# Luce Choice Axiom, Quantal Response Equilibrium

How is the Quantal response equilibrium model influenced by the Luce choice axiom since Quantal Response equilibrium takes into account the rationality parameter ß and I dont see the mention of rationality perturbation in Luce Choice Axiom? And it would be of great help if anyone could post an example application of the Luce Choice Axiom. Thank You.

• See Daniel T. Jessie and Donald G. Saari (2015) "From the Luce Choice Axiom to the Quantal Response Equilibrium", Journal of Mathematical Psychology Feb 22, 2018 at 20:00
• Hey! I came across this journal, but I don't have access to it, do you know any other resources? Feb 23, 2018 at 11:41
• You'll have to check with the librarian at your institution to see if there's a way to gain access to the article. A quick Google search also brings up other articles. But those seem more about the empiricality of QRE than what you're after. Feb 23, 2018 at 17:23

The effect of LCA is to endow each alternative with an intrinsic level of likelihood that is independent of the particular set from which it is chosen. Mathematically speaking, in Luce’s formulation, the choice axiom implies the existence of a weight function $v(A)$ for an alternative $A$ in which the probability of selecting $A$ can be written as $$P_T(A)=\frac{e^{v(A)}}{\sum e^{v(B)}}\tag{1}$$ where the sum in the denominator is taken over all alternatives in $T$ (Luce, 1959). Because function $v(\cdot)$ does not depend upon the set $T$, it defines an intrinsic weight of the alternative. An important special case is when $v(\cdot)$ is a linear function; here Eq. (1) defines the multinomial logit model.
Regarding Quantal Response Equilibrium (QRE) [in a $2\times2$ normal form game, where Row player chooses between Top and Bottom]:
Instead of perfectly observing the payoffs for a given strategy, assume that the subjects perceive the expected value plus a stochastic term $\frac{\epsilon}{\lambda}$, where $\epsilon$ is a random variable and $\lambda$ is a magnitude parameter. In this framework, Row strategically chooses Top if $$E(\text{Top})+\frac{\epsilon_1}{\lambda}>E(\text{Bottom})+\frac{\epsilon_2}{\lambda} \;\Leftrightarrow\; \lambda[E(\text{Top})-E(\text{Bottom})]>\epsilon_1-\epsilon_2. \tag{4}$$ That is, Row’s choice of a probabilistic strategy combines the actual expected values with a measure of how it is recognized. A common assumption about $\epsilon_i$ is that they are independent and identically distributed type-I extreme value random variables. In the general case of several strategies, the probability $p_i$ that strategy $s_i$ is chosen is given by $$p_i=\frac{e^{\lambda E(s_i)}}{\sum_j e^{\lambda E(s_j)}}.\tag{5}$$ This expression is identical to the LCA multinomial logit expression in Eq. (1) with being a strategy’s expected payoff, with an emphasis on preserving the LCA structure.
The $\frac{\epsilon}{\lambda}$ form of the stochastic term in Eq. (4) makes it reasonable to treat $\lambda$ as a measure of “rationality”. This is because with small $\lambda$ values, the random terms dominate; indeed, with $\lambda=0$ (Eq. (5)), the subjects are completely insensitive to differences in expected value, so strategies are selected from a uniform distribution. In contrast, the effect of the random terms diminish as $\lambda\to\infty$, which means that the subjects become increasingly responsive to payoff differences; in the limit they reach the Nash equilibrium.