# Anisotropic Kelvin Transform

Francesco Esposito∗, Giuseppe Riey∗, Berardino Sciunzi∗, Domenico Vuono∗
∗Dipartimento di Matematica e Informatica, UNICAL, Ponte Pietro Bucci 31B,
87036 Arcavacata di Rende, Cosenza, Italy  francesco.esposito, giuseppe.riey,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: August 28, 2024; Date: August 28, 2024)

###### Abstract.

In this paper we introduce and study a new Kelvin-type transform in the
anisotropic setting. In particular, we deal with semilinear and quasilinear
anisotropic elliptic problems in the entire space.

###### Key words and phrases:

Kelvin transform, anisotropic elliptic equations, Finsler geometry

###### 2020 Mathematics Subject Classification:

35A30, 35B06, 35J62, 35J75

The authors are members of INdAM. F. Esposito, G. Riey and B. Sciunzi are
partially supported by PRIN project 2017JPCAPN (Italy): Qualitative and
quantitative aspects of nonlinear PDEs.

##  1\. Introduction

The aim of this paper is to consider elliptic partial differential equations
involving the Finsler Laplace operator:

$-\Delta^{H}u:=-\operatorname{div}(H(\nabla u)\nabla H(\nabla u))\,,$

where $H$ is a Finsler norm (see assumption $(h_{H})$ below), and for
$H(\xi)=|\xi|$, the operator $\Delta^{H}$ reduces to the classical Laplacian.
Elliptic equations involving the anisotropic operators arise as Euler-Lagrange
equations of Wulff-type energy functionals.

In the first part of the present work, we shall consider solutions to the
equation

(1.1) $-\Delta^{H}u=f(x)\qquad\text{ in }\mathbb{R}^{N},\quad N\geq 2\,,$

which is the Euler-Lagrange equation of the _Wulff-type energy functional_

(1.2) $\mathcal{W}(u)=\int_{\mathbb{R}^{N}}\left[\frac{H(\nabla
u)^{2}}{2}-f(x)u\right]\,\text{d}x\,.$

As usual, a solution has to be understood in the weak distributional meaning,
i.e. $u\in H^{1}_{loc}(\mathbb{R}^{N})$ is a weak solution of (1.1) if

(1.3) $\int_{\mathbb{R}^{N}}H(\nabla u)\langle\nabla H(\nabla
u),\nabla\varphi\rangle\,\,\text{d}x\,=\,\int_{\mathbb{R}^{N}}f(x)\varphi\,\,\text{d}x,\quad\forall\varphi\in
C^{1}_{c}(\mathbb{R}^{N})\,.$

We address here this deep issue and we succeed in providing a suitable Kelvin-
type transformation in the Riemannian case,

($h_{M}$) $H(\xi)=\sqrt{\langle M\xi,\xi\rangle}$

for some symmetric and positive definite matrix $M$. In this setting, we
define the anisotropic Kelvin-type transformation by

$T_{H}:\mathbb{R}^{N}\setminus\\{0\\}\rightarrow\mathbb{R}^{N}\setminus\\{0\\},\quad
T_{H}(\xi):=\frac{\nabla H(\xi)}{H(\xi)},$

and consequently

$\hat{u}:=\frac{u\circ T_{H}}{H^{N-2}}\quad\text{in
}\mathbb{R}^{N}\setminus\\{0\\},$

see Section 3 for more details.

######  Remark 1.1.

Although our main results will be proved in the Riemannian case ($h_{M}$), a
part of our construction holds for a general Finsler norm. This is why many
preliminary results will be stated in full generality and also we shall try to
use the language of Finsler geometry, where it is possible. This will also
highlight the geometric idea that is behind our construction.

With such construction at hand, we prove the following.

######  Theorem 1.2.

Let $H$ be given by ($h_{M}$). If $u\in H^{1}_{loc}(\mathbb{R}^{N})$ is a
locally bounded solution of (1.1), with $f\in L^{2}_{loc}(\mathbb{R}^{N})$,
then $\hat{u}$ belongs to $D^{1,2}$ far from the origin, and weakly solves the
dual equation

(1.4) $-\Delta^{H^{\circ}}\hat{u}=\frac{f\circ T_{H}}{H^{N+2}}\quad\text{in
}\mathbb{R}^{N}\setminus\\{0\\},$

where $H^{\circ}$ is the dual norm.

######  Remark 1.3.

Let us stress the fact that the result in Theorem 1.2 could be also obtained
by using a suitable change of variable together with the techniques known for
the standard Laplace operator. Such a proof would not be so easier or shorter.
This led us to provide a proof that emphasizes the geometrical properties of
the anisotropic Kelvin transformation.

As observed by Ciraolo, Figalli and Roncoroni in [11], a Kelvin-type approach
doesn’t work, in general, in the anisotropic context. This also happen in the
Euclidean framework when $p\neq 2$. As we shall show adding an example in
Remark 3.4, a pure invariance of the equation is not expected for general
Finsler norms, even in the semilinear case $p=2$.

To the best of our knowledge this is the first result in this direction in the
anisotropic framework. Let us also mention the work of Monti and Morbidelli
[23] for a non-trivial application of the Kelvin transformation for equations
involving the Grushin operators.

The last part of our paper is devoted to study the following quasilinear
elliptic equation:

(1.5) $-\Delta^{H}_{N}\text{ }u=g(x)\qquad\text{in }\mathbb{R}^{N},\quad N\geq
3\,,$

where $\Delta^{H}_{N}\text{ }u:=\operatorname{div}(H^{N-1}(\nabla u)\nabla
H(\nabla u))$ is the Finsler $N$-Laplace operator, and $g\in
L^{\frac{N}{N-1}}_{loc}(\mathbb{R}^{N})$.

As pointed out above, we say that $u\in W^{1,N}_{loc}(\mathbb{R}^{N})$ is a
weak solution of (1.1) if

(1.6) $\int_{\mathbb{R}^{N}}H^{N-1}(\nabla u)\langle\nabla H(\nabla
u),\nabla\varphi\rangle\,\,\text{d}x\,=\,\int_{\mathbb{R}^{N}}g(x)\varphi\,\,\text{d}x,\quad\forall\varphi\in
C^{1}_{c}(\mathbb{R}^{N})\,.$

In the Euclidean framework, i.e. if $H(\xi)=|\xi|$, we refer to the seminal
paper by W. Chen and C. Li [9], where the, making use of the Kelvin
transformation, classified solutions to the Liouville equation ($g=e^{u}$) via
an adaptation of the moving plane method. As remarked above, in quasilinear
case the situation is much more involved and we refer to the work of P.
Esposito [17] for a classification result of the Liouville equation for
$p=N>2$, with the use of the Kelvin transform, but with a different approach
to the one carried out in [9].

######  Theorem 1.4.

Let $H$ be given be ($h_{M}$). If $u\in W^{1,N}_{loc}(\mathbb{R}^{N})$ is a
locally bounded solution of (1.5), with $g\in
L^{\frac{N}{N-1}}_{loc}(\mathbb{R}^{N})$, then $u^{*}:=u\circ T_{H}$ weakly
solves the dual equation

(1.7) $-\Delta^{H^{\circ}}_{N}u^{*}=\frac{g\circ T_{H}}{H^{2N}}\quad\text{in
}\mathbb{R}^{N}\setminus\\{0\\},$

where $H^{\circ}$ is the dual norm.

We point out that under suitable assumptions on $f$ and $g$, due to the
results in [2, 14, 22, 27], it is possible to assume that solutions of (1.1)
and (1.5) are of class $\mathcal{C}^{1,\alpha}$.

Concluding the introduction, let us mention that anisotropic operators arise
in Finsler geometry, see [6, 10, 21, 25, 26], as modelization e.g. of material
science [8, 19], relativity [3], biology [1], image processing [15, 24].

In the following we devote Section 2 to the introduction of the main
geometrical notion used in the paper. In Section 3 we study the Kelvin-type
transformation, defined above, and we prove our main results.

##  2\. Notation and preliminary results

The aim of this section is to introduce some notation and to recall technical
results about anisotropic elliptic operator that will be involved in the proof
of the main results.

In all the paper generic fixed and numerical constants will be denoted by
$\mathcal{C}$ (with subscript in some cases) and they will vary from line to
line. For $a,b\in\mathbb{R}^{N}$ we denote by $a\otimes b$ the matrix whose
entries are $(a\otimes b)_{ij}=a_{i}b_{j}$. We remark that for any
$v,w\in\mathbb{R}^{N}$ it holds that:

$\langle a\otimes b\ v,w\rangle=\langle b,v\rangle\langle a,w\rangle.$

Moreover, we denote with $I$ the identity matrix of order $N$.

