# Quantitative Korovkin theorems for sublinear, monotone and strongly
translatable operators in $L_{p}([0,1]),1\leq p\leq+\infty$

Sorin G. Gal  Department of Mathematics and Computer Science
University of Oradea
University Street No. 1, Oradea, 410087, Romania
and Academy of Romanian Scientists, Splaiul Independentei nr. 54, 050094,
Bucharest, Romania<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Constantin P.
Niculescu  Department of Mathematics, University of Craiova
Craiova 200585, Romania and The Institute of Mathematics of the Romanian
Academy, Bucharest, Romania<EMAIL_ADDRESS>
(Date: September 28, 2022)

###### Abstract.

By extending the classical quantitative approximation results for positive and
linear operators in $L^{p}([0,1]),1\leq p\leq+\infty$ of Berens and DeVore in
1978 and of Swetits and Wood in 1983 to the more general case of sublinear,
monotone and strongly translatable operators, in this paper we obtain
quantitative estimates in terms of the second order and third order moduli of
smoothness, in Korovkin type theorems. Applications to concrete examples are
included and an open question concerning interpolation theory for sublinear,
monotone and strongly translatable operators is raised.

###### Key words and phrases:

Korovkin type theorems, monotone operator, sublinear operator, weakly
nonlinear operator, $C([0,1])$-space, $L_{p}([0,1])$-space, $1\leq
p\leq+\infty$, second order modulus of smoothness, $L_{p}$-modulus of
smoothness of orders 1 and 2, quantitative estimates.

###### 2000 Mathematics Subject Classification:

41A35, 41A36, 41A63

##  1\. Introduction

Korovkin’s theorem [15], [16] provides a very simple test of convergence to
the identity for any sequence $(T_{n})_{n}$ of positive linear operators that
map $C\left([0,1]\right)$ into itself: the occurrence of this convergence for
the functions $e_{0}(x)=1,e_{1}(x)=x$ and $e_{2}(x)=x^{2}$. In other words,
the fact that

$\lim_{n\rightarrow\infty}T_{n}(f)=f\text{\quad uniformly on }[0,1]$

for every $f\in C\left([0,1]\right)$ reduces to the status of the three
aforementioned functions. Due to its simplicity and usefulness, this result
has attracted a great deal of attention leading to numerous generalizations.
Part of them are included in the authoritative monograph of Altomare and
Campiti [6] and the excellent survey of Altomare [2]. See [3], [4], [5], for
some very recent contributions.

In the case of positive and linear operators, quantitative estimates in terms
of the uniform norm and modulus of continuity for the classical Korovkin’s
theorems were obtained by Shisha and Mond in [17], and for the case of $L_{p}$
spaces with $1\leq p<\infty$ were obtained by Berens and DeVore in [7] and by
Swetits and Wood in [18].

Korovkin’s theorem was extended to the framework of sublinear and monotone
operators acting on function spaces defined on appropriate subsets $K$ of
$\mathbb{R}^{N}$ in Gal and Niculescu [8], [10], [11] and [12]. Also, a
quantitative estimate in terms of the uniform norm and uniform modulus of
continuity in the operator version of Korovkin type theorems for the case of
monotone and sublinear operators, was obtained in Gal and Niculescu [13].

The goal of the present paper is to obtain quantitative estimates in
Korovkin’s theorems in terms of uniform norm and the second order modulus of
smoothness in $C([0,1])$ and in terms of $L_{p}$ norm and second and third
$L_{p}$ moduli of smoothness, with $1\leq p\leq\infty$. Our results are based
on the extensions of some classical results for positive and linear operators
of Berens and DeVore in [7] and of Swetits and Wood in [18], to the more
general frame of sublinear, monotone and strongly translatable operators.

Section 2 contains some preliminaries on weakly nonlinear and monotone
operators. The main results for $p=1$ and $p=\infty$ are proved in Section 3
by adapting the results for positive and linear operators in Berens and DeVore
[7] to sublinear monotone and strongly translatable operators. The case
$1<p<\infty$ of Theorem 3 in Berens-DeVore [7] is based on an interpolation
technique for linear continuous operators. Therefore an extension of Theorem 3
in Berens-DeVore [7] in our new frame, would require an extension of this
technique to sublinear monotone and strongly translatable operators. But
because for the moment it seems that such a theory is not known, the
quantitative approximation results for $1<p<\infty$ are obtained in Section 4
by adapting the results for positive and linear operators in Swetits and Wood
[18], to sublinear monotone and strongly translatable operators. Section 5
presents some applications of the main results obtained to some concrete
examples. Section 6 raises as open question the extension of the interpolation
technique to sublinear monotone and strongly translatable operators.

