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Rock-Paper-Scissors (Intransitive Cyclic Dominance)

Prime #
1153
Origin domain
Information Theory
Subdomain
game theory → Information Theory

Core Idea

A relation over a set of options is intransitive when "A beats B" and "B beats C" do not entail "A beats C" — and in its strongest form a cycle closes: A beats B, B beats C, C beats A. No option is globally dominant; the ranking is circular rather than linear. The structural consequence is decisive: any attempt to pick "the best" collapses, because there is no best, and the system's behavior is governed by the cycle's dynamics — mixed strategies, rotation, coexistence — rather than by convergence to an apex.

The load-bearing structural content is that ranking is a property of the relation, not of the options. The folk game that lends the pattern its name is merely the most familiar instance of a topological fact about binary relations: the beats-graph contains a directed cycle, and a directed cycle admits no consistent linear order. From this single fact a great deal follows without further detail. There is no pure-strategy equilibrium; the unique equilibrium is mixed, assigning each option positive weight. Under population dynamics, all types coexist at positive frequency and their shares oscillate, because any monoculture is invadable by the type that beats it. And any procedure that requires a linear order — single-winner voting, fully consistent pairwise rating, optimization toward "the best" — will produce arbitrary or unstable results when applied to a cyclic relation. The discipline the prime imposes is to ask not "which is best?" but "who beats whom?", and to distinguish genuine intransitivity, present in the payoff graph, from apparent intransitivity arising from sampling noise or shifting context.

How would you explain it like I'm…

The Beating Circle

In rock-paper-scissors, rock beats scissors, scissors beats paper, and paper beats rock — so nothing is the best, it goes in a circle. You can't pick a "winner" move because each one loses to another. The only smart plan is to mix it up and not always play the same thing.

No Best Move

Sometimes "beats" goes in a circle instead of a straight line: A beats B, B beats C, but C beats A — like rock, paper, scissors. When that happens there's no single best choice, because every option loses to something. That circle is called intransitive: "A beats B" and "B beats C" do NOT mean "A beats C." The big idea is that winning is a property of the matchup, not of any one option. So instead of asking "which is best?", you ask "who beats whom?" — and the smart move is to mix your choices, because if you always played one thing, someone could always pick the thing that beats it.

Intransitive Cyclic Dominance

A relation over a set of options is intransitive when "A beats B" and "B beats C" don't entail "A beats C" — and in its strongest form a cycle closes: A beats B, B beats C, C beats A. No option is globally dominant; the ranking is circular, not linear. The decisive consequence is that any attempt to pick "the best" collapses, because there is no best, and behavior is governed by the cycle's dynamics — mixed strategies, rotation, coexistence — rather than convergence to an apex. The load-bearing point is that ranking is a property of the relation, not of the options: the familiar game is just the most recognizable instance of a topological fact — the beats-graph contains a directed cycle, and a directed cycle admits no consistent linear order. From that one fact much follows: no pure-strategy equilibrium (the only equilibrium is mixed, weighting every option); under population dynamics all types coexist and their shares oscillate, since any monoculture is invadable by whatever beats it; and any procedure needing a linear order — single-winner voting, consistent pairwise rating — gives arbitrary or unstable results. The discipline is to ask "who beats whom?" rather than "which is best?", and to tell genuine intransitivity in the payoff graph from apparent intransitivity from sampling noise or shifting context.

 

A relation over a set of options is intransitive when "A beats B" and "B beats C" do not entail "A beats C" — and in its strongest form a cycle closes: A beats B, B beats C, C beats A. No option is globally dominant; the ranking is circular rather than linear. The structural consequence is decisive: any attempt to pick "the best" collapses, because there is no best, and the system's behavior is governed by the cycle's dynamics — mixed strategies, rotation, coexistence — rather than by convergence to an apex. The load-bearing structural content is that ranking is a property of the relation, not of the options. The folk game that lends the pattern its name is merely the most familiar instance of a topological fact about binary relations: the beats-graph contains a directed cycle, and a directed cycle admits no consistent linear order. From this single fact a great deal follows without further detail. There is no pure-strategy equilibrium; the unique equilibrium is mixed, assigning each option positive weight. Under population dynamics, all types coexist at positive frequency and their shares oscillate, because any monoculture is invadable by the type that beats it. And any procedure that requires a linear order — single-winner voting, fully consistent pairwise rating, optimization toward "the best" — will produce arbitrary or unstable results when applied to a cyclic relation. The discipline the prime imposes is to ask not "which is best?" but "who beats whom?", and to distinguish genuine intransitivity, present in the payoff graph, from apparent intransitivity arising from sampling noise or shifting context.

