# Quantal response equilibrium

Quantal response equilibrium
A solution concept in game theory
Relationships
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, 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), 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

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

$P_{ij} = \frac{\exp(\lambda EU_{ij}(P_{-i}))}{\sum_k{\exp(\lambda EU_{ik}(P_{-i}))}}$

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

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 irrational", 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. This provides an additional advantage of AQRE over perfectly rational solution concepts. Since every path is followed with some probability, there is no concern about defining beliefs "off the equilibrium path".

## Critiques

### Free parameter

LQRE has the free parameter λ. As λ→∞, LQRE→Nash equilibrium, so LQRE will always be at least as good a fit as Nash equilibrium. Changes in the parameter can result in large changes to equilibrium behavior.

However, the theory is incomplete without describing where λ comes from. Estimates of λ from experiments can vary significantly. Sometimes this variance seems to be a result of individual characteristics (for instance, λ sometimes increases with learning). Other times it appears that λ varies from game to game.