##  2\. Weakly nonlinear operators acting on ordered Banach spaces

The goal of this section is to describe a class of nonlinear operators which
provides a convenient framework for the extension of Berens and DeVore’s
results in [7].

Given a metric space $X,$ we attach to it the vector lattice $\mathcal{F}(X)$
of all real-valued functions defined on $X$, endowed with the metric $d$ and
the pointwise ordering.

Suppose that $X$ and $Y$ are two metric spaces and $E$ and $F$ are
respectively ordered vector subspaces (or subcones of the positive cones) of
$\mathcal{F}(X)$ and $\mathcal{F}(Y)$ and that $\mathcal{F}(X)$ contains the
unity. An operator $T:E\rightarrow F$ is said to be a _weakly nonlinear_ if it
satisfies the following two conditions:

  1. (SL)

(_Sublinearity_) $T$ is subadditive and positively homogeneous, that is,

$T(f+g)\leq T(f)+T(g)\quad\text{and}\quad T(\alpha f)=\alpha T(f)$

for all $f,g$ in $E$ and $\alpha\geq 0;$

  2. (TR)

(_Translatability_) $T(f+\alpha\cdot 1)=T(f)+\alpha T(1)$ for all functions
$f\in E$ and all numbers $\alpha\geq 0.$

In the case when $T$ is _unital_ (that is, $T(1)=1)$ the condition of
translatability takes the form

$T(f+\alpha\cdot 1)=T(f)+\alpha 1,$

for all $f\in E$ and $\alpha\geq 0.$

A stronger condition than translatability is

  1. (TR∗)

(_Strong translatability_) $T(f+\alpha\cdot 1)=T(f)+\alpha T(1)$ for all
functions $f\in E$ and all numbers $\alpha\in\mathbb{R}.$

The last condition occurs naturally in the context of Choquet’s integral,
being a consequence of what is called there the property of _comonotonic
additivity_ , that is,

  1. (CA)

$T(f+g)=T(f)+T(g)$ whenever the functions $f,g\in E$ are comonotone in the
sense that

$(f(s)-f(t))\cdot(g(s)-g(t))\geq 0\text{\quad for all }s,t\in X.$

See [10] and [9], as well as the references therein.

In this paper we are interested in those called weakly nonlinear, which verify
the following condition:

  1. (M)

(_Monotonicity_) $f\leq g$ in $E$ implies $T(f)\leq T(g)$ for all $f,g$ in
$E.$

######  Remark 1.

If $T$ is a weakly nonlinear and monotone operator, then

$T(\alpha\cdot 1)=\alpha\cdot T(1)\text{\quad for all }\alpha\in\mathbb{R}.$

Indeed, for $\alpha\geq 0$ the property follows from positive homogeneity.
Suppose now that $\alpha<0.$ Since $T(0)=0$ and $-\alpha>0$, by
translatability it follows that $0=T(0)=T(\alpha\cdot 1+(-\alpha\cdot
1))=T(\alpha\cdot 1)+(-\alpha)T(1)$, which implies $T(\alpha)=\alpha T(1)$.

Ergodic theory, harmonic analysis, probability theory and Choquet’s theory of
integration offer numerous examples of monotone sublinear and strongly
translatable operators, see, e.g., Gal and Niculescu [10], [9], [11].

Suppose that $E$ and $F$ are two ordered Banach spaces and $T$ $:E\rightarrow
F$ is an operator (not necessarily linear or continuous).

If $T$ is positively homogeneous, then

$T(0)=0.$

As a consequence,

$-T(-f)\leq T(f)\text{\quad for all }f\in E$

and every positively homogeneous and monotone operator $T$ maps positive
elements into positive elements, that is,

(2.1) $Tf\geq 0\text{\quad for all }f\geq 0.$

Therefore, for linear operators the property (2.1) is equivalent to
monotonicity.

Every sublinear operator is convex and a convex operator is sublinear if and
only if it is positively homogeneous.