Structural Signature

the set of three or more options — the binary beats-relation over them — the intransitivity (failure of A>B, B>C ⇒ A>C) — the closed directed cycle in the beats-graph — the absence of any consistent linear order — the mixed-equilibrium-with-rotation dynamics that replace convergence

The pattern is present when each of the following holds:

  • A set of options. Three or more alternatives are compared — morphs, candidates, teams, standards, strategies.
  • A beats-relation. A binary "beats" or "dominates" relation is defined over pairs of options, locating ranking in the relation rather than in any intrinsic property of the options.
  • Intransitivity. The relation fails transitivity: "A beats B" and "B beats C" do not entail "A beats C."
  • A closed cycle. In its strongest form the beats-graph contains a directed cycle (A beats B beats C beats A), so dominance loops rather than terminating.
  • No linear order. A directed cycle admits no consistent linear ranking; there is no globally dominant option and no apex to converge to.
  • Rotation dynamics. Because no pure strategy dominates, the unique equilibrium is mixed, all types coexist at positive frequency, shares oscillate, and any monoculture is invadable by the type that beats it.

These compose so that the single topological fact — the cycle — yields, without further payoff detail, a package of consequences: no pure-strategy equilibrium, persistent coexistence, oscillating shares, and the incoherence of any procedure requiring a linear order. The discipline is to ask "who beats whom?" rather than "which is best?", and to separate genuine intransitivity in the payoff graph from apparent intransitivity due to noise or shifting context.

What It Is Not

  • Not competition. Competition is rivalry over a scarce resource; rock-paper-scissors is a specific topology of the beats-relation — a closed directed cycle. Competition can be transitive (a clear pecking order) or cyclic; only the cyclic case is this prime (see competition).
  • Not equilibrium. An equilibrium is a rest state. The cyclic structure has a mixed equilibrium but its dynamics are rotation and oscillation, not convergence to a pure rest point. The prime is about why no pure equilibrium exists, not about settling (see equilibrium).
  • Not oscillation. Oscillation describes a variable cycling in time. Rock-paper-scissors is the relational cause — an intransitive beats-graph — that can produce oscillating population shares. The oscillation is a downstream consequence, not the structure itself.
  • Not trade_offs. A trade-off ranks options along trading axes, still yielding a (Pareto) order. The defining feature of cyclic dominance is the failure of any consistent order at all — no scalar reproduces the cycle, which trade-off framing presupposes.
  • Not attractor_selection_and_basin_control. Despite embedding-nearness, that prime concerns which of several stable states a system settles into. Rock-paper-scissors specifically lacks a stable apex to settle into; its signature is persistent coexistence, not basin capture.
  • Common misclassification. Fitting a single-number rating (Elo-style) to a genuinely cyclic competitive field. No consistent scalar can reproduce a cycle, so the ranking will contradict head-to-head results. The tell: do consistent pairwise predictions follow from any per-option score? If a cycle exists, they cannot.

Broad Use

The same cyclic-dominance topology recurs across substrates that share nothing but the shape of their beats-relation. In game theory, rock-paper-scissors itself has a unique Nash equilibrium at the uniform mixed strategy, because no pure choice dominates. In ecology, the side-blotched lizard's three mating morphs cycle in dominance, and bacterial toxin-producer, sensitive, and resistant strains maintain coexistence by cyclic competition rather than competitive exclusion. In social choice, Condorcet's voting paradox shows that majorities can prefer A over B, B over C, and C over A, so no Condorcet winner exists even when every individual is rational. In sports and competition, stylistic matchups produce reliable cycles in which Team A beats Team B beats Team C beats Team A, defying any linear power-ranking. And in markets and technology, three competing standards or business models can form a triangle in which each is vulnerable to the next — generalist beats specialist beats super-specialist beats generalist — with no stable winner. In every case the cycle alone, independent of the full payoff detail, predicts the qualitative outcome: no fixation, persistent coexistence, and oscillating shares.

