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Nash equilibrium assumes flawless best response: the slightly better option is chosen every time. Real players lean toward better options but slip. One dial, from pure noise to pure best response, spans the range, and the data usually sit in the middle.

Quantal response equilibrium

Level-k relaxed how deeply players reason. Quantal response relaxes how sharply they act. Instead of always best-responding, players pick better actions with higher probability, and equilibrium is where these noisy choices are mutually consistent. A single rationality parameter tunes it from random to Nash.

From earlier lessons

Two quick ones before we start.

Against either opponent move, A pays 3 and B pays 2. Which action maximizes expected payoff against any mix?

Level-k models relax the assumption about players' depth of...

Predict, then play

One action carries a small expected-payoff advantage. Slide the rationality dial from fully noisy to sharp and watch how often the better action is chosen.

Why did that happen?

Take a guess, then reveal the reason

People are payoff-sensitive but noisy: a better option is more attractive, so it is chosen more, but small mistakes remain. The rationality parameter measures how sensitive: at zero, payoffs are ignored and choice is a coin flip; as it grows, the better action dominates, and in the limit you recover exact best response. And beliefs stay consistent: each player’s noisy choices are a best noisy-response to the others’, so this is still an equilibrium, just a smoothed one.

The math

Definition, quantal response equilibrium

In a logit QRE, a player chooses action $a$ with probability proportional to $\exp(\lambda \cdot EU(a))$ , where $EU(a)$ is its expected payoff against the other players’ choice probabilities and $\lambda$ is the precision parameter. Equilibrium requires all of these noisy responses to be mutually consistent. At $\lambda = 0$ all actions are equally likely. As precision grows, limit points are Nash equilibria, though games can have more than one QRE branch.

For two actions with expected-payoff difference $d = EU(A) - EU(B)$ , the probability of $A$ is

P(A) = \frac{1}{1 + e^{-\lambda d}}.

This formula is one player’s logit response, not yet a QRE. In the displayed coordination game, if the other player chooses A with probability $q$ , then $d=2q-(1-q)=3q-1$ . A symmetric QRE solves the fixed-point equation

p = \frac{1}{1+e^{-\lambda(3p-1)}}.

The game solves this consistency condition for both players as the precision slider moves.

Work it out

Close the loop between two noisy responses

At $\lambda=0$ , every payoff is multiplied by zero, so both players choose A with probability $1/2$ . At a positive $\lambda$ , start from a conjecture $q$ about the other player’s probability of A. Your payoff advantage is $3q-1$ , so your logit response is $1/(1+e^{-\lambda(3q-1)})$ .

Equilibrium requires your resulting probability to equal the probability used in the other player’s calculation. Solving those two response equations together gives the probabilities shown in the game. Raising $\lambda$ makes choices more sensitive to expected payoff and moves the displayed branch toward the payoff-dominant Nash equilibrium $(A,A)$ .

Key idea

Quantal response equilibrium replaces perfect best response with noisy, payoff-sensitive choice governed by one rationality dial. It spans random play to Nash and fits real, mistake-prone behaviour in between.

Check yourself

Check yourselfapply

In a logit QRE, what does the rationality parameter lambda control?

Check yourselfanalyse

Why is QRE still an equilibrium concept and not just random play?

Check yourselfbrings back: mixed strategies

How does a QRE mix differ from a mixed Nash equilibrium?

Sources

McKelvey, R. D., and Palfrey, T. R. (1995). Quantal response equilibria for normal form games. Games and Economic Behavior, 10(1), 6-38.