# Uniqueness of the inverse conductivity problem once-differentiable complex
conductivities in three dimensions

Ivan Pombo

(June 2022)

###### Abstract

We prove uniqueness of the inverse conductivity problem in three dimensions
for complex conductivities in $W^{1,\infty}$.

We apply quaternionic analysis to transform the inverse problem into an
inverse Dirac scattering problem, as established in two dimensions by Brown
and Uhlmann.

This is a novel methodology that allows to extend the uniqueness result from
once-differentiable real conductivities to complex ones.

##  1 Introduction

Let $\gamma\in W^{1,\infty}(\Omega)$ be the complex-valued conductivity
defined in a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{3}$ and given
by $\gamma=\sigma+i\omega\epsilon$ where $\sigma$ is the electrical
conductivity and satisfies $\sigma\geq c>0$, $\epsilon$ is the electrical
permittivity and $\omega$ is the current frequency.

Given a boundary value $f\in H^{1/2}(\partial\Omega)$ we can determine the
respective electrical potential $u\in H^{1}(\Omega)$ by uniquely solving

$\displaystyle\begin{cases}\nabla\cdot(\gamma\nabla u)=0,\text{ in }\Omega,\\\
\left.u\right|_{\partial\Omega}=f.\end{cases}$ (1)

This is the so-called conductivity equation which describes the behavior of
the electrical potential, $u$, in a conductive body when a voltage potential
is applied on the boundary, $f$.

In 1980, A.P. Calderón, [11] introduced the problem of whether one can recover
uniquely a conductivity $\sigma\in L^{\infty}(\Omega)$ from the boundary
measurements, i.e., from the Dirichlet-to-Neumann map

$\displaystyle\Lambda_{\sigma}:\,H^{1/2}(\partial\Omega)$
$\displaystyle\rightarrow H^{-1/2}(\partial\Omega),$ (2) $\displaystyle f$
$\displaystyle\mapsto\sigma\left.\frac{\partial
u}{\partial\nu}\right|_{\partial\Omega}$

which connects the voltage and electrical current at the boundary. The normal
derivative exists as an element of $H^{-1/2}(\partial\Omega)$ by

$\displaystyle\langle\Lambda_{\sigma}f,g\rangle=\int_{\Omega}\sigma\nabla
u\cdot\nabla v\,dx$ (3)

where $v\in H^{1}(\Omega)$ with $\left.v\right|_{\partial\Omega}=g$ and $u$
solves (1).

In the same paper, Calderón was able to prove that the linearized problem at
constant real conductivities has a unique solution. Thereafter, many authors
extended is work into global uniqueness results. Sylvester and Uhlmann [31]
used ideas of scattering theory, namely the exponential growing solutions of
Faddeev [14] to obtain global uniqueness in dimensions $n\geq 3$ for smooth
conductivities. Using this foundations the uniqueness for lesser regular
conductivities was further generalized for dimensions $n\geq 3$ in the works
of ([1], [7], [8], [12], [13], [18], [22], [24], [27]). Currently, the best
know result is due to Haberman [17] for conductivities $\gamma\in
W^{1,3}(\Omega)$. The reconstruction procedure for $n\geq 3$ was obtained in
both [22] and [25] independently. As far as we are aware, there seems to be no
literature concerning reconstruction for conductivities with less than two
derivatives.

In two dimensions the problem seems to be of a different nature and tools of
complex analysis were used to establish uniqueness. Nachman [23] obtained
uniqueness and a reconstruction method for conductivities with two
derivatives. The uniqueness result was soon extend for once-differentiable
conductivities in [9] and a corresponding reconstruction method was obtained
in [20]. In 2006, Astala and Päivärinta [3] gave a positive answer Calderón’s
problem for $\sigma\in L^{\infty}(\Omega),\,\sigma\geq c>0$, by providing the
uniqueness proof through the reconstruction process.

All of this definitions can be extended to the complex-conductivity case with
the assumption $\text{Re }\gamma\geq c>0$, in particular, we can define the
Dirichlet-to-Neumann as above $\Lambda_{\gamma}$.

In this scenario, the first works was done in two-dimensions by Francini [15],
by extending the work of Brown and Uhlmann [9] in two-dimensions proving
uniqueness for small frequencies $\omega$ and $\gamma\in W^{2,\infty}$.
Afterwards, Bukgheim influential paper [10] proved the general result in two-
dimensions for complex-conductivities in $W^{2,\infty}$. He reduced the (1) to
a Schrödinger equation and shows uniqueness through the stationary phase
method (based on is work many extensions followed in two-dimensions [2], [4],
[26]). Recently, by mixing techniques of [9] and [10], Lakshtanov, Tejero and
Vainberg obtained [21] uniqueness for Lipschitz complex-conductivities in
$\mathbb{R}^{2}$. In [28], the author followed up their work to show that it
is possible to reconstruct complex-conductivity with a jump at least in a
certain set of points.

In three dimensions, the uniqueness results presented in [31] and [24] hold
for twice-differentiable complex-conductivities in $W^{2,\infty}$, but there
was no reconstruction process presented. Nachman’s reconstruction method in
three dimensions [22] was used in [19] to reconstruct complex conductivities
from boundary measurements. Even though the Nachman’s proof was presented only
for real conductivities, the paper [29] structures the proof in order to show
the result holds for complex-conductivities. As far as we aware, the works
with lower regularity require real-conductivities.

In this paper our interest resides in Calderón’s problem for once-
differentiable complex-conductivities in three-dimensions. The aim is to prove
the following theorem:

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{3}$ a bounded Lipschitz domain, $\gamma_{i}\in
W^{1,\infty}(\Omega),\,i=1,\,2$ be two complex-valued conductivities with
$\textnormal{Re }\gamma_{i}\geq c>0.$

$\text{If }\Lambda_{\gamma_{1}}=\Lambda_{\gamma_{2}},\text{ then
}\gamma_{1}=\gamma_{2}.$

Our work basis itself on a transformation of (1) into a Dirac system of
equation in three-dimensions with the help of quaternions. In this scenario,
we obtain a potential $q$ that we want to determine from boundary data. The
main ideas follow the work of Brown and Uhlmann [9] for real conductivities in
two-dimensions and Lakshtanov, Vainberg and Tejero [21], as well as the
authors work [28], for complex-conductivities.

In this paper, we provide a novel reconstruction of the bounded potential $q$
from the boundary data, but we are yet to be able to establish a relation
between this boundary data and the Dirichlet-to-Neumann map. This is
essentially to answer Calderón problem for Lipschitz complex conductivities,
but the lack of a well-suited Poincaré lemma that fits the quaternion
structure does not allow such a simple work as in 2D.

##  2 Minimalistic lesson of Quaternions

We present the basis of the quaternionic framework we will use for our work.
Let $\mathbb{R}_{(2)}$ be the real universal Clifford Algebra over
$\mathbb{R}^{2}$. By definition, it is generated as an algebra over
$\mathbb{R}$ by the elements $\\{e_{0},e_{1},e_{2}\\}$, where $e_{1},e_{2}$ is
a basis of $\mathbb{R}^{2}$ with $e_{i}e_{j}+e_{j}e_{i}=-2\delta_{ij}$, for
$i,\,j=1,\,2$ and $e_{0}=1$ is the identity and commutes with the basis
elements. This algebra has dimension 4 and is identified with the algebra of
the quaternions, $\mathbb{H}$. As such it holds $e_{3}=e_{1}e_{2}$. In the
following, we refer to this algebra as the quaternions. An element of the
quaternions can be written as:

$\displaystyle x=x_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3},$ (4)

where $x_{j},\,j=0,...,3$ are real. We define the quaternionic conjugate
$\bar{x}$ of an element $x$ as

$\displaystyle\bar{x}=x_{0}-x_{1}e_{1}-x_{2}e_{2}-x_{3}e_{3}.$ (5)

Let $x,\,y\in\mathbb{H}$, we write $xy$ for the resulting quaternionic
product. The product $\bar{x}y$ defines a Clifford valued inner product on
$\mathbb{H}$. Further, we have $\overline{xy}=\bar{y}\bar{x}$ and the
conjugate of the conjugate of quaternion is that same quaternion. Let
$x\in\mathbb{H}$ then $\text{Sc}\,x=x_{0}$ denotes the scalar of $x$ and
$\text{Vec}\,x=x-\text{Sc}\,x$. The scalar of a Clifford inner product
$\text{Sc}(\bar{x}y)$ is the usual inner product in $\mathbb{R}^{4}$ for
$x,\,y$ identified as vectors.

With this inner product $\mathbb{H}$ is an Hilbert space and the resulting
norm is the usual Euclidean norm.

In order to introduce some of the concepts we also extend the real quaternions
to complex quaternions $\mathbb{C}_{2}=\mathbb{C}\otimes\mathbb{H}$. Here, we
use the same generators $(e_{1},e_{2})$ as above, with the same multiplication
rules, however, the coefficients of the quaternion can be complex-valued. That
is, $\lambda\in\mathbb{C}\otimes\mathbb{H}$ may be written as

$\displaystyle\lambda=\lambda_{0}+\lambda_{1}e_{1}+\lambda_{2}e_{2}+\lambda_{3}e_{3},\quad\lambda_{j}\in\mathbb{C},\,j=0,...,3$
(6)

or still as

$\displaystyle\lambda=x+iy,\quad x,\,y\in\mathbb{H}.$ (7)

Due to the complexification we can still take another conjugation, to which we
define has Hermitian conjugation and denote it by
$\overline{\cdot}^{\dagger}$. Explicitly, for
$\lambda\in\mathbb{C}\otimes\mathbb{H}$ one has

$\displaystyle\bar{\lambda}^{\dagger}=\lambda_{0}^{c}-\lambda_{1}^{c}e_{1}-\lambda_{2}^{c}e_{2}-\lambda_{3}^{c}e_{3},$
(8)

where $\cdot^{c}$ denotes complex conjugation, or

$\displaystyle\bar{\lambda}^{\dagger}=\bar{x}-i\bar{y}.$ (9)

Similarly, one can introduce an associated inner product and norm in
$\mathbb{C}\otimes\mathbb{H}$ by means of this conjugation:

$\displaystyle\langle\lambda,\mu\rangle=\text{Sc}\left(\bar{\lambda}^{{\dagger}}\mu\right);\quad|\lambda|_{\mathbb{C}_{2}}=\sqrt{\text{Sc}\left(\bar{\lambda}^{{\dagger}}\lambda\right)}.$
(10)

For ease of notation, we also define for $\lambda\in\mathbb{C}_{2}$ the
complex conjugation as

$\displaystyle\bar{\lambda}^{c}=\lambda_{0}^{c}+\lambda_{1}^{c}e_{1}+\lambda_{2}^{c}e_{2}+\lambda_{3}^{c}e_{3}.$
(11)

Now, we can also introduce Quaternion-valued functions
$f:\mathbb{R}^{3}\rightarrow\mathbb{C}_{2}$ written as
$f=f_{0}+f_{1}e_{1}+f_{2}e_{2}+f_{3}e_{3},$ where
$f_{j}:\mathbb{R}^{3}\rightarrow\mathbb{C}$.

The Banach spaces $L^{p},\,W^{n,p}$ of $C_{2}$-valued functions are defines by
requiring that each component is in such space. On $L^{2}(\Omega)$ we
introduce the $C_{2}$-valued inner product

$\displaystyle\langle f,g\rangle=\int_{\Omega}\bar{f}^{\dagger}(x)g(x)\,dx.$
(12)

Analogously to the Wirtinger derivatives in complex analysis, we have the
Cauchy-Riemann operators under $(x_{0},\,x_{1},\,x_{2})$ coordinates of
$\mathbb{R}^{3}$ defined as

$\displaystyle D=\partial_{0}+e_{1}\partial_{1}+e_{2}\partial_{2},$ (13)

where $\partial_{j}$ is the derivative with respect to the $x_{j},\,j=0,1,2$
variable; and

$\displaystyle\bar{D}=\partial_{0}-e_{1}\partial_{1}-e_{2}\partial_{2}.$ (14)

The vector part of the Cauchy-Riemann operator is designated as Dirac
operator. It holds that $D\bar{D}=\Delta$ where $\Delta$ is the Laplacian.

We designate any function $f$ fulfilling $Df=0$ as a monogenic function,
analogous to the holomorphic functions in complex analysis.

###  2.1 A bit of Operator Theory

Let $\Omega$ be a bounded domain and $f:\Omega\rightarrow\mathbb{C}_{2}$. All
the results in this subsection were taken out from the classical book on
quaternionic analysis of Gürlebeck and Sprössig [16]

The Cauchy-Riemann operator has a right-inverse in the form

$\displaystyle\left(Tf\right)(x)=-\frac{1}{\omega}\int_{\Omega}\frac{\overline{y-x}}{|y-x|^{3}}f(y)\,dy,\text{
for }x\in\Omega,$ (15)

where $E(x,y)=-\frac{1}{\omega}\frac{\overline{y-x}}{|y-x|^{3}}$ is the
generalized Cauchy kernel and $\omega=4\pi$ stands for the surface area of the
unit sphere in $\mathbb{R}^{3},$ that is, $DTf=f.$ This operator acts from
$W^{k,p}(\Omega)$ to $W^{k+1,p}(\Omega)$ with $1<p<\infty$ and
$k\in\mathbb{N}_{0}$.

Furthermore, we introduce the boundary integral operator for
$x\notin\partial\Omega$

$\displaystyle\left(F_{\partial\Omega}f\right)(x)=\frac{1}{\omega}\int_{\partial\Omega}\frac{\overline{y-x}}{|y-x|^{3}}\alpha(y)f(y)\,dS(y),$
(16)

where $\alpha(y)$ is the outward pointing normal unit vector to
$\partial\Omega$ at $y$. We get the well-known Borel-Pompeiu formula

$\displaystyle\left(F_{\partial\Omega}f\right)(x)+\left(TDf\right)(x)=f(x)\text{
for }x\in\Omega.$

Obviously, $DF_{\partial\Omega}=0$ holds through this formula it it holds that
$F_{\partial\Omega}$ acts from $W^{k-\frac{1}{p},p}(\partial\Omega)$ into
$W^{k,p}(\Omega)$, for $k\in\mathbb{N}$ and $1<p<\infty$.

One of the other well-known results we will need for our work is the Plemelj-
Sokhotzki formula is obtaining by taking the trace of the boundary integral
operator.

First we introduce an operator over the boundary of $\Omega$.

######  Proposition 2.1.

If $f\in W^{k,p}(\partial\Omega)$, then there exists the integral

$\displaystyle\left(S_{\partial\Omega}f\right)=\frac{1}{2\pi}\int_{\partial\Omega}\frac{\overline{y-x}}{|y-x|^{3}}\alpha(y)f(y)\,dS(y)$
(17)

for all points $x\in\Omega$ in the sense of Cauchy principal value.

Furthermore, the operator $S_{\partial\Omega}$ is continuous in
$W^{k,p}(\partial\Omega)$, for $1<p<\infty,\,k\in\mathbb{N}$.

From this the Plemelj-Sokhotzki formula is given as:

######  Theorem 2.2.

Let $f\in W^{k,p}(\partial\Omega)$ where by taking the non-tangential limit we
have:

$\displaystyle\lim_{\begin{subarray}{c}x\rightarrow x_{0},\\\
x\in\Omega,\,x_{0}\in\partial\Omega\end{subarray}}\left(F_{\partial\Omega}f\right)(x)=\frac{1}{2}\left(f(x_{0})+\left(S_{\partial\Omega}f\right)(x_{0})\right).$

One of the corollaries concerns the limit to the boundary acting as a
projector. That is,

######  Corollary 2.3.

The operator $P_{\partial\Omega}$ denoting the projection onto the space of
all $\mathbb{H}-$valued functions which may be monogenicaly extended into the
domain $\Omega$.

