---
slug: gaming_proof_impossibility
name: Gaming-Proof Impossibility
one_liner: "For a rich enough space of outcomes, no rule can be simultaneously non-trivial, fair, and immune to strategic manipulation — the negative, no-such-rule-exists dual of constructive mechanism_design along the can-a-rule-be-engineered axis."
type: prime
schema_version: 2
status: density_v2_draft
categories: [economic, mathematical, social]
origin_domain: mechanism_design_and_social_choice
alternate_origin_domains: [voting_theory, computer_security, contract_and_law]
aliases: [strategy_proofness_impossibility, incentive_compatibility_limit, manipulation_theorem, no_dominant_strategy_rule]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-01 dualization, sub-flavor 1c) — the negation/impossibility dual of `mechanism_design` along the constructive-vs-impossibility (can-a-rule-be-engineered) axis. Discovered in the op-01 full sweep 2026-07-03. Verified absent via search_prime + slug check (no gaming_proof / impossibility / strategy_proofness prime present). Collision risk: overlaps the Gibbard–Satterthwaite / Arrow named-result family — the theorem-vs-prime question below is the live concern. STATUS: HOLD — pending owner curation."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). Did NOT go through Phase-A triage. SI/SF/similarity grades are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT yet written."
- "Phase-C action: re-similarity-check against mechanism_design, no_free_lunch, conservation-law/impossibility genus primes, and any existing arrow_impossibility / trade-off primes; at incorporation draw the dual edge to mechanism_design and a possible genus edge to an impossibility-result / conservation family; sync one_liner to a v1 draft per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 3
  domain_breadth: 3
  structural_abstraction: 4
  transfer_evidence: 3
  reasoning: 'The structure — "over a rich enough outcome space, a fixed rule cannot jointly satisfy non-triviality, a fairness/efficiency criterion, and immunity to manipulation by self-interested inputs" — is a medium-neutral impossibility statement about rules, so structural abstraction is high (4). It recurs beyond its social-choice home: strategy-proofness limits in auction and matching design, the impossibility of a perfectly non-gameable metric (Goodhart-flavored), and the general security intuition that any checkable rule an adversary can see can be played to. But much of the sharpest evidence clusters in economics/voting and is proved as specific named theorems rather than felt as one structural pattern, so domain breadth and transfer evidence are a cautious 3 and the composite is 3.'
similarity_to_nearest_existing_prime:
  slug: mechanism_design
  similarity: 0.55
structural_framed:
  label: mixed-structural
  aggregate: 0.3
  criteria: {vocab_travels: 0.3, evaluative_weight: 0.4, institutional_origin: 0.3, human_practice_bound: 0.2, import_vs_recognize: 0.4}
  rationale: 'The core claim is a formal impossibility about rules and strategic inputs, which reads structural; but the load-bearing predicates ("fair", "non-dictatorial", "manipulation") carry evaluative weight and the canonical results are stated in a social-choice / voting idiom, so evaluative_weight and import_vs_recognize lift it off zero. It is not human-practice-bound in principle (a metric or protocol can be gamed with no humans), which keeps it from reading fully framed. Net: mixed-structural.'
phase_b_passes_completed: []
---
# Gaming-Proof Impossibility

## Core Idea

Gaming-proof impossibility is the structural claim that *no fixed rule can simultaneously be non-trivial, satisfy a fairness/efficiency criterion, and be immune to manipulation by the self-interested inputs it evaluates* — once the space of possible outcomes is rich enough. It is one pole of a single axis, *can a rule be engineered to achieve a target*, whose opposite pole is `mechanism_design` (start from a desired outcome and search rule-space for a game whose equilibrium implements it). Mechanism design is the constructive, yes-here-is-a-rule pole; gaming-proof impossibility is the negative, no-such-rule-exists pole. Where the designer asks "what rules make honest, aligned behavior an equilibrium?", this prime states the ceiling on that search: for sufficiently expressive domains, the three properties one most wants — that the rule actually discriminates between outcomes (non-triviality), that it respects a fairness or efficiency standard, and that no participant can gain by misreporting or manipulating (strategy-proofness / incentive compatibility) — cannot all hold at once.