Let $H$ be a function belonging to $C^{2}(\mathbb{R}^{N}\setminus\\{0\\})$.
$H$ is said a “ _Finsler norm_ ” if it satisfies the following assumptions:

  * $(h_{H})$

    1. (i)

$H(\xi)>0\quad\forall\xi\in\mathbb{R}^{N}\setminus\\{0\\}$;

    2. (ii)

$H(s\xi)=|s|H(\xi)\quad\forall\xi\in\mathbb{R}^{N}\setminus\\{0\\},\,\forall
s\in\mathbb{R}$;

    3. (iii)

$H$ is _uniformly elliptic_ , that means the set
$\mathcal{B}_{1}^{H}:=\\{\xi\in\mathbb{R}^{N}:H(\xi)<1\\}$ is _uniformly
convex_

(2.8)
$\exists\Lambda>0:\quad\langle\hbox{D}^{2}H(\xi)v,v\rangle\geq\Lambda|v|^{2}\quad\forall\xi\in\partial\mathcal{B}_{1}^{H},\;\forall
v\in\nabla H(\xi)^{\bot}.$

Now we provide some remarks on Finsler norms.

######  Remark 2.1.

A set is said uniformly convex if the principal curvatures of its boundary are
all strictly positive. We point out that these kind of sets are also called “
_strongly convex_ ”, which is stronger a condition than the usual “ _strict
convexity_ ”. Moreover, assumption (iii) is equivalent to assume that
$\hbox{D}^{2}(H^{2})$ is definite positive.

######  Remark 2.2.

Actually, in literature the definition of “ _Finsler norm_ ” is not unique. In
differential geometry, it is usual to define this class of norms satisfying
directly the assumption that $\hbox{D}^{2}(H^{2})$ is definite positive. The
last one allows to define the fundamental tensor, that is a very useful tool
in order to show suitable properties of geometrical objects like curvatures,
distances and so on. However, in many framework, it is enough to require that
$\mathcal{B}_{1}^{H}$ is only strictly convex, and not strongly convex.
Therefore, it is usual to give a definition of Finsler norm without the
requirement (iii) (see for instance [5, 6, 18]), but with the only assumption
that $H^{2}$ has to be strictly convex. For example, for $p\neq 2$, the
$p$-norm in $\mathbb{R}^{N}$ is not uniformly elliptic, because its unit ball
$\mathcal{B}_{1}^{H}$ is not strongly convex but only strictly convex. For our
computations we need the stronger condition (iii) and we choose a particular
class of Finsler norms satisfying it.

Since $H$ a norm in $\mathbb{R}^{N}$, we immediately get that there exists
$c_{1},c_{2}>0$ such that:

(2.9) $c_{1}|\xi|\leq H(\xi)\leq
c_{2}|\xi|,\,\quad\forall\,\xi\in\mathbb{R}^{N}.$

The _dual norm_ $H^{\circ}:\mathbb{R}^{N}\to[0,+\infty)$ is defined as:

$H^{\circ}(x)=\sup\\{\langle\xi,x\rangle:H(\xi)\leq 1\\}.$

It is easy to prove that $H^{\circ}$ is also a Finsler norm and it has the
same regularity properties of $H$. Moreover it follows that
$(H^{\circ})^{\circ}=H$. For $r>0$ and $\overline{x}\in\mathbb{R}^{N}$ we
define:

$\mathcal{B}^{H}_{r}(\overline{x})=\\{x\in\mathbb{R}^{N}:H(x-\overline{x})\leq
r\\}$

and

$\mathcal{B}^{H^{\circ}}_{r}(\overline{x})=\\{x\in\mathbb{R}^{N}:H^{\circ}(x-\overline{x})\leq
r\\}.$

For simplicity, when $\overline{x}=0$, we set:
$\mathcal{B}^{H}_{r}=\mathcal{B}^{H}_{r}(0),\,\mathcal{B}^{H^{\circ}}_{r}=\mathcal{B}^{H^{\circ}}_{r}(0)$.
In literature $\mathcal{B}^{H}_{r}$ and $\mathcal{B}^{H^{\circ}}_{r}$ are also
called “Wulff shape” and “Frank diagram” respectively.

We recall that, since $H$ is a differentiable and $1$-homogeneous function, it
holds the Euler’s characterization result, i.e.

(2.10) $\langle\nabla
H(\xi),\xi\rangle=H(\xi)\qquad\forall\,x\in\mathbb{R}^{N},$

and

(2.11) $\nabla
H(t\xi)=\hbox{sign}(t)H(\xi)\qquad\forall\,\xi\in\mathbb{R}^{N},\,\forall
t\in\mathbb{R},$

and the same is true for $H^{\circ}$. Moreover, there holds following
identities:

(2.12) $H(\nabla H^{\circ}(x))=1=H^{\circ}(\nabla H(x))$

and

(2.13) $H(x)\nabla H^{\circ}(\nabla H(x))=x=H^{\circ}(x)\nabla H(\nabla
H^{\circ}(x)),$

and we refer the reader to [6] for a complete proof. For more details on
Finsler geometry see for instance [4, 6, 20].

As pointed out in Remark 1.1, a part of our construction hold for generic
Finsler norms, but the main results are proved in the Riemannian case
($h_{M}$). In this case

$H^{\circ}(\xi)=\sqrt{\langle M^{-1}\xi,\xi\rangle}.$

We refer to [12, 13, 18] for a discussion on the geometry of the Riemannian
case and for interesting related results.

##  3\. Anisotropic Kelvin transform

The aim of this section is to prove Theorem 1.2. In order to achieve our
claim, we introduce the formal definition of Kelvin transformation in the
context of Finsler geometry.

######  Definition 3.1.

Let $H$ be a Finsler norm. We define the map
$T_{H}:\mathbb{R}^{N}\setminus\\{0\\}\rightarrow\mathbb{R}^{N}\setminus\\{0\\}$
as

(3.14) $T_{H}(\xi)=\frac{\nabla H(\xi)}{H(\xi)}.$

Note that, when $H(\xi)=|\xi|$, $T_{H}$ coincides with the usual Euclidean
Kelvin transform. Hence, we observe that it holds the following result.

######  Proposition 3.2.

The map $T_{H}$ is a global diffeomorphism with inverse given by
$T_{H^{\circ}}$.

###### Proof.

We show that, if $y=T_{H}(x)$, then $x=T_{H^{\circ}}(y)$. By (2.11),(2.12) and
(2.13) we have:

(3.15) $\begin{split}T_{H}(T_{H^{\circ}}(y))&=\nabla H\left(\frac{\nabla
H^{\circ}(y)}{H^{\circ}(y)}\right)H^{-1}\left(\frac{\nabla
H^{\circ}(y)}{H^{\circ}(y)}\right)\\\ &=\nabla H(\nabla
H^{\circ}(y))H^{\circ}(y)H^{-1}(\nabla H^{\circ}(y))\\\ &=H^{\circ}(y)\nabla
H(\nabla H^{\circ}(y)))=y\,.\end{split}$

∎

Now, we denote by $\hbox{D}T_{H}$ the Jacobian matrix of $T_{H}$, namely:

$\hbox{D}T_{H}=\frac{1}{H^{2}}\left[H\hbox{D}^{2}H-\nabla H\otimes\nabla
H\right]=\frac{1}{H^{2}}\left[\hbox{D}\left(\frac{1}{2}\nabla(H^{2})\right)-2\nabla
H\otimes\nabla H\right]\,.$

In particular, if $H$ satisfies ($h_{M}$), we have

(3.16) $\hbox{D}T_{H}(x)=\frac{1}{H(x)^{2}}\left[M-\frac{2}{H(x)^{2}}Mx\otimes
Mx\right]\,.$

Moreover, we set $J(x)=|\det\hbox{D}T_{H}(x)|$. Using these notations, we are
ready to prove the following key lemma.