Then, this projection may be represented as

$P_{\partial\Omega}=\frac{1}{2}\left(I+S_{\partial\Omega}\right).$

The proofs of this results and others to follow in our proofs may be found in
[16].

Now we are ready to start constructing our work on the inverse conductivity
problem.

##  3 Inverse Dirac scattering problem

Transforming our conductivity equation into another type of equation also
changes the inverse problem we are concerned. We transform it into a system of
equations based on the Cauchy-Riemann operator $D$ (also called Dirac operator
in some contexts) and thus we need to solve the inverse Dirac scattering
problem first and only afterwards we care about the inverse conductivity
problem.

Let $u$ be a solution to (1) for some boundary function. We define

$\displaystyle\phi=\gamma^{1/2}\left(\,\bar{D}u,Du\right)^{T},$

remark that $\gamma^{1/2}$ is well-defined since it is contained in
$\mathbb{C}^{+}$. Then, $\phi$ solves the system

$\displaystyle\begin{cases}D\phi_{1}&=\phi_{2}q_{1},\\\
\bar{D}\phi_{2}&=\phi_{1}q_{2},\end{cases}\quad\text{ in }\mathbb{R}^{3}.$
(18)

where $q_{1}=-\frac{1}{2}\frac{\bar{D}\gamma}{\gamma}$ and
$q_{2}=-\frac{1}{2}\frac{D\gamma}{\gamma}$.

This transformation arises as follows:

$\displaystyle D\phi_{1}$
$\displaystyle=D\left(\gamma^{1/2}\bar{D}u\right)=D\gamma^{1/2}\bar{D}u+\gamma^{1/2}\Delta
u$ $\displaystyle=D\gamma^{1/2}\bar{D}u-\gamma^{-1/2}\nabla\gamma\cdot\nabla
u$
$\displaystyle=D\gamma^{1/2}\bar{D}u-\frac{1}{2}\gamma^{-1/2}\left(D\gamma\bar{D}u+\text{D}u\bar{D}\gamma\right)$
$\displaystyle=-\frac{1}{2}\left(\gamma^{1/2}Du\right)\frac{\bar{D}\gamma}{\gamma}=\phi_{2}q_{1}$

Carefully, we can extend our potential to the outside by setting $\gamma\equiv
1$ outside of $\Omega$, which lead us to treat the study the equation in
$\mathbb{R}^{3}$.

###  3.1 Exponentially Growing Solutions

We devise new exponentially growing solutions from the classical ones used in
three dimensions. In most literature works, the exponential behavior is
defined through the function $e^{x\cdot\zeta},$ with $\zeta\in\mathbb{C}^{3}$
fulfilling $\zeta\cdot\zeta=0$. However, in our scenario this function does
not fulfill $De^{ix\cdot\zeta}=0$, which brings the simplicity in all of the
literature works.

Since we know that it is harmonic we can generate a monogenic function through
it. Let $\zeta\in\mathbb{C}^{3}$ such that
$\zeta\cdot\zeta:=\zeta_{0}^{2}+\zeta_{1}^{2}+\zeta_{2}^{2}=0$, then it holds

$\displaystyle\Delta e^{x\cdot\zeta}=0\Leftrightarrow
D\left(\bar{D}e^{x\cdot\zeta}\right)=0\equiv
D\left(e^{x\cdot\zeta}\,\bar{\zeta}\right)$

where now $\zeta$ is also defined as a quaternion through
$\zeta=\zeta_{0}+e_{1}\zeta_{1}+e_{2}\zeta_{2}\in\mathbb{C}_{2}.$ Thus the
function $E(x,\zeta)=e^{x\cdot\zeta}\,\bar{\zeta}$ is monogenic. This also
arises from the choice of $\zeta$, since
$\zeta\,\bar{\zeta}=\zeta_{0}^{2}+\zeta_{1}^{2}+\zeta_{2}^{2}=0$.

We make a clear statement of when $\zeta$ is a complex-quaternion or complex-a
vector, but in most cases it is clear from context: it is a vector if it is in
the exponent and a quaternion otherwise.

We assume the following asymptotic behaviour for $\phi$:

$\displaystyle\phi_{1}=e^{x\cdot\zeta}\bar{\zeta}\mu_{1},$ (19)
$\displaystyle\phi_{2}=e^{x\cdot\bar{\zeta}^{c}}\bar{\zeta}^{c}\mu_{2}$ (20)

Setting $\tilde{\mu}_{1}=\bar{\zeta}\mu_{1}$ and
$\tilde{\mu}_{2}=\bar{\zeta}^{c}\mu_{2}$ we have the equations:

$\displaystyle\begin{cases}D\tilde{\mu}_{1}&=e^{-x\cdot\left(\zeta-\,\bar{\zeta}^{c}\right)}\tilde{\mu}_{2}q_{1}\\\
\bar{D}\tilde{\mu}_{2}&=e^{x\cdot\left(\zeta-\,\bar{\zeta}^{c}\right)}\tilde{\mu}_{1}q_{2}\\\
\end{cases}$ (21)

Further, we assume $\tilde{\mu}\rightarrow\begin{pmatrix}1\\\ 0\end{pmatrix}$
as $|x|\rightarrow\infty.$ These system of equations will lead us to an
integral equation from which we can extract interesting behaviour for
$\zeta\rightarrow\infty$.

The main point of this subsection is to demonstrate how we can obtain the
system of integral equations related with (21). Here, the approach is similar
to [21], but we need to be careful due to the non-commutative nature of
quaternions.

Recall, that $DT=\bar{D}\,\bar{T}=I$ (in appropriate spaces). Hence, applying
this to (21) it holds:

$\displaystyle\begin{cases}\tilde{\mu}_{1}=1+T\left[e^{-x\cdot\left(\zeta-\bar{\zeta}^{c}\right)}\tilde{\mu}_{2}q_{1}\right]\\\
\\\
\tilde{\mu}_{2}=\overline{T}\left[e^{x\cdot\left(\zeta-\bar{\zeta}^{c}\right)}\tilde{\mu}_{1}q_{2}\right]\end{cases}$

Thus, we can obtain two integral equations with respect to their function:

$\displaystyle\begin{cases}\tilde{\mu}_{1}=1+T\left[e^{-x\cdot\left(\zeta-\bar{\zeta}^{c}\right)}\bar{T}\left[e^{x\cdot\left(\zeta-\bar{\zeta}^{c}\right)}\tilde{\mu}_{1}q_{2}\right]q_{1}\right]\\\
\\\
\tilde{\mu}_{2}=\bar{T}\left[e^{x\cdot\left(\zeta-\bar{\zeta}^{c}\right)}q_{2}\right]+\bar{T}\left[e^{x\cdot\left(\zeta-\bar{\zeta}^{c}\right)}T\left[e^{-x\cdot\left(\zeta-\bar{\zeta}^{c}\right)}\tilde{\mu}_{2}q_{1}\right]q_{2}\right]\end{cases}$
$\displaystyle\begin{cases}\tilde{\mu}_{1}=1+M^{1}\tilde{\mu}_{1}\\\
\tilde{\mu}_{2}=\overline{T}\left[e^{x\cdot\left(\zeta-\overline{\zeta}^{\mathbb{C}}\right)}q_{2}\right]+M^{2}\tilde{\mu}_{2}\end{cases}\Leftrightarrow\begin{cases}[I-M^{1}](\tilde{\mu}_{1}-1)=M^{1}1\\\
[I-M^{2}](\tilde{\mu}_{2})=\bar{T}\left[e^{x\cdot\left(\zeta-\bar{\zeta}^{\mathbb{C}}\right)}q_{2}\right]\end{cases}$
(22)

Our objective now is to study the uniqueness and existence of this equations,
we approach this task by proving that $M^{j},\,j=1,\,2$ are contractions.

Instead of working with all possible $\zeta\in\mathbb{C}_{(2)}$ fulfilling
$\zeta\bar{\zeta}=0$, we choose them for $k\in\mathbb{R}^{3}$ as

$\zeta=k^{\perp}+i\frac{k}{2},\quad k^{\perp}\cdot k=0$

and $k^{\perp}$ can be algorithmically found.

We now describe our space of functions in terms of the space variable and
$k\in\mathbb{R}^{3}$ as

$\displaystyle S=L^{\infty}_{x}(L^{p}_{k}(|k|>R))$ (23)

where $R>0$ is a constant. In this space we prove that the operators
$M^{1},\,M^{2}$ are indeed contractions:

######  Lemma 3.1.

Let $p>2$. Then

$\lim_{R\rightarrow\infty}\|M^{j}\|_{S}=0.$

To further study the system (22), we also need to show that the right-hand
side is in $S$ for an $R$ large enough:

######  Lemma 3.2.

Let $p>2$. Then there exists $R>0$ such that

$\displaystyle M^{1}1\in S,$ (24)
$\displaystyle\bar{T}\left[e^{x\cdot\left(\zeta-\bar{\zeta}^{\mathbb{C}}\right)}q_{2}\right]\in
S$ (25)

The above Lemmas imply the existence and uniqueness of
$(\tilde{\mu}_{1},\tilde{\mu}_{2})$ solving the system (22) with respect to
the potential $q$. This is essential for the reconstruction procedure we show
up next.

###  3.2 Reconstruction from scattering data

In this section, we are mixing ideas from [21] and [22] with quaternionic
theory to obtain the potential from the scattering data.

Starting from Clifford-Green theorem

$\displaystyle\int_{\Omega}\left[g(x)\left(\bar{D}f(x)\right)+\left(g(x)\bar{D}\right)f(x)\right]\,dx=\int_{\partial\Omega}g(x)\overline{\eta(x)}f(x)\,dS_{x}$

and using $g(x;i\xi+\zeta)=(i\xi+\zeta)e^{-x\cdot(i\xi+\zeta)}$ for
$\xi\in\mathbb{R}^{3}$ such that $(i\xi+\zeta)\cdot(i\xi+\zeta)=0$. This
implies that $g\bar{D}=0.$ Thus we define the scattering data as:

$\displaystyle
h(\xi,\zeta)=\left(i\xi+\zeta\right)\int_{\partial\Omega}e^{-x\cdot(i\xi+\zeta)}\overline{\eta(x)}\phi_{2}(x,\zeta)\,dx$
(26)

Applying now Clifford-Green theorem we obtain another form for the scattering
data:

$\displaystyle h(\xi,\zeta)$
$\displaystyle=\left(i\xi+\zeta\right)\int_{\Omega}e^{-x\cdot(i\xi+\zeta)}\bar{D}\phi_{2}(x,\zeta)\,dx$
$\displaystyle=(i\xi+\zeta)\int_{\Omega}e^{-ix\cdot\xi}\left(e^{-x\cdot\zeta}\phi_{1}(x,\zeta)\right)q_{2}(x)\,dx,\quad\text{by
}\overline{\text{D}}\phi_{2}=\phi_{1}q_{2}$
$\displaystyle=(i\xi+\zeta)\int_{\Omega}e^{-ix\cdot\xi}\left(\zeta\mu_{1}(x,\zeta)\right)q_{2}(x)\,dx$
$\displaystyle=i\xi\int_{\Omega}e^{-ix\cdot\xi}\tilde{\mu}_{1}(x,\zeta)q_{2}(x)\,dx,\quad\textnormal{since
}\bar{\zeta}\zeta=0$
$\displaystyle=i\xi\hat{q}_{2}({\xi})+i\xi\int_{\Omega}e^{-ix\cdot\xi}\left[\tilde{\mu}_{1}(x,\zeta)-1\right]q_{2}(x)\,dx.$

Thus, we have:

$\displaystyle\hat{q}_{2}(\xi)=\frac{h(\xi,\zeta)}{i\xi}-\int_{\Omega}e^{-ix\cdot\xi}\left[\tilde{\mu}_{1}(x,\zeta)-1\right]q_{2}(x)\,dx$
(27)

This is yet not enough to reconstruct the potential, since the integral acts
as a residual in the reconstruction and requires data that we technically do
not have. Therefore, we integrate everything over an annulus in $k$

$\displaystyle\int_{R<|k|<2R}\frac{\hat{q}_{2}(\xi)}{|k|^{3}}\,dk$
$\displaystyle=\frac{1}{i\xi}\int_{R<|k|<2R}\frac{h(\xi,\zeta(k))}{|k|^{3}}\,dk$
$\displaystyle-\int_{R<|k|<2R}\frac{1}{|k|^{3}}\int_{\Omega}e^{-ix\cdot\xi}\left[\tilde{\mu}_{1}(x,\zeta(k))-1\right]q_{2}(x)\,dx,$
(28)

since the potential does not depend on $k$ it can be taken out of the integral
and taking the limit as $R\rightarrow\infty$ leads the second integral on the
right to decay to zero, obtaining a reconstruction formula.

######  Theorem 3.3.

Let $\Omega\subset\mathbb{R}^{3}$ a bounded Lipschitz domain, $q\in
L^{\infty}(\Omega)$ be a complex-valued potential obtained through a
conductivity $\gamma\in W^{1,\infty}(\Omega),\,\text{Re }\gamma\geq c>0.$
Then, we can reconstruct the potential from

$\displaystyle\hat{q}_{2}(\xi)=\lim_{R\rightarrow\infty}\frac{C}{i\xi}\int_{R<|k|<2R}\frac{h(\xi,\zeta(k))}{|k|^{3}}\,dk,$
(29)

where $C=\frac{1}{4\pi\ln(2)}$.

###### Proof.

The scattering data is defined from the solutions of the Dirac system (22) and
therefore it holds that $\tilde{\mu}_{1}-1\in S$. Starting from (3.2) we
obtain by integrating the right-hand side for any $\xi\in\mathbb{R}^{3}$:

$\displaystyle 4\pi\ln 2\,\hat{q}_{2}(\xi)$
$\displaystyle=\frac{1}{i\xi}\int_{R<|k|<2R}\frac{h(\xi,\zeta(k))}{|k|^{3}}\,dk$
$\displaystyle-\int_{R<|k|<2R}\frac{1}{|k|^{3}}\int_{\Omega}e^{-ix\cdot\xi}\left[\tilde{\mu}_{1}(x,\zeta(k))-1\right]q_{2}(x)\,dx$

Let $p>2$ and $1/p+1/q=1$. We estimate the last integral:

$\displaystyle\Bigg{|}$
$\displaystyle\int_{R<|k|<2R}\frac{1}{|k|^{3}}\int_{\Omega}e^{-ix\cdot\xi}\left[\tilde{\mu}_{1}(x,\zeta(k))-1\right]q_{2}(x)\,dx\Bigg{|}\leq$
$\displaystyle\leq\int_{R<|k|<2R}\frac{1}{|k|^{3}}\int_{\Omega}\left|e^{-ix\cdot\xi}\left[\tilde{\mu}_{1}(x,\zeta(k))-1\right]q_{2}(x)\right|\,dx$
$\displaystyle\leq
C_{\Omega}\|q\|_{\infty}\int_{R<|k|<2R}\frac{1}{|k|^{3}}\sup_{x}\left|\tilde{\mu}_{1}(x,\zeta(k))-1\right|\,dk$
$\displaystyle\leq
C_{\Omega}\|q\|_{\infty}\left[\int_{R<|k|<2R}\frac{1}{|k|^{3q}}\,dk\right]^{1/q}\left[\int_{R<|k|<2R}\sup_{x}\left|\tilde{\mu}_{1}(x,\zeta(k))-1\right|^{p}\,dk\right]^{1/p}$
$\displaystyle\leq
C_{\Omega}\|q\|_{\infty}\|\tilde{\mu}_{1}-1\|_{S}\left[\int_{R<|k|<2R}\frac{1}{|k|^{3q}}\,dk\right]^{1/q}$

Taking the limit as $R\rightarrow 0$ the integral that is left goes to zero
which implies the desired decay to zero and leaves us with our reconstruction
formula. ∎

Now, in order to connect the functions that solve the electrical conductivity
equation (1) and the solutions to the Dirac equation (18), which are
exponential growing, we introduce the following result:

######  Proposition 3.4.

Let $\Omega$ be a bounded domain in $\mathbb{R}^{3}$. Let
$\phi=(\phi_{1},\phi_{2})$ be a solution of the Dirac system (18) for a
potential $q\in L^{\infty}(\Omega)$ associated with the complex-conductivity
$\gamma\in W^{1,\infty}(\Omega).$

If $\phi_{1}=\bar{\phi}_{2}$ then there exists a unique solution $u$ of:

$\displaystyle\begin{cases}\bar{D}u=\gamma^{-1/2}\phi_{1},\\\
Du=\gamma^{-1/2}\phi_{2}.\end{cases}$ (30)

Further, this function fulfills the conductivity equation

$\nabla\cdot\left(\gamma\nabla u\right)=0\text{ in }\Omega.$

Let us recall the main theorem, that we are now able to prove with all these
pieces we assembled.