The load-bearing invariant is a *tension among three properties over a rich domain*, not any single property alone. Restrict the domain (only two alternatives, single-peaked preferences, quasilinear utility with money) and gaming-proof rules exist; expand it to unrestricted preferences over three or more outcomes and the impossibility bites. The characteristic payoff of naming this pole is diagnostic honesty: it tells a designer *in advance* that a proposed "un-gameable, fair, fully general" rule is chasing a contradiction, so the real work is choosing which property to relax — accept manipulability, narrow the domain, weaken fairness, add outside enforcement (payments, randomization, verification) that steps outside the pure-rule frame, or tolerate the manipulation and predict it. The cost of the pole is that it is a *negative* result: it forecloses without constructing, and the space of near-miss escapes (approximate strategy-proofness, restricted domains, money-augmented mechanisms) is exactly where the designer's remaining freedom lives.

As one pattern it recurs wherever a rule must aggregate or evaluate strategically-supplied inputs. In **social choice and voting**, the Gibbard–Satterthwaite result and Arrow's theorem are the sharp forms: any non-dictatorial voting rule over three or more alternatives is manipulable, and no aggregation rule meets a small set of fairness axioms at once. In **mechanism design and auctions**, the impossibility that no dominant-strategy, efficient, budget-balanced, individually-rational mechanism exists in general (Myerson–Satterthwaite) is the same ceiling. In **metric design and management**, the intuition that any performance measure exposed to the agents it scores becomes gameable is the Goodhart-flavored shadow of this pole. In **computer security and adversarial settings**, any checkable rule an adversary can inspect can in principle be played to, so "un-gameable and fully general and useful" is over-constrained. Across these, the recognizable move is the same: enumerate the properties you want the rule to have, notice the domain is rich, and conclude that some property must give.

## Notes

**Provenance.** Produced by the op-01 dualization operator, sub-flavor 1c (negation / impossibility dual), from input prime `mechanism_design`. The operator inverted the constructive "search rule-space for a game implementing a target" into its negative pole "for rich enough domains no such rule exists," which lands squarely on the Gibbard–Satterthwaite / Arrow / Myerson–Satterthwaite family. Convergence in the sweep: the impossibility pole is a well-attested structural companion of mechanism design, so the route is a natural single-flavor find rather than a multi-route convergence.

**For owner evaluation (keep/cut).** The case FOR a prime: there is a genuine, domain-general *shape* here — "over a rich enough space, non-triviality + a fairness criterion + manipulation-immunity are jointly unsatisfiable" — and it is the clean dual of an existing accepted prime, so keeping it completes a constructive/impossibility pair on the rule-engineering axis (a lattice-gap / pair-completion signal worth weighing). The primary risk, flagged in the manifest, is the **theorem-vs-prime question**: Gibbard–Satterthwaite, Arrow, and Myerson–Satterthwaite are *named results*, and this candidate may be a family of theorems dressed as a prime rather than a substrate-independent structure. The keep decision turns on whether the impossibility generalizes cleanly beyond social choice (security metrics, Goodhart, any adversarially-inspected rule) — if it does, it is a prime in the same way `no_free_lunch` or a conservation-law impossibility is; if the crisp content lives only in the voting/auction axioms, it is better recorded as results attached to `mechanism_design`. Secondary consideration: check whether an existing impossibility/trade-off or conservation genus already subsumes it, and whether a Goodhart / metric-gaming prime is the better home for the management-facing instances. Genus candidate to weigh at incorporation: a general "impossibility result" family.

**At incorporation.** Re-similarity-check against `mechanism_design` (dual edge), `no_free_lunch`, any existing Arrow/impossibility or trade-off primes, a possible Goodhart/metric-gaming prime, and a conservation-law / impossibility genus. If kept, draw the dual edge to `mechanism_design` on the constructive-vs-impossibility axis and a genus edge to whatever impossibility-result family is adopted. Confirm the prime does not collapse into "manipulation of metrics" alone (that is narrower). Sync the one_liner to a v1 draft per CLAUDE.md before merge, keeping both files' `one_liner` identical.