######  Lemma 3.3.

If $H$ is a Finsler norm that satisfy ($h_{M}$), then

(3.17) $J(x)=\frac{\det{M}}{H(x)^{2N}}\quad\forall
x\in\mathbb{R}^{N}\setminus\\{0\\}.$

###### Proof.

Let $L$ be a diagonal matrix with entries $d_{1},\ldots,d_{N}$ and denote by
$\sqrt{L}$ the diagonal matrix whose entries are
$\sqrt{d_{1}},\ldots,\sqrt{d_{N}}$. Suppose $H(x)=\sqrt{\langle Lx,x\rangle}$.
In view of (3.16) we have:

(3.18)
$\begin{split}\hbox{D}T_{H}(x)&=\frac{1}{H(x)^{2}}\left(L-\frac{2}{\langle
Lx,x\rangle}Lx\otimes Lx\right)=\\\
&=\frac{1}{H(x)^{2}}\sqrt{L}\left(I-\frac{2}{\langle\sqrt{L}x,\sqrt{L}x\rangle}\sqrt{L}x\otimes\sqrt{L}x\right)\sqrt{L}\end{split}$

Since

(3.19) $\det\left(I-\frac{2}{|y|^{2}}y\otimes y\right)=1,\quad\forall
y\in\mathbb{R}^{N}\setminus\\{0\\},$

we get (3.17) in the case $L$ diagonal. Let $H$ be of the form ($h_{M}$). The
spectral theorem ensures that it holds the decomposition $C^{t}MC=L$, where
$C$ and $D$ are respectively an orthogonal and diagonal matrices. Having in
mind this fact, we have

(3.20)
$C^{t}\hbox{D}T_{H}(x)C=\frac{1}{H(x)^{2}}\sqrt{L}\left(I-\frac{2}{\langle\sqrt{L}C^{t}x,\sqrt{L}C^{t}x\rangle}\sqrt{L}C^{t}x\otimes\sqrt{L}C^{t}x\right)\sqrt{L}.$

By (3.19) we get the thesis. ∎

Assumption ($h_{M}$) is crucial in the previous lemma, and this result is
essential also in the proof of Theorem 1.2 and Theorem 1.4. Indeed, our
approach doesn’t work without Lemma 3.3, as pointed out in the next remark.

######  Remark 3.4.

In the proof of Theorem 1.2 it will be crucial the fact that $H(x)^{2N}J(x)$
is constant, see (3.17). For a generical Finsler norm this property is not
true. For example, if we consider the norm

$H(x)=\sqrt[4]{x_{1}^{4}+3x_{1}^{2}x_{2}^{2}+x_{2}^{4}}$

for $x=(x_{1},x_{2})\in\mathbb{R}^{2}$, then it follows that $H(x)^{2N}J(x)$
is not constant, even if $H$ satisfies all the conditions (i)-(ii)-(iii)
stated above in the definition of a Finsler norm.

Now, in the same spirit of the seminal paper of Caffarelli, Gidas and Spruck
[7], given a function $u(x)$, we define the action of the Kelvin transform on
$u$ as follows

(3.21) $\hat{u}(y):=\frac{1}{H(y)^{N-2}}u(T_{H}(y)),$

for every $y\in\mathbb{R}^{N}\setminus\\{0\\}$.

######  Remark 3.5.

It is easy to check that

(3.22) $u(x):=\frac{1}{H^{\circ}(x)^{N-2}}\hat{u}(T_{H^{\circ}}(x)),$

for every $x\in\mathbb{R}^{N}\setminus\\{0\\}$. In fact, using (2.12), by
(3.21) we have:

(3.23)
$\begin{split}u(x)&=\hat{u}\left(T_{H^{\circ}}(x)\right)H\left(T_{H^{\circ}}(x)\right)^{N-2}=\hat{u}\left(T_{H^{\circ}}(x)\right)H\left(\frac{\nabla
H^{\circ}(x)}{H^{\circ}(x)}\right)^{N-2}\\\
&=\hat{u}\left(T_{H^{\circ}}(x)\right)\frac{1}{H^{\circ}(x)^{N-2}}H(\nabla
H^{\circ}(x))^{N-2}=\frac{\hat{u}\left(T_{H^{\circ}}(x)\right)}{H^{\circ}(x)^{N-2}}\,.\end{split}$

Now we are ready to prove the first result of our paper.

###### Proof of Theorem 1.2.

We use $\psi\in C^{1}_{c}(\mathbb{R}^{N}\setminus\\{0\\})$ as test function in
(1.3). Then $\varphi(x):=\psi(T_{H^{\circ}}(x))$ can also be used as test
function in (1.3), obtaining:

(3.24) $\begin{split}\int_{\mathbb{R}^{N}}H(\nabla u(x))\langle\nabla H(\nabla
u(x)),\hbox{D}T_{H^{\circ}}(x)\nabla\psi(T_{H^{\circ}}(x))\rangle\,&\,\text{d}x\\\
=\int_{\mathbb{R}^{N}}f(x)\psi(T_{H^{\circ}}(x))\,&\,\text{d}x.\end{split}$

Using the change of variable $y=T_{H^{\circ}}(x)$, (3.24) becomes

(3.25) $\begin{split}\int_{\mathbb{R}^{N}}H(\nabla
u(T_{H}(y)))\langle\hbox{D}T_{H^{\circ}}(T_{H}(y))\nabla H(\nabla
u(T_{H}(y))),\nabla\psi(y)\rangle J(y)\,&\,\text{d}y\,\\\
=\int_{\mathbb{R}^{N}}f(T_{H}(y))\psi(y)J(y)\,&\,\text{d}y.\end{split}$

Under our assumption ($h_{M}$) we deduce that

(3.26) $\begin{split}H(\nabla
u(T_{H}(y)))&\hbox{D}T_{H^{\circ}}(T_{H}(y))\nabla H(\nabla u(T_{H}(y)))\\\
&=H(y)^{4}H^{\circ}(\hbox{D}T_{H}(y)\nabla u(T_{H}(y)))\nabla
H^{\circ}(\hbox{D}T_{H}(y)\nabla u(T_{H}(y))).\end{split}$

If we set $u^{*}(y):=u(T_{H}(y))$ and $f^{*}(y):=f(T_{H}(y))$, by using (3.26)
and Lemma 3.3, (3.25) becomes:

(3.27)
$\begin{split}\int_{\mathbb{R}^{N}}\frac{1}{H(y)^{2N-4}}H^{\circ}(\nabla
u^{*}(y))\langle\nabla H^{\circ}(\nabla
u^{*}(y)),\nabla\psi(y)\rangle\,&\,\text{d}y\\\
=\int_{\mathbb{R}^{N}}\frac{1}{H(y)^{2N}}f^{*}(y)\psi(y)\,&\,\text{d}y.\end{split}$

Let $\phi\in C^{1}_{c}(\mathbb{R}^{N}\setminus\\{0\\})$ be such that
$\psi(y)=H(y)^{N-2}\phi(y)$. Then $\psi$ can be used as test function in
(3.27), getting:

(3.28)
$\begin{split}&\int_{\mathbb{R}^{N}}\frac{1}{H(y)^{N-2}}H^{\circ}(\nabla
u^{*}(y))\langle\nabla H^{\circ}(\nabla u^{*}(y)),\nabla\phi(y)\rangle\\\
&\quad-\int_{\mathbb{R}^{N}}H^{\circ}(\nabla u^{*}(y))\langle\nabla
H^{\circ}(\nabla
u^{*}(y)),\nabla\left(\frac{1}{H(y)^{N-2}}\right)\rangle\phi(y)\,\,\text{d}x\\\
&=\int_{\mathbb{R}^{N}}\frac{1}{H(y)^{N+2}}f^{*}(y)\phi(y)\,\,\text{d}y\end{split}$

Recalling (2.11), the fact that $H$ is a $1$-homogeneous function and that

$\nabla\hat{u}=\nabla\left(\frac{1}{H^{N-2}}\right)u^{*}+\frac{1}{H^{N-2}}\nabla
u^{*},$

we get

(3.29)
$\begin{split}&\int_{\mathbb{R}^{N}}H^{\circ}\left(\nabla\hat{u}(y)-\nabla\left(\frac{1}{H(y)^{N-2}}\right)u^{*}\right)\times\\\
&\qquad\times\langle\nabla
H^{\circ}\left(\nabla\hat{u}(y)-\nabla\left(\frac{1}{H(y)^{N-2}}\right)u^{*}\right),\nabla\phi(y)\rangle\,\,\text{d}y\\\
&\quad-\int_{\mathbb{R}^{N}}H^{\circ}(\nabla u^{*}(y))\langle\nabla
H^{\circ}(\nabla
u^{*}(y)),\nabla\left(\frac{1}{H(y)^{N-2}}\right)\rangle\phi(y)\,\,\text{d}y\\\
&=\int_{\mathbb{R}^{N}}\frac{1}{H(y)^{N+2}}f^{*}(y)\phi(y)\,\,\text{d}y\end{split}$

By ($h_{M}$), we deduce that $H^{\circ}\nabla H^{\circ}$ is linear and
symmetric with respect to the Euclidean inner product of $\mathbb{R}^{N}$,
i.e.