The _norm_ of a continuous sublinear operator $T:E\rightarrow F$ can be
defined via the formulas

$\displaystyle\left\|T\right\|$
$\displaystyle=\inf\left\\{\lambda>0:\left\|T\left(f\right)\right\|\leq\lambda\left\|f\right\|\text{
for all }f\in E\right\\}$ $\displaystyle=\sup\left\\{\left\|T(f)\right\|:f\in
E,\text{ }\left\|f\right\|\leq 1\right\\}.$

A sublinear operator may be discontinuous, but when it is continuous, it is
Lipschitz continuous. More precisely, if $T:E\rightarrow F$ is a continuous
sublinear operator, then

$\left\|T\left(f\right)-T(g)\right\|\leq
2\left\|T\right\|\left\|f-g\right\|\text{\quad for all }f\in E.$

All sublinear and monotone operators are Lipschitz continuous, as stated by
the following result.

######  Theorem 1.

Every sublinear and monotone operator $T$ $:E\rightarrow F$ verifies the
inequality

$\left|T(f)-T(g)\right|\leq T\left(\left|f-g\right|\right)\text{\quad for all
}f,g\in E$

and thus it is Lipschitz continuous with Lipschitz constant equals to
$\left\|T\right\|,$ that is,

$\left\|T(f)-T(g)\right\|\leq\left\|T\right\|\left\|f-g\right\|\text{\quad for
all }f,g\in E.$

See [12] for details. Theorem 1 is a generalization of a classical result of
M. G. Krein concerning the continuity of positive linear functionals. See [1].

######  Remark 2.

The above weakly nonlinear operators were also called in [10] as Choquet
operators. Notice that the classical integral representation of such operators
were generalized by using the Choquet-Bochner integral of a real-valued
function with respect to a vector capacity in Gal and Niculescu [9].

##  3\. Main results for $p=\infty$ and $p=1$

For $r\geq 0$ and $1\leq p\leq+\infty$, let us denote by $W_{p}^{r}([0,1])$
the Sobolev space of all functions $f\in L_{p}([0,1])$ such that the
derivatives $f^{(\alpha)}$ exist (in the Sobolev sense) and are in
$L_{p}([0,1])$, for all $\alpha\in\\{0,1,...,r\\}$, endowed with the norm

$\|f\|_{r,p}=\max_{j=0,1,...,r}\|f^{(j)}\|_{p}.$

Also, if $T:C([0,1])\to C([0,1])$ is a monotone sublinear operator, denote

$\lambda_{p}=\max\\{\|T(e_{0})-e_{0}\|_{p},\|T(e_{1})-e_{1}\|_{p},\|T(-e_{1})+e_{1}\|_{p},\|T(e_{2})-e_{2}\|_{p}\\}.$

The first main result is the following.

######  Theorem 2.

Let $1\leq p\leq+\infty$ and $T:C([0,1])\to C([0,1])$ be a sublinear monotone
and strongly translatable operator. If $f\in W^{\infty}_{2}$, then

(3.1) $\|f-T(f)\|_{p}\leq C\cdot\|f\|_{2,\infty}\cdot\lambda_{p},$

with $C>0$ an absolute constant.

###### Proof.

We will adapt the considerations for the positive linear operator in the proof
of Theorem 1 in [7], to the properties of the monotone sublinear and strongly
translatable operators.

Firstly, let us consider that $1\leq p<+\infty$.

Choose $\varepsilon=\frac{1}{k}$, $k\in\mathbb{N}$ and write $(0,1)$ as an
union of $k$ subntervals $I_{i}$, pairwise disjoint, of lengths
$\leq\frac{1}{k}$. For each $i\in\\{1,...,k\\}$ let $\xi_{i}$ be the center of
$I_{i}$ and define

$l_{i}(x)=f(\xi_{i})+f^{\prime}(\xi_{i})(x-\xi_{i}).$

Reasoning exactly as for the relation $(2.3)$ in the paper of Berens-DeVore,
we arrive at the estimate

