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$\DeclareMathOperator\marg{marg}$I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data.

Consider two sets $\mathcal{A}=\{a_1,\dots,a_n\}$ and $\mathcal{Y}\subseteq\mathbb{R}$. Let $\alpha\in\Delta(\mathcal{A})$ and $\nu\in\Delta(\mathcal{Y})$ be two probability distribution. Assume that both distributions have full support $\nu$ has at least finite first moment.

Let $$\Gamma(\mathcal{A}\times\mathcal{Y})=\{\gamma\in\Delta(\mathcal{A}\times\mathcal{Y})\ |\ \marg_\mathcal{A}\gamma=\alpha,\; \marg_\mathcal{Y}\gamma=\nu\}$$ be the set of couplings between $\alpha$ and $\nu$. Observe that, since $\alpha$ assigns positive probability mass to each $a\in\mathcal{A}$, for each $\gamma\in\Gamma(\alpha,\nu)$ and each $a\in\mathcal{A}$ it is well defined a conditional probability distribution $\gamma|a$ which is the law of a random variable $Y|A=a$ obtained considering a random vector $(A,Y)$ distributed according to $\gamma$ and then conditioning on the event $A=a$.

In this setting, my problem is the following.

Given a measurable function $u:\mathcal{Y}\to\mathbb{R}$, an element $a\in\mathcal{A}$ and two probability distributions $\alpha\in\Delta(\mathcal{A}), \nu\in\Delta(\mathcal{Y})$ solve the following maximization problem

$$V_a(\alpha,\nu)\equiv\max_{\gamma\in\Gamma(\alpha,\nu)}\mathbb{E}_{\gamma|a}[u]$$

In principle, this seems similar to an optimal transport problem in that its domain is that of couplings between two probability distributions. The point is that the objective is not among those which, to the best of my knowledge, are traditionally considered in that setting: indeed, it depends on the coupling $\gamma$ only via the conditional $\gamma|a$ and one is not really minimizing a cost ($u$ is defined on $\mathcal{Y}$ only).

Have you ever encountered this kind of problem? Any help or reference would be greatly appreciated.

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    $\begingroup$ I'm trying to understand the problem a bit better: It seems the value $V_a(\alpha, \nu)$ does not really depend on $\alpha$, but only on $\alpha(a)$? Also, if I don't miss anything, the solution seems quite simple: Basically $\gamma \mid a$ fills the part of $\nu$ where $u$ has the largest values. This means $\gamma \mid a$ simply has to equal the (scaled) restriction of $\nu$ to a set $A$, where the set $A$ is such that $\nu(A) = \alpha(a)$ and $u(x) \geq u(y)$ for all $x \in A \cap supp(\nu), y \in A^C \cap supp(\nu)$. $\endgroup$
    – Steve
    Commented Oct 2, 2023 at 19:29
  • $\begingroup$ @Steve I think you are completely right, and indeed that's the (simple) solution I am obtaining. One can prove that with the bathtub principle. I am trying to frame the problem as OT to see if I can generalize it a bit (e.g. considering a function $u(a,y)$) and more in general to see weather the connection reveals something interesting $\endgroup$
    – Francesco Bilotta
    Commented Oct 3, 2023 at 20:22
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    $\begingroup$ I see. I haven't seen this kind of problem. It seems slightly related to weak optimal transport (see the paper "Kantorovich duality for general transport costs and applications" by Gozlan et al), where the cost depends on $\gamma \mid a$. In contrast to your problem, one still integrates over $\alpha$. So weak OT are problems of the form $\inf_{\gamma \in \Gamma(\alpha, \nu)} \int C(a, \gamma \mid a) \alpha(da)$ (choosing $C(a, \gamma \mid a) = \int c(a, b) \gamma(db \mid a)$ reduces to normal OT). $\endgroup$
    – Steve
    Commented Oct 4, 2023 at 18:35

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