3 · QRE vs mixed Nash — sensitivity, not indifference

Mixed-strategy Nash also outputs probabilities, so the two are often confused. The logic could not be more different.

A mixed Nash requires indifference: you randomise 60/40 only if both actions have exactly equal expected payoff — otherwise you'd purify. The probabilities are pinned down by making your opponent indifferent, which produces famously odd comparative statics (your mix depends on their payoffs, not yours).

QRE requires payoff sensitivity: with EU(A) = 10 and EU(B) = 8, B keeps positive probability — smaller than A's, by an amount governed by λ. The probability ordering follows the payoff ordering, always (this is one of the regular-QRE axioms). Nothing needs to be indifferent to anything.

The practical consequence: QRE's predicted frequencies respond smoothly to payoff changes — which is exactly what makes the susceptibility χ = dσ*/dh a well-defined instrument (explainer 7 measures its symmetry). Nash's correspondence is piecewise-constant-then-jumping; there is no meter to build on it.