(3.2) $|f(x)-l_{i}(x)|\leq\frac{1}{2}\|f\|_{2,\infty}|x-\xi_{i}|^{2}.$

Then, since for each $i\in\\{1,...,k\\}$ and almost all $x\in I_{i}$ we have

$T(f)(x)-f(x)=T(f)(x)-T(l_{i})(x)+T(l_{i})(x)-l_{i}(x)+l_{i}(x)-f(x)$

and taking into account the property of $T$ in Theorem 1 too, it follows

$|T(f)(x)-f(x)|\leq T(|f-l_{i}|)(x)+|T(l_{i})(x)-l_{i}(x)|+|l_{i}(x)-f(x)|$

and consequently for almost everywhere $x\in[0,1]$ we obtain

$|T(f)(x)-f(x)|$
$\leq\sum_{i=1}^{k}T(|f-l_{i}|)(x)\chi_{I_{i}}(x)+\sum_{i=1}^{k}|T(l_{i})(x)-l_{i}(x)|\chi_{I_{i}}(x)+\sum_{i=1}^{k}|l_{i}(x)-f(x)|\chi_{I_{i}}(x)$
$:=S_{1}(x)+S_{2}(x)+S_{3}(x).$

By inequality (3.2), for almost everywhere $x\in[0,1]$, we obtain

$S_{1}(x)\leq\frac{1}{2}|\|f\|_{2,\infty}T(|t-\xi_{i}|^{2})(x).$

But by the sublinearity of $T$, we get

$T(|t-\xi_{i}|^{2})(x)=T[e_{2}(t)+2(-e_{1})(t)\xi_{i}+\xi_{i}^{2})e_{0}(t)](x)$
$\leq T(e_{2})(x)+2\xi_{i}T(-e_{1})(x)+\xi_{i}^{2}T(e_{0})(x)$
$=T(e_{2})(x)-e_{2}(x)+2\xi_{i}[T(-e_{1})(x)+e_{1}(x)]+\xi_{i}^{2}[T(e_{0})(x)-e_{0}(x)]+(x-\xi_{i})^{2}$
$\leq|T(e_{2})(x)-e_{2}(x)|+2|T(-e_{1})(x)+e_{1}(x)|+|T(e_{0})(x)-e_{0}(x)|+(x-\xi_{i})^{2},$

which immediately implies (as in the proof for the estimate (2.6) in Berens
and DeVore [7])

$\|S_{1}\|_{p}\leq\frac{1}{2}\leq\|f\|_{2,\infty}(4\lambda_{p}+\varepsilon^{2}).$

For the estimate of $S_{2}(x)$, for each $i\in\\{1,...,k\\}$ and almost all
$x\in I_{i}$, we get

$|T(l_{i})(x)-l_{i}(x)|=|T[f(\xi_{i})-f^{\prime}(\xi_{i})\xi_{i}+f^{\prime}(\xi_{i})e_{1}(t)](x)-f(\xi_{i})+f^{\prime}(\xi_{i})\xi_{i}-xf^{\prime}(\xi_{i})|$
$=|(f(\xi_{i})-f^{\prime}(\xi_{i})\xi_{i})[T(e_{0})(x)-1]+T(f^{\prime}(\xi_{i})e_{1})(x)-xf^{\prime}(\xi_{i})|$
$\leq(\|f\|_{\infty}+\|f^{\prime}\|_{\infty})|T(e_{0})(x)-1|+|T(f^{\prime}(\xi_{i})e_{1})(x)-xf^{\prime}(\xi_{i})|.$

Now, if $f^{\prime}(\xi_{i})\geq 0$, then

$|T(f^{\prime}(\xi_{i})e_{1})(x)-xf^{\prime}(\xi_{i})|=f^{\prime}(\xi_{i})|T(e_{1})(x)-x|\leq\|f^{\prime}\|_{\infty}|T(e_{1})(x)-x|$

and if f $f^{\prime}(\xi_{i})<0$, then

$|T(f^{\prime}(\xi_{i})e_{1})(x)-xf^{\prime}(\xi_{i})|$
$=|T(-f^{\prime}(\xi_{i})(-e_{1}))(x)+x(-f^{\prime}(\xi_{i}))|=-f^{\prime}(\xi_{i})|T(-e_{1})(x)+x|$
$\leq\|f^{\prime}\|_{\infty}|T(-e_{1})(x)+x|.$