$\langle H^{\circ}(x)\nabla H^{\circ}(x),y\rangle=\langle x,H^{\circ}(y)\nabla
H^{\circ}(y)\rangle\quad\forall x,y\in\mathbb{R}^{N}.$

Hence, we have

(3.30)
$\begin{split}&\int_{\mathbb{R}^{N}}H^{\circ}(\nabla\hat{u}(y))\langle\nabla
H^{\circ}(\nabla\hat{u}(y)),\nabla\phi(y)\rangle\,\,\text{d}y\\\
&\;-\int_{R^{N}}H^{\circ}\left(\nabla\left(\frac{1}{H(y)^{N-2}}\right)\right)\langle\nabla
H^{\circ}\left(\nabla\left(\frac{1}{H(y)^{N-2}}\right)\right),\nabla(u^{*}(y)\phi(y))\rangle\,\,\text{d}y\,\\\
&=\int_{R^{N}}\frac{1}{H(y)^{N+2}}f^{*}(y)\phi(y)\,\,\text{d}y.\end{split}$

Integrating by parts (3.30), we get

(3.31)
$\begin{split}&\int_{\mathbb{R}^{N}}H^{\circ}(\nabla\hat{u}(y))\langle\nabla
H^{\circ}(\nabla\hat{u}(y)),\nabla\phi(y)\rangle\,\,\text{d}y\\\
&\quad+\int_{\mathbb{R}^{N}}\operatorname{div}\left(H^{\circ}\left(\nabla\left(\frac{1}{H(y)^{N-2}}\right)\right)\nabla
H^{\circ}\left(\nabla\frac{1}{H(y)^{N-2}}\right)\right)u^{*}(y)\phi(y)\,\,\text{d}y\\\
&=\int_{\mathbb{R}^{N}}\frac{1}{H(y)^{N+2}}f^{*}(y)\phi(y)\,\,\text{d}y.\end{split}$

Since $w(x):=H(x)^{2-N}$ is an $H^{\circ}$-harmonic function, namely
$\Delta^{H^{\circ}}w=0$, we get the thesis.

∎

To conclude, with the same spirit of the previous proof, we prove our last
result.

###### Proof of Theorem 1.4.

Let us consider $\psi\in C^{1}_{c}(\mathbb{R}^{N}\setminus\\{0\\})$ and we set
$\varphi:=\psi(T_{H^{\circ}})$. We note that $\varphi$ is a good test function
and substituting in (1.5), we get

(3.32) $\begin{split}\int_{\mathbb{R}^{N}}H(\nabla u(x))^{N-1}\langle\nabla
H(\nabla
u(x)),\hbox{D}T_{H^{\circ}}(x)\nabla\psi(T_{H^{\circ}}(x))\rangle\,&\,\text{d}x\\\
=\int_{\mathbb{R}^{N}}g(x)\psi(T_{H^{\circ}}(x))\,&\,\text{d}x.\end{split}$

We set $y=T_{H^{\circ}}(x)$, and using this variable change (3.32) becomes

(3.33) $\begin{split}\int_{\mathbb{R}^{N}}H(\nabla
u(T_{H}(y)))^{N-1}\langle\hbox{D}T_{H^{\circ}}(T_{H}(y))\nabla H(\nabla
u(T_{H}(y))),\nabla\psi(y)\rangle J(y)\,&\,\text{d}y\,\\\
=\int_{\mathbb{R}^{N}}g(T_{H}(y))\psi(y)J(y)\,&\,\text{d}y.\end{split}$

If we set $g^{*}(y):=g(T_{H}(y))$, and recalling (3.26) and Lemma 3.3, (3.33)
becomes:

(3.34) $\begin{split}\int_{\mathbb{R}^{N}}\frac{H^{N-2}(\nabla
u(T_{H}(y)))}{H(y)^{2N-4}}H^{\circ}(\nabla u^{*}(y))\langle\nabla
H^{\circ}(\nabla u^{*}(y)),\nabla\psi(y)\rangle\,&\,\text{d}y\\\
=\int_{\mathbb{R}^{N}}\frac{g^{*}(y)\psi(y)}{H(y)^{2N}}\,&\,\text{d}y.\end{split}$

Under assumption ($h_{M}$), it is easy to check

(3.35)
$H^{\circ}(DT_{H}(y)\xi)=\frac{H(\xi)}{H^{2}(y)}\quad\forall\xi\in\mathbb{R}^{N}.$

By (3.35) we get the thesis. ∎

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# Strategy-Proof Auctions through Conformal Prediction

Roy Maor Lotan
Department of Electrical and Computer Engineering
Technion - Israel Institute of Technology
Haifa, Israel
<EMAIL_ADDRESS>
&Inbal Talgam-Cohen
Department of Computer Science
Technion - Israel Institute of Technology
Haifa, Israel
<EMAIL_ADDRESS>
&Yaniv Romano
Department of Electrical and Computer Engineering
Department of Computer Science
Technion - Israel Institute of Technology
Haifa, Israel
<EMAIL_ADDRESS>
Correspondence Author

###### Abstract

Auctions are key for maximizing sellers’ revenue and ensuring truthful bidding
among buyers. Recently, an approach known as _differentiable economics_ based
on deep learning shows promise in learning optimal auction mechanisms for
multiple items and participants. However, this approach has no guarantee of
_strategy-proofness_ at test time. Strategy-proofness is crucial as it ensures
that buyers are incentivized to bid their true valuations, leading to optimal
and fair auction outcomes without the risk of manipulation. Building upon
_conformal prediction_ , we introduce a novel approach to achieve strategy-
proofness with rigorous statistical guarantees. The key novelties of our
method are: (i) the formulation of a regret prediction model, used to
quantify—at test time—violations of strategy-proofness; and (ii) an _auction
acceptance rule_ that leverages the predicted regret to ensure that for a new
auction, the data-driven mechanism meets the strategy-proofness requirement
with high probability (e.g., 99%). Numerical experiments demonstrate the
necessity for rigorous guarantees, the validity of our theoretical results,
and the applicability of our proposed method.

##  1 Introduction

Auction design represents a fundamental component of economic theory, holding
significant practical relevance in various industries and the public sector
for organizing the sale of products and services (Milgrom, 2004; Krishna,
2009). Notable examples include sponsored search auctions by search engines
such as Google, and auctions on platforms such as eBay. Within the standard
_independent private valuations_ model, each bidder possesses a _valuation
function_ —the true values they are willing to pay—over subsets of items,
independently drawn from possibly distinct distributions. The auctioneer is
presumed to have knowledge of these value distributions, a crucial factor in
auction design. However, a primary challenge arises from the private nature of
valuations, as bidders might not disclose their true valuations, complicating
the auction process.

The pursuit of optimal auction mechanisms has been a subject of great interest
to economics for many years. Myerson auction’s (Myerson, 1981) stands out as
the optimal mechanism for auctioning a single item to several bidders. In
contexts where multiple items are auctioned to a single bidder, the concept of
optimality has been largely delineated (Manelli and Vincent, 2006; Pavlov,
2011; Daskalakis et al., 2015; Kash and Frongillo, 2016). However, when it
comes to auctioning multiple items to various bidders, optimal mechanisms have
been identified only in simplified scenarios, such as those discussed by Yao
(2017). Indeed, the existence of an optimal mechanism in the multi-bidder
multi-items setting remains an open question even in simple scenario such as
two or more bidders and several items.

The predictive power of deep neural network (DNN) models, coupled with the
quest for analytical outcomes, drive the research by Dütting et al. (2019) to
a data-driven solution (see also (Dütting et al., 2024)). While this data-
driven approach is not guaranteed to maximize revenue or be entirely strategy-
proof, empirical evidence shows that the resulting deep learning solutions are
close to recapturing the outcome of well-known optimal auction design, when
exist. Although the DNN model approximations of Dütting et al. (2019) for
optimal auction mechanisms aren’t guaranteed to maximize revenue or be
entirely strategy-proof, they come close to recapturing the optimal mechanisms
of established auctions.

###  1.1 The Lack of Strategy-proofness in Data-Driven Auctions

Strategy-proofness in essence relates closely to the concept of regret in
auctions, a crucial metric that indicates whether the outcomes of a given
auction mechanism, or any other model, are strategy-proof. Specifically, a
mechanism with low regret ensures that bidders have little incentive to
deviate from their true valuations. In other words, a low-regret mechanism
encourages truthful bidding, which is a cornerstone of strategy-proofness
(Dütting et al., 2015; Milgrom, 2004; Myerson, 1981). The importance of low
regret in auctions cannot be overstated. In a practical sense, low regret
translates to higher revenue for the auctioneer and more efficient allocation
of resources among bidders. Moreover, it fosters trust in the auction
mechanism, as participants can be confident that truthful bidding is in their
best interest (Krishna, 2009).