Clarity

The prime makes visible that ranking is a property of the relation, not of the options. As soon as a reasoner looks for the cycle, they stop asking "which is best?" and start asking "who beats whom?" — which exposes why interventions that presuppose a linear order, such as seeding a tournament, choosing a single pure strategy, or declaring a category winner, can be incoherent. The frame also separates genuine intransitivity, structurally present in the payoff graph, from apparent intransitivity produced by sampling noise or a shifting context, so that a cyclic pattern in data can be correctly attributed either to the relation's topology or to measurement artifact. The clarifying force is to relocate the question from the items being ranked to the structure of the relation ranking them.

Manages Complexity

Once the cycle is identified, a great deal of system behavior follows without any further detail. No pure equilibrium exists; all types coexist at positive frequency; population shares oscillate; and any monoculture is vulnerable to invasion by its predator-type. The analyst does not need the full payoff matrix to predict these qualitative outcomes — the cycle alone is enough. This is a sharp compression: a potentially complicated system of competing options reduces to a single topological feature of its beats-graph, from which the absence of a winner and the presence of rotation are immediately derivable. The complexity payoff is that the impossibility results — no linear order, no stable monoculture — come for free with the recognition of the cycle, sparing the practitioner the futile search for a best option that the structure forbids.

Abstract Reasoning

The signature lets a reasoner reason about an entire class of systems that share a cyclic beats-relation, independent of substrate — genes, ballots, plays, market share. It also clarifies what cannot be done: because no consistent linear ordering exists, any procedure requiring one will produce arbitrary or unstable results, so the prime functions as an impossibility theorem as much as a pattern. The reasoning concerns the topology of a binary relation over three or more options, which is why it is indifferent to what the options are: the same conclusions — mixed equilibrium, coexistence, oscillation, invadability of monocultures — follow wherever the directed cycle is present. To recognize the cycle is to know in advance both what the system will do (rotate) and what one cannot make it do (settle on a single winner).

Knowledge Transfer

The transferable content is a recognition and an intervention vocabulary that travel across domains: identify the cycle, choose a mixed strategy rather than a pure one, predict oscillation rather than convergence, and design for cyclic balance when monoculture is dangerous. From ecology to markets, the persistence of three rivals with no clear winner suggests cyclic dominance rather than market inefficiency, and the right response is not to pick one but to stabilize the cycle or to break it by changing the relation. From voting to model evaluation, intransitive results — a model beating another on one benchmark and losing on another in a cycle — imply there is no single ranking, only profiles. And from games to institutional design, deliberately constructing a cyclic capability triangle, in which three checks each constrain the next, prevents any single actor from dominating.

The structural roles map across substrates. The options are the morphs, the candidates, the teams, the standards, or the strategies; the beats-relation is the mating dominance, the majority preference, the stylistic matchup, or the competitive vulnerability; the cycle is the directed loop that closes the relation; and the mixed equilibrium with oscillation is the rotation of morph frequencies, the absence of a Condorcet winner, or the coexistence of rival standards. An ecologist watching three lizard morphs rotate in frequency, a social-choice theorist confronting a preference cycle with no Condorcet winner, and an institutional designer building three mutually-checking powers are reasoning about the same object: a beats-relation whose cyclic topology forbids a winner and mandates coexistence. The diagnostic — does the beats-relation close into a cycle, and if so is it genuine or an artifact? — travels unchanged across games, ecology, social choice, sports, and markets. Because the response repertoire is the same everywhere — play mixed, expect rotation, design for or against the cycle — a practitioner who has handled intransitive dominance in one domain can import the whole discipline into a domain that knows the pattern only by its folk-game nickname.

Examples

Formal/abstract

The game itself, treated as a formal payoff matrix, exhibits the whole package. The set of options is {rock, paper, scissors}; the beats-relation is rock beats scissors, scissors beats paper, paper beats rock. Intransitivity is immediate — rock beats scissors and scissors beats paper, but rock does not beat paper — and the beats-graph is a closed directed cycle. From this single topological fact, and with no further payoff detail beyond symmetry, the consequences follow by derivation: there is no pure-strategy equilibrium, because whatever pure choice you commit to, your opponent has a pure best-response that beats it; the unique Nash equilibrium is mixed at (⅓, ⅓, ⅓), assigning every option positive weight. Replace the two-player game with a large population whose members play their type against random opponents, and the same cycle yields rotation dynamics: a population of mostly rock is invaded by paper, a population of mostly paper is invaded by scissors, and the frequencies oscillate in a closed orbit, all three types coexisting at positive frequency with no fixation. The impossibility result is the sharp payoff: any procedure demanding a linear ranking — "which throw is best?" — is incoherent here, because a directed cycle admits no consistent linear order. The intervention the prime licenses is to stop seeking a best option and instead play the mixed strategy, or, if one controls the rules, to break the cycle by altering the relation.

