The reciprocity meter — ℛ¶
What it reads. How asymmetric the equilibrium give-and-take between players is:
zero exactly when the response is fully reciprocal; values above 1 are possible and meaningful — they say the antisymmetric (circulating) part of the response dominates the symmetric part, as it does in matching pennies (ℛ = 1.2 at λ = 1.2),
where \(\chi^{\mathrm{eq}} = (I - SB)^{-1}S\) is the equilibrium susceptibility — the response of everyone's play to a small payoff perturbation, after all strategic feedback settles.
Why it means something (tier: derived — see the claims ledger, entries R1/N1/N2). The reciprocity transfer (Result 2): \(\chi^{\mathrm{eq}}\) is symmetric iff \(S(B-B^\top)S = 0\), i.e. iff the normalised game has zero harmonic component. Strategic feedback neither creates nor destroys reciprocity — so the observable response inherits the symmetry of the unobservable payoff operator exactly. Consequences:
- ℛ = 0 exactly on potential games (where QRE is a Gibbs equilibrium, Onsager-style reciprocity holds);
- ℛ grows with harmonic content. The λ-free property is the symmetry statement, not the magnitude: whether ℛ = 0 (reciprocity holds ⟺ potential) does not depend on λ, which is what answers the Haile–Hortaçsu–Kosenok critique — there is no noise parameter that can absorb an asymmetry. The magnitude of ℛ does vary with λ (approximately ∝ λ at small λ; findings F-0002), so cross-system comparisons of ℛ levels must hold λ fixed or report it;
- operationally, ℛ is estimable from cross-agent pass-through asymmetry: how much player \(i\) moves when \(j\)'s costs move, versus the reverse — no payoff knowledge required.
How it is computed. All algebra on the tangent space via an explicit Helmert basis (a rank-deficiency slip here fakes criticality); the resolvent is shared with implicit differentiation and the spectral phase locator. See strataq.finite.response.reciprocity.
Calibration state. Gate finite.response.reciprocity:
| Reading | Requirement | Artifact |
|---|---|---|
| 5 exact potential games | ℛ < 10⁻¹⁰ | reciprocity_potential.json |
| RPS family + matching pennies | ℛ > 0.1 | reciprocity_harmonic.json |
| 2,000-game α sweep at fixed λ = 1.2 | Spearman ρ(ℛ, α) > 0.9, bootstrap CI | reciprocity_alpha_sweep.json |
| χ^eq vs finite differences | agreement to 10⁻⁶ on 50 games | chi_fd_agreement.json |
The sweep holds λ fixed, so λ cannot drive the correlation; ρ < 1 reflects genuine cross-game structure at equal α (e.g. RPS-3 and RPS-5 both sit at α = 1 but read different ℛ). Reported readings are rank-order evidence, not a functional law.
All artifacts regenerate from a fixed seed via uv run python -m experiments.reciprocity_calibration (make reproduce).
Limitations, stated once. ℛ is a property of the normalised game at a specific QRE point; near criticality (distance_to_criticality below the configured threshold) magnitudes of χ are unreliable and the API flags rather than reports. Empirical estimation from pass-through (Stage 3) inherits the identification caveats of the demand stage, not of ℛ itself.