Theorem 1.1 Let $\Omega\subset\mathbb{R}^{3}$ a bounded Lipschitz domain,
$\gamma_{i}\in W^{1,\infty}(\Omega),\,i=1,\,2$ be two complex-valued
conductivities with $\textnormal{Re }\gamma_{i}\geq c>0.$

$\text{If }\Lambda_{\gamma_{1}}=\Lambda_{\gamma_{2}},\text{ then
}\gamma_{1}=\gamma_{2}.$

###### Proof.

Due to Theorem 3.3, one only needs to show that the scattering data $h$ for
$|k|>>1$ is uniquely determined by the Dirichlet-to-Neumann map
$\Lambda_{\gamma}$. Due to the lack of Poincaré Lemma in our current framework
in quaternionic analysis with the $D$ and $\bar{D}$ operator, than a new
technique is required to obtain a similar proof to [9], for example.

For such, let us start with two conductivities $\gamma_{1},\,\gamma_{2}$ in
$W^{1,\infty}(\Omega)$ for $\Omega$ a bounded domain. By hypothesis
$\Lambda_{\gamma_{1}}=\Lambda_{\gamma_{2}}$ and thus by [29] we have
$\left.\gamma_{1}\right|_{\partial\Omega}=\left.\gamma_{2}\right|_{\partial\Omega}$.

Further, we can extend $\gamma_{j},\,j=1,2$ outside $\Omega$ in such a way
that in $\mathbb{R}^{3}\setminus\Omega$ and $\gamma_{j}-1\in
W^{1,\infty}_{\text{comp}}(\mathbb{R}^{3})$. Let
$q_{j},\phi^{j},\mu^{j},h_{j},\,j=1,2$ be the potential and the solution in
(18), the function in (19), and the scattering data in (26) all associated
with the conductivity $\gamma_{j}$.

Due to the scattering formulation at the boundary $\partial\Omega$, then we
just want to know if $\phi^{1}=\phi^{2}$ on $\partial\Omega$ when $|k|>>1$.

First, by Proposition 3.4, we know that there exists an $u_{1}$ such that

$\phi^{1}=\gamma_{1}^{1/2}(\bar{D}u_{1},Du_{1})^{T},$

which is a solution to

$\displaystyle\nabla\cdot(\gamma_{1}\nabla u_{1})=0\text{ in }\mathbb{R}^{3}.$

Now, let us define $u_{2}$ by

$\displaystyle u_{2}=\begin{cases}u_{1}&\text{ in
}\mathbb{R}^{3}\setminus\Omega,\\\ u&\text{ in }\Omega.\end{cases}$

where $\hat{u}$ is the solution to the Dirichlet problem

$\displaystyle\begin{cases}\nabla\cdot\left(\gamma_{2}\nabla u\right)=0&\text{
in }\Omega,\\\ u=u_{1}&\text{ on }\partial\Omega.\end{cases}$

Let $g\in C^{\infty}_{c}(\mathbb{R}^{3}).$ Then,

$\displaystyle\int_{\mathbb{R}^{3}}\gamma_{2}\nabla u_{2}\nabla g\,dx$
$\displaystyle=\int_{\mathbb{R}^{3}\setminus\Omega}\gamma_{1}\nabla
u_{1}\nabla g\,dx+\int_{\Omega}\gamma_{2}\nabla\hat{u}\nabla g\,dx$
$\displaystyle=-\int_{\partial\Omega}\Lambda_{\gamma_{1}}\left[\left.u_{1}\right|_{\partial\Omega}\right]g\,ds_{x}+\int_{\partial\Omega}\Lambda_{\gamma_{2}}\left[\left.u\right|_{\partial\Omega}\right]g\,ds_{x}$
$\displaystyle=0.$

Hence, $u_{2}$ is the solution of $\nabla\cdot\left(\gamma_{2}\nabla
u_{2}\right)=0$ in $\mathbb{R}^{3}$. Further, the following function

$\psi^{2}=\gamma_{2}^{1/2}\left(\bar{D}u_{2},Du_{2}\right)^{T}$

is the solution of (18) where the potential is given by $\gamma_{2}$.

Furthermore, $\psi^{2}$ has the asymptotics of $\phi^{1}$ in
$\mathbb{R}^{3}\setminus\Omega$, thus by Lemma 3.1 and 3.2 it will be the
unique solution of the respective integral equation of (18). Thus, $\psi^{2}$
will be equal $\phi^{2}$ when $|k|>R$. Since, on the outside
$\psi^{2}\equiv\phi^{1}$. Then we obtain:

$\displaystyle\phi^{1}=\phi^{2}\text{ in }\mathbb{R}^{3}\setminus\Omega.$

In particular, we have equality at the boundary $\partial\Omega$. So, this
implies that if the Dirichlet-to-Neumann maps are equal the respective
scattering data will also be the same. Thus, the Dirichlet-to-Neumann map
uniquely determines the potential $q$.

From the definition of $q$, we can uniquely determine the conductivity
$\gamma$ up to a constant, which in the end is defined by
$\left.\gamma\right|_{\partial\Omega}$ which is uniquely determined by the
Dirichlet-to-Neumann map $\Lambda_{\gamma}$. ∎

##  4 Auxiliary Proofs

Proof of Lemma 3.1.

Let us assume, without loss of generality, that $f$ is a scalar function.
Further, we present the proof for $M^{1}$, since for $M^{2}$ it follows
analogously.

Recall, that we choose $\zeta\in\mathbb{C}_{(2)}$ with respect to
$k\in\mathbb{R}_{(2)}$ as

$\zeta=k^{\perp}+i\frac{k}{2},\quad k^{\perp}\cdot k=0.$

In vector form, this leads to $\zeta-\zeta^{c}=ik$ which implies the following
deductions:

$\displaystyle M^{1}f(x)$
$\displaystyle=\int_{\mathbb{R}^{3}}e^{-w\cdot\left(\zeta-\bar{\zeta}^{c}\right)}\frac{\overline{x-w}}{|x-w|^{3}}\int_{\mathbb{R}^{3}}e^{y\cdot\left(\zeta-\bar{\zeta}^{c}\right)}\frac{w-y}{|w-y|^{3}}f(y)q_{2}(y)\,dy\,q_{1}(w)\,dw$
$\displaystyle=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{-iw\cdot
k}\frac{\overline{x-w}}{|x-w|^{3}}e^{iy\cdot
k}\frac{w-y}{|w-y|^{3}}f(y)q_{2}(y)q_{1}(w)\,dwdy$
$\displaystyle=\int_{\mathbb{R}^{3}}A(x,y;k)f(y)\,dy,$

where

$\displaystyle A(x,y;k)=\int_{\mathbb{R}^{3}}e^{-i(w-y)\cdot
k}\frac{\overline{x-y}}{|x-y|^{3}}\frac{w-y}{|w-y|^{3}}q_{2}(y)q_{1}(w)\,dw.$

Due to the compact support of the potential $q_{2}$, it holds that $A$ has
compact support on the second variable.

Let us now apply the norm in terms of $k$ to it:

$\displaystyle\|Mf(x,\cdot)\|_{L^{p}(|k|>R)}$
$\displaystyle=\left[\int_{|k|>R}|Mf(x,\zeta)|^{p}\,d\sigma_{\zeta}\right]^{1/p}$
$\displaystyle=\left[\int_{|k|>R}\left|\int_{\Omega}A(x,y;k)f(y)\,dy\right|^{p}\,d\sigma_{k}\right]^{1/p}$
$\displaystyle\leq\int_{\Omega}\left[\int_{|k|>R}|A(x,y;k)f(y)|^{p}\,d\sigma_{k}\right]^{1/p}dy$
$\displaystyle\leq\int_{\Omega}\sup_{k}\left|A(x,y;k)\right|\,dy\,\|f\|_{S}.$

In order to complete the proof we show that the first integral goes to zero as
$R\rightarrow\infty$.

Let $A^{s}$ be given with the extra factor $\alpha(s|x-w|)\alpha(s|w-y|)$ in
the integrand, where $\alpha\in C^{\infty}$ is $1$ outside a neighborhood of
the origin and $0$ inside a smaller neighborhood of it.

Since,

$\int_{B_{1}(0)}\int_{B_{1}(0)}\frac{1}{|w|^{2}}\frac{1}{|w-y|^{2}}\,dw\,dy,$

it holds that for any $\epsilon>0$ there exists an $s>0$ such that:

$\displaystyle\int_{\Omega}|A-A^{s}|\,dy<\epsilon.$

Further, we denote $A^{s_{0},n}$ the function $A^{s_{0}}$ with potentials
$q_{1},\,q_{2}$ replaced by their $L^{1}$ smooth approximation
$Q_{1}^{n},\,Q_{2}^{n}\in C^{\infty}$. Since the other factors are bounded it
holds

$\int_{\Omega}|A^{s_{0}}-A^{s_{0},n}|\,dy<\epsilon.$

Now it is enough to show that $A^{s_{0},n}\rightarrow 0$ as $|k|\rightarrow 0$
uniformly!

All integrands inside of it will be in $C^{\infty}$ and uniformly bounded,
thus by Riemann-Lebesgue the result follows. ∎

Proof of Lemma 3.2. Once again recall that
$\zeta=\left(k^{\perp}+i\frac{k}{2}\right)$ for $k\in\mathbb{R}^{3}$. First we
show that $M^{1}1\in S$. We have

$\displaystyle M^{1}1=\int_{\Omega}\int_{\Omega}e^{-iw\cdot
k}\frac{\overline{x-w}}{|x-w|^{3}}\frac{w-y}{|w-y|^{3}}e^{iy\cdot
k}q_{2}(y)q_{1}(w)\,dy\,dw,$

and applying the $L^{p}$ norm in $k$ followed with Minkowski integral
inequality we obtain

$\displaystyle\left[\int_{|k|>R}|M^{1}1|^{p}dk\right]^{1/p}\leq\int_{\Omega}\frac{|q_{1}(w)|}{|x-w|^{2}}\left[\int_{|k|>R}\left|\int_{\Omega}e^{iy\cdot
k}\frac{w-y}{|w-y|^{3}}q_{2}(y)dy\right|^{p}dk\right]^{1/p}dw$

The inner most integral resembles a Fourier transform, hence, applying the
Hausdorff-Young inequality for $p>2$ we have

$\displaystyle\left[\int_{|k|>R}\left|\int_{\Omega}e^{iy\cdot
k}\frac{w-y}{|w-y|^{3}}q_{2}(y)\,dy\right|^{p}\,dk\right]^{1/p}\leq\left[\int_{\Omega}\frac{|q_{2}(y)|^{p^{\prime}}}{|w-y|^{2p^{\prime}}}\,dy\right]^{1/p^{\prime}}<C\|q_{2}\|_{\infty},$

where the last inequality follows quickly by Young’s convolution inequality
and Riesz type estimate of the kernel.

Therefore, by the same Riesz type estimate it holds:

$\displaystyle\left[\int_{|k|>R}|M^{1}1|^{p}\,dk\right]^{1/p}\leq
C\|q_{2}\|_{\infty}\int_{\Omega}\frac{|q_{1}(w)|}{|x-w|^{2}}\,dw\leq
C^{\prime}\|q_{2}\|_{\infty}\|q_{1}\|_{\infty}.$

To complete the proof we need to show statement (25). Similarly, to the above
proof, we have by Hausdorff-Young Inequality, Young’s convolution inequality
and a Riesz type estimate the following:

$\displaystyle\left[\int_{|k|>R}\left|\int_{\mathbb{R}^{3}}e^{iy\cdot
k}\frac{x-y}{|x-y|^{3}}q_{2}(y)\,d\sigma_{y}\right|^{p}\,d\sigma_{k}\right]^{1/p}$
$\displaystyle\leq\left[\int_{\mathbb{R}^{3}}\left|\frac{x-y}{|x-y|^{3}}q_{2}(y)\right|^{p^{\prime}}\,d\sigma_{y}\right]^{1/p^{\prime}}$
$\displaystyle\leq C\|q_{2}\|_{\infty}$

∎

We need the following auxiliary result for the proof of Proposition 3.4.

######  Lemma 4.1.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{3}$.

If $h$ is a scalar-valued and harmonic function that fulfills

$\textnormal{Vec}(S_{\partial\Omega}h)=0,$

then $\left.h\right|_{\partial\Omega}$ is constant.

###### Proof.

First, note that $I+S_{\partial\Omega}=P_{\partial\Omega}$ is a projector and
by Proposition 2.5.12 and Corollary 2.5.15 of [16] it holds that
$P_{\partial\Omega}h$ is the boundary value of a monogenic function in
$\Omega$.

Since $h$ is a scalar-valued function it holds that

$\displaystyle P_{\partial\Omega}h$
$\displaystyle=\text{Sc}(P_{\partial\Omega}h)+\text{Vec}(P_{\partial\Omega}h)$
$\displaystyle=(h+\text{Sc}_{\partial\Omega}h)+\text{Vec}(S_{\partial\Omega}h).$

Let $w=(h+\text{Sc}_{\partial\Omega}h)$ and
$v=\text{Vec}(S_{\partial\Omega}h)$. Now, we denote $f$ as the monogenic
extension of $P_{\partial\Omega}h$ in $\Omega$, as such, the boundary values
of $f$ fulfill $\text{tr}f=w+v$. Note that by hypothesis we have that
$\left.v\right|_{\partial\Omega}=0$.

Hence, $f$ is also an harmonic function, which implies that the scalar and
vector components are harmonic.

$\displaystyle\begin{cases}\Delta(\text{Vec}f)=0,\\\
\left.\text{Vec\,}f\right|_{\partial\Omega}=0.\end{cases}$

By a mean value theorem or a maximum principle it holds that $\text{Vec}f=0$.
Due to this and $f$ being monogenic we obtain that $Df=0\Leftrightarrow
D(\text{Re}f)=0$. Thus, $\text{Re}f=c$ since $D$ is a quaternionic operator.