Mapped back: the three throws are the options, the beats-rule is the beats-relation, the closed loop is the cycle, and the (⅓,⅓,⅓) mixed equilibrium with population rotation is the rotation dynamics — every signature role instantiated.

Applied/industry

Two real substrates carry the identical topology. First, the side-blotched lizard in evolutionary ecology has three male mating morphs distinguished by throat color, and their reproductive success forms a cycle: the aggressive territory-holder is beaten by the cooperative sneaker strategy, the sneaker is beaten by the mate-guarder, and the mate-guarder is beaten by the aggressive territory-holder. The options are the three morphs, the beats-relation is reproductive dominance, and because it closes into a cycle, no morph fixes: field observations show the morph frequencies oscillate over generations in exactly the rotation the structure predicts, each morph's rise sowing the conditions for the morph that beats it. An ecologist who recognizes the cycle knows in advance that competitive exclusion will not occur and that conservation of all three is the stable outcome — coexistence is mandated, not accidental. Second, social choice: Condorcet's paradox shows that a rational electorate can hold majority preferences A over B, B over C, and C over A simultaneously. The options are the candidates, the beats-relation is pairwise majority preference, and the cycle means no Condorcet winner exists — so any single-winner voting rule applied to this profile produces a result that depends on arbitrary features like agenda order, and a designer who detects the cycle knows the "pick the best candidate" framing is structurally incoherent and must instead report the preference profile or fix a procedure rule transparently. The same recognition underwrites institutional design that deliberately builds a cyclic check-triangle so no single power dominates.

Mapped back: the lizard morphs and the candidates are the options; reproductive dominance and majority preference are the beats-relations; the closed loops are the cycles; and oscillating morph frequencies and the absent Condorcet winner are the rotation-and-impossibility consequences — the same cyclic topology across ecology and social choice.

Structural Tensions

T1 — Genuine versus Apparent Intransitivity (measurement). The cycle must live in the actual payoff graph, but observed intransitivity can be an artifact of sampling noise, shifting context, or aggregating heterogeneous sub-populations each of which is transitive. The prime explicitly warns of this but cannot, from the data alone, tell you which you have. Failure mode: building a rotation strategy around a "cycle" that was a measurement artifact and dissolves on more data, or missing a real cycle masked by noise. Diagnostic: is the beats-relation stable across resampling and contexts? A cycle that survives is genuine; one that flickers is apparent.

T2 — Pairwise versus Global Structure (scopal). Intransitivity is a property of pairwise relations, but real systems mix transitive and cyclic substructure — a mostly-rankable field with a local cycle, or nested cycles. Treating the whole as either cleanly transitive or cleanly cyclic mis-models it. Failure mode: declaring "no best exists" from one local cycle when a near-total order holds elsewhere, or forcing a global ranking that a local cycle quietly corrupts. Diagnostic: is the cycle the global topology or an embedded subgraph? Map the full beats-graph before concluding either order or rotation governs.

T3 — Cyclic Dynamics versus Cycle Collapse (temporal). The prime predicts persistent coexistence and oscillating shares — but real cyclic-dominance systems can spiral: amplitude grows until one type hits zero and the cycle collapses to a fixed point, an outcome the idealized rotation hides. Stability of the cycle is not guaranteed by its existence. Failure mode: assuming all three types will persist (and conserving them) when the dynamics are actually heteroclinic and trending toward extinction of one. Diagnostic: are the oscillation amplitudes bounded or growing? Coexistence requires a stabilizing mechanism the bare cycle does not supply.

