Learn · 02 / 10

2 · The fixed point — everyone's noise responds to everyone's noise

Softmax response (explainer 1) is a one-player idea. The strategic leap is circular: your rival's distribution over prices determines your expected payoffs; your softmax over those payoffs is your distribution; that determines their expected payoffs; and so on:

σB→EUA→σA→EUB→σB→⋯\sigma_B \to EU_A \to \sigma_A \to EU_B \to \sigma_B \to \cdots

A quantal response equilibrium is where this loop stops moving: every player's distribution is the softmax of the payoffs induced by everyone else's distribution. Existence follows from Brouwer; at λ = 0 it is the uniform centroid; as λ grows it traces a path (a branch) that generically terminates at a Nash equilibrium.

Computing it is its own subject. strataq ships three solvers that must agree (and are tested against each other and against Gambit's homotopy tracer to 10⁻⁸–10⁻¹³): damped iteration (simple, fine at moderate λ), magnetic mirror descent (last-iterate convergence even where naive iteration would cycle), and a pseudo-arclength tracer that follows the whole branch — including through bifurcations, where the number of equilibria changes and the most interesting physics of this project begins.

So what would you do differently?

Your best move depends on theirs, and theirs on yours. Any planning process that fixes their behaviour first and optimises against it is answering a different question than the one you have.

Watch that assumption cost money over a hundred rounds →
▶ try it · the loop, settling — click inside the triangle
in-browser · goldens-checked
rockpaperscissors
rationality λ = 2.00
0.2 · noise20 · sharp

harmonic: trajectories spiral around the centre — the circulation the dissipation meter measures. The quantal response equilibrium is where the flow stops moving — watch it drift from the uniform centre (λ small) toward a Nash equilibrium (λ large).