description</th> </tr> </thead><tbody><tr> <td>Equideterminacy</td> <td>$$x \Leftrightarrow y$</td> <td>Both$x$,$y$$determined identically for all roles</td> </tr> <tr> <td>Contextual Equiv.</td> <td>$$x \approx y$</td> <td>$x$,$y$$remain equidetermined in all additive (sum-based) contexts; greatest congruence refining equideterminacy</td> </tr> </tbody></table></div> <p>For well-founded games, contextual equivalence coincides with the classic relation$$\sim$ derived from Conway&#39;s partial-order semantics, but for loopy and non-stopper-sided games, $\sim$is not an equivalence; only$\approx$$is robust (<a href="/papers/1107.1351" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Honsell et al., 2011</a>).</p> <p>Examples of non-stopper-sided games include:</p> <ul> <li>$$a = (\{b\}, \varnothing)$,$b = (\varnothing, \{a\})$:$a$non-losing for Left only,$b$$for Right only, each sitting in unique determinacy regions.</li> <li>$$c = (\{c\}, \{c\})$: Both players can perpetually return to$c$$(pure draw), both have non-losing strategies, and under$$\approx$the games$a$,$b$,$c$$are pairwise inequivalent.</li> </ul> <h2 class='paper-heading' id='the-diminishing-weight-metric-and-topological-partition'>4. The Diminishing-Weight Metric and Topological Partition</h2> <p>A real-valued metric$$wd$ (diminishing-weight distance) on short canonical-form games is constructed by measuring the &quot;cost&quot; of editing moves in the game&#39;s digraphs $D(G)$, where altering an edge at digraph distance$d$from the root costs$2^{-d}$.$(\mathcal{C}, wd)$is a metric space; the discrete nature of$C$$(finite canonical games) means only loopy games, including non-stopper-sided, may arise as closure points via Cauchy sequences.</p> <p>For loopy games$$G$,$wd(G, J)$$is extended as the infimum of distances over infinite <a href="https://www.emergentmind.com/topics/directed-acyclic-graph-dag-formalism" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">DAG</a> representatives$$H \approx_{\text{tree}} G$ (<a href="/papers/2601.10574" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Burke et al., 15 Jan 2026</a>). <a href="https://www.emergentmind.com/topics/sidling-sequences" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Sidling sequences</a> yield Cauchy sequences converging to (possibly loopy) non-stoppers, with explicit geometric rate of convergence.</p> <p>However, non-stopper-sided loopy games such as Bach&#39;s Carousel and $tis = \{tisn|\}$,$tisn = \{|tis\}$resist such approximation: no Cauchy sequence in$\mathcal{C}$$converges to their class; thus, they reside outside the closure$$\overline{\mathcal{C}}$. This partitions loopy games into those &quot;approximable&quot; by short games and those, especially prototypical non-stopper-sided games, that are not.</p> <h2 class='paper-heading' id='contextual-equivalence-sums-and-classification-obstacles'>5. Contextual Equivalence, Sums, and Classification Obstacles</h2> <p>Sum in the hypergame context ($x + y$) extends Conway&#39;s operation commutatively and associatively (up to coalgebraic equality): $x + y = \left( \{ x^L + y \mid x^L \in X^L \} \cup \{ x + y^L \mid y^L \in Y^L \}, \{ x^R + y \mid x^R \in X^R \} \cup \{ x + y^R \mid y^R \in Y^R \} \right)$$Non-losing strategies are closed under sum: if Left has non-losing strategies on$$x$and$y$$, these can be concatenated to form a non-losing strategy on$$x+y$$(<a href="/papers/1107.1351" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Honsell et al., 2011</a>).</p> <p>Contextual equivalence$$\approx$$is defined as closure under additive contexts, and is the greatest congruence refining equideterminacy. In ordinary games, this agrees with the equivalence induced by the partial order, but for non-stopper-sided loopy games,$$\sim$$fails to be robust, indicating the intensification of equivalence complexity with the introduction of loops and non-stopper-sidedness.</p> <p>A significant obstacle remains: in the impartial (symmetric options) case, a generalization of Sprague–Grundy theory with a fixpoint characterization$$\gamma: J \to \mathrm{Ord} \cup \{\infty_K\}$ is possible, but no comparable fixpoint method, canonical classification, or Grundy-like assignment is known for the partizan, non-stopper-sided loopy case. The lack of a &quot;single mex&quot; analogue for $P(X) \times P(X)$$functor means the combinatorial semantics are not fully captured (<a href="/papers/1107.1351" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Honsell et al., 2011</a>).</p> <h2 class='paper-heading' id='topological-and-structural-phenomena'>6. Topological and Structural Phenomena</h2> <p>The topological structure induced by$$wd$$admits several phenomena unique or particularly notable in the loopy/non-stopper-sided context (<a href="/papers/2601.10574" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Burke et al., 15 Jan 2026</a>):</p> <ul> <li>$$(\mathcal{C}, wd)$$is discrete: Each canonical short game is isolated by a positive-radius ball.</li> <li>$$\mathcal{C}$$(the set of canonical short games) is countable and separable but not compact; unboundedness arises, for instance, via the sequence of nimbers$$*n$$.</li> <li>Closure phenomena: Some plums and non-stoppers are approachable as$$wd$-limit points, but no non-stopper-sided games of Bach&#39;s Carousel (and similar structures) admit such approximants.</li> <li>Addition anomalies: The sum of two Cauchy sequences may fail to be Cauchy, signalling complex algebraic-topological interplay in the non-wellfounded landscape.</li> </ul> <p>Table: Partition of Loopy Games by $wd$$-Closure (<a href="/papers/2601.10574" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Burke et al., 15 Jan 2026</a>)</p> <div class='overflow-x-auto max-w-full my-4'><table class='table border-collapse w-full' style='table-layout: fixed'><thead><tr> <th>Subclass</th> <th>$$\in\overline{\mathcal{C}}$$?</th> <th>Examples</th> </tr> </thead><tbody><tr> <td>Plums</td> <td>Yes</td> <td>on, over, upon, dud</td> </tr> <tr> <td>Non-stoppers (some)</td> <td>Yes</td> <td>$$\chi$,$\psi$, dud</td> </tr> <tr> <td>Non-stopper-sided</td> <td>No</td> <td>Bach&#39;s Carousel, tis, tisn</td> </tr> </tbody></table></div> <p>The existence of robust barriers—rooted in dominated-option obstructions—explains the resistance of non-stopper-sided loopy games to finite canonical approximation.</p> <h2 class='paper-heading' id='open-questions-and-research-directions'>7. Open Questions and Research Directions</h2> <p>Several open questions remain concerning non-stopper-sided loopy games, as articulated in recent research (<a href="/papers/2601.10574" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Burke et al., 15 Jan 2026</a>):</p> <ul> <li>Is every plum tree (a certain class of loopy games) contained in $\overline{\mathcal{C}}$$?</li> <li>Does any non-stopper-sided loopy game lie in$$\overline{\mathcal{C}}$$, or is approximation universally obstructed in this class?</li> <li>What intrinsic combinatorial or coalgebraic property distinguishes non-stoppers inside$$\overline{\mathcal{C}}$ from those outside?

Additionally, the lack of canonical forms, generalized Grundy semantics, and a classification theory for partizan non-stopper-sided loopy games is a major direction for future work. The coalgebraic approach is foundational but the algebraic, topological, and combinatorial subtleties of these games remain a prominent area of investigation (Honsell et al., 2011, Burke et al., 15 Jan 2026).