T4 — No-Best versus Meta-Strategy (sign/direction). "There is no globally dominant option" holds at the object level, yet at the meta level a mixed strategy (the uniform Nash) is optimal, and against a predictable opponent an exploitative pure strategy can dominate. The prime's "no best" is true of options but false of policies-over-options. Failure mode: concluding that strategy is hopeless because no move dominates, missing that the equilibrium mixture and opponent-exploitation are the real decision variables. Diagnostic: is the opponent playing the equilibrium mixture (then you cannot do better than mix) or exploitably (then a counter-cycle move wins)?

T5 — Cycle as Pathology versus Cycle as Design (scopal). A cyclic structure is a failure for any procedure needing a linear order (single-winner voting, optimization) but a feature where you want no dominant power (checks-and-balances, biodiversity, anti-monopoly). The same topology is something to detect-and-avoid or to deliberately engineer. Failure mode: trying to "resolve" a cyclic check-triangle into a hierarchy, destroying the very non-domination it provided. Diagnostic: does the task require selecting a best (cycle is the problem) or preventing dominance (cycle is the solution)? The structure is identical; the desideratum decides.

T6 — Relation Locus versus Option Properties (coupling). The load-bearing claim is that ranking lives in the relation, not in intrinsic properties of the options — so attempts to score options independently (assign each a strength rating) will fail wherever the beats-relation is genuinely cyclic, because no consistent scalar reproduces a cycle. Failure mode: fitting Elo-style or single-number ratings to a cyclic competitive field, producing a ranking that contradicts the head-to-head results and mis-seeds tournaments. Diagnostic: do consistent pairwise predictions follow from any single per-option score? If a cycle exists, they cannot — the relation is irreducible to option properties.

Structural–Framed Character

Rock-paper-scissors sits at the structural pole of the structural–framed spectrum — a clean structural zero, every diagnostic pointing the same way. Its content is a pure relational fact about a binary beats-relation: the relation closes into a directed cycle, so no consistent linear order exists, no option dominates, and the system is governed by mixed equilibrium and rotation rather than convergence to an apex. The folk-game name is merely the most familiar label for what is at bottom a topological claim about directed graphs.

Every diagnostic reads structural. The home vocabulary travels freely: the intransitive-cycle structure describes lizard mating morphs, Condorcet voting paradoxes, stylistic sports matchups, and competing market standards, each told in its own field's words (reproductive dominance, majority preference, competitive vulnerability) with no game-theoretic lexicon imported — what travels is the bare relation "A beats B beats C beats A," not any home frame. It carries no inherent approval or disapproval: a cycle is neither good nor bad until you specify the task, which is exactly why the entry's tension T5 must note that the same topology is a pathology for single-winner selection and a feature for checks-and-balances. Its origin is formal and relational — the topology of a binary relation over three or more options — with no appeal to human institutions, and it runs in genomes and bacterial strains with no agent or norm present, so it is thoroughly human-practice-independent. And invoking it merely recognizes a cycle already present in the beats-graph rather than importing an interpretive overlay. Even the institutional-origin criterion reads zero: although "rock-paper-scissors" is a folk-game nickname, the structural content is the bare binary-relation topology, which has no home discipline that owns it. On every diagnostic, it reads structural.

Substrate Independence

Rock-paper-scissors (intransitive cyclic dominance) is a highly substrate-independent prime — composite 5 / 5 on the substrate-independence scale. Its content is purely relational — a beats-relation that closes into a cycle rather than a ranking, so no option dominates and the system is governed by rotation and coexistence — and that is a bare property of a binary-relation topology, carrying no domain commitments. Hence structural abstraction is a full 5: the signature is medium-neutral, recognized the moment a non-transitive dominance loop appears. Domain breadth sits at 4 rather than 5: the pattern has substantive instances in game theory and social choice (Condorcet cycles in voting), behavioral and microbial ecology (the classic three-strain E. coli and side-blotched-lizard cycles where each morph beats one and loses to another), sports and competitive matchups, and market or strategy dynamics — a strong spread, though slightly narrower than the universal mathematical primes. Transfer evidence is 4: the cyclic-dominance model is concretely documented across games, ecology, and social choice, with the same topology carrying, even if a few applications stay closer to illustration. A bare relational signature plus broad attested instances gives a composite 5.

