7 · Reciprocity — the same number both ways

Here is a measurement you can do on any strategic system without knowing its payoffs: poke player 1, read player 2; poke player 2, read player 1. Nudge one player's incentives by a small observable amount — a toll on a road, a cost shock to one firm — let everyone re-equilibrate, and record how much everyone else moved.

The matrix of all such readings is the equilibrium susceptibility $\chi^{\mathrm{eq}} = (I - SB)^{-1}S$: the response of the whole system's play to a payoff perturbation, after strategic feedback settles. In physics, the symmetry of such a matrix is Onsager reciprocity, and it is a signature of thermodynamic equilibrium.

The result that makes this a meter (Result 2, tier: derived): the strategic feedback loop neither creates nor destroys reciprocity. $\chi^{\mathrm{eq}}$ is symmetric exactly when the game's normalised payoff structure has zero harmonic (circulating) component — that is, exactly when the game is a potential game. One would generically expect a resolvent $(I-SB)^{-1}$ to scramble symmetry; it doesn't. So the observable response matrix inherits the symmetry of the unobservable payoff operator, and the reciprocity defect $\mathcal{R} = \|\chi - \chi^\top\| / \|\chi + \chi^\top\|$ is an operational test of potentiality requiring no payoff knowledge — and whether it reads zero doesn't depend on λ, so there is no noise parameter that can absorb an asymmetry.

Calibration, measured: on exact potential games — including logit route choice on the real Sioux Falls road network, where the potential is known analytically — ℛ reads $10^{-16}$: zero to machine precision. On rock–paper–scissors it reads 0.69; on matching pennies 1.2 (yes, above 1 — ℛ is a norm ratio, and a value above 1 means the circulating part of the response dominates the reciprocal part).

What ℛ is not (measured the hard way): its magnitude grows with λ, and in the near-harmonic regime it stops tracking dissipation — we proposed a fix and refuted it ourselves. ℛ answers one question exactly — "is this system potential?" — at every λ and every α. For "how hard is it circulating?" you need the entropy-production meter. Two instruments, two questions.

(Interactive controls in the Lab: poke-player selector, poke size, the two cross-readings side by side, ℛ gauge tracking the α slider.)