# Optimal joint coupling of all probability measures on a 3 point space

I am looking for any remotely related reference for the following problem, for which I have not the least clue what techniques would be useful.

Consider a discrete probability space $\Omega = \{x, y, z\}$, and the space of all probability measures on it, $\mathcal{P}(\Omega)$, which can be identified with the standard 2-simplex $\Sigma_2 \subset \mathbb{R}^3_+$.

Let $H: \mathcal{P}(\Omega) \times (\Gamma, \gamma) \to \Omega$ be a map satisfying $\gamma[H(\mu, \cdot) = w] = \mu(w)$ for all $\mu \in \mathcal{P}(\Omega)$ and $w \in \Omega$, where $(\Gamma, \gamma)$ is some probability space. $H$ can be thought of as a joint/simultaneous coupling of all probability measures on $\Omega$, with the underlying probability space given by $\Gamma$.

One example of $H$ is the jointly independent coupling, where $\Gamma$ is given by the direct product $\Sigma_2^{\Sigma_2}$, with sigma algebra consisting of cylindrical sets that are nontrivial only on finitely many coordinates.

A more interesting one is given as follows. First identify $\mathcal{P}(\Omega)$ with $\Sigma_2$ via $\mu' \mapsto \sigma' := (\mu'(x), \mu'(y), \mu'(z))$. Let $\Gamma = \Sigma_2$ and define $$H_2(\mu, \sigma') = \arg\max_{w \in \Omega} [\mu(w) - \mu'(w)],$$

which is well-defined up to a $\Gamma$-set of measure $0$. Then it is easy to check that $(H_2, \Sigma_2, \lambda)$, where $\lambda$ is the normalized Lebesgue measure restricted to $\Sigma_2$, satisfies the simultaneous coupling condition listed in the beginning. I would like to show that $(H_2, \Sigma_2, \lambda)$ is optimal in the sense that it achieves the infimum value of the following functional of triples $(H, \Gamma, \gamma)$:

$$\Phi((H, \Gamma, \gamma)) := \int_{\mu \in \mathcal{P}(\Omega)} \int_{\nu \in \mathcal{P}(\Omega)} \gamma(\{\sigma \in \Gamma: H(\mu, \sigma) = H(\nu, \sigma)\}) \lambda(d\mu) \lambda(d\nu),$$

where $\lambda$ is the Lebesgue measure restricted to $\Gamma_2 \cong \mathcal{P}(\Omega)$, and normalized to be a probability measure.

Alternatively, show that $H_2$ is Pareto optimal in the sense that for any other admissible triple $(H, \Gamma, \gamma)$, the condition $$\gamma(\{\sigma \in \Gamma: H(\mu, \sigma) = H(\nu, \sigma)\}) \ge \lambda(\{\sigma \in \Sigma_2: H_2(\mu, \sigma) = H_2(\nu, \sigma)\}), ~~~ \forall ~ \mu, \nu \in \mathcal{P}(\Omega)$$ implies that there is a measure preserving isomorphism mod 0, $\psi: (\Sigma_2, \lambda) \to (\Gamma, \gamma)$, such that $H_2(\mu, \sigma) = H(\mu, \psi(\sigma))$, for all $\mu \in \mathcal{P}(\Omega)$ and $\sigma \in \Sigma_2$.

This has been solved in a joint paper with a colleague that is forthcoming. The key is to consider an interpolating probability measure $\rho$ of $\mu$ and $\nu$ defined by $\rho_i = \mu_i \nu_i / \sum_j \max\{\mu_j, \nu_j\}$.