A critical issue in deploying data-driven mechanisms for auctions,
particularly those based on DNNs, is the lack of statistical guarantees during
test time for new auctions. While these models are trained to minimize regret
on average, leading to low-regret outcomes in many cases, they occasionally
produce predictions for allocations and payments that grossly violate the low-
regret requirement in unseen scenarios. This limitation is visualized in
Figure (1), illustrating that data-driven auction mechanism (RegretNet) fails
to attain low-regret levels. While the average regret across many scenarios
tends to be low, the presence of high regret values in the tail of the
distribution is evident in the red zone in Figure 1(a). Figure 1(b)
underscores this issue further by showing an increase in the maximum regret as
the number of auctions evaluated grows, suggesting that the model’s
reliability diminishes as it encounters a wider variety of auction contexts.

(a) Distribution of actual regret for RegretNet 2x3 model across 10,000
auctions.

(b) Maximum regret value obtained by RegretNet 2x3 as function of the number
of auctions.

Figure 1: The left and right panels illustrate the variability in regret
outcomes of the RegretNet 2x3 model (Dütting et al., 2019). While the average
regret remains generally low, the model outcome occasional exceeding a
reasonable regret level (e.g, 0.03), highlighting the challenges in
maintaining consistent performance across various auctions. For additional
insights into the experimental configuration of this model, please refer to
Section 4.

The discussion above reveals a significant limitation of data-driven auction
design: there is no assurance that the regret will remain low with any level
of confidence once the model is deployed in real-world scenarios. While the
model may perform well on historical data or in controlled settings, its
performance in novel or dynamic auction environments is uncertain. This
undesired behaviour could result in substantial revenue losses for the
auctioneer and unfair outcomes for the bidders, undermining the trust and
efficiency of the auction system. The primary goal of this paper is to address
this challenge.

###  1.2 Related Work

Given the theoretical and practical importance of strategy-proofness, the
following lines of work tackle the strategy-proofness challenge in auctions:
data-driven methods for approximate strategy-proofness,111Approximate
strategy-proofness refers here to the estimation of low regret outcomes,
rather than to notions of $\epsilon$ equilibria. and parameterization of
allocation menus to optimize auctions.

The first line of work includes several studies that continue Dütting et al.
(2015) in exploring data-driven methods to achieve strategy-proofness in
auctions, notably Curry et al. (2020), Peri et al. (2021), and Rahme et al.
(2021b, a); Ivanov et al. (2022). The downside of these innovative approaches
is that they only guarantee approximate adherence to strategy-proofness. Our
research aims to overcome this limitation by introducing a mechanism to
statistically validate and enforce strategy-proofness over new auctions—at
test time—thus enhancing reliability in auction outcomes.

Another line of work tackles the challenge differently by focusing on the
parameterization of allocation menus that are guaranteed to be strategy-proof.
This area was recently advanced by Curry et al. (2023) and Duan et al. (2023),
who optimize the parameters of affine maximizer auctions (AMA) (Roberts,
1979). While AMA-based approaches align closely with strategy-proof
principles, they often achieve lower revenue compared to standard DNN-based
approaches since they cannot capture many auction formats. They require
sophisticated architecture, and for the most part they suffer from scalability
issues, as discussed by Duan et al. (2023), indicating a trade-off with
strategy-proofness.

In contrast to these existing methodologies, our approach is highly adaptable
to a variety of auction formats and environments. Moreover, our method is
designed to be model-agnostic, allowing it to be applied seamlessly with any
black-box model.

###  1.3 Our Contributions

We take a different approach to enhance the reliability of data-driven auction
mechanisms without altering the underlying auction model. In essence, we
bridge the disciplines of conformal prediction and auction design by adopting
the conformal prediction method to rigorously evaluate and ensure the
strategy-proofness of data-driven auction mechanisms. Our technique integrates
a predictive model that estimates future regret levels in auction outcomes,
which are unknown at test time. Armed with such a regret prediction model, we
offer a new approach to maintain low-regret levels with the following
properties:

  * •

Statistical Guarantees: We provide statistical guarantees for controlling the
level of regret at test time. Our approach ensures that, for any desired
confidence level (e.g., 99%), the maximum regret over new auctions will not
exceed a specified threshold (e.g., 0.05). This contribution addresses a
critical gap in existing literature by offering a quantifiable measure of
reliability for data-driven auction mechanisms. Furthermore, unlike prior
methods that offer only approximate assurances of strategy-proofness, our work
introduces precise statistical guarantees at test time, providing a level of
certainty and robustness previously unattainable in the field.

  * •

Black-Box Approach: Our algorithm operates as a black box, eliminating the
need to modify the baseline auction model or retrain it. We leverage the
pretrained model, preserving its original architecture and efficiency while
enhancing its reliability in new auction environments. Importantly, our
approach can be applied universally across any baseline auction model without
enforcing architectural modifications or requiring prior knowledge of the
auction’s specifics, unlike AMA-based methods which requires structural
adjustments.

Our method functions within the confines of specified statistical confidence
levels, ensuring that auctions maintain predefined low-regret levels across
different scenarios. By implementing this predictive oversight, we
significantly increase the robustness and transparency of auctions, thereby
mitigating risks associated with dynamic and unpredictable auction
environments.

##  2 Background

###  2.1 Problem Setup

We consider an auction environment comprising of a set of $n$ bidders, denoted
by $N=\\{1,\ldots,n\\}$, and a set of $m$ items, denoted by
$M=\\{1,\ldots,m\\}$. Each bidder $i\in N$ possesses a private valuation
$v_{i}\geq 0$ which maps item sets $S\subseteq M$ to their value
$v_{i}(S)\in\mathbb{R}_{\geq 0}$ for the bidder. Valuation $v_{i}$ is drawn
from a distribution $P_{i}$ over the set of possible valuations $V_{i}$.

Participants in the auction submit bids, which might not necessarily reflect
their true valuations, to the auction mechanism. The auction mechanism,
denoted by $f=(g,p)$, consists of a pair of allocation and payment rules
$g(b)$ and $p(b)$, where $b=(b_{1},b_{2},\ldots,b_{n})$ represents the bid
profile of all bidders. The mechanism aims to allocate the $m$ items to the
bidders while charging each bidder a certain payment, often with the objective
of revenue maximization.

The utility of bidder $i$, denoted by $u_{i}(v_{i};b)$, is defined as the
difference between their valuation of the allocated items and the payment
made, i.e., $u_{i}(v_{i};b)=v_{i}(g_{i}(b))-p_{i}(b)$. Here, $g_{i}$ and
$p_{i}$ represent the allocation and payment for bidder $i$ given the bid
profile $b$.

To effectively frame strategy-proof auctions as a deep learning problem, we
aim to minimize regret while considering bidder utility. An auction mechanism
$f=(g,p)$ can be parameterized as
$\hat{f}(b;\theta)=[g^{\theta}(b),p^{\theta}(b)]$, where $\theta$ represents
the trainable parameters. The utility function for bidder $i$ is then:

$u_{i}^{\theta}(v_{i};b)=v_{i}(g_{i}^{\theta}(b))-p_{i}^{\theta}(b).$ (1)

Using this utility function, a bidder’s regret is defined in the literature as
the difference in utility between bidding truthfully and bidding to maximize
utility. Let $v_{-i}$ denote the valuation profile excluding bidder $i$. Then
the regret function $\operatorname{\textbf{rgt}}$ as a function of the private
valuations $v$ is given by:

$\operatorname{\textbf{rgt}}_{i}(v)=\max_{b_{i}\in
V_{i}}u_{i}^{\theta}(v_{i};(b_{i},v_{-i}))-u_{i}^{\theta}(v_{i};(v_{i},v_{-i})).$
(2)

Denote that $\operatorname{\textbf{rgt}}$ without a specific index represents
the maximum value of $\operatorname{\textbf{rgt}}_{i}$ for all $i\in n$.

The optimization problem for finding a feasible, strategy-proof, and revenue-
maximizing auction can be formulated as minimizing the expected negated total
payment, subject to the constraints of no regret and _individual rationality_
(IR):

$\displaystyle\min_{\theta\in\mathbb{R}^{d}}$
$\displaystyle\quad\underset{v\sim P}{\mathbb{E}}\left[-\sum_{i\in
N}p_{i}^{\theta}(v)\right]$ [REVENUE] (3) s.t.
$\displaystyle\operatorname{\textbf{rgt}}_{i}(v)=0,\hskip 7.39755pt\forall
i\in N,v$ [REGRET] $\displaystyle u_{i}(v)\geq 0,\hskip 15.6491pt\forall i\in
N,v$ [IR] $\displaystyle\sum_{i\in N}g_{i,j}(v)\leq 1,\forall j\in M$
$\displaystyle\text{[FEASIBLE]}.$

###  2.2 Conformal Prediction

We now dive into the method of _conformal prediction_ , which we intend to
employ for controlling regret levels within data-driven auction mechanisms.

