Computing expectation of logit error conditional on choice

Question

Consider a standard multinomial logit choice model. A consumer chooses a good $j$ from a choice set $J$ by choosing the good with the highest realized utility where the utility of good $j$ is given by

$$u_j = -p_j + \varepsilon_j,$$ where $p_j$ is the price of good $j$ and $\varepsilon_j$ is iid type-I extreme value. I am interested in finding a closed-form formula for the expected value of $\varepsilon_j$ conditional on $j$ being chosen--that is, conditional on, for all $k\in J \neq j$, $$-p_j + \varepsilon_j \geq -p_k + \varepsilon_k.$$ Has such a formula been found?

Thanks!

Answer 1

Let $X_j = v_j + e_j$ with $e_j$'s being IID type I extreme. Define

$$\hat X= \max \{ X_1,...,X_J\},$$

and let $\hat X_j$ be the variables $X_j$ conditional on being the max. Then the invariance property states that

$$\hat X,\hat X_1,...,\hat X_J \sim F^*,$$

all have the same distribution $F^*$. They, therefore, have the same expectation. It follows that

$$\mathbb E[v_j + e_j\lvert j = j^*] = \mathbb E[\hat X],$$

where $j^*\in \arg \max_j \{ X_1,...,X_J\}$. Hence

$$\mathbb E[\hat X] - v_j = \mathbb E[e_j \lvert j=j^*],$$

where $v_j$ is known and there is an analytical closed form for $\mathbb E[\hat X]$ as the standard log-sum expression.

Here is a simulation in R displaying the property of invariance

library(evd)
v_1 <- 1
v_2 <- 2

N <- 100000
Z <- matrix(rgumbel(2*N),nrow=2)
W <- Z + c(v_1,v_2)
index1 <- W[1,]>W[2,]
index2 <- W[2,]>W[1,]

mean(W[1,index1])
mean(W[2,index2])
0.5772 + log(sum(exp(c(v_1,v_2))))