# Quantal response equilibrium

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Quantal response equilibrium
A solution concept in game theory
Relationship
Superset of Nash equilibrium, Logit equilibrium
Significance
Proposed by Richard McKelvey and Thomas Palfrey
Used for Non-cooperative games
Example Traveler's dilemma

Quantal response equilibrium (QRE) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey,[1][2] it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.

In a quantal response equilibrium, players are assumed to make errors in choosing which pure strategy to play. The probability of any particular strategy being chosen is positively related to the payoff from that strategy. In other words, very costly errors are unlikely.

The equilibrium arises from the realization of beliefs. A player's payoffs are computed based on beliefs about other players' probability distribution over strategies. In equilibrium, a player's beliefs are correct.

## Application to data

When analyzing data from the play of actual games, particularly from laboratory experiments, particularly from experiments with the matching pennies game, Nash equilibrium can be unforgiving. Any non-equilibrium move can appear equally "wrong", but realistically should not be used to reject a theory. QRE allows every strategy to be played with non-zero probability, and so any data is possible (though not necessarily reasonable).

## Logit equilibrium

The most common specification for QRE is logit equilibrium (LQRE). In a logit equilibrium, player's strategies are chosen according to the probability distribution:

${\displaystyle P_{ij}={\frac {\exp(\lambda EU_{ij}(P_{-i}))}{\sum _{k}{\exp(\lambda EU_{ik}(P_{-i}))}}}}$

${\displaystyle P_{ij}}$ is the probability of player ${\displaystyle i}$ choosing strategy ${\displaystyle j}$. ${\displaystyle EU_{ij}(P_{-i}))}$ is the expected utility to player ${\displaystyle i}$ of choosing strategy ${\displaystyle j}$ under the belief that other players are playing according to the probability distribution ${\displaystyle P_{-i}}$.

Of particular interest in the logit model is the non-negative parameter λ (sometimes written as 1/μ). λ can be thought of as the rationality parameter. As λ→0, players become "completely non-rational", and play each strategy with equal probability. As λ→∞, players become "perfectly rational", and play approaches a Nash equilibrium.

## For dynamic games

For dynamic (extensive form) games, McKelvey and Palfrey defined agent quantal response equilibrium (AQRE). AQRE is somewhat analogous to subgame perfection. In an AQRE, each player plays with some error as in QRE. At a given decision node, the player determines the expected payoff of each action by treating their future self as an independent player with a known probability distribution over actions. As in QRE, in an AQRE every strategy is used with nonzero probability.

## Critiques

### Non-falsifiability

Work by Haile et al. has shown that QRE is not falsifiable in any normal form game, even with significant a priori restrictions on payoff perturbations.[3] The authors argue that the LQRE concept can sometimes restrict the set of possible outcomes from a game, but may be insufficient to provide a powerful test of behavior without a priori restrictions on payoff perturbations.

## References

1. ^ McKelvey, Richard; Palfrey, Thomas (1995). "Quantal Response Equilibria for Normal Form Games". Games and Economic Behavior. 10: 6–38. doi:10.1006/game.1995.1023.
2. ^ McKelvey, Richard; Palfrey, Thomas (1998). "Quantal Response Equilibria for Extensive Form Games". Experimental Economics. 1: 9–41. doi:10.1007/BF01426213.
3. ^ Haile, Philip A.; Hortaçsu, Ali; Kosenok, Grigory (2008). "On the Empirical Content of Quantal Response Equilibrium". American Economic Review. 98 (1): 180–200. doi:10.1257/aer.98.1.180.

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