The method of conformal prediction (Papadopoulos et al., 2002; Vovk et al.,
2005; Lei and Wasserman, 2014), offers a generic approach, applicable to any
predictive model, for quantifying prediction uncertainty. In recent years,
conformal prediction has increasingly gained traction across various domains,
demonstrating its broad utility and effectiveness in practical settings
(Angelopoulos et al., 2023). This method leverages a subset of holdout
_calibration data_ to quantify the prediction error of the model. In turn,
this is used to construct a range of plausible predictions—a _prediction
interval_ —that covers the unknown test outcome with high probability.

In what follows, we describe how to construct prediction intervals for
regression problems, as this task is tightly connected to our goal of reliably
predicting regret values. Let $x$ represent the input features to the model,
$y$ denote the true output, and $\hat{f}(x)$ be the predicted output generated
by the model. The process of conformal prediction for general inputs and
outputs can be outlined as follows: Initially, a pre-trained model is used to
identify a heuristic notion of uncertainty, or model’s prediction error.
Subsequently, a score function denoted by $s(x,y)$ is defined to quantify the
level of uncertainty or disagreement between the predicted and actual values,
with larger scores indicating a worse agreement. The next step involves
calculating $\hat{q}$, the quantile of the calibration scores derived from
known input-output pairs, based on a specified confidence level $\alpha$. This
quantile serves as a threshold to assess the reliability of new predictions.
For new test inputs, a prediction interval is formed, which includes all
potential output values $y$ that satisfy the condition
$s(x_{\text{test}},y)\leq\hat{q}$. This interval represents the range of
plausible output values given the input and the model’s uncertainty. This
interval, denoted by $\mathcal{I}$, contains the range of outputs that are
considered plausible given the input and the model’s uncertainty.

The key property of conformal prediction is that it guarantees a certain level
of coverage. This means that the true output will fall within the prediction
set with a specified probability, regardless of the choice of score function
or the underlying distribution of the data. This provides a formal assurance
of the reliability of the predictions made by the model.

######  Theorem 2.1 (Conformal Coverage Guarantee; Vovk, Gammerman, and
Saunders 1999).

Assume that the calibration set
${\mathcal{D}_{cal}=\\{x_{i},y_{i}\\}}_{i=1}^{n}$ and the test point $(x_{\rm
test},y_{\rm test})$ are i.i.d. samples. Then, with $\hat{q}$ and
$\mathcal{I}(x_{\rm test})$ defined as above, the following coverage guarantee
holds:

$\mathbb{P}\Big{[}y_{\rm test}\in\mathcal{I}(x_{\rm test})\Big{]}\geq
1-\alpha.$ (4)

Standard conformal prediction cannot be applied “as is”; rather, the unique
nature of the auction domain should be taken into account. The reason is that
in the auction domain there is no “label”: the regret
$\operatorname{\textbf{rgt}}$ depends on both the bid $b$ and the model itself
$\hat{f}$. This necessitates a tailored adaptation of the method, where the
calibration set consists of bid-regret pairs based on the model’s performance,
and the first step is regret estimation (see Section 3). This modification is
inevitable in order to integrate conformal prediction into auctions.

##  3 Methods

Our method involves three critical components to maintain the reliability of
auction outcomes. First, we introduce a data-driven regret model designed to
estimate the unknown regret associated with a new auction (Section 3.1. Next,
we demonstrate how to calibrate the predicted regret to establish an
acceptance rule, ensuring that the auction outcome adheres to the desired
maximal regret level at any specified confidence level (Section 3.2. Lastly,
in Section 4 we apply this acceptance rule to a new test point, resulting in a
data-driven auction mechanism that operates within predefined regret bounds
with high probability, thus preserving its integrity and strategic robustness.

###  3.1 Regret Estimation

Since the regret of a new auction is unknown, we first introduce a _regret
estimation model_. In its most general form, we formulate it as a neural
network $\hat{r}(b;\theta_{r})$, parameterized by $\theta_{r}$. This model
predicts $\operatorname{\textbf{rgt}}^{\theta}(b)$—the outcome regret for a
given auction, where $\theta$ represents the parameters of the auction model
$\hat{f}$. To fit the regret model on the training set $\mathcal{D}_{train}$,
we minimize the following objective function:

$\displaystyle\hat{\theta}_{r}=\operatorname*{arg\,min}_{\theta_{r}\in\mathbb{R}^{d}}\sum_{b\in\mathcal{D}_{train}}\|r(b;\theta_{r})-\operatorname{\textbf{rgt}}^{\theta}(b)\|_{1}.$
(5)

We propose two different approaches to design the regret estimation model
$\hat{r}(b;\theta_{r})$. In the _shared backbone setup_ illustrated in Figure
2(a), the regret model is integrated into the existing allocation and payment
backbone, providing a unified structure. The advantage of this approach is
that it leverages the backbone features of the auction model to better predict
the regret. However, this setup requires access to the auction model
parameters and modifies the training procedure. In the second _black-box_
approach, the auction model is treated as a black-box, avoiding the need to
access the auction model’s internal parameters. This design choice is
visualized in Figure 2(b), where the regret estimation model is a standalone
neural network.

(a) Shared backbone regret estimation

(b) Black-box auction model regret estimation

Figure 2: Regret estimation model setups

###  3.2 Model Calibration

In this section we present a data-driven auction acceptance rule that selects
auctions with low regret outcomes. Let
$\operatorname{\ell_{\operatorname{\textbf{rgt}}}}$ be the maximal requested
regret level of new auctions. Naïvely, one might consider a straightforward
approach to accept new auctions according to the following rule:
$\hat{r}(b;\hat{\theta}_{r})\leq\operatorname{\ell_{\operatorname{\textbf{rgt}}}}$
(accept if their regret is below the maximal level). However, recognizing the
inaccuracies inherent in any prediction task, the naïve selection rule
presented above can mistakenly accept auctions whose _actual_ regret
$\operatorname{\textbf{rgt}}$ is much above the desired regret level. This
discussion emphasizes the importance of the calibration procedure presented in
this section, which rigorously bridges the gap between the estimated regret
$r(b;\hat{\theta}_{r})$ and the actual, unknown regret
$\operatorname{\textbf{rgt}}$. In essence, we will show how to obtain a
corrected threshold on the output of $r(b;\hat{\theta}_{r})$, ensuring the
regret of the auction outcome obtained from the auction model does not exceed
the requested maximal regret level
$\operatorname{\ell_{\operatorname{\textbf{rgt}}}}$ with high probability
$1-\alpha$.

To achieve such a calibrated threshold, we employ conformal prediction.
However, it is important to emphasize that conformal prediction can not be
applied in a straightforward manner due to the distinctive characteristics of
auction settings. Here, the “label”—the predicted regret—that is required to
calibrate the regret model is a function of both the bid and the predictive
auction model. Consequently, this necessitates a customized adaptation of the
conformal prediction method, involving a calibration set composed of pairs of
bids and their associated regrets as determined by the model’s outcomes.

We now turn to formulate the _Auction Acceptance Rule_ function. Given the
auction model $\hat{f}$, a requested regret level
$\operatorname{\ell_{\operatorname{\textbf{rgt}}}}$, and a control level
$\alpha$, we define the auction acceptance rule function
$\mathcal{T}_{\alpha}(b,\operatorname{\ell_{\operatorname{\textbf{rgt}}}})$
for a bid vector $b$ as follows:

$\mathcal{T}_{\alpha}(b,\operatorname{\ell_{\operatorname{\textbf{rgt}}}})=\begin{cases}\hat{f}({b}),&\hat{r}<\operatorname{\ell_{\operatorname{\textbf{rgt}}}}-q_{\alpha},\\\
\;\;\;\emptyset,&\text{otherwise}.\end{cases}$ (6)

The function is determined by the estimated regret $\hat{r}$ and the
calibration threshold $q_{\alpha}$, which is derived from the calibration set
$\mathcal{D}_{cal}$ at the confidence level $\alpha$. The output of the
function
$\mathcal{T}_{\alpha}(b,\operatorname{\ell_{\operatorname{\textbf{rgt}}}})$ is
the prediction of the auction model $\hat{f}({b})$ if the estimated regret
$\hat{r}$ is less than the difference between the requested regret level
$\operatorname{\ell_{\operatorname{\textbf{rgt}}}}$, and the correction value
$q_{\alpha}$ derived from our calibration method. Otherwise, the function
$\mathcal{T}_{\alpha}$ yields an empty set $\emptyset$, indicating that the
predicted auction does not meet the desired regret level. This acceptance rule
ensures that the auction mechanism only operates within the predefined regret
bounds.