Consequently, the boundary values are also constant, which means that $w=c$ in
$\partial\Omega$. Since, $\text{Sc}(S_{\partial\Omega}h)$ is an averaging
operator it holds that $h=c$. ∎

Proof of Proposition 3.4

Suppose that $(u,v)$ are solutions to the following equations:

$\displaystyle\begin{cases}\bar{D}u=\gamma^{-1/2}\phi_{1}\\\
Dv=\gamma^{-1/2}\phi_{2}.\end{cases}$

From applying the operator $D$ and $\bar{D}$ to the first and second equation
respectively, we obtain from $\phi_{2}=\overline{\phi}_{1}^{\mathbb{H}}$ and
$q_{2}=\overline{q}_{1}^{\mathbb{H}}$ the following:

$\displaystyle\Delta u$
$\displaystyle=D(\gamma^{-1/2}\phi_{1})=D(\gamma^{-1/2})\phi_{1}+\gamma^{-1/2}D\phi_{1}$
$\displaystyle=-\frac{1}{2}\gamma^{-3/2}(D\gamma\phi_{1})+\gamma^{-1/2}\phi_{2}q_{1}$
$\displaystyle=\gamma^{-1/2}\left[q_{2}\phi_{1}+\phi_{2}q_{1}\right]=\gamma^{-1/2}\left[\overline{q}_{1}^{\mathbb{H}}\phi_{1}+\overline{\phi}_{1}^{\mathbb{H}}q_{1}\right]$
$\displaystyle=\gamma^{-1/2}\text{Sc
}(\overline{\phi}_{1}^{\mathbb{H}}q_{1}).$

and

$\displaystyle\Delta v$
$\displaystyle=\bar{D}(\gamma^{-1/2}\phi_{2})=\bar{D}(\gamma^{-1/2})\phi_{2}+\gamma^{-1/2}\bar{D}\phi_{2}$
$\displaystyle=-\frac{1}{2}\gamma^{-3/2}(\bar{D}\gamma)\phi_{2}+\gamma^{-1/2}\phi_{1}q_{2}$
$\displaystyle=\gamma^{-1/2}\left[q_{1}\phi_{2}+\phi_{1}q_{2}\right]=\gamma^{-1/2}\left[q_{1}\overline{\phi}_{1}^{\mathbb{H}}+\phi_{1}\overline{q}_{1}^{\mathbb{H}}\right]$
$\displaystyle=\gamma^{-1/2}\text{Sc
}(\phi_{1}\overline{q}_{1}^{\mathbb{H}}).$

The first thing to notice is that both equations imply that $u$ and $v$ must
be scalar-valued functions.

Further, notice that

$\displaystyle\Delta(u-v)$ $\displaystyle=\gamma^{-1/2}\left[\text{Sc
}(\overline{\phi}_{1}^{\mathbb{H}}q_{1})-\text{Sc
}(\phi_{1}\overline{q}_{1}^{\mathbb{H}})\right]$
$\displaystyle=\gamma^{-1/2}\left[\text{Sc
}(\overline{\phi}_{1}^{\mathbb{H}}q_{1})-\text{Sc
}(\overline{q_{1}\overline{\phi}_{1}^{\mathbb{H}}})\right]=0.$

Therefore, $h=u-v$ is an harmonic function. Our objective is to show that
$h\equiv 0$, thus showing that $u=v$.

For such, let us consider the theory of integral transforms in quaternionic
analysis. We have

$\displaystyle u$
$\displaystyle=\bar{T}(\gamma^{-1/2}\phi_{1})+\overline{F}_{\partial\Omega}(\gamma^{-1/2}\phi_{1})\text{
and }$ $\displaystyle u$
$\displaystyle=\bar{T}(\gamma^{-1/2}\phi_{1})+\overline{F}_{\partial\Omega}(u),$

which implies that

$\overline{F}_{\partial\Omega}(\gamma^{-1/2}\phi_{1})=\overline{F}_{\partial\Omega}u.$

Analogously, we obtain

$F_{\partial\Omega}(\gamma^{-1/2}\phi_{2})=F_{\partial\Omega}v.$

Here, we can extrapolate from the first equation and from $u$ being scalar-
valued that

$\displaystyle\overline{\gamma^{-1/2}\phi_{1}}F_{\partial\Omega}=F_{\partial\Omega}u$
$\displaystyle\Leftrightarrow\gamma^{-1/2}\phi_{2}F_{\partial\Omega}=F_{\partial\Omega}u.$

Applying the operator $F_{\partial\Omega}$ on the other side, we obtain:

$\displaystyle F_{\partial\Omega}^{2}u$
$\displaystyle=F_{\partial\Omega}(\gamma^{-1/2}\phi_{2})F_{\partial\Omega}\text{
and }$ $\displaystyle F_{\partial\Omega}^{2}v$
$\displaystyle=F_{\partial\Omega}(\gamma^{-1/2}\phi_{2})F_{\partial\Omega}$
$\displaystyle\Rightarrow
F_{\partial\Omega}^{2}h=F_{\partial\Omega}^{2}(u-v)=0.$

If we take the trace on both sides, the operator becomes a projector thus we
obtain $\text{tr }F_{\partial\Omega}h=0$.

Now, through the Sokhotski-Plemelj formula we obtain:

$\displaystyle\text{tr
}F_{\partial\Omega}h=\left.h\right|_{\partial\Omega}+S_{\partial\Omega}h=0,\text{
at }\partial\Omega.$

Since $h$ is a scalar-valued function that we decompose this formulation with
the scalar and vector part to obtain two conditions:

$\displaystyle\begin{cases}&h+\text{Sc}(S_{\partial\Omega}h)=0\\\
&\text{Vec}(S_{\partial\Omega}h)=0.\end{cases}$

Through the second condition and Lemma 4.1 we have that $h$ is constant over
$\partial\Omega$.

Now, given that $h$ is a scalar constant, the first condition reduces to:

$\displaystyle h(1+\text{Sc}(S_{\partial\Omega}1))=0$

By [16] we obtain that $1+\text{Sc}(S_{\partial\Omega}1)=1/2$ in
$\partial\Omega$. Therefore, we conclude that $h\equiv 0$ in $\partial\Omega$.
Given that $h$ is harmonic, this immediately implies that $h=0$ in $\Omega$.

Therefore, we obtain $u=v$, and therefore there exists a unique solution to
the initial system through the $T$ and $F_{\partial\Omega}$ operators in
$\Omega$.

To finalize, we only need to show that $u$ fulfills the conductivity equation
in $\Omega$.

Bringing the first equation to light

$\bar{D}u=\gamma^{-1/2}\phi_{1},$

changing the side of the conductivity we get $\gamma^{1/2}\bar{D}u=\phi_{1}$
and applying the $D$ operator to both sides now brings

$\displaystyle D\left(\gamma^{1/2}\bar{D}u\right)=D\phi_{1}$
$\displaystyle\Leftrightarrow\quad$ $\displaystyle
D\left(\gamma^{1/2}\right)\bar{D}u+\gamma^{1/2}\Delta u=\phi_{2}q_{1}$
$\displaystyle\Leftrightarrow\quad$ $\displaystyle
D\left(\gamma^{1/2}\right)\bar{D}u+\gamma^{1/2}\Delta
u=\gamma^{-1/2}Du\frac{1}{2}\frac{\bar{D}\gamma}{\gamma}$
$\displaystyle\Leftrightarrow\quad$
$\displaystyle\frac{1}{2}\gamma^{1/2}D\gamma\bar{D}u+\gamma^{1/2}\Delta
u+\frac{1}{2}Du\frac{\bar{D}\gamma}{\gamma^{1/2}}=0$
$\displaystyle\Leftrightarrow\quad$ $\displaystyle\nabla\gamma\cdot\nabla
u+\gamma\Delta u=0\Leftrightarrow\nabla\cdot\left(\gamma\nabla u\right)=0$

∎

As such, we conclude our proof of uniqueness for complex-conductivities in
$W^{1,\infty}(\Omega)$ from the Dirichlet-to-Neumann map $\Lambda_{\gamma}$.
Notice that (29) even provides a reconstruction formula, but as mentioned in
the previous section it is very unstable for computational purposes.

## References

  * [1] Alessandrini, G. (1988). Stable determination of conductivity by boundary measurements. Applicable Analysis, 27(1-3), 153-172.
  * [2] Astala, K., Faraco, D., Rogers, K. M. (2013). Unbounded potential recovery in the plane. Ann. Sci. Éc. Norm. Supér.(4).
  * [3] Astala, K., Päivärinta, L. (2006). Calderón’s inverse conductivity problem in the plane. Annals of Mathematics, 265-299.
  * [4] Blåsten, E., Imanuvilov, O. Y., Yamamoto, M. (2015). Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Problems & Imaging, 9(3), 709-723.
  * [5] Beals, R. (1985). Multidimensional inverse scatterings and nonlinear partial differential equations. In Proc. Symp. Pure Math. (Vol. 43, pp. 45-70).
  * [6] Borcea, L. (2002). Electrical impedance tomography. Inverse problems, 18(6), R99.
  * [7] Brown, R. M. (1996). Global uniqueness in the impedance-imaging problem for less regular conductivities. SIAM Journal on Mathematical Analysis, 27(4), 1049-1056.
  * [8] Brown, R. M., Torres, R. H. (2003). Uniqueness in the inverse conductivity problem for conductivities with 3/2 derivatives in $L^{p},p>2n$. Journal of Fourier Analysis and Applications, 9(6), 563-574.
  * [9] Brown, R. M., Uhlmann, G. A. (1997). Uniqueness in the inverse conductivity problem for non-smooth conductivities in two dimensions. Communications in partial differential equations, 22(5-6), 1009-1027.
  * [10] Bukhgeim, A. L. (2008). Recovering a potential from Cauchy data in the two-dimensional case.
  * [11] Calderón, A. P. (2006). On an inverse boundary value problem. Computational & Applied Mathematics, 25, 133-138.
  * [12] Caro, P., Rogers, K. M. (2016). Global uniqueness for the Calderón problem with Lipschitz conductivities. In Forum of Mathematics, Pi (Vol. 4). Cambridge University Press.
  * [13] Chanillo, S. (1990). A problem in electrical prospection and an n-dimensional Borg-Levinson theorem. Proceedings of the American Mathematical Society, 108(3), 761-767.
  * [14] Faddeev, L. D. (1965). Growing solutions of the Schrödinger equation, Dokl. Akad. Nauk SSSR, 165, 514–517.
  * [15] Francini, E. (2000). Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map. Inverse Problems, 16(1), 107.
  * [16] Gürlebeck, K., Sprössig, W. (1989). Quaternionic Analysis and Elliptic Boundary Value Problems. nternational Series of Numerical Mathematics. Birkhäuser Basel.
  * [17] Haberman, B. (2015). Uniqueness in Calderón’s problem for conductivities with unbounded gradient. Communications in Mathematical Physics, 340(2), 639-659.
  * [18] Haberman, B., Tataru, D. (2013). Uniqueness in Calderón’s problem with Lipschitz conductivities. Duke Mathematical Journal, 162(3), 497-516.
  * [19] Hamilton, S. J., Isaacson, D., Kolehmainen, V., Muller, P. A., Toivainen, J., Bray, P. F. (2021). 3D Electrical Impedance Tomography reconstructions from simulated electrode data using direct inversion $\mathbf{t}^{\rm{{\textbf{exp}}}}$ and Calderón methods. Inverse Problems & Imaging.
  * [20] Knudsen, K., Tamasan, A. (2003). Reconstruction of less regular conductivities in the plane. Communications in Partial Differential Equations, 1, 28.
  * [21] Lakshtanov, E., Tejero, J., Vainberg, B. (2017). Uniqueness in the inverse conductivity problem for complex-valued Lipschitz conductivities in the plane. SIAM Journal on Mathematical Analysis, 49(5), 3766-3775.
  * [22] Nachman, A. I. (1988). Reconstructions from boundary measurements. Annals of Mathematics, 128(3), 531-576.
  * [23] Nachman, A. I. (1996). Global uniqueness for a two-dimensional inverse boundary value problem. Annals of Mathematics, 71-96.
  * [24] Nachman, A., Sylvester, J., Uhlmann, G. (1988). An n-dimensional Borg-Levinson theorem. Communications in Mathematical Physics, 115(4), 595-605.
  * [25] Novikov, R. G. (1988). Multidimensional inverse spectral problem for the equation $-\Delta\psi+(v(x)-Eu(x))\psi=0$. Functional Analysis and Its Applications, 22(4), 263-272.
  * [26] Novikov, R. G., Santacesaria, M. (2011). Global uniqueness and reconstruction for the multi-channel Gelfand–Calderón inverse problem in two dimensions. Bulletin des Sciences Mathematiques, 135(5), 421-434.
  * [27] Päivärinta, L., Panchenko, A., Uhlmann, G. (2003). Complex geometrical optics solutions for Lipschitz conductivities. Revista Matematica Iberoamericana, 19(1), 57-72.
  * [28] Pombo, I. (2020). CGO-Faddeev approach for complex conductivities with regular jumps in two dimensions. Inverse Problems, 36(2), 024002.
  * [29] Pombo, I. (2021). Reconstructions from boundary measurements: complex conductivities. arXiv preprint arXiv:2112.09894.
  * [30] Rosén, A. (2019). Geometric multivector analysis. Springer International Publishing.
  * [31] Sylvester, J., Uhlmann, G. (1987). A global uniqueness theorem for an inverse boundary value problem. Annals of mathematics, 153-169


# Distributions of Posterior Quantiles via Matching

Anton Kolotilin and Alexander Wolitzky

(Date: 28th August 2024)

###### Abstract.

We offer a simple analysis of the problem of choosing a statistical experiment
to optimize the induced distribution of posterior medians, or more generally
$q$-quantiles for any $q\in(0,1)$. We show that all implementable
distributions of the posterior $q$-quantile are implemented by a single
experiment, the _$q$ -quantile matching experiment_, which pools pairs of
states across the $q$-quantile of the prior in a positively assortative
manner, with weight $q$ on the lower state in each pair. A dense subset of
implementable distributions of posterior $q$-quantiles can be uniquely
implemented by perturbing the $q$-quantile matching experiment. A linear
functional is optimized over distributions of posterior $q$-quantiles by
taking the optimal selection from each set of $q$-quantiles induced by the
$q$-quantile matching experiment. The $q$-quantile matching experiment is the
only experiment that simultaneously implements all implementable distributions
of the posterior $q$-quantile.

_JEL Classification:_ C61, D72, D82

_Keywords:_ quantiles, statistical experiments, overconfidence,
gerrymandering, persuasion

Kolotilin: School of Economics, UNSW Business School.
Wolitzky: Department of Economics, MIT

##  1\. Introduction

Several problems of recent economic interest amount to characterizing the set
of distributions of posterior quantiles that can be induced by some
statistical experiment, or to finding a distribution in this set that
maximizes some objective. These problems include _apparent overconfidence_
(Benoît and Dubra, 2011)—e.g., what distributions of medians of individuals’
beliefs about their own abilities are consistent with Bayesian updating?;
_partisan gerrymandering_ (Friedman and Holden 2008; Kolotilin and Wolitzky
2020b)—e.g., what is the highest distribution of district median voters
attained by any districting plan?; and _quantile persuasion_ (Kolotilin and
Wolitzky, 2020a)—e.g., what experiment maximizes the expected action of a
receiver who minimizes the expected absolute deviation of her action from the
unknown state of the world?111Yang and Zentefis (2024) explore these and other
applications. Kolotilin and Wolitzky (2020b) consider a more general
gerrymandering model, which reduces to optimizing the distribution of
posterior quantiles in a special case. Kolotilin and Wolitzky (2020a)
introduce quantile persuasion as a special case of a more general persuasion
model, which is further developed in Kolotilin, Corrao, and Wolitzky (2023).

