Quantal Response Equilibrium

An equilibrium where players best-respond noisily with choice probabilities increasing in expected payoff.

Ask the Game theory assistant1 min read · Updated September 9, 2026

Definition

In a QRE with logit errors, player $i$ chooses $s_i$ with probability $\exp(\lambda u_i(s_i,\sigma_{-i})) / \sum_{s_i'}\exp(\lambda u_i(s_i',\sigma_{-i}))$ , where $\lambda\ge 0$ is the precision parameter.

The fixed-point condition: $\sigma_i^*(s_i) = \Pr(\text{choose } s_i \mid u_i(\cdot,\sigma_{-i}^*))$ under a stochastic choice model.

Intuition

Players are more likely to choose better strategies but make errors with probability decreasing in payoff differences.

As $\lambda\to\infty$ , QRE converges to a Nash equilibrium; as $\lambda\to 0$ , it converges to uniform randomization.

Worked example

McKelvey-Palfrey (1995) showed QRE fits laboratory data from the centipede game and signaling games far better than Nash.

QRE explains why subjects in zero-sum games mix with slight deviations from the exact Nash prediction.

The math

The logit QRE is the most common specification; nested families include power-function and payoff-monotone QRE.

Structural estimation of $\lambda$ from experimental data measures the degree of strategic sophistication.

Where it is used

Behavioral game theory, experimental economics, and estimation of bounded rationality.

Used to predict behavior in auctions, contests, and coordination games.

Go deeper

GuideFind a Nash Equilibrium with Best Responses

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