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This question is about the mathematical foundations of the continuous logit model, as derived in McFadden (1976) (https://eml.berkeley.edu/reprints/mcfadden/math_theory.pdf) and Ben-Akiva et al (1985) (https://www.sciencedirect.com/science/article/abs/pii/0191260785900226).

Suppose I have a set of choices $\Omega$, which is uncountable (e.g. $\Omega = [0,1]$). Let the utility of choice $j \in I$ be $$ u(j) = V_j + b_x^{-1} x_j + b_\epsilon^{-1}\epsilon_j $$ where $x$ and $\epsilon$ have standard Gumbel distributions. All characteristics $(V_j, x_j, \epsilon_j)$ are distributed independently from each other and across choices $j \in \Omega$.

The choice density function conditional on $V_j$ and $x_j$ is then $$ p(j | V_j, x_j) = \frac{\exp \left[b_\epsilon(V_j + b_x^{-1}x_j)\right]}{\int_\Omega \exp \left[b_\epsilon(V_{j'} + b_x^{-1}x_{j'})\right] \, dj'} = \frac{\exp (b_\epsilon V_j) \exp (b_\epsilon b_x^{-1}x_j)}{\int_\mathbb{R} \int_\mathbb{R} \exp (b_\epsilon V) \exp(b_\epsilon b_x^{-1}x) \, dF_V dF_x} \\ = \frac{\exp (b_\epsilon V_j)}{\int_\mathbb{R} \exp (b_\epsilon V) \, dF_V} \frac{\exp (b_\epsilon b_x^{-1}x_j)}{\int_\mathbb{R} \exp(b_\epsilon b_x^{-1}x) \, dF_x}$$ where I have used the independence of $V$ and $x$. The choice density conditional on just $V_j$ is then $$ p(j|V_j) = \frac{\exp (b_\epsilon V_j)}{\int_\mathbb{R} \exp (b_\epsilon V) \, dF_V}. $$ This choice density (and consequently the distribution of $V_{j^*}$ that the individual receives from optimal choice $j^*$) depends on the variance of $\epsilon$ across $\Omega$ through the $b_\epsilon$ term, but not on the variance of $x$, seemingly because it is independently distributed from $V$.

However, if I instead compute the choice density conditional on $V_j$ and $\epsilon_j$ and repeat the same process, then the choice density conditional on just $V_j$ is instead $$ p(j|V_j) = \frac{\exp (b_x V_j)}{\int_\mathbb{R} \exp (b_x V) \, dF_V}. $$ In this case, the variance of $x$ across $\Omega$ matters, but the variance of $\epsilon$ cancels out through its independence from $V$.

But, at least according to my reading of the derivations in the articles linked, these should be two methods of deriving the same function. What explains the difference? My hunch is that there's a measure-theoretic foundation for the initial setup, and that in each derivation, I'm implicitly making different assumptions about the random variables $x_j$ and $\epsilon_j$. I haven't been able to work this out formally though.

Any assistance at all would be greatly appreciated!

Edit

In deriving $p(j|V_j,x_j)$, I have rewritten the denominator in this way: $$ \int_\Omega \exp \left[b_\epsilon(V_{j'} + b_x^{-1}x_{j'})\right] \, dj' = \int_\mathbb{R} \int_\mathbb{R} \exp (b_\epsilon V) \exp(b_\epsilon b_x^{-1}x) \, dF_V dF_x. $$

I think that in doing so, I am assuming that the sample distribution of the realisations $\{(V_j, x_j)\}_{j \in \Omega}$ is identical to the bivariate population distribution for $(V, x)$ (Sun (2006), Podczeck (2010) explain when this assumption is valid). I think that I do not make the same assumption with respect to the $\epsilon_j$. If I did so, it would imply that for every possible realisation $(V, x, \epsilon)$ in the support of the joint distribution, there exists an alternative $j \in \Omega$ such that $(V_j, x_j, \epsilon_j) = (V, x, \epsilon)$. In this case, there can never be an optimal choice, because the support of the distribution - and therefore utility - is unbounded. Could this assumption explain why the variance of $\epsilon$ affects the choice density in the first derivation, but not in the second?

Note that the set-ups in McFadden (1976) and Ben-Akiva et al (1985) seem somewhat different and therefore do not require this assumption. I construct the distribution of $(V,x)$ over the opportunity space as a sample realisation of a continuum of random variables $\{V_j,x_j\}_{j \in \Omega}$. They instead let $(V(j),x(j))$ be a deterministic function on $\Omega$, and also assume a probability measure on $\Omega$. In this way, they construct the distribution of $(V,x)$ directly.

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