We provide a simple solution for this problem. For any $q\in(0,1)$, there is a
single experiment—the _$q$ -quantile matching experiment_—that simultaneously
implements all implementable distributions of the posterior $q$-quantile. For
example, if the state is uniformly distributed on $[0,1]$ and the relevant
quantile is the median, the $q$-quantile matching experiment is the _median-
matching experiment_ that, whenever the true state is $\theta\in[0,1/2]$,
reveals only that the state is either $\theta$ or $1/2+\theta$.222To our
knowledge, the median-matching experiment first appears in Kolotilin and
Wolitzky (2020a). It is closely related to the median one-to-one matching
introduced by Kremer and Maskin (1996) and further studied by Legros and
Newman (2002).  In general, the $q$-quantile matching experiment pools pairs
of states across the $q$-quantile of the prior in a positively assortative
manner, with weight $q$ on the lower state in each pair.

$0$$\frac{1}{4}$$\frac{1}{2}$$\frac{3}{4}$$1$$0$$\frac{1}{4}$$\frac{1}{2}$$\frac{3}{4}$$1$$F$$\overline{H}$$\underline{H}$$x$$H$
Figure 1. Implementable Distributions of Posterior Medians _Notes:_ When the
prior $F$ is uniform on $[0,1]$, $\overline{H}$ and $\underline{H}$ are the
lowest and highest implementable distributions of posterior medians. A
distribution $H$ is implementable iff $\underline{H}\leq H\leq\overline{H}$.
Optimizing a linear functional over distributions of posterior medians
requires taking the optimal selection from each horizontal dotted line. For
example, the blue (red) dots are the optimal selections for an increasing
(decreasing) objective function.

To see why the $q$-quantile matching experiment implements all implementable
distributions of the posterior $q$-quantile, consider again the median-
matching experiment with a uniform state. When the experiment reveals that the
state is $\theta$ or $1/2+\theta$ with equal probability, every value
$x\in[\theta,1/2+\theta]$ is a posterior median. In particular, the median-
matching experiment simultaneously implements (i) the distribution
$\underline{H}(x)=\max\\{0,2x-1\\}$, (ii) the distribution
$\overline{H}(x)=\min\\{2x,1\\}$, and (iii) every distribution $H$ satisfying
$\underline{H}\leq H\leq\overline{H}$. Conversely, simple Markov-type
inequalities imply that every implementable distribution is bounded by
$\underline{H}$ and $\overline{H}$. Moreover, the set of implementable
distributions of unique posterior quantiles is essentially the same: any
desired selection from each set of $q$-quantiles induced by the $q$-quantile
matching experiment can be uniquely selected by assigning it an extra
$\varepsilon$ probability. Finally, optimizing a linear functional over
distributions of posterior quantiles simply requires taking the optimal
selection from each set of $q$-quantiles induced by the $q$-quantile matching
experiment. See Figure 1 for an illustration of our results.333Similar figures
in the literature include Figure 1 of Owen and Grofman (1988), Figure 2 of
Kamenica and Gentzkow (2011), and Figure 3 of Yang and Zentefis (2024).

We also show that the $q$-quantile matching experiment is the _unique_
experiment that implements all implementable distributions of the posterior
$q$-quantile. To see why, consider again a uniform state, and compare the
median-matching experiment with the _negative assortative matching_ experiment
that, whenever the true state is $\theta\in[0,1/2]$, reveals only that the
state is either $\theta$ or $1-\theta$. The negative assortative matching
experiment simultaneously implements the lowest and highest distributions of
the posterior median, $\overline{H}$ and $\underline{H}$, but it does not
implement all intermediate distributions, such as the distribution $H_{1/2}$
given by $H_{1/2}(x)=\overline{H}(x)$ for $x<1/4$, $H_{1/2}(x)=1/2$ for
$x\in[1/4,3/4)$, and $H_{1/2}(x)=\underline{H}(x)$ for $x\geq 3/4$. Indeed,
the negative assortative matching experiment induces posteriors with medians
between $1/4$ and $3/4$ when the true state lies between $1/4$ and $3/4$,
while $H_{1/2}$ assigns probability $0$ to these medians.

The current paper is closely related to Benoît and Dubra (2011) and Yang and
Zentefis (2024). Both of these papers establish results that are very similar
to our Theorem 1 (albeit Benoît and Dubra do so for discrete experiments with
finitely many induced posteriors). Our main contribution is to introduce the
$q$-quantile matching experiment, which yields a much simpler proof of Theorem
1, as well as new results (Theorems 2 and 3).

##  2\. Implementable Distributions of Posterior Quantiles

This section shows that the $q$-quantile matching experiment implements all
implementable distributions of the posterior $q$-quantile.

Let $\Theta=[\underline{\theta},\overline{\theta}]\subset\mathbb{R}$, with
$\underline{\theta}<\overline{\theta}$, be a compact state space; let
$C(\Theta)$ be the set of continuous functions on $\Theta$; let
$\Delta(\Theta)$ be the set of cumulative distribution functions on $\Theta$,
endowed with the weak⋆ topology, and let $\Delta(\Delta(\Theta))$ be the set
of probability measures on $\Delta(\Theta)$. Recall that $G\in\Delta(\Theta)$
is a non-decreasing, right-continuous function satisfying
$G(\underline{\theta})\geq 0$ and $G(\overline{\theta})=1$. Let $\delta_{x}$,
with $x\in\Theta$, denote the degenerate distribution at $x$, so that
$\delta_{x}(\theta)=\mathbf{1}\\{\theta\geq x\\}$.

Fix a prior distribution $F\in\Delta(\Theta)$ and a quantile of interest
$q\in(0,1)$. Following Kamenica and Gentzkow (2011), define an _experiment_ as
a distribution $\tau\in\Delta(\Delta(\Theta))$ of posterior distributions
$G\in\Delta(\Theta)$ such that $\int G\mathrm{d}\tau(G)=F$. For each posterior
$G$, define the set of _$q$ -quantiles_ of $G$ as

$X(G)=\\{x\in\Theta:G(x^{-})\leq q\leq G(x)\\},$

where $G(x^{-})$ denotes the left limit $\lim_{\theta\uparrow x}G(\theta)$,
with the convention $G(\underline{\theta}^{-})=0$. In addition, for each $G$,
define its generalized inverse $G^{-1}$ as

$G^{-1}(p)=\inf\\{\theta\in\Theta:G(\theta)\geq p\\},\quad\text{for all
$p\in[0,1]$}.$

That is, $G^{-1}(p)$ is the smallest $p$-quantile of $G$.

To define the $q$-quantile matching experiment, let $\omega$ be uniformly
distributed on $[0,1]$, and for each $\omega\in[0,q]$, let $G=G_{\omega}$ be
the distribution that assigns probability $q$ to $F^{-1}(\omega)$ and assigns
probability $1-q$ to $F^{-1}(q+(1-q)\omega/q)$. The _$q$ -quantile matching
experiment_ is defined as an experiment $\tau^{\star}$ such that for
$\tau^{\star}$-almost all $G$, there exists $\omega\in[0,q]$ such that
$G=G_{\omega}$.444For example, when $F$ is atomless, we can let
$\omega=F(\theta)$, so that the $q$-quantile matching experiment induces
posteriors that assign probability $q$ to $\theta$ and assign probability
$1-q$ to $F^{-1}(q+(1-q)F(\theta)/q)$, for $\theta\in[0,F^{-1}(q)]$. Formally,
$\tau^{\star}$ is defined by

$\tau^{\star}(M)=\int_{0}^{q}\mathbf{1}\\{q\delta_{F^{-1}(\omega)}+(1-q)\delta_{F^{-1}(q+\frac{1-q}{q}\omega)}\in
M\\}\frac{\mathrm{d}\omega}{q},\quad\text{for all $M\subset\Delta(\Theta)$}.$

While all of our results hold for general $F$ and $q$, we will provide
intuition only for the uniform-median case where $F$ is uniform on $[0,1]$ and
$q=1/2$. The uniform-median case is essentially without loss of generality, up
to a change of variables, because the distribution function of
$\theta=F^{-1}(\omega)$ is $F$ if $\omega$ is uniformly distributed on
$[0,1]$.

A distribution $H$ of $q$-quantiles is _implemented_ by an experiment $\tau$
if there exists a (measurable) selection $\chi$ from $X$ such that the
distribution of $\chi(G)$ induced by $\tau$ is $H$. A distribution $H$ of
$q$-quantiles is _uniquely implemented_ by an experiment $\tau$ if $H$ is
implemented by $\tau$ and $X(G)$ is a singleton for $\tau$-almost all $G$. Let
$\mathcal{H}$ and $\mathcal{H}^{\star}$ be the sets of implementable and
uniquely implementable distributions of $q$-quantiles.

The following theorem characterizes $\mathcal{H}$ and $\mathcal{H}^{\star}$.

######  Theorem 1.

The following hold:

  1. (1)

$\mathcal{H}=\\{H\in\Delta(\Theta):\underline{H}\leq H\leq\overline{H}\\}$,
where $\underline{H}(x)=\max\\{0,{(F(x)-q)}/{(1-q)}\\}$ and
$\overline{H}(x)=\min\\{{F(x)}/{q},1\\}$ for all $x\in\Theta$.

  2. (2)

Every $H\in\mathcal{H}$ is implemented by $\tau^{\star}$.

  3. (3)

If $F$ has a positive density on $\Theta$ then $\mathcal{H}$ is the closure of
$\mathcal{H}^{\star}$. In particular, for any objective function $V\in
C(\Theta)$, we have

$\sup_{H\in\mathcal{H}^{\star}}\int_{\Theta}V(x)\mathrm{d}H(x)=\max_{H\in\mathcal{H}}\int_{\Theta}V(x)\mathrm{d}H(x).$
(1)

Figure 1 illustrates the set $\mathcal{H}$. The intuition for Theorem 1 is
straightforward. First, by simple Markov-type inequalities, any implementable
$H$ must satisfy $\underline{H}\leq H\leq\overline{H}$. For example, if the
posterior median is less than $x$ with probability $p$, then $\theta$ must be
less than $x$ with probability at least $p/2$. When $F(x)=x$, this implies
that $p\leq 2x$, so the probability that the posterior median is less than $x$
is at most $\min\\{2x,1\\}=\overline{H}(x)$.555This argument is closely
related to Kamenica and Gentzkow’s 2011 “prosecutor-judge” example. As in
their example, the key observation is that if the prior probability of an
event (e.g., the event that $\theta\leq x$) is $x$, then the maximum
probability that the posterior probability of this event is at least $1/2$ is
$\min\\{2x,1\\}$.

Conversely, to see that any $H$ satisfying $\underline{H}\leq
H\leq\overline{H}$ is implementable, consider the median-matching experiment
$\tau^{\star}$ that induces only posteriors $G_{\theta}$ that assign equal
probability to some $\theta\in[0,1/2]$ and to $1/2+\theta$. The set of medians
of such a posterior is $X(G_{\theta})=[\theta,1/2+\theta]$. At the same time,
$H\leq\overline{H}$ implies that $H^{-1}(2\theta)\geq\theta$, and
$H\geq\underline{H}$ implies that $H^{-1}(2\theta)\leq 1/2+\theta$, so we have
$H^{-1}(2\theta)\in[\theta,1/2+\theta]$. Thus,
$\chi(G_{\theta})=H^{-1}(2\theta)$ is a selection from $X(G_{\theta})$.
Finally, the distribution of $\chi(G_{\theta})$ induced by $\tau^{\star}$ is
$H$, because the states that induce medians below $x$ under $\tau^{\star}$
with selection $\chi(G_{\theta})$ are precisely those in $[0,H(x)/2]$ and
$[1/2,1/2+H(x)/2]$, and the measure of these states is $H(x)$.

Finally, for unique implementation, for any $e\in(0,1]$ and any implementable
and absolutely continuous $H$ with density $h$, we can explicitly construct a
modification of the median-matching experiment $\tau^{\star}_{e}$ that
uniquely implements the distribution $(1-e)H+eF$ of medians, by making every
posterior $G_{\theta}$ a convex combination of the median matching
distribution $(\delta_{\theta}+\delta_{1/2+\theta})/2$ and the degenerate
distribution $\delta_{H^{-1}(2\theta)}$ at the unique median
$H^{-1}(2\theta)\in[\theta,1/2+\theta]$. Intuitively, for each
$\theta\in[0,1/2]$, $\tau^{\star}_{e}$ induces posteriors $G_{\theta}$ and
$G_{H(\theta)/2}$ with probabilities $1-e$ and $e$; similarly, for each
$\theta\in(1/2,1]$, $\tau^{\star}_{e}$ induces posteriors $G_{\theta-1/2}$ and
$G_{H(\theta)/2}$ with probabilities $1-e$ and $e$. Then posterior medians in
$[x,x+\mathrm{d}x]$ are induced at
$\theta\in[{H(x)}/{2},{H(x+\mathrm{d}x)}/{2}]$ with probability $1-e$, at
$\theta\in[1/2+H(x)/2,{1/2+H(x+\mathrm{d}x)/2}]$ with probability $1-e$, and
at $\theta\in[x,x+\mathrm{d}x]$ with probability $e$. Since
$H(x+\mathrm{d}x)=H(x)+h(x)\mathrm{d}x$, the density of the posterior median
$x$ multiplied by the posterior at $x$ is equal to
$(1-e)h(x)({\delta_{{H(x)}/{2}}}+{\delta_{{1}/{2}+{H(x)}/{2}}})/2+e\delta_{x}$,
as required. To complete the proof of Theorem 1, we provide a simple argument
showing that any distribution in $\mathcal{H}$ can be approximated by uniquely
implementable distributions $(1-e)H+eF$.666A complete characterization of the
set $\mathcal{H}^{\star}$ remains an open problem. Two observations are that
$\mathcal{H}^{\star}$ is a proper subset of $\mathcal{H}$ (as $\underline{H}$
and $\overline{H}$ do not belong to $\mathcal{H}^{\star}$), and that not all
uniquely implementable distributions can be implemented by our modification of
$q$-quantile matching. For example, $H=\delta_{1/2}$ is uniquely implemented
by complete pooling but not by our modification of median matching.

The literature contains several close antecedents of Theorem 1. Friedman and
Holden (2008) study partisan gerrymandering with a finite number of
legislative districts. Benoît and Dubra (2011) study testing for
overconfidence with a finite number of bins in a self-ranking experiment. In
our notation, Friedman and Holden and Benoît and Dubra consider discrete
experiments with finitely many induced posteriors. Friedman and Holden show
that a discrete version of $\underline{H}$ is the highest implementable
distribution of posterior medians. Benoît and Dubra show that the set of
uniquely implementable distributions of posterior medians is a discrete
version of the set $\\{H\in\Delta(\Theta):\underline{H}<H<\overline{H}\\}$. In
a general setting with possibly infinitely many induced posteriors in the
contexts of quantile persuasion and partisan gerrymandering, respectively,
Kolotilin and Wolitzky (2020a) and Kolotilin and Wolitzky (2020b) show that
$\underline{H}$ is the highest implementable distribution of posterior
medians. Finally, in a general setting, Yang and Zentefis (2024) show that the
set of implementable distributions of posterior medians is
$\\{H\in\Delta(\Theta):\underline{H}\leq H\leq\overline{H}\\}$, and also
construct a dense subset of distributions that are uniquely
implementable.777To establish results similar to our Theorem 1, Yang and
Zentefis characterize the extreme points of the set
$\\{H\in\Delta(\Theta):\underline{H}\leq H\leq\overline{H}\\}$. As recently
emphasized by Kleiner, Moldovanu, and Strack (2021), characterizing a convex
set by its extreme points can be useful for establishing a given property of
the set. In contrast, we show that directly characterizing the set of
implementable distributions of posterior quantiles is much easier than
characterizing the extreme points of this set. Relative to Benoît and Dubra
and Yang and Zentefis, Theorem 1 shows that the $q$-quantile matching
experiment implements every $H\in\mathcal{H}$, and also yields a much simpler
proof.

