Learn · 09 / 10

9 · The one-price objection — the strongest case against all of this

The objection, in its owner's own words: "In reality you still have to choose one price. So what's the use of a probability? You then tend to increase λ and land on the same result as Nash."

This objection is correct — for the question it asks. If your objective is "what price should I charge tomorrow, given a payoff model I trust," then compute expected profit for each price and take the argmax. QRE used as a prescription adds nothing; it contains Nash as its λ → ∞ limit, and cranking λ is just a slow way of taking an argmax. Nothing in this project disputes that, and no panel in this app will pretend otherwise.

The reframe — and it is a reframe, not a rebuttal — is that the useful output was never a randomised own price. It is two other things:

1. The distribution over your rivals. The expected profit you just maximised — expected over what? Over the competitor's behaviour. A point prediction of the rival ("they will charge £1.74") is empirically terrible; observed play is dispersed. QRE supplies the strategically consistent distribution of rival behaviour — everyone responding probabilistically to everyone else's probabilities, at a precision λ estimated from data rather than assumed. Your argmax is then taken against a credible average instead of a brittle point. That is why every optimisation endpoint in this system returns the competitor distribution alongside the recommended price: the distribution is the deliverable.

2. A positive model of the market you're in. When £1.74 is estimated best, firms in scanner data choose it — say — 61% of the time, not 100%. A model whose predicted frequencies match observed frequencies lets you estimate λ (how sharply this market responds to incentives), test whether conduct looks competitive or coordinated, and measure the system-level quantities the rest of this app is about — reciprocity, dissipation, distance to criticality. None of those are available to a model that says P(best)=1P(\text{best}) = 1.

So: prescription — the objection wins; prediction and measurement — the distribution wins. The honest division of labour, stated once, here.

(Interactive panel: the same market shown twice — "argmax against a point rival" vs "argmax against the QRE rival distribution" — with the profit difference and its sensitivity to the rival assumption.)

So what would you do differently?

If your rival really does set one price and never move, none of this buys you anything and you should say so. Check that first; it takes a minute and it is the cheapest way to avoid a wrong model.

Run the check →
▶ try it · argmax against a point rival vs the QRE distribution
in-browser · goldens-checked
how noisy is the rival? (their λ) = 1.00
0.1 · noise20 · sharp
rival as a POINT prediction recommend £1.70
£1.70
1.00
£1.72
0.00
£1.74
0.00
£1.76
0.00
£1.78
0.00
my expected profit per price
£1.70
50.0
£1.72
30.0
£1.74
35.0
£1.76
40.0
£1.78
45.0
rival as the QRE DISTRIBUTION recommend £1.70
£1.70
0.99
£1.72
0.00
£1.74
0.00
£1.76
0.00
£1.78
0.01
my expected profit per price
£1.70
50.2
£1.72
30.5
£1.74
35.6
£1.76
40.6
£1.78
45.4
cost of the point assumption: 0.00 profit

Both sides take an argmax — the objection is right that you charge ONE price. What differs is the rival model the argmax is taken against. Slide the rival's λ: when they are noisy, the point prediction is badly wrong and the distribution earns its keep; as λ → ∞ the two recommendations converge and the objection wins on its own terms. The distribution is the deliverable.