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TERM-077: ordinal potential: of a negotiation over a finite set of options in a window of time, an ordering of the options that every move winning one of its conflicts climbs; it exists exactly when the wins never cycle, which is ordinal extendibility, that some complete ordering agrees with every win, and not observed comparability, since pairs never contested are not compared; and it is fixed only up to a class: every strictly increasing transform of one is one, and on a sparse record several orderings can be

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Properties7
StatusActive
Version2
Tagsmathematics, game-theory, emotion-and-affect
Lineformal-apparatus
Used byCLAIM-434 — On a finite set of outcomes, some function rises with every move that wins one of a negotiation's conflicts exactly when no chain of winning moves returns to where it began, and the length of the longest such chain ending at an outcome is one such function; a perfectly transitive record of which side won each pair, in a complete tournament contesting every pair once, leaves a cyclic share of (n − 2)/3n under the Hodge decomposition, a baseline that does not carry to sparse records; the Hodge gradient can reverse a comparison the wins settle, netting a record erases alternation, a record fixes a class of potentials rather than one, a coherent record is not yet one aligned with any single perspective's ordering, a best-fitting ordering of a cycle is no Lyapunov function, the sign a class of potentials gives can be set-valued, and an ordering fitted to a record overstates its agreement there, CLAIM-440 — An organization's appraisal is the negotiation among its perspectives, a common currency is that negotiation's admitting an ordering of the options it faces, and goal-conduciveness valence is a regime in which an ordering of the organization's own states by their bearing on its maintenance prevails in the negotiation; so valence presupposes contested rankings, not frustrated demands, its sign is whether the organization runs ahead of or behind a reference course along the potential ordering, its intensity is how far the negotiation's decisions follow that ordering, and ambivalence, where no ordering prevails, differs from neutrality, where one prevails and the organization keeps to its course, CLAIM-448 — An ordinal potential describes only the winning-move relation it is computed on; where an organization's legibility regime selects which contests are admitted or recorded, an acyclic record, even a persistent potential, can be the product of selective contestation, so observed coherence is not evidence that the organization's perspectives are integrated, and an intervention that improves its self-governance may at first increase the disagreement observed, CLAIM-451 — When an organization's record of winning moves cycles, the failure is organizational nonintegrability, distinct from a perspective's own preference instability and from switching of the governing regime; it becomes organizational tension only when realized as costly reversal against demands the organization maintains, and cognitive dissonance only when that tension is detected and appraised as aversive; learning can relieve it by accommodation, by entrenched precedence or by cheap periodic coordination, which observed coherence does not tell apart, and ambivalence and nonintegrability entail neither one another
Refined byTERM-078 — persistent potential: an ordinal potential of a negotiation that is the same, up to its class, across successive windows of time, that decides the conflicts it bears on (priority), and towards which the dynamics climb again after a perturbation (resumption); exact when no move outside a perturbation descends it, and then a Lyapunov function of the one-move-at-a-time dynamics, and approximate when some do, and then no Lyapunov function, short of a drift condition that needs a cardinal potential and further assumptions
Cited as “uses” byCLAIM-482 — An organization's aggregate revealed preference exists exactly when its record of winning moves never cycles, and where it exists it is fitted to one regime of authority, so an intervention on authority that changes the winning moves changes the fitted preference and the preference does not predict the change

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Definition

Let a negotiation face the finite options X in a window W, and let x → y when, in W, a move from x to y won one of its conflicts. An ordinal potential of the negotiation on W is a strict ordering ≻ of X (or any function P : X → ℝ) with y ≻ x whenever x → y.

  • Existence. One exists exactly when no chain of wins in W returns to where it began; the length of the longest chain ending at x is one (CLAIM-434, part 1). The relation of wins need only be acyclic: pairs never contested in W need not be compared, so completeness is not required. A complete, transitive relation is the special case in which every pair was contested.
  • Approximation. Where the wins cycle, the negotiation’s distance from having one is the least fraction of its wins whose reversal leaves them acyclic, a minimum feedback arc set; the orderings that attain it are its best-fitting orderings. This is ordinal throughout: it counts wins and reverses none of them by margin.
  • The class. An ordinal potential is never unique. Every strictly increasing transform of one is one; and where pairs were not contested, orderings that rank those pairs differently are all potentials of the same record (CLAIM-434, part 7). What a record fixes is the class, and a claim that a particular ordering is the potential is a claim that needs an intervention contesting the pairs that separate it from the others.

The term names the first of four senses that “common currency” ran together. They are not one chain of implications. The first two are evidential, about what the record establishes, and the second implies the first. The last two are representational, about what structure a representation has, and the fourth implies the third. The pairs do not imply across: a complete scalar utility can exist with many pairs never observed, so the third does not give the second.

  1. ordinal extendibility: the negotiation has an ordinal potential, so some complete ordering agrees with every win (on a finite set, an acyclic relation extends to a linear order: Szpilrajn’s theorem in its finite form, which part 1 of CLAIM-434 constructs). It establishes that a shared ordering could account for the record, not that the record contains one. If the only win is A over B, three complete orderings fit and C’s place is not observed (scratch/formal/valence.py, case 11);
  2. ordinal comparability: every pair was contested and the wins are complete and transitive, so the record itself orders all the options;
  3. scalar currency: the ordering has a real-valued representation, which a finite ordering always has and a graded lexicographic one on a continuum does not (CLAIM-436, parts 1 and 2);
  4. cardinal trade-off structure: differences in the representation mean something, so exchange rates between options are fixed up to an affine transform. Only here are margins intensities, and only here does the Hodge decomposition’s least-squares potential measure what it says (THEORY-894).

The record’s “common currency”, in the owner’s sense, needs only the first: the existence of a compatible extension, not an observed shared evaluation of every option. Anything inferred about pairs never contested is inferred about the class, not observed. The first level was named “ordinal comparability” in a first revision and renamed “ordinal extendibility” after a second review of the pull request, which pointed out that acyclicity licenses only the weaker inference.

What it is not

  • Not the Hodge gradient. The gradient of the Hodge decomposition is a least-squares fit to margins, a cardinal object. It can rank B above A when A won their contest: margins A over B by 1, B over C by 10, A over C by 1 give scores 2/3, 3 and −11/3 (CLAIM-434, part 4; scratch/formal/valence.py, case 1). Its cyclic share is positive on that record although the wins are transitive.
  • Not a currency. On a finite set the two coincide at each instant; across instants and on continua they do not (CLAIM-436).
  • Not vacuous only by accident. A negotiation with no contested pairs in W has an empty relation of wins, which is acyclic, so every ordering is trivially a potential. The term allows this; terms built on it, such as goal-conduciveness valence (TERM-076), must exclude it.
  • Not persistent. A potential on each window says nothing about whether it is the same potential across windows. That is TERM-078.
  • Not a potential game. Monderer and Shapley’s generalized ordinal potential (THEORY-896) is defined over players’ unilateral deviations; this term needs only the relation of wins.

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