##  3\. Optimal Distributions of Posterior Quantiles

This section uses the $q$-quantile matching experiment to characterize the
distributions of posterior $q$-quantiles that optimize a continuous linear
functional.

######  Theorem 2.

Let $V\in C(\Theta)$. Then $H$ (uniquely) maximizes $\int V(x)\mathrm{d}H(x)$
on $\mathcal{H}$ iff $H^{-1}(p)$ (uniquely) maximizes $V$ on
$[\overline{H}^{-1}(p),\underline{H}^{-1}(p)]$ for (almost) all $p\in[0,1]$.
Consequently, the value of the maximization problem is

$\max_{H\in\mathcal{H}}\int_{\Theta}V(x)\mathrm{d}H(x)=\int_{0}^{1}\max\\{V(x):\,x\in[\overline{H}^{-1}(p),\underline{H}^{-1}(p)]\\}\mathrm{d}p.$
(2)

Conceptually, Theorem 2 follows easily from Theorem 1. Since the median
matching experiment $\tau^{\star}$ implements all implementable distributions
of medians, optimization just requires selecting an optimal median
$\chi(G_{\theta})\in\operatorname*{arg\,max}_{x\in[\theta,1/2+\theta]}V(x)$
for each posterior $G_{\theta}$ induced by $\tau^{\star}$, as illustrated in
Figure 1. The value of the maximization problem is thus
$2\int_{0}^{1/2}\max_{x\in[\theta,1/2+\theta]}V(x)\mathrm{d}\theta$, and a
distribution $H$ of medians is optimal iff
$H^{-1}(2\theta)\in\operatorname*{arg\,max}_{x\in[\theta,1/2+\theta]}V(x)$ for
all $\theta\in[0,1/2]$. That is, optimal solutions can be obtained by
pointwise maximization without any ironing procedure.

In general, by Theorem 1, for each $H\in\mathcal{H}$ and $p\in[0,1]$, we have
$\overline{H}^{-1}(p)\leq H^{-1}(p)\leq\underline{H}^{-1}(p)$. If we consider
the relaxed problem of finding a measurable function
$J:[0,1]\rightarrow\Theta$ to

$\displaystyle\text{maximize }\int_{0}^{1}V(J(p))\mathrm{d}p$
$\displaystyle\text{subject to }\overline{H}^{-1}(p)\leq
J(p)\leq\underline{H}^{-1}(p),\quad\text{for all }p\in[0,1],$

one solution is

$J^{\star}(p)=\min\operatorname*{arg\,max}\\{V(x):\,x\in[\overline{H}^{-1}(p),\underline{H}^{-1}(p)]\\},\quad\text{for
all }p\in[0,1].$

This function $J^{\star}$ is monotone; moreover, the proof of Theorem 2 shows
that there exists $H^{\star}\in\Delta(\Theta)$ such that
$J^{\star}=H^{\star-1}$, so $H^{\star}$ solves the optimization problem (2).

The closest antecedent to Theorem 2 is Corollary 4 of Yang and Zentefis
(2024), which solves the maximization problem (2) in the special cases where
$V$ is quasi-concave or quasi-convex. The solution follows immediately from
Theorem 2. To see how, suppose that $V$ is quasi-concave with a maximum at
$x^{\star}\in[0,1]$. For each interval $[\theta,1/2+\theta]$, it is optimal to
select $x^{\star}$ if $x^{\star}\in[\theta,1/2+\theta]$, $\theta$ if
$x^{\star}<\theta$, and $1/2+\theta$ if $x^{\star}>1/2+\theta$. This induces
the distribution of posterior medians

$\displaystyle H(x)=\begin{cases}\underline{H}(x),&x<x^{\star},\\\
\overline{H}(x),&x\geq x^{\star}.\end{cases}$ (3)

Next, suppose that $V$ is quasi-convex with $V(x^{\star})=V(1/2+x^{\star})$
for some $x^{\star}\in[0,1/2]$. Then, for each interval $[\theta,1/2+\theta]$,
it is optimal to select $\theta$ if $x^{\star}>\theta$ and $1/2+\theta$ if
$x^{\star}<\theta$. This induces the distribution of posterior medians

$\displaystyle H(x)=\begin{cases}\overline{H}(x),&x<x^{\star},\\\
2x^{\star},&x\in[x^{\star},\frac{1}{2}+x^{\star}),\\\
\underline{H}(x),&x\geq\frac{1}{2}+x^{\star}.\end{cases}$ (4)

From the perspective of optimization, it is natural to ask whether each
extreme point of $\mathcal{H}$ is exposed, meaning that it is the unique
maximizer in $\mathcal{H}$ of $\int V(x)\mathrm{d}H(x)$ for some $V\in
C(\Theta)$. It turns out that some extreme points are not exposed. To see
this, note that in the uniform-median case the distribution
$H^{\star}=(\delta_{1/4}+\delta_{1/2})/2$ is an extreme point of
$\mathcal{H}$, as there are no distinct $H_{1},H_{2}\in\mathcal{H}$ such that
$H^{\star}=(H_{1}+H_{2})/2$. By Theorem 2, if $H^{\star}$ maximizes $\int
V(x)\mathrm{d}H(x)$ on $\mathcal{H}$ for some $V\in C(\Theta)$, then
$V(1/4)\geq V(x)$ for all $x\in[\theta,1/2+\theta]$ and all
$\theta\in[0,1/4]$, and, similarly, $V(1/2)\geq V(x)$ for all
$x\in[\theta,1/2+\theta]$ and all $\theta\in[0,1/2]$. Thus, $V(1/4)=V(1/2)\geq
V(x)$ for all $x\in[0,1]$. But then the distribution
$\delta_{1/2}\in\mathcal{H}$ also maximizes $\int V(x)\mathrm{d}H(x)$, which
shows that $H^{\star}$ is not an exposed point of $\mathcal{H}$.888The
distribution $H^{\star}$ does uniquely maximize $\int V(x)\mathrm{d}H(x)$ for
$V=2\cdot\mathbf{1}\\{x=1/4\\}+\mathbf{1}\\{x=1/2\\}$, which is upper semi-
continuous but not continuous. An open question is whether each extreme point
of $\mathcal{H}$, characterized in Theorem 1 of Yang and Zentefis (2024), is
the unique maximizer of $\int V(x)\mathrm{d}H(x)$ for some upper-
semicontinuous $V$. This is a weaker property than exposedness, as the usual
theory of exposed points relies on continuity.

##  4\. Unique Properties of the Quantile Matching Experiment

Theorem 1 shows that the $q$-quantile matching experiment simultaneously
implements all implementable distributions of posterior $q$-quantiles. We now
show that it is the unique experiment to do so. For simplicity, in this
section we assume that $F$ has a positive density on $\Theta$.

We actually establish the stronger result that the $q$-quantile matching
experiment is the unique experiment that simultaneously implements all optimal
distributions for strictly quasi-convex objective functions: that is, all
distributions of the form of equation (4).

######  Theorem 3.

The $q$-quantile matching experiment $\tau^{\star}$ is the unique experiment
$\tau$ that, for each $p\in[0,1]$, implements the distribution
$H_{p}\in\mathcal{H}$ given by

$\begin{gathered}H_{p}(x)=\begin{cases}\overline{H}(x),&x<\underline{x}_{p},\\\
p,&x\in[\underline{x}_{p},\overline{x}_{p}),\\\
\underline{H}(x),&x\geq\overline{x}_{p},\end{cases}\\\ \text{where}\ \
\underline{x}_{p}=F^{-1}(qp)\ \ \text{and}\ \
\overline{x}_{p}=F^{-1}(q+(1-q)p).\end{gathered}$ (5)

In other words, for any experiment $\tau\neq\tau^{\star}$, there is some
$p\in[0,1]$ such that $\tau$ does not implement $H_{p}$. For example, in the
uniform-median case, the negative assortative matching experiment does not
implement $H_{1/2}$, as noted in the introduction.

An immediate corollary of Theorem 3 is that the $q$-quantile matching
experiment is the unique experiment that minimizes the maximum regret of a
designer who chooses an experiment $\tau$ before learning her objective $V$,
but chooses a selection $\chi$ after learning $V$. Formally, for each
experiment $\tau\in\Delta(\Delta(\Theta))$ and each objective $V\in C(\Theta)$
define the designer’s _regret_ as

$r(\tau,V)=\max_{H\in\mathcal{H}}\int_{\Theta}V(x)\mathrm{d}H(x)-\sup_{H\in\mathcal{H}}\left\\{\int_{\Theta}V(x)\mathrm{d}H(x):\,\text{$H$
is implemented by $\tau$}\right\\}.$

Note that $r(\tau,V)\geq 0$ for all $\tau$ and $V$. Say that a set of possible
objective functions $\mathcal{V}\subset C(\Theta)$ is _rich_ if, for all
$x_{0},x_{1}\in\Theta$, there exists a strictly quasi-convex $V\in\mathcal{V}$
with $V(x_{0})=V(x_{1})$. We then have the following result.

######  Corollary 1.

If $\mathcal{V}$ is rich then the $q$-quantile matching experiment
$\tau^{\star}$ is the unique experiment $\tau$ such that $r(\tau,V)=0$ for all
$V\in\mathcal{V}$.

##  5\. Discussion

We conclude by discussing implications of our results in the gerrymandering
context and mentioning some directions for future research.

A standard model of gerrymandering indexes voters by a one-dimensional
partisan type $\theta\in\Theta=[\underline{\theta},\bar{\theta}]$ with
population distribution $F$, and views a _district_ $G\in\Delta(\Theta)$ as a
distribution over voter types. A _districting plan_
$\tau\in\Delta(\Delta(\Theta))$ is then a distribution over districts that
satisfies the constraint $\int G\mathrm{d}\tau(G)=F$, reflecting the legal
requirement that all voters must be assigned to equipopulous districts. The
outcome of an election held under a districting plan then depends on the
realization of an aggregate shock, whose realization $\rho$ is drawn from some
distribution $R$ on $\mathbb{R}$. In particular, party 1 wins a district $G$
when the aggregate shock takes value $\rho$ iff $\int
v(\theta,\rho)\mathrm{d}G(\theta)\geq 1/2$, where $v(\theta,\rho)$ is party
1’s vote share among type-$\theta$ voters at aggregate shock $\rho$.999This
formulation follows Kolotilin and Wolitzky (2020b), which generalizes the
earlier models of Owen and Grofman (1988), Friedman and Holden (2008), and Gul
and Pesendorfer (2010). The connection between gerrymandering and
distributions of posterior quantiles is also discussed by Yang and Zentefis
(2024).

An important special case of this model arises when
$v(\theta,\rho)=\mathbf{1}\\{\theta\geq\rho\\}$, so there is no residual
uncertainty about individual voter behavior conditional on the aggregate
shock. In this “no individual uncertainty” case (which approximates the model
of Friedman and Holden 2008), party 1 wins district $G$ at aggregate shock
$\rho$ iff the median of $G$, $\chi(G)$, exceeds $\rho$. The distribution of
the parties’ seat shares under a districting plan $\tau$ is thus determined by
the distribution of posterior medians under $\tau$: in particular, the
designer wins each district $G$ with probability $R(\chi(G))$. Consequently,
under _partisan gerrymandering_ , where the districting plan is designed to
maximize party 1’s expected seat share—so that the designer’s utility from
creating a district with a median voter of type $x$ is $V(x)=R(x)$—the
resulting districting plan is $\underline{H}$, the highest implementable
distribution of medians (Kolotilin and Wolitzky 2020b, Proposition 3).

While partisan gerrymandering has attracted much attention, other practically
relevant models include _bipartisan gerrymandering_ —where incumbent
politicians of both parties design a districting plan to maximize the security
of their own districts—and _non-partisan gerrymandering_ —where an independent
districting commission designs a districting plan to make districts as
competitive as possible.101010On bipartisan gerrymandering, see, e.g.,
Issacharoff (2002), Persily (2002), and Konishi and Pan (2020). The objective
of creating competitive districts has not been studied in the literature as
far as we know, but it is an established districting criterion in several US
states: see, e.g., https://www.ncsl.org/redistricting-and-
census/redistricting-criteria. The results of the current paper have
implications for optimal bipartisan and non-partisan gerrymandering (without
individual uncertainty). In particular, it seems natural that a bipartisan
designer’s utility for creating a district with a median voter of type $x$
should be quasi-convex in $x$, since districts with low or high medians are
safer for the stronger party in that district: i.e., the stronger party’s
probability of winning a district with median $x$, $\max\\{R(x),1-R(x)\\}$, is
quasi-convex in $x$. By Theorem 2, optimal bipartisan districting plans are
then given by (4). Conversely, a non-partisan designer’s utility for creating
a district with median $x$ may be quasi-concave in $x$, since districts with
moderate medians are more competitive: i.e., the weaker party’s probability of
winning a district with median $x$, $\min\\{R(x),1-R(x)\\}$, is quasi-concave
in $x$. Optimal non-partisan districting plans are then given by (3). More
generally, Theorem 1 characterizes all implementable distributions of the
parties’ seat shares; and Theorem 3 shows that the only districting plan that
simultaneously implements all such distributions is the median-matching plan
that forms districts by pairing voter types across the median of $F$ in a
positively assortative manner.

In the gerrymandering context, characterizing the set of implementable
distributions of seat shares with both aggregate and individual uncertainty is
an open question. More generally, this question corresponds to characterizing
the set of implementable distributions of posterior statistics that
interpolate between the median (the relevant statistic when
$v(\theta,\rho)=\mathbf{1}\\{\theta\geq\rho\\}$) and the mean (the relevant
statistic when $v(\theta,\rho)=\theta-\rho$). Benoît and Dubra (2011), Yang
and Zentefis (2024), and the current paper characterize implementable
distributions of posterior medians (or other quantiles); Blackwell (1953),
Strassen (1965), and Kolotilin (2018) characterize implementable distributions
of posterior means. The problem of characterizing optimal distributions of
intermediate statistics—the analogous problem to that of Theorem 2 in the
current paper—is studied in the gerrymandering context by Kolotilin and
Wolitzky (2020b), and in general by Kolotilin, Corrao, and Wolitzky (2023). It
is currently unknown whether a useful analog of Theorem 1 (for quantiles) or
Strassen’s theorem (for means) exists for intermediate statistics.

##  6\. Proofs

###### Proof of Theorem 1.

Consider any experiment $\tau\in\Delta(\Delta(\Theta))$ and any measurable
selection $\chi(G)$ from $X(G)$. Let $H$ be the distribution of $\chi(G)$
induced by $\tau$. Then, for each $x\in\Theta$, we have

$\displaystyle F(x)=\int G(x)\mathrm{d}\tau(G)=\int\mathbf{1}\\{G(x)\geq
q\\}G(x)\mathrm{d}\tau(G)+\int\mathbf{1}\\{G(x)<q\\}G(x)\mathrm{d}\tau(G)$
$\displaystyle\geq\int\mathbf{1}\\{G(x)\geq
q\\}q\mathrm{d}\tau(G)\geq\int\mathbf{1}\\{\chi(G)\leq
x\\}q\mathrm{d}\tau(G)=qH(x),$

showing that $H\leq\overline{H}$. A symmetric argument shows that
$H\geq\underline{H}$.