Algorithm 1 details the calibration process to align estimated regrets with
actual outcomes, ensuring auction acceptance aligns with the auctioneer’s risk
tolerance. The algorithm starts by computing
$e_{i}=\operatorname{\textbf{rgt}}(b_{i})-\hat{r}(b_{i})$: the difference
between the actual and the estimated regrets for each bid vector $b_{i}$ of
the calibration dataset. Then, we compute the empirical quantile $q_{\alpha}$
of these scores, at the desired confidence level $\alpha$. This quantile forms
our threshold, which is used to set the corrected cutoff in the acceptance
rule $\mathcal{T}_{\alpha}$.

Algorithm 1 Regret Calibration

1: Input: $\hat{f}$ \- ML model,
$\operatorname{\ell_{\operatorname{\textbf{rgt}}}}$ \- requested regret level,
$\alpha$ \- requested control level. $\mathcal{D}_{cal}$ \- previously unseen
calibration set.

2: Output: $\mathcal{T}_{\alpha}$ \- acceptance rule function.

3: Initialize an empty list $R$ to store residual errors.

4: for each bid vector $b_{i}$ in $\mathcal{D}_{cal}$ do

5: Compute the estimated regret $\hat{r}_{i}$ using the ML model $\hat{f}$ for
$b_{i}$.

6: Compute the actual regret $r_{i}$ for $b_{i}$.

7: Calculate the residual error $e_{i}=r_{i}-\hat{r}_{i}$ and append it to
$R$.

8: end for

9: Sort the list $R$ in ascending order.

10: Determine the index
$k_{\alpha}=\lceil|\mathcal{D}_{cal}|\cdot(n+1)(1-\alpha)/{n}\rceil$ to find
the $1-\alpha$ quantile $q_{\alpha}$.

11: Set the calibration threshold $q_{\alpha}=R[k_{\alpha}]$.

12: Define the acceptance rule function
$\mathcal{T}_{\alpha}(b,\operatorname{\ell_{\operatorname{\textbf{rgt}}}})$ as
per Equation 6 using $q_{\alpha}$.

13: return $\mathcal{T}_{\alpha}$.

Following Equation (6), the $q_{\alpha}$ adjusts the acceptance of auctions by
taking into account the prediction error of the regret model $\hat{r}$.
Ideally, if the regret model were perfect such that
$\hat{r}(b)=\operatorname{\textbf{rgt}}(b)$, we would get $q_{\alpha}=0$.
However, since perfect predictions are unattainable, $q_{\alpha}$ rigorously
compensates for the regret model’s prediction inaccuracies.

The procedure we present is formalized by the following theorem, which
confirms the validity of our acceptance rule $\mathcal{T}_{\alpha}$:

######  Theorem 3.1.

Suppose the calibration set $\mathcal{D}_{cal}$ and a new auction (test point)
$b$ are i.i.d. Let $\mathcal{T}_{\alpha}$ be the acceptance rule in Equation
(6) of a given auction model $\hat{f}$, obtained from Algorithm 1. Then, the
probability that the regret of an auction accepted by $\mathcal{T}_{\alpha}$
exceeds level $\operatorname{\ell_{\operatorname{\textbf{rgt}}}}$ is at most
$\alpha$. Formally:

$\mathbb{P}\left[(\operatorname{\textbf{rgt}}(b)>\operatorname{\ell_{\operatorname{\textbf{rgt}}}})\
\textup{and}\
(\mathcal{T}_{\alpha}(b,\operatorname{\ell_{\operatorname{\textbf{rgt}}}})\neq\emptyset)\right]\leq\alpha.$
(7)

###### Proof.

Denote by $r$ the actual regret $\operatorname{\textbf{rgt}}(b)$ of
$\hat{f}(b)$ and by $\hat{r}$ the predicted regret $\hat{r}(b)$. With these
notations, we can express the probability of interest as follows:

$\displaystyle\mathbb{P}\left[(r>\operatorname{\ell_{\operatorname{\textbf{rgt}}}})\
\textup{and}\
(\mathcal{T}_{\alpha}(b,\operatorname{\ell_{\operatorname{\textbf{rgt}}}})\neq\emptyset)\right]=$
$\displaystyle\mathbb{P}\left[(r>\operatorname{\ell_{\operatorname{\textbf{rgt}}}})\>\cap\>(\mathcal{T}_{\alpha}(b,\operatorname{\ell_{\operatorname{\textbf{rgt}}}})\neq\emptyset)\right]$
$\displaystyle=$
$\displaystyle\mathbb{P}\left[(r>\operatorname{\ell_{\operatorname{\textbf{rgt}}}})\>\cap\>(\hat{r}<\operatorname{\ell_{\operatorname{\textbf{rgt}}}}-q_{\alpha})\right]$
\\\ by definition of $\mathcal{T}_{\alpha}$ $\displaystyle=$
$\displaystyle\mathbb{P}\left[(r>\operatorname{\ell_{\operatorname{\textbf{rgt}}}})\>\cap\>(\operatorname{\ell_{\operatorname{\textbf{rgt}}}}>\hat{r}+q_{\alpha})\right]$
$\displaystyle=$ $\displaystyle\mathbb{P}\left[r>\hat{r}+q_{\alpha}\right]$
\\\ intersecting events $\displaystyle=$
$\displaystyle\mathbb{P}\left[r-\hat{r}>q_{\alpha}\right]$ $\displaystyle=$
$\displaystyle 1-\mathbb{P}\left[r-\hat{r}<q_{\alpha}\right]$ \\\ complement
event $\displaystyle\leq$ $\displaystyle 1-(1-\alpha)$ \\\ by Theorem 2.1
$\displaystyle=$ $\displaystyle\alpha.$

∎

The theorem establishes that the probability of the actual regret exceeding a
specified threshold, while the auction outcome is accepted by the acceptance
rule function, is strictly limited by the control level $\alpha$. For
instance, if we set $\alpha=0.01$ and request a regret level of
$\operatorname{\ell_{\operatorname{\textbf{rgt}}}}=0.025$, then the actual
regret of auctions accepted by $\mathcal{T}_{\alpha}$ will not exceed 0.025
with probability 99%.

##  4 Experiments

In this section, we empirically evaluate our proposal and focus on auctions
with configurations of (bidders $\times$ items) sets of size $2\times 2$,
$2\times 3$, and $3\times 5$. The training process is based on the RegretNet
implementation by Dütting et al. (2019) (PyTorch version by Stein et al.
(2023)), which is used as the data-driven auction model. Throughout the
experiments, we refer to our conformal method as ConfRegretNet. That is, we
apply the acceptance rule $\mathcal{T}_{\alpha}$ to the output of the auction
model $\hat{f}(b)=\left[\hat{g}(b),\hat{p}(b)\right]$. We fit our regret
estimation model $\hat{r}$ within the shared backbone architecture setup,
illustrated in 2(a). For further implementation details, please refer to
Section A in the Appendix.

The performance of ConfRegretNet model is evaluated in terms of revenue,
empirical regret, and maximum regret. Table 1 summarizes the results for
different auction configurations.

Table 1: Summary of the average revenue and empirical regret, with their
standard deviations in parentheses, for both RegretNet and ConfRegretNet
models across $2\times 2$, $2\times 3$, and $3\times 5$ auction settings. The
test and the calibration sets are of size 10,000 examples. The ‘Requested Max
Regret Level‘ column indicates the requested regret used in ConfRegretNet to
control the regret at $99\%$ confidence ($\alpha=0.01$). The ‘Accepted
Auctions‘ column shows the percentage of accepted auctions by ConfRegretNet
based on the regret threshold. the ‘Max Regret‘ column refers the max regret
obtained at test time.
Auction Setting | Revenue |  Empirical Regret |  Requested Max Regret |  Max Regret |  Acceptance Auctions
---|---|---|---|---|---
RegretNet $2\times 2$ | 0.93 (0.33) | 0.007 (0.006) | - | 0.039 | -
ConfRegretNet $2\times 2$ | 0.92 (0.32) | 0.004 (0.003) | 0.025 | 0.020 | 89.23%
RegretNet $2\times 3$ | 1.41 (0.38) | 0.011 (0.014) | - | 0.164 | -
ConfRegretNet $2\times 3$ | 1.38 (0.39) | 0.007 (0.005) | 0.035 | 0.032 | 94.4%
RegretNet $3\times 5$ | 2.93 (0.40) | 0.023 (0.012) | - | 0.121 | -
ConfRegretNet $3\times 5$ | 2.67 (0.38) | 0.009 (0.007) | 0.055 | 0.055 | 93.4%

The results indicate that our auction mechanism model achieves competitive
revenue across different auction settings while maintaining the requested
levels of empirical maximum regret.

