1 · Softmax and λ — where the randomness comes from

Suppose two petrol stations can each charge one of five prices, and given your rival's behaviour, your expected profits are:

Nash says: charge £1.74, with probability one. A 2-point profit difference and a 13-point difference get the same treatment — total.

Quantal response says instead: choose each price with probability rising in its payoff,

$$P(p) \;=\; \frac{e^{\lambda\,EU(p)}}{\sum_{p'} e^{\lambda\,EU(p')}},$$

which is exactly the softmax of the payoffs at precision λ. At λ = 0 every price is equally likely; as λ → ∞ the distribution collapses onto the best response, and Nash reappears as the zero-temperature limit. In between, the magnitude of the payoff difference matters: £1.76 (2 points behind) keeps real probability; £1.70 (13 behind) nearly none.

What λ is, and is not. Mathematically λ converts payoff differences into log-odds: two prices 5 apart in profit have odds $e^{5\lambda}$. It has a rational-inattention microfoundation — the inverse shadow price of Shannon information (Matějka–McKay 2015) — so "low λ" can mean information is expensive here, not people are dumb here. And it is not scale-free: pay in pennies instead of pounds and λ must shrink by 100 to describe the same behaviour. Every strataq output therefore reports both lam and lambda_normalised = λ × payoff range.

Where does the randomness live? Not necessarily in a coin flip. Write the payoff as $\tilde U(p) = EU(p) + \varepsilon_p$ with $\varepsilon$ the things the modeller cannot see — private forecasts, manager idiosyncrasy, stale competitor data. If $\varepsilon$ is extreme-value distributed, the observed frequencies of a deterministic decision-maker are exactly the softmax above. The firm picks one price; we, who don't see its ε, predict a distribution.

(Interactive controls in the Lab: payoff bars → probability bars; λ slider from 0 to ∞; watch the collapse to Nash.)