For the converse, we first note that the median-matching experiment
$\tau^{\star}$ is well-defined because $\int G\mathrm{d}\tau^{\star}(G)=F$:
indeed, for all $\theta\in\Theta$, we have

$\displaystyle\int G(\theta)\mathrm{d}\tau^{\star}(G)$
$\displaystyle=\int_{0}^{q}\left(q\delta_{F^{-1}(\omega)}+(1-q)\delta_{F^{-1}(q+\frac{1-q}{q}\omega)}\right)\left(\theta\right)\frac{\mathrm{d}\omega}{q}$
$\displaystyle=\begin{cases}\int_{0}^{F(\theta)}q\frac{\mathrm{d}\omega}{q},&F(\theta)<q,\\\
\int_{0}^{q}q\frac{\mathrm{d}\omega}{q}+\int_{0}^{\frac{q}{1-q}(F(\theta)-q)}(1-q)\frac{\mathrm{d}\omega}{q},&F(\theta)\geq
q,\end{cases}\>=F(\theta),$

where the second equality holds because $F^{-1}(\omega)\leq\theta$ iff
$\omega\leq F(\theta)$, and $F^{-1}(q+\frac{1-q}{q}\omega)\leq\theta$ iff
$\omega\leq\frac{q}{1-q}(F(\theta)-q)$. Note also that, for each
$\omega\in[0,q]$, the set of $q$-quantiles of $G_{\omega}$ is
$X(G_{\omega})=[F^{-1}(\omega),F^{-1}(q+\frac{1-q}{q}\omega)]$.

Now fix a distribution $H\in\Delta(\Theta)$ satisfying $\underline{H}\leq
H\leq\overline{H}$. Note that, for each $\omega\in[0,q]$, since
$H\leq\overline{H}$, we have
$H^{-1}(\frac{\omega}{q})\geq\overline{H}^{-1}(\frac{\omega}{q})=F^{-1}(\omega)$;
and, since $H\geq\underline{H}$, we have
$H^{-1}(\frac{\omega}{q})\leq\underline{H}^{-1}(\frac{\omega}{q})=F^{-1}(q+\frac{1-q}{q}\omega)$.
Thus, $H^{-1}(\frac{\omega}{q})\in X(G_{\omega})$. We can therefore define a
selection $\chi(G)$ from $X(G)$ by letting $\chi(G)=H^{-1}(\frac{\omega}{q})$
in the $\tau^{\star}$-almost sure event that $G=G_{\omega}$ for some
$\omega\in[0,q]$, and (for concreteness) letting $\chi(G)=\min X(G)$
otherwise. Finally, the distribution of $\chi(G)$ induced by $\tau^{\star}$ is
$H$, because, for all $x\in\Theta$, we have

$\displaystyle\int\mathbf{1}\\{\chi(G)\leq
x\\}\mathrm{d}\tau^{\star}(G)=\int\mathbf{1}\\{H^{-1}(\tfrac{\omega}{q})\leq
x\\}\frac{\mathrm{d}\omega}{q}=\int_{0}^{qH(x)}\frac{\mathrm{d}\omega}{q}=H(x).$

For unique implementation, assume that $F$ has a positive density on
$\Theta=[\underline{\theta},\overline{\theta}]$. Fix any $H\in\mathcal{H}$.
Consider a sequence of partitions of $\Theta$ given by
$\theta_{i,n}=\underline{\theta}+(\overline{\theta}-\underline{\theta})\frac{i}{2^{n}}$,
with $i\in\\{0,1,\dots,2^{n}\\}.$ Define a sequence $H_{n}\in\Delta(\Theta)$
by

$H_{n}(x)=H(\theta_{i-1,n})\frac{F(\theta_{i,n})-F(x)}{F(\theta_{i,n})-F(\theta_{i-1,n})}+H(\theta_{i,n})\frac{F(x)-F(\theta_{i-1,n})}{F(\theta_{i,n})-F(\theta_{i-1,n})},$

for all $i\in\\{1,\dots,2^{n}\\}$ and all $x\in[\theta_{i-1,n},\theta_{i,n}]$.
Note that $H_{n}$ is well-defined, because $F$ is strictly increasing on
$\Theta$. Since $H\in\mathcal{H}$, we have $H_{n}\in\mathcal{H}$. Moreover,
$H_{n}$ has a simple density function $h_{n}$ with respect to $F$, given by

$h_{n}(x)=\frac{H(\theta_{i,n})-H(\theta_{i-1,n})}{F(\theta_{i,n})-F(\theta_{i-1,n})},$

for all $i\in\\{1,\dots,2^{n}\\}$ and all $x\in(\theta_{i-1,n},\theta_{i,n})$.

Next, for each $e\in(0,1]$ and each $n$, there exists an experiment
$\tau^{\star}_{e,n}\in\Delta(\Delta(\Theta))$ satisfying the following two
properties. First, for $\tau^{\star}_{e,n}$-almost all $G$, there exists
$x\in\Theta$ such that $G=G^{x}$ where

$G^{x}=\frac{(1-e)h_{n}(x)(q\delta_{F^{-1}(qH_{n}(x))}+(1-q)\delta_{F^{-1}(q+(1-q)H_{n}(x))})+e\delta_{x}}{(1-e)h_{n}(x)+e}.$

This implies that $X(G^{x})$ is the singleton $\\{x\\}$, because $e>0$ and
$F^{-1}(qH_{n}(x))\leq x\leq F^{-1}(q+(1-q)H_{n}(x))$ (which holds because
$\underline{H}(x)\leq H_{n}(x)\leq\overline{H}(x)$). Second, the distribution
of unique quantiles $\chi(G^{x})=x$ induced by $\tau^{\star}_{e,n}$ is
$(1-e)H_{n}+eF$.

Formally, $\tau^{\star}_{e,n}$ is defined by

$\displaystyle\tau^{\star}_{e,n}(M)=\int_{0}^{1}\mathbf{1}\left\\{G^{x}\in
M\right\\}\left((1-e)h_{n}(x)+e\right)\mathrm{d}F(x),\quad\text{for all
$M\subset\Delta(\Theta)$}.$

Note that $\tau^{\star}_{e,n}$ is a well-defined experiment because $\int
G\mathrm{d}\tau^{\star}_{e,n}(G)=F$. Indeed, for all $\theta\in\Theta$, we
have

$\displaystyle\int G(\theta)\mathrm{d}\tau^{\star}_{e,n}(G)$
$\displaystyle=\int_{0}^{1}((1-e)h_{n}(x)(q\delta_{F^{-1}(qH_{n}(x))}+(1-q)\delta_{F^{-1}(q+(1-q)H_{n}(x))})+e\delta_{x})\left(\theta\right)\mathrm{d}F(x)$
$\displaystyle=(1-e)\begin{cases}\int_{0}^{H_{n}^{-1}(\frac{F(\theta)}{q})}q\mathrm{d}H_{n}(x),&F(\theta)<q,\\\
\int_{0}^{H_{n}^{-1}(1)}q\mathrm{d}H_{n}(x)+\int_{0}^{H_{n}^{-1}(\frac{F(\theta)-q}{1-q})}(1-q)\mathrm{d}H_{n}(x),&F(\theta)\geq
q,\end{cases}$
$\displaystyle\quad+e\int_{0}^{\theta}\mathrm{d}F(x)=F(\theta),$

where the second equality holds because $F^{-1}(qH_{n}(x))\leq\theta$ iff
$x\leq H_{n}^{-1}(\frac{F(\theta)}{q})$, and
$F^{-1}(q+(1-q)H_{n}(x))\leq\theta$ iff $x\leq
H_{n}^{-1}(\frac{F(\theta)-q}{1-q})$; and the third equality holds because
$H_{n}(H_{n}^{-1}(p))=p$ for all $p\in[0,1]$, by continuity of $H_{n}$ (which
holds because $H_{n}$ has a density with respect to $F$ and $F$ has a density
with respect to the Lebesgue measure). Finally, the distribution of unique
quantiles $\chi(G)$ induced by $\tau^{\star}_{e,n}$ is $(1-e)H_{n}+eF$,
because, for all $y\in\Theta$, we have

$\displaystyle\int\mathbf{1}\\{\chi(G)\leq
y\\}\mathrm{d}\tau^{\star}_{e,n}(G)=\int_{0}^{y}((1-e)h_{n}(x)+e)\mathrm{d}F(x)=(1-e)H_{n}(y)+eF(y).$

Now fix $V\in C(\Theta)$. By continuity of $V$ and compactness of $\Theta$,
for all $\varepsilon>0$, there exists $N\in\mathbb{N}$ such that (i)
$|V(x)-V(y)|\leq\varepsilon$ for all $x,y\in[\theta_{i-1,n},\theta_{i,n}]$,
all $i\in\\{1,\dots.,2^{n}\\}$, and all $n\geq N$, and (ii)
$e|V(x)-V(y)|\leq\varepsilon$ for all $x,y\in\Theta$ and all
$e\in(0,\frac{1}{N}]$. Then, for all $n\geq N$ and $e\in(0,\frac{1}{N}]$, we
have

$\displaystyle\left|\int V(x)\mathrm{d}H(x)-\int
V(x)\mathrm{d}((1-e)H_{n}+eF)(x)\right|$ $\displaystyle\leq(1-e)\left|\int
V(x)\mathrm{d}(H-H_{n})(x)\right|+e\left|\int
V(x)\mathrm{d}(H-F)(x)\right|\leq\varepsilon+\varepsilon.$

Since this holds for any $V\in C(\Theta)$, it follows that
$(1-\frac{1}{n})H_{n}+\frac{1}{n}F$ converges weakly to $H$. In turn, since we
have seen that the experiment $\tau^{\star}_{1/n,n}$ uniquely implements
$(1-\frac{1}{n})H_{n}+\frac{1}{n}F$, we conclude that $\mathcal{H}^{\star}$ is
dense in $\mathcal{H}$. Finally, $\mathcal{H}$ is compact, as $\Delta(\Theta)$
is compact by Theorem 15.11 in Aliprantis and Border (2006), and $\mathcal{H}$
is the intersection over $x\in\Theta$ of the closed subsets
$\mathcal{H}_{x}:=\\{H\in\Delta(\Theta):\underline{H}(x)\leq
H(x)\leq\overline{H}(x)\\}$ of $\Delta(\Theta)$. Thus, the closure of
$\mathcal{H}^{\star}$ is $\mathcal{H}$, and hence (1) holds for any $V\in
C(\Theta)$. ∎

###### Proof of Theorem 2.

For each $H\in\Delta(\Theta)$, we have

$\int_{\Theta}V(x)\mathrm{d}H(x)=\int_{0}^{1}V(H^{-1}(p))\mathrm{d}p.$

Recall that

$J^{\star}(p)=\min\operatorname*{arg\,max}\\{V(x):\,x\in[\overline{H}^{-1}(p),\underline{H}^{-1}(p)]\\},\quad\text{for
all }p\in[0,1].$

Since $J^{\star}$ is defined as the minimum selection from the
$\operatorname*{arg\,max}$, it follows that (i) $J^{\star}$ is non-decreasing
(and hence measurable), because $\overline{H}^{-1}$ and $\underline{H}^{-1}$
are non-decreasing, (ii) $J^{\star}$ is left-continuous, because
$\overline{H}^{-1}$ and $\underline{H}^{-1}$ are left-continuous and $V\in
C(\Theta)$, (iii) $J^{\star}(1)\leq\overline{\theta}$, because
$\underline{H}^{-1}(1)\leq\overline{\theta}$, and (iv)
$J^{\star}(0)=\underline{\theta}$, because
$\overline{H}^{-1}(0)=\underline{H}^{-1}(0)=\underline{\theta}$. This implies
that $J^{\star}=H^{\star-1}$, where $H^{\star}\in\Delta(\Theta)$ is given by

$H^{\star}(x)=\sup\\{p\in[0,1]:\,J^{\star}(p)\leq x\\},\quad\text{for all
}x\in\Theta.$

Moreover, since $\overline{H}^{-1}\leq J^{\star}\leq\underline{H}^{-1}$, it
follows that $H^{\star}\in\mathcal{H}$, so $H^{\star}$ solves the original
problem, and its value coincides with the value of the relaxed problem,
yielding (2). Consequently, $H\in\mathcal{H}$ maximizes $\int
V(x)\mathrm{d}H(x)$ on $\mathcal{H}$ iff $H^{-1}(p)$ maximizes $V$ on
$[\overline{H}^{-1}(p),\underline{H}^{-1}(p)]$ for almost all $p\in[0,1]$.
Moreover, by continuity of $V$ and left-continuity of $\overline{H}^{-1}$ and
$\underline{H}^{-1}$, $H^{-1}(p)$ maximizes $V$ on
$[\overline{H}^{-1}(p),\underline{H}^{-1}(p)]$ for almost all $p\in[0,1]$ iff
it does so for all $p\in[0,1]$.