From these estimates it immediately follows that

$|T(l_{i})(x)-l_{i}(x)|$
$\leq(\|f\|_{\infty}+\|f^{\prime}\|_{\infty})[|T(e_{0})(x)-1|+|T(e_{1})(x)-x|+|T(-e_{1})(x)+x|]$
$\leq
2\|f\|_{1,\infty}[|T(e_{0})(x)-1|+|T(e_{1})(x)-x|+|T(-e_{1})(x)+x|+|T(e_{2})(x)-x^{2}|],$

which leads to

$\|S_{2}\|_{p}\leq 2\|f\|_{1,p}\lambda_{p}.$

Finally, from (3.2) it easily follows that

$\|S_{3}\|_{p}\leq\frac{1}{2}\|f\|_{2,\infty}\varepsilon^{2}.$

Collecting now all the estimates for $S_{1},S_{2},S_{3}$ and taking into
account that $\varepsilon>0$ is arbitrary small, we arrive at the estimate in
the statement.

The case $p=+\infty$ easily follows by using the above lines of proof in the
case when $1\leq p<+\infty$. ∎

If we define the $r$th order modulus of smoothness in $L_{p}([0,1])$, $1\leq
p\leq\infty$, by

$\omega_{r,p}(f;\delta)=\sup_{|h|\leq\delta}\|\Delta_{h}^{r}\|_{p},$

where $\Delta_{h}^{r}(x)=\sum_{j=0}^{r}(-1)^{j}{r\choose j}f(x+(r-j)h)$, we
can state the following result which is an analogue of Theorem 2 in Berens and
DeVore [7].

######  Theorem 3.

(i) If $T:C([0,1])\to C([0,1])$ is a sublinear monotone and strongly
translatable operator, then for all $f\in C([0,1])$ we have

$\|f-T(f)\|_{\infty}\leq
C\\{\|f\|_{\infty}\lambda_{\infty}+\omega_{2,\infty}(f;\lambda_{\infty}^{1/2})\\},$

where $C$ depends only on the norm of $T$.

(ii) Let $p=1$ and $T:L_{1}([0,1])\to L_{1}([0,1])$ be a monotone sublinear
and strongly translatable operator. Then for all $f\in L_{1}([0,1])$ we have

$\|f-T(f)\|_{1}\leq
C\\{\|f\|_{1}\lambda_{1}+\omega_{3,1}(\lambda_{1}^{1/3})\\},$

where $C$ depends only on the norm of $T$.

###### Proof.

The proof is based on the above Theorem 2 and clearly that it is identical
with the proof of Theorem 2 in Berens and DeVore [7], based also on Lemma 1 in
the same paper.

For the reader’s convenience, we sketch below the proof. For $1\leq
p\leq+\infty$ and $r\in\mathbb{N}$, let us consider the $K$ functional

$K_{r,p}(f;t)=\inf\\{\|f-g\|_{p}+t\|g\|_{p};g\in W_{p}^{r}([0,1])\\},t>0.$

It is well-known the fact (see, e.g., relation (3.1) in [7]) that we have

(3.3) $K_{r,p}(t;t)\leq C(t^{r}\|f\|_{p}+\omega_{r,p}(f;t)),$

with $C>0$ depending only on $r$.

(i) For $p=\infty$, from the estimate in Theorem 2 in this paper combined with
(3.3), we easily get

$\|f-T(f)\|_{\infty}\leq C_{T}\cdot K_{2,\infty}(f;\lambda_{\infty})\leq
C_{T}\\{\lambda_{\infty}\|f\|_{\infty}+\omega_{2,\infty}(f;\lambda_{\infty}^{1/2})\\}.$

(ii) For $p=1$, Lemma 1 in [7] states that for all $f\in W_{1}^{1}$ we have
$\|f\|_{\infty}\leq\|f\|_{1,1}$. From here, from the estimate in Theorem2 and
from (3.3), we obtain

$\|g-T(g)\|_{1}\leq
C\|g\|_{2,\infty}\lambda_{1}\leq\|g\|_{3,1}\lambda_{1},\mbox{ for each }g\in
W_{1}^{3}$

and consequently

$\|f-T(f)\|_{1}\leq C_{T}\cdot K_{3,1}(f;\lambda_{1})\leq
C_{T}(\lambda_{1}\|f\|_{1}+\omega_{3,\infty}(f;\lambda_{1}^{1/3})).$

∎

##  4\. Main results for $1<p<\infty$

Due to the reason mentioned at the end of Introduction, in this section we
adapt the proofs in Swetits-Wood [18], to the case of sublinear monotone and
strongly translatable operators.

