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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!

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1 Answer 1

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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))))  
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    $\begingroup$ Thanks! For future reference, the "invariance property" of logit in this answer is discussed e.g. in Lindberg, Eriksson, and Mattsson. "Invariance of achieved utility in random utility models" 1995 Environment and Planning. $\endgroup$
    – Smithey
    Nov 28, 2021 at 0:59
  • $\begingroup$ You are welcome. Yes, invariance property is mentioned in many places. For a newer reference see for example: Fosgerau, Weibull, Lindberg, Mattson (2018) A note on the invariance of the distribution of the maximum. $\endgroup$ Nov 28, 2021 at 7:51

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