5 · Gibbs and potential games — where the physics is exact

Take a congestion game: three resources, each slower the more players use it. This game has a special property: there is a single function Φ over joint choices — Rosenthal's potential — such that every player's incentive to switch is exactly the change in Φ. The game is, in a precise sense, all of us rolling downhill on one shared landscape.

Now let players revise logit-style: at random moments, one player re-picks an action with probabilities softmax(λ · payoffs). In a potential game this revision process is exactly heat-bath dynamics on Φ, and its stationary distribution is exactly the Gibbs measure $\pi \propto e^{\lambda\Phi}$ — λ playing inverse temperature, Nash appearing as the zero-temperature limit. This is not an analogy; strataq verifies it to $2.5\times10^{-16}$ on congestion games as a permanent regression test.

Everything thermal-equilibrium follows for free: detailed balance, zero probability current, zero entropy production, symmetric (Onsager) response. That is why potential games are the programme's calibration standard — every meter must read exactly zero on them, and does, including on logit route choice over a real road network, where the potential (the Beckmann integral) is known analytically.

And where it is not exact. Break the potential — mix in a harmonic component like rock–paper–scissors — and there is no landscape any more: the same revision dynamics still settle into a stationary state, but one that circulates and dissipates (explainer 6). The physics language stays honest by staying tiered: Gibbs statements are exact for potential games; for everything else what survives is the non-equilibrium machinery, which is where the interesting readings live.