  • Composite substrate independence — 5 / 5
  • Domain breadth — 4 / 5
  • Structural abstraction — 5 / 5
  • Transfer evidence — 4 / 5

Relationships to Other Primes

One-hop neighborhood: parents above, mutual partners to the right, children below.Rock-Paper-Scissors …subsumption: CompetitionCompetition

Parents (1) — more general patterns this builds on

  • Rock-Paper-Scissors (Intransitive Cyclic Dominance) is a kind of, typical Competition

    The file: competition imposes no constraint on the beats-graph topology and can be transitive (a pecking order); rock_paper_scissors is 'the specific subcase where the beats-relation closes into a directed cycle.' A specialization of competition by relation-topology.

Path to root: Rock-Paper-Scissors (Intransitive Cyclic Dominance) → Competition

Neighborhood in Abstraction Space

Rock-Paper-Scissors (Intransitive Cyclic Dominance) sits in a sparse region of abstraction space (77th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely rather than landing on a neighbor.

Family — Strategic Interaction & Markets (38 primes)

Nearest neighbors

Computed from structural-signature embeddings · 2026-06-14

Not to Be Confused With

Rock-paper-scissors is most readily confused with competition, since both involve options that beat or lose to one another. The distinction is that competition is a substantive relation — rivalry over scarce resources, advantage, or survival — whereas rock-paper-scissors is a topological claim about the shape that relation can take. Competition imposes no constraint on the structure of who-beats-whom: it is entirely compatible with a clean transitive pecking order (the strongest competitor beats all others, the next beats all but the strongest, and so on), in which case a globally dominant option exists and the system converges on it. Rock-paper-scissors is the specific subcase where the beats-relation closes into a directed cycle, so that no consistent linear order exists and no option dominates. The practitioner consequence is sharp: under transitive competition the right move is to identify and back the dominant option, while under cyclic dominance that move is structurally incoherent — there is no best to back, and the correct response is to play a mixed strategy, expect rotation, and either stabilize or break the cycle. A reasoner who treats all competition as implicitly transitive will keep searching for "the strongest competitor" in a cyclic field and mistake the resulting instability for measurement error or market inefficiency, when it is the signature of the cycle's topology.

A second confusion is with equilibrium, because the prime does have a unique equilibrium (the uniform mixed strategy) and the word invites the expectation of a rest state. But the rock-paper-scissors structure is precisely a case where the interesting behavior is non-convergent: at the population level the dynamics are rotation and oscillation, with shares cycling in a closed orbit and any monoculture invadable by the type that beats it. The equilibrium here is a knife-edge mixed point, not an attractor the system settles into the way a stable equilibrium would be. The contrast with equilibrium as a prime is that equilibrium foregrounds the rest state — the balance of forces with no net change — whereas rock-paper-scissors foregrounds the impossibility of a pure rest state and the rotation that replaces it. Confusing the two leads to expecting a cyclic-dominance system to settle down ("it will reach equilibrium and stay there"), when in fact its characteristic trajectory is perpetual oscillation, and the only "equilibrium" is a mixed strategy that no individual playing a pure type ever occupies.

A third worthwhile contrast is with oscillation, because the prime's most visible consequence is oscillating population shares, and one might mistake the consequence for the structure. Oscillation is a temporal pattern — a variable rising and falling over time — and can arise from many mechanisms (feedback delays, predator-prey coupling, resonance) that have nothing to do with intransitive dominance. Rock-paper-scissors is the relational cause of one particular kind of oscillation: the cyclic topology of a beats-graph drives the rotation of type frequencies. The distinction matters because the remedy differs. If the oscillation is a relational consequence of cyclic dominance, you cannot damp it by the usual oscillation-control moves (adding negative feedback, tuning gains) without changing the underlying beats-relation; the oscillation is structurally entailed by the cycle and will persist as long as the cycle does. A reasoner who sees only "the shares are oscillating" and reaches for generic oscillation-damping will fail, because the lever is the topology of the relation, not the dynamics of a single variable.

These distinctions matter because each neighbor, mistaken for the prime, prescribes the wrong response. Confusing rock-paper-scissors with competition sends you hunting for a dominant option that the cycle forbids; confusing it with equilibrium sends you waiting for a settling that will never come; and confusing it with oscillation sends you damping a rotation whose true cause is relational topology. The prime's distinctive contribution — ranking lives in the relation, the relation can close into a cycle, and a cycle entails no-winner-plus-rotation — is exactly what none of these neighbors supplies.

Solution Archetypes

No catalogued solution archetypes reference this prime yet.