To validate the effectiveness of our approach, we further analyze the behavior
of auctions accepted and rejected by ConfRegretNet in Figure 4. Specifically,
we examine whether the auctions that are not accepted by our model are indeed
those that exceed the specified regret threshold, and whether the accepted
auctions maintain regret levels within the acceptable range. This analysis is
crucial to ensure that our model not only controls regret effectively but also
tends not to reject auctions that could have been acceptable. Figure 4
presents the precise regret versus the predicted regret. It is observed that
there is a high correlation between the two, indicating that our model has
learned to predict the regret accurately. This explains the behavior we
observe in Figure 4, showing that our method tends to reject the auctions with
high regret.

Figure 3: The distribution of regret values of the accepted and rejected
auctions. The request regret level is marked by the red line.

Figure 4: Joint distribution and histograms of the regret
$\operatorname{\textbf{rgt}}$ and the predicted regret $\hat{r}(b)$ of
ConfRegretNet $2\times 2$

To further illustrate the performance of our model, Figure 5 presents the
distribution of regret across different auction settings. The distributions of
regret for both RegretNet and ConfRegretNet are notably similar, despite
ConfRegretNet being trained with an additional regret model. This similarity
in distributions suggests that ConfRegretNet maintains the desirable
properties of the original baseline RegretNet. Notably, ConfRegretNet ensures
that all auctions remain within the specified regret level, as highlighted by
the red line.

(a) Auction setup 2x2

(b) Auction setup 2x3

(c) Auction setup 3x5

Figure 5: The distribution of regret values corresponding to RegretNet and
ConfRegretNet across different auction settings: $2\times 2$, $2\times 3$, and
$3\times 5$. The test and the calibration sets are of size 1,000 examples
each. The red line indicates the requested regret level
$\operatorname{\ell_{\operatorname{\textbf{rgt}}}}$ of each setup

To further demonstrate the flexibility of our approach, we now conduct
experiments in which we treat the original RegretNet model as a black-box (see
Table 2) and focus on training the separate regret model. The results outlined
below demonstrate the validity of our approach and our ability to perform well
even without having access to the auction model’s parameters.

Table 2: Summary of the average revenue and empirical regret, with their
standard deviations in parentheses, for both RegretNet and ConfRegretNet
$2\times 2$ models applied within the black-box settings as illustrated in
Figure 2(b). The test and the calibration sets are of size 10,000 examples.
Auction Setting | Revenue |  Empirical Regret |  Requested Max Regret |  Max Regret |  Acceptance Auctions
---|---|---|---|---|---
RegretNet $2\times 2$ | 0.93 (0.33) | 0.007 (0.006) | - | 0.039 | -
ConfRegretNet $2\times 2$ | 0.92 (0.31) | 0.004 (0.005) | 0.025 | 0.023 | 93.7%

Overall, the results validate the effectiveness of our approach in designing
auctions that balance revenue maximization with strategy-proofness, making
them suitable for various real-world applications.

##  5 Discussion

#### Limitations.

The usefulness of our rejection approach is affected by the predictive power
of the underlying regret estimation model. If this model has poor predictive
capabilities, our algorithm may make more rejections to ensure that the
maximal regret will be controlled. Future research could benefit from
exploring various learning schemes to predict the unknown regret more
effectively. Another limitation is the i.i.d. assumption we make, which might
be violated in practice. As a future research direction, it would be important
to quantify the effect of violation of this assumption on the resulting
maximal regret level, possibly by taking inspiration from Barber et al.
(2023). Additionally, an extension of the current framework to accommodate
multi-objective calibration would be valuable. For example, this approach
would enable auctioneers to simultaneously manage regret levels alongside
other auction considerations and properties.

#### Broader impact.

Our approach enhances the efficiency and reliability of data-driven auctions,
thereby benefiting sellers and buyers alike by ensuring truthful bidding and
optimal auction outcomes. However, there are potential negative societal
impacts, such as the risk of reliance on automated systems that may reinforce
existing biases. Addressing these concerns is crucial to ensure that the
deployment of such mechanisms contributes positively to society.

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##  Appendix A Architectural and Training Details

The experiments were conducted on a high-performance computing setup
consisting of an Ubuntu 20.04.6 LTS operating system, powered by 96 Intel(R)
Xeon(R) Gold CPUs running at 2.40 GHz. The machine was equipped with 16 Nvidia
A40 GPUs and 512 GB of RAM, providing ample computational resources for
training and evaluating our models. The software environment was based on
PyTorch 2.1, running under Python 3.11.5, which ensured compatibility with the
(current) latest available libraries and frameworks.

The architecture of our models includes ReLU activations for both the
allocation and payment networks, as well as the score model if exists. To
ensure that each item’s allocations sum to one, we apply a softmax activation
to the allocation network. For the payment network, a sigmoid activation
function is used to constrain the payments to a fraction between zero and one,
enforcing individual rationality (IR).

Models are trained on 700,000 sample valuation profiles $b_{i}\sim U[0,1]$,
divided into batches of 1024, using the Adam optimizer with an initial
learning rate of $1e^{-3}$. The training process spans several epochs, with
the exact number determined based on Table 3.

An additional term for regret minimization is incorporated into the training
process, similar to the augmented Lagrangian method used in RegretNet. The
weight for this regret term is periodically updated to adjust the emphasis on
regret minimization in the loss function. Furthermore, the value of the
parameter $\rho$, which influences the regret term, is also periodically
updated during training.

The agents’ regret is optimized in an inner loop, with specific parameters
detailed in the last three rows of Table 3. For a comprehensive understanding
of the loss function derivation and further details on the training procedure,
we refer the reader to the original RegretNet paper by Dütting et al. (2019).

Table 3: Hyperparameters. We provide training hyperparameters for all auction
sizes we study.
Auction setup (Bidders x Items) | 2$\times$2 | 2$\times$3 | 3$\times$5
---|---|---|---
Epochs | 10 | 30 | 20
Train Batch Size | 128 | 128 | 128
Initial Learning Rate | 0.001 | 0.001 | 0.001
Number of Hidden Layers | 5 | 5 | 5
Hidden Layer size | 100 | 100 | 100
$\rho$ Update Period (Epochs) | 2 | 2 | 2
Lagrange Weight Update Period (Iters) | 100 | 100 | 100
Initial $\rho$ | 1 | 1 | 1
$\rho$ Increment | 10 | 5 | 1
Initial Lagrange Weight | 5 | 5 | 5
Misreport Learning Rate (Training) | 0.1 | 0.1 | 0.1
Misreport Iterations (Training) | 25 | 25 | 25
Misreport Initializations (Training) | 10 | 10 | 10

##  Appendix B Additional Experiments

In this section, we provide the reader with additional experimental results
and findings of the 2x2 auction setup.

#### Performance of the regret estimation model

Our analysis investigates the relationship between predicted regret,
$\hat{r}$, and actual regret $\operatorname{\textbf{rgt}}$, to evaluate our
model’s ability to recognize its limitations. As expected, we observed a
strong correlation between these two variables despite noticeable differences
in their distributions (Figure 6). This correlation indicates that the model
effectively predicts the regret values. Figure 4 illustrates the joint
distribution of predicted and actual regret, further supporting this finding.

(a) Linear scale.

(b) Logarithmic scale.

Figure 6: The distribution of predicted regret $\hat{r}(b)$ and actual regret
$\operatorname{\textbf{rgt}}(b)$, of RegretNet model.

#### The effect of the requested max regret level on the number of rejections

As the auctioneer enforces stricter conformal regret levels, the number of
auctions rejected increases due to not meeting these stringent criteria.
Figure 7 shows a clear trend of rising auction rejections as the requested
regret level decreases. This relationship highlights the trade-off between the
regret level of the auction mechanism and its utility; a lower regret level
ensures greater strategy-proofness but results in more auctions being
excluded.

Figure 7: The relationship between the number of auctions rejected by the
model and the requested conformal regret level
$\operatorname{\ell_{\operatorname{\textbf{rgt}}}}$.