Cyclic Payoff Table¶
Artifact — instantiates Cyclic Dominance Counterbalancing
A payoff table used to compare pairwise outcomes and expose nontransitive relationships that a single score would hide.
A Cyclic Payoff Table records, for every ordered pair of options, the signed magnitude of the outcome — how much one option gains or loses when it meets another — so that intransitive structure surfaces not as arrows but as numbers that refuse to add up into a consistent ranking. Its defining move is that it records by how much, not just who wins: where a beats-map says only that A defeats B, the payoff table says A nets +40 against B, and it is the by-how-much that exposes a cycle a single aggregate score would quietly launder into a leaderboard. When the row totals of a table like this cannot be ordered — every option out-earns one rival and is out-earned by another — the ring is proven in the ledger's own arithmetic.
Example¶
A snack brand runs three shelf strategies and cannot decide which is "best": deep-discount, loyalty-bundle, and premium-hold. The category team stops arguing and builds a payoff table from head-to-head store-week trials, entering net margin per store-week for each matchup. Deep-discount beats premium-hold by about +$120 a week (it steals shoppers who won't pay full price). Premium-hold beats loyalty-bundle by about +$90 (the bundle's giveaways erode margin faster than they add volume). And loyalty-bundle beats deep-discount by about +$70 (it keeps customers who would otherwise churn to a competitor once the discount ends). Each cell carries a context tag — "holds in a soft economy; reverses when shopper budgets loosen."
Summed as row totals, the numbers go nowhere: no strategy has the highest total against the field, because each decisively out-earns exactly one rival and loses to another. The table's payoff is not a winner but a magnitude-backed loop — proof that whoever crowns one strategy on its best matchup is reading a single cell and ignoring the ring the money traces.
How it works¶
- Enter signed magnitudes per ordered pair. Each off-diagonal cell holds the net outcome of row-option versus column-option in shared units (margin, points, expected value), with a sign so gains and losses are directly comparable.
- Net the reciprocal cells. A-vs-B and B-vs-A are reconciled so the table is internally consistent and the advantage's size is unambiguous.
- Tag each cell with its context. The magnitude is recorded together with the condition under which it holds, because the same matchup can flip sign when the environment changes.
- Read the row totals against the ring. Sum each row; when the totals cannot be consistently ordered, the loop is confirmed in magnitude, not merely in direction.
Tuning parameters¶
- Payoff unit — raw outcome, margin, or risk-adjusted value. Richer units capture what actually matters but are harder to measure consistently across matchups.
- Sign and zero-point — where "even" sits. A mis-set zero can turn a near-tie into an apparent decisive win, faking a cycle.
- Aggregation window — how many encounters each cell averages. Longer windows steady the numbers but blur a matchup that is itself shifting.
- Context stratification — one table or one per condition. Splitting reveals sign reversals but can shatter a real cycle into unreadable slivers.
- Confidence weighting — whether thin-sample cells are down-weighted or flagged, so a lucky streak doesn't anchor a whole edge.
When it helps, and when it misleads¶
Its strength is that magnitudes separate a decisive counter from a coin-flip and show that the cycle is structural, not a rounding artifact — a near-tie and a blowout look identical on a beats-map but are worlds apart here, and the numbers make the loop impossible to wave away as opinion.
Its failure mode is that a table full of numbers invites a spurious cycle: small-sample magnitudes wobble, and three noisy edges can trace a ring that vanishes with more data — exactly the trap of nontransitive dice, where A beats B beats C beats A on average yet the effect is a fragile property of the specific distributions.[n1] The classic misuse is the mirror error: averaging every cell into one "expected payoff" score and picking the maximum, which re-flattens the ring the table exists to reveal. The guarding discipline is to treat the magnitudes as claims with error bars — down-weight thin cells, keep context tags visible, and hand the artifact to a pairwise audit before betting on the loop.
How it implements the components¶
counter_relation_matrix— the signed-magnitude cells are the relation matrix's numeric body: every pairwise edge with its size, sign, and evidence weight.dominance_cycle_map— because both directions are netted and summed, the loop appears as row totals that cannot be consistently ordered; the cycle is drawn in arithmetic.context_condition_vector— each cell is stored with the condition under which its magnitude holds, so a context-driven sign reversal is captured rather than averaged away.
It does not curate which options belong in the set — the repertoire and the direction-only map are Beats-Relation Matrix's job — and it never says what to play when; that rotation logic belongs to Countermove Rotation Playbook.
Related¶
- Instantiates: Cyclic Dominance Counterbalancing — supplies the magnitude-backed relation data the rest of the repertoire's governance prices its decisions against.
- Consumes: Beats-Relation Matrix — the direction map is the skeleton this table fills in with signed numbers.
- Sibling mechanisms: Beats-Relation Matrix · Countermove Rotation Playbook · Portfolio Minimum-Viability Rule · Pairwise Dominance Audit · Adaptive Mix Review · Sentinel Option Trial · Nontransitive Scenario Simulation
Editorial Notes¶
Form Classification¶
Form family: Representation, Specification & Plan
Rationale: Cyclic Payoff Table operates as a non-executable information artifact that externalizes static or prospective structure because it a payoff table used to compare pairwise outcomes and expose nontransitive relationships that a single score would hide.
Independent corroboration: The frozen evidence defines Cyclic Payoff Table as 'A payoff table used to compare pairwise outcomes and expose nontransitive relationships that a single score would hide', so its operative form is Representation, Specification & Plan.
Nearest alternative: Analysis, Modeling & Optimization — The payoff table is the persistent comparative artifact; any numerical netting supplies its cells.
Review outcome: Independent reviewer agreement; medium confidence.
Origin Attribution¶
Primary origin: Mathematics
Origin pattern: Cross-disciplinary synthesis
Present-day reach: Universal
Rationale: Mathematical game theory is primary because the artifact is a strategic-form payoff matrix extended to reveal nontransitive pairwise relations. Economics is a materially co-formative lineage: modern game theory was founded jointly as a mathematical theory of economic and social organization.
Related originating lineages:
- Economics & Finance — Economic game theory institutionalized payoff matrices for comparing interacting strategies and interpreting their utilities.
Review resolution: Mathematical game theory is primary because the artifact is a strategic-form payoff matrix extended to reveal nontransitive pairwise relations. Economics is a materially co-formative lineage: modern game theory was founded jointly as a mathematical theory of economic and social organization.
Review outcome: Researched adjudication after independent review; high confidence.
Sources consulted:
- Stanford Encyclopedia of Philosophy: Game Theory
- von Neumann and Morgenstern: Theory of Games and Economic Behavior
Notes¶
[n1] Nontransitive (intransitive) dice — sets of dice, such as Efron's, where die A beats die B, B beats C, and C beats A on average, despite each die being a fixed object with no memory. They are the standard demonstration that a cyclic win-relation can be real yet is a delicate property of the exact payoff distributions, which is why small-sample magnitudes can fake a cycle that fuller data dissolves. ↩