For the proof of the main result we need the following auxiliary result.

######  Lemma 1.

Let $(L_{n})_{n}$ be a uniformly bounded sequence of sublinear, strongly
translatable and monotone operators from $L_{p}([0,1])$ into $L_{p}([0,])$,
where $1<p<\infty$. Let $L_{p}^{(2)}([0,1])$ be the space of those functions
$f\in L_{p}([0,1])$ with $f^{\prime}$ absolutely continuous and
$f^{\prime\prime}\in L_{p}([0,1])$. Let us denote
$\mu_{n}=\|T_{n}((e_{1}-x)^{2})(x)\|_{\infty}$ and

$t_{n,p}=(\max\\{\|L(e_{0})-e_{0}\|_{p},\|T_{n}(|e_{1}-x|)(x)\|_{p},\mu_{n}\\})^{1/2}.$

If $\lim_{n\to\infty}t_{n,p}=0$, then for any $f\in L_{p}^{(2)}([0,1])$, we
have

$\|T_{n}(f)-f\|_{p}\leq
M_{p}^{\prime}(\|f\|_{p}+\|f^{\prime\prime}\|_{p})t^{2}_{n,p},$

where $M^{\prime}_{p}>0$ is independent of $f$ and $n$.

###### Proof.

Let $f\in L_{p}^{(2)}([0,1])$ and assume that $f$ has been extended outside of
$[0,1]$ so that $f^{\prime\prime}(x)=0$ if $x\notin[0,]$.

By

$T_{n}(f)(x)-f(x)=T_{n}(f)(x)-f(x)T_{n}(e_{0})(x)+f(x)T_{n}(e_{0})(x)-f(x)$
$=T_{n}(f)(x)-f(x)T_{n}(e_{0})(x)+f(x)(T_{n}(e_{0})(x)-1),$

we obtain

(4.1) $\|T_{n}(f)(x)-f(x)\|_{p}$
$\leq\|T_{n}(f)(x)-f(x)T_{n}(e_{0})(x)\|_{p}+\|f\|_{\infty}\cdot\|T_{n}(e_{0})(x)-e_{0}(x)\|_{p}.$

For $t,x\in[0,1]$, we have

$f(t)-f(x)=f(t)-f(x)e_{0}(t)=f^{\prime}(x)(t-x)+\int_{x}^{t}(t-u)f^{\prime\prime}(u)du$

which by applying to the both members $T_{n}$, implies

$T_{n}(f)(x)-f(x)T_{n}(e_{0})(x)=T_{n}\left[f^{\prime}(x)(e_{1}-x)+\int_{x}^{t}(t-u)f^{\prime\prime}(u)du\right].$

Taking the absolute value, by Theorem 1 and by the sublinearity of $T_{n}$, we
get

$|T_{n}(f)(x)-f(x)T_{n}(e_{0})(x)|=\left|T_{n}\left[f^{\prime}(x)(e_{1}-x)+\int_{x}^{t}(t-u)f^{\prime\prime}(u)du\right]\right|$
$\leq
T_{n}\left[|f^{\prime}(x)|\cdot|e_{1}-x|+|\int_{x}^{t}(t-u)f^{\prime\prime}(u)du|\right]$
$\leq\|f^{\prime}\|_{\infty}T_{n}(|e_{1}-x|)(x)+T_{n}\left[\left|\int_{x}^{t}(t-u)f^{\prime\prime}(u)du\right|\right].$

This implies

(4.2) $\|T_{n}(f)(x)-f(x)T_{n}(e_{0})(x)\|_{p}$
$\leq\|f^{\prime}\|_{\infty}\|T_{n}(|e_{1}-x|)(x)\|_{p}+\left\|T_{n}\left[\left|\int_{x}^{t}(t-u)f^{\prime\prime}(u)du\right|\right](x)\right\|_{p}.$

Reasoning exactly as in Swetits-Wood [18], proof of Lemma 2, page 87 (that is
by using the Hardy-Littlewood majorant), we get

$\left\|T_{n}\left[\left|\int_{x}^{t}(t-u)f^{\prime\prime}(u)du\right|\right](x)\right\|_{p}\leq
K_{p}\cdot\|T_{n}((e_{1}-x)^{2})(x)\|_{\infty}\cdot\|f^{\prime\prime}\|_{p}.$

