Everything about Quantal Response Equilibria totally explained
Quantal response equilibria (
QRE) are a
game-theoretical formulation. First introduced by
Richard McKelvey and
Thomas Palfrey, it provides an equilibrium notion with
bounded rationality. QRE isn't an equilibrium refinement, and it can give significantly different results than
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 shouldn't 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. In a logit equilibrium, player's strategies are chosen according to the probability distribution:
.
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.
Further Information
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