Next, if $V$ has a unique maximum on
$[\overline{H}^{-1}(p),\underline{H}^{-1}(p)]$ for almost all $p\in[0,1]$,
then $J^{\star}$ is the unique solution of the relaxed problem that satisfies
properties (i)–(iv), and hence $H^{\star}$ is the unique solution of the
original problem. Conversely, if there exists a non-negligible set
$P\subset[0,1]$ such that $V$ has multiple maxima on
$[\overline{H}^{-1}(p),\underline{H}^{-1}(p)]$ for each $p\in P$, then there
are multiple solutions of the relaxed problem that satisfy properties
(i)–(iv). For example, $\hat{J}$ defined as the maximum selection from the
$\operatorname*{arg\,max}$ also solves the relaxed problem, and so does
$\hat{J}^{\star}$ defined by $\hat{J}^{\star}(p)=\hat{J}(p^{-})$ for all
$p\in(0,1]$ and $\hat{J}^{\star}(0)=\underline{\theta}$. But, by construction,
$\hat{J}^{\star}$ satisfies properties (i)–(iv) and is not equal to
$J^{\star}$. Then $\hat{J}^{\star}=\hat{H}^{\star-1}$ where
$\hat{H}^{\star}\in\Delta(\Theta)$ is given by

$\hat{H}^{\star}(x)=\sup\\{p\in[0,1]:\,\hat{J}^{\star}(p)\leq
x\\},\quad\text{for all }x\in\Theta.$

Thus, $\hat{H}^{\star}\neq H^{\star}$ also solves the original problem. ∎

###### Proof of Theorem 3.

Suppose that an experiment $\tau\in\Delta(\Delta(\Theta))$ implements all
$H_{p}$. Fix any $p\in[0,1]$. Since $\tau$ implements $H_{p}$, there exists a
measurable selection $\chi_{p}(G)$ from $X(G)$ such that the distribution of
$\chi_{p}(G)$ induced by $\tau$ is $H_{p}$. Since $F$ has a density on
$\Theta$, we have

$\displaystyle qp=F(\underline{x}_{p})=\int
G(\underline{x}_{p})\mathrm{d}\tau(G)=\int\mathbf{1}\\{G(\underline{x}_{p})\geq
q\\}G(\underline{x}_{p})\mathrm{d}\tau(G)$
$\displaystyle+\int\mathbf{1}\\{G(\underline{x}_{p})<q\\}G(\underline{x}_{p})\mathrm{d}\tau(G)\geq\int\mathbf{1}\\{G(\underline{x}_{p})\geq
q\\}q\mathrm{d}\tau(G)$
$\displaystyle\geq\int\mathbf{1}\\{\chi_{p}(G)\leq\underline{x}_{p}\\}q\mathrm{d}\tau(G)=qH_{p}(\underline{x}_{p})=qp,$

so all inequalities hold with equality. Thus,
$\tau(G(\underline{x}_{p})=0)=1-p$, $\tau(G(\underline{x}_{p})=q)=p$, and
$\tau(\chi_{p}(G)\leq\underline{x}_{p})=p$. A symmetric argument yields
$\tau(G(\overline{x}_{p})=q)=1-p$, $\tau(G(\overline{x}_{p})=1)=p$, and
$\tau(\chi_{p}(G)>\overline{x}_{p})=1-p$. Next, since $G(\underline{x}_{p})=0$
and $G(\overline{x}_{p})=1$ imply that
$\underline{x}_{p}<\chi_{p}(G)\leq\overline{x}_{p}$, it follows that
$\tau(G(\underline{x}_{p})=0,\,G(\overline{x}_{p})=1)=0$, because

$\tau(\underline{x}_{p}<\chi_{p}(G)\leq\overline{x}_{p})=1-\tau(\chi_{p}(G)\leq\underline{x}_{p})-\tau(\chi_{p}(G)>\overline{x}_{p})=1-p-(1-p)=0.$

So,
$\tau(G(\underline{x}_{p})=0,\,G(\overline{x}_{p})=q)=\tau(G(\underline{x}_{p})=0)-\tau(G(\underline{x}_{p})=0,\,G(\overline{x}_{p})=1)=1-p$
and
$\tau(G(\underline{x}_{p})=q,\,G(\overline{x}_{p})=1)=\tau(G(\overline{x}_{p})=1)-\tau(G(\underline{x}_{p})=0,\,G(\overline{x}_{p})=1)=p$.
In sum,

$\begin{aligned} \tau(G(\underline{x}_{p})=0,\,G(\overline{x}_{p})=q)&=1-p,\\\
\tau(G(\underline{x}_{p})=q,\,G(\overline{x}_{p})=1)&=p,\end{aligned}\qquad\text{for
all $p\in[0,1]$.}$ (6)

We now show that (6) yields $\tau=\tau^{\star}$. Let
$X_{0}=[\underline{x}_{0},\overline{x}_{0}]=[\underline{\theta},F^{-1}(q)]$
and
$X_{1}=[\underline{x}_{1},\overline{x}_{1}]=[F^{-1}(q),\overline{\theta}]$.
For each experiment $\tilde{\tau}\in\Delta(\Delta(\Theta))$, define a joint
distribution function $I_{\tilde{\tau}}:X_{0}\times X_{1}\to[0,1]$ by

$I_{\tilde{\tau}}(x_{0},x_{1})=\tilde{\tau}(G=q\delta_{\theta_{0}}+(1-q)\delta_{\theta_{1}},\,\theta_{0}\in[\underline{x}_{0},x_{0}],\,\theta_{1}\in[\underline{x}_{1},x_{1}]).$

To prove that $\tau=\tau^{\star}$, it suffices to show that
$I_{\tau}(x_{0},x_{1})=I_{\tau^{\star}}(x_{0},x_{1})$ for all
$(x_{0},x_{1})\in X_{0}\times X_{1}$, with
$I_{\tau}(\overline{x}_{0},\overline{x}_{1})=I_{\tau^{\star}}(\overline{x}_{0},\overline{x}_{1})=1$.
Fix any $(x_{0},x_{1})\in X_{0}\times X_{1}$, and let
$\hat{p}=\tfrac{F(x_{0})}{q}\wedge\tfrac{F(x_{1})-q}{1-q}$. First, by
definition of $\tau^{\star}$, we have

$I_{\tau^{\star}}(x_{0},x_{1})=\int_{0}^{q}\mathbf{1}\\{F^{-1}(\omega)\leq
x_{0},\,F^{-1}(q+\tfrac{1-q}{q}\omega)\leq
x_{1}\\}\frac{\mathrm{d}\omega}{q}=\hat{p},$

with $I_{\tau^{\star}}(\overline{x}_{0},\overline{x}_{1})=1$, because
$F(\overline{x}_{0})=F(F^{-1}(q))=q$ and
$F(\overline{x}_{1})=F(\overline{\theta})=1$. Second, by (6) and definition of
$I_{\tau}$, we have

$\displaystyle\hat{p}=\tau(G(\underline{x}_{\hat{p}})=q,\,G(\overline{x}_{\hat{p}})=1)\leq
I_{\tau}(x_{0},x_{1})\leq\tau((G(\underline{x}_{\hat{p}}),G(\overline{x}_{\hat{p}}))\neq(0,q))=1-(1-\hat{p}),$

showing that $I_{\tau}(x_{0},x_{1})=\hat{p}=I_{\tau^{\star}}(x_{0},x_{1})$. ∎

###### Proof of Corollary 1.

Consider any experiment $\tau\neq\tau^{\star}$. By Theorem 3, there exists
$p\in[0,1]$ such that $\tau$ does not implement $H_{p}$. Since $\mathcal{V}$
is rich, there exists a continuous and strictly quasi-convex $V\in\mathcal{V}$
with $V(\underline{x}_{p})=V(\overline{x}_{p})$. Then
$\underline{x}_{\tilde{p}}$ uniquely maximizes $V$ on
$[\underline{x}_{\tilde{p}},\overline{x}_{\tilde{p}}]$ for all
$\tilde{p}\in[0,p)$, and $\overline{x}_{\tilde{p}}$ uniquely maximizes $V$ on
$[\underline{x}_{\tilde{p}},\overline{x}_{\tilde{p}}]$ for all
$\tilde{p}\in(p,1]$. By Theorem 2, $H_{p}$ uniquely maximizes $\int
V(x)\mathrm{d}H(x)$ on $\mathcal{H}$.

Suppose for contradiction that $r(\tau,V)=0$. Then there exists a sequence
$H^{n}\in\mathcal{H}$ implemented by $\tau$ such that $\int
V(x)\mathrm{d}H^{n}(x)\to\int V(x)\mathrm{d}H_{p}(x)$. Since $\mathcal{H}$ is
weak⋆ compact, passing to a subsequence if necessary, we can assume that
$H^{n}\to\hat{H}\in\mathcal{H}$. Note that $\hat{H}=H_{p}$, because $H_{p}$
uniquely maximizes $\int V(x)\mathrm{d}H(x)$ on $\mathcal{H}$. In sum, there
exists a sequence of measurable selections $\chi^{n}(G)$ from $X(G)$ such that
$H^{n}(x)=\tau(\chi^{n}(G)\leq x)$ and $H^{n}(x)\to H_{p}(x)$ for all
$x\in\Theta$.

Next, we show that $\tau$ cannot simultaneously satisfy the following three
conditions

$\displaystyle\tau(G(\underline{x}_{\tilde{p}})=0)=1-\tilde{p}\quad$
$\displaystyle\text{and}\quad\tau(G(\underline{x}_{\tilde{p}})=q)=\tilde{p},\quad\text{for
all $\tilde{p}\in[0,p]$},$ (7)
$\displaystyle\tau(G(\overline{x}_{\tilde{p}})=q)=1-\tilde{p}\quad$
$\displaystyle\text{and}\quad\tau(G(\overline{x}_{\tilde{p}})=1)=\tilde{p},\quad\text{for
all $\tilde{p}\in[p,1]$},$ (8)
$\displaystyle\tau(G(\underline{x}_{p})=0,\,G(\overline{x}_{p})=q)=1-p\quad$
$\displaystyle\text{and}\quad\tau(G(\underline{x}_{p})=q,\,G(\overline{x}_{p})=1)=p,$
(9)

as otherwise $\tau$ would implement $H_{p}$. Indeed, if $\tau$ satisfies
(7)–(9), we can define a selection $\chi(G)$ from $X(G)$ by letting
$\chi(G)=\theta_{0}$ if $\theta_{0}<\underline{x}_{p}$ and
$\chi(G)=\theta_{1}$ if $\theta_{0}>\underline{x}_{p}$ in the $\tau$-almost
sure event that $G=q\delta_{\theta_{0}}+(1-q)\delta_{\theta_{1}}$ for some
$\theta_{0}\in[\underline{x}_{0},\overline{x}_{0}]$ and
$\theta_{1}\in[\underline{x}_{1},\overline{x}_{1}]$. Then the distribution of
$\chi(G)$ induced by $\tau$ is $H_{p}$, because, by (7), for all
$\tilde{p}\leq p$, we have

$\displaystyle\tau(\chi(G)\leq\underline{x}_{\tilde{p}})=\tau(G(\underline{x}_{\tilde{p}})=q)=\tilde{p}=\tfrac{F(\underline{x}_{\tilde{p}})}{q}=H_{p}(\underline{x}_{\tilde{p}}),$

and, by (8) and (9), for all $\tilde{p}\geq p$, we have

$\displaystyle\tau(\chi(G)\leq\overline{x}_{\tilde{p}})=\tau(G(\overline{x}_{\tilde{p}})=1)+\tau(G(\underline{x}_{p})=q,\,G(\overline{x}_{\tilde{p}})=q)=\tilde{p}=\tfrac{F(\overline{x}_{\tilde{p}})-q}{1-q}=H_{p}(\overline{x}_{\tilde{p}}).$

Finally, we show that if at least one of conditions (7)–(9) fails, then
$H^{n}\nrightarrow H_{p}$. First, if (7) fails at some $\tilde{p}\in[0,p]$,
then there exists $\varepsilon>0$ such that

$\displaystyle F(\underline{x}_{\tilde{p}})$ $\displaystyle=\int
G(\underline{x}_{\tilde{p}})\mathrm{d}\tau(G)=\int\mathbf{1}\\{G(\underline{x}_{\tilde{p}})\geq
q\\}G(\underline{x}_{\tilde{p}})\mathrm{d}\tau(G)+\int\mathbf{1}\\{G(\underline{x}_{\tilde{p}})<q\\}G(\underline{x}_{\tilde{p}})\mathrm{d}\tau(G)$
$\displaystyle\geq\varepsilon+\int\mathbf{1}\\{G(\underline{x}_{\tilde{p}})\geq
q\\}q\mathrm{d}\tau(G)\geq\varepsilon+\int\mathbf{1}\\{\chi^{n}(G)\leq\underline{x}_{\tilde{p}}\\}q\mathrm{d}\tau(G)=\varepsilon+qH^{n}(\underline{x}_{\tilde{p}}),$

so $H^{n}(\underline{x}_{\tilde{p}})\nrightarrow
H_{p}(\underline{x}_{\tilde{p}})$. Similarly, if (8) fails at some
$\tilde{p}\in[p,1]$, then $H^{n}(\overline{x}_{\tilde{p}})\nrightarrow
H_{p}(\overline{x}_{\tilde{p}})$. Finally, if (7) and (8) hold, but (9) fails,
then there exists $\varepsilon>0$ such that

$\displaystyle H^{n}(\overline{x}_{p})-H^{n}(\underline{x}_{p})$
$\displaystyle=\tau(\underline{x}_{p}<\chi^{n}(G)\leq\overline{x}_{p})\geq\tau(G(\underline{x}_{p})=0,\,G(\overline{x}_{p})=1)$
$\displaystyle\geq\varepsilon>0=H^{p}(\overline{x}_{p})-H^{p}(\underline{x}_{p}),$

so $H^{n}(\underline{x}_{p})\nrightarrow H_{p}(\underline{x}_{p})$ or
$H^{n}(\overline{x}_{p})\nrightarrow H_{p}(\overline{x}_{p})$. ∎

## References

  * (1)
  * Aliprantis and Border (2006) Aliprantis, C. D., and K. Border (2006): _Infinite Dimensional Analysis: A Hitchhiker’s Guide_. Springer.
  * Benoît and Dubra (2011) Benoît, J.-P., and J. Dubra (2011): “Apparent Overconfidence,” _Econometrica_ , 79, 1591–1625.
  * Blackwell (1953) Blackwell, D. (1953): “Equivalent Comparisons of Experiments,” _Annals of Mathematical Statistics_ , 24, 265–272.
  * Friedman and Holden (2008) Friedman, J. N., and R. T. Holden (2008): “Optimal Gerrymandering: Sometimes Pack, but Never Crack,” _American Economic Review_ , 98(1), 113–44.
  * Gul and Pesendorfer (2010) Gul, F., and W. Pesendorfer (2010): “Strategic Redistricting,” _American Economic Review_ , 100(4), 1616–1141.
  * Issacharoff (2002) Issacharoff, S. (2002): “Gerrymandering and Political Cartels,” _Harvard Law Review_ , 116, 593–648.
  * Kamenica and Gentzkow (2011) Kamenica, E., and M. Gentzkow (2011): “Bayesian Persuasion,” _American Economic Review_ , 101, 2590–2615.
  * Kleiner, Moldovanu, and Strack (2021) Kleiner, A., B. Moldovanu, and P. Strack (2021): “Extreme Points and Majorization: Economic Applications,” _Econometrica_ , 89(4), 1557–1593.
  * Kolotilin (2018) Kolotilin, A. (2018): “Optimal Information Disclosure: A Linear Programming Approach,” _Theoretical Economics_ , 13, 607–636.
  * Kolotilin, Corrao, and Wolitzky (2023) Kolotilin, A., R. Corrao, and A. Wolitzky (2023): “Persuasion and Matching: Optimal Productive Transport,” _ArXiv Working Paper_ , https://arxiv.org/abs/2206.09164.
  * Kolotilin and Wolitzky (2020a) Kolotilin, A., and A. Wolitzky (2020a): “Assortative Information Disclosure,” _SSRN Working Paper_ , https://ssrn.com/abstract=3616781.
  * Kolotilin and Wolitzky (2020b) (2020b): “The Economics of Partisan Gerrymandering,” _SSRN Working Paper_ , https://ssrn.com/abstract=3698690.
  * Konishi and Pan (2020) Konishi, H., and C.-Y. Pan (2020): “Partisan and Bipartisan Gerrymandering,” _Journal of Public Economic Theory_ , 22, 1183–1212.
  * Kremer and Maskin (1996) Kremer, M., and E. Maskin (1996): “Wage Inequality and Segregation by Skill,” _NBER Working Paper_ , https://www.nber.org/papers/w5718.
  * Legros and Newman (2002) Legros, P., and A. F. Newman (2002): “Monotone Matching in Perfect and Imperfect Worlds,” _Review of Economic Studies_ , 69(4), 925–942.
  * Owen and Grofman (1988) Owen, G., and B. Grofman (1988): “Optimal Partisan Gerrymandering,” _Political Geography Quarterly_ , 7(1), 5–22.
  * Persily (2002) Persily, N. (2002): “In Defense of Foxes Guarding Henhouses: The Case for Judicial Acquiescence to Incumbent-Protecting Gerrymanders,” _Harvard Law Review_ , 116, 649–683.
  * Strassen (1965) Strassen, V. (1965): “The Existence of Probability Measures with Given Marginals,” _Annals of Mathematical Statistics_ , 36, 423–439.
  * Yang and Zentefis (2024) Yang, K. H., and A. Zentefis (2024): “The Economics of Monotone Function Intervals,” _ArXiv Working Paper_ , https://arxiv.org/abs/2302.03135.