Therefore, (4.2) becomes

(4.3) $\|T_{n}(f)(x)-f(x)T_{n}(e_{0})(x)\|_{p}$
$\leq\|f^{\prime}\|_{\infty}\cdot\|\left[T_{n}(|e_{1}-x|)(x)\right]\|_{p}+K_{p}\cdot\|T_{n}((e_{1}-x)^{2})(x)\|_{\infty}\cdot\|f^{\prime\prime}\|_{p}.$

From (4.1), it immediately follows

(4.4)
$\|T_{n}(f)(x)-f(x)\|_{p}\leq\|f\|_{\infty}\cdot\|T_{n}(e_{0})(x)-e_{0}(x)\|_{p}$
$+\|f^{\prime}\|_{\infty}\cdot\|T_{n}(|e_{1}-x|)(x)\|_{p}+K_{p}\cdot\|T_{n}((e_{1}-x)^{2})(x)\|_{\infty}\cdot\|f^{\prime\prime}\|_{p}.$

Using now Theorem 3.1 in [14], from (4.4) we immediately arrive to

(4.5) $\|T_{n}(f)(x)-f(x)\|_{p}\leq
C^{\prime}_{p}(\|f\|_{p}+\|f^{\prime\prime}\|_{p})t_{n,p}^{2}.$

∎

The main result of this section is the following.

######  Theorem 4.

Let $(T_{n})_{n}$ be a uniformly bounded sequence of strongly translatable
sublinear and monotone operators from $L_{p}([0,1])$ into $L_{p}([0,1])$,
where $1<p<\infty$, and denote $\mu_{n}=\|T_{n}((e_{1}-x)^{2})(x)\|_{\infty}$,

$t_{n,p}=(\max\\{\|T_{n}(e_{0})-e_{0}\|_{p},\|T_{n}(|e_{1}-x|)(x)\|_{p},\mu_{n}\\})^{1/2},$

Supposing that $\lim_{n\to\infty}t_{n,p}=0$, for any $f\in L_{p}([0,])$, we
have

$\|T_{n}(f)-f\|_{p}\leq M_{p}[t^{2}_{n,p}\|f\|_{p}+\omega_{2,p}(f;t_{n,p})],$

where $M_{p}>0$ is independent of $f$ and $n$.

###### Proof.

Let $f\in L_{p}([0,1])$ and $g\in L_{p}^{(2)}([0,1])$. By Theorem 1, for each
$T_{n}$ we have

$\|T_{n}(f)-T_{n}(g)\|_{p}\leq\|T_{n}\|\cdot\|f-g\|_{p},\mbox{ for all }f,g\in
L_{p}([0,1])\leq R_{p}\cdot\|f-g\|_{p},$

where $\|T_{n}\|\leq R_{p}$ (with $R_{p}$ independent of $n$) for all
$n\in\mathbb{N}$, from the uniform boundedness of the sequence $(T_{n})_{n}$.

Then, by

$L_{n}(f)-f=L_{n}(f)-L_{n}(g)+L_{n}(g)-g+g-f,$

passing to absolute value and to $\|\cdot\|_{p}$, we immediately get (by Lemma
1 too)

$\|L_{n}(f)-f\|_{p}\leq\|L_{n}(f)-L_{n}(g)\|_{p}+\|L_{n}(g)-g\|_{p}+\|g-f\|_{p}$
$\leq(1+R_{p})\|f-g\|_{p}+M^{\prime}_{p}\cdot
t^{2}_{n,p}(\|g\|_{p}+\|g^{\prime\prime}\|_{p}).$

Taking here the infimum after $g\in L_{p}^{(2)}([0,1])$ and taking into
account relations $(2.1)$ and $(2.3)$ on the page 88 in Swetits-Wood [18], we
arriveat the desired conclusion. ∎

######  Remark 3.

Since the calculation of $L_{n}(|e_{1}-x|)(x)$ is difficult, we may estimate
it by a simpler quantity in calculation, by using the Hölder’s inequality in
Theorem 3 of Gal and Niculescu [10]. It follows

$T_{n}(|e_{1}-x|)(x)\leq(T_{n}((e_{1}-x)^{2})(x)T_{n}(e_{0})(x))^{1/2}\leq
C^{1/2}(T_{n}((e_{1}-x)^{2})(x))^{1/2},$

where $C>0$ is a constant independent of $n$ which comes from the uniform
boundedness of the sequence $(T_{n})_{n}$.

This implies

$\|T_{n}(|e_{1}-x|)(x)\|_{p}\leq
C^{1/2}\|\left[T_{n}((e_{1}-x)^{2})(x)\right]^{1/2}\|_{p}.$

By this remark, we immediately arrive at the following result.

######  Corollary 1.

Let $(T_{n})_{n}$ be a uniformly bounded sequence of strongly translatable
sublinear and monotone operators from $L_{p}([0,1])$ into $L_{p}([0,1])$,
where $1<p<\infty$, and denote $\mu_{n}=\|T_{n}((e_{1}-x)^{2})(x)\|_{\infty}$,

$s_{n,p}=(\max\\{\|T_{n}(e_{0})-e_{0}\|_{p},\|[T_{n}((e_{1}-x)^{2})(x)]^{1/2}\|_{p},\mu_{n}\\})^{1/2},$

Supposing that $\lim_{n\to\infty}s_{n,p}=0$, for any $f\in L_{p}([0,1])$, we
have

$\|T_{n}(f)-f\|_{p}\leq M_{p}[s^{2}_{n,p}\|f\|_{p}+\omega_{2,p}(f;s_{n,p})],$

where $M_{p}>0$ is independent of $f$ and $n$.

######  Remark 4.

If $(T_{n})_{n}$ is a sequence of strongly translatable sublinear and monotone
operators from $L_{p}([0,1])$ into $L_{p}([0,1])$ which satisfies the
conditions

$\lim_{n\to\infty}\|T_{n}(e_{0})-e_{0}\|_{\infty}=0,\lim_{n\to\infty}\|T_{n}(-e_{1})+e_{1}\|_{\infty}=0,\lim_{n\to\infty}\|T_{n}(e_{2})-e_{2}\|_{\infty}=0,$

then $\lim_{n\to\infty}s_{n,p}=0$.

Indeed, this is immediate from the relations

$0\leq T_{n}((e_{1}-x)^{2})(x)=T_{n}[e_{2}-2xe_{1}+x^{2}e_{0}](x)$ $\leq
T_{n}(e_{2})(x)+2xT_{n}(-e_{1})(x)+x^{2}T_{n}(e_{0})(x)\to 0\mbox{ as
}n\to\infty$

and

$\|[T_{n}((e_{1}-x)^{2})(x)]^{1/2}\|_{p}\leq\|[T_{n}((e_{1}-x)^{2})(x)]^{1/2}\|_{\infty}.$

##  5\. Applications

In this section we apply the previous results to some concrete cases.

######  Example 1.

Let us consider the sequence of Bernstein operators $B_{n}:C([0,1])\rightarrow
C\left([0,1]\right),$ defined by the formulas

$B_{n}(f)(x)=\sum_{k=0}^{n}p_{n,k}(x)f(k/n).$

The operators $T_{n}:C([0,1])\rightarrow C\left([0,1]\right)$ given by

$T_{n}(f)=\max\left\\{B_{n}(f),B_{n+1}(f)\right\\}$

are sublinear, monotone and strongly translatable. Known computations imply
that

$B_{n}(e_{0})(x)=1,B_{n}(-e_{1})(x)=-e_{1}(x),B_{n}(e_{1})=e_{1}(x),B_{n}(e_{2})(x)=e_{2}(x)+\frac{x(1-x)}{n}.$

These imply $T_{n}(e_{0})=e_{0}$, $T_{n}(-e_{1})=-e_{1}$, $T_{n}(e_{1})=e_{1}$
and

$T_{n}(e_{2})(x)=\max\\{B_{n}(e_{2})(x),B_{n+1}(e_{2})(x)\\}=e_{2}(x)+\frac{x(1-x)}{n}.$

Since for $p=\infty$ we immediately get

$\lambda_{n,p}=\|x(1-x)/n\|_{p}\leq\frac{1}{4n},$

by Theorem 3, (i), it follows the estimate

$\|f-T_{n}(f)\|_{\infty}\leq
C\left[\|f\|_{\infty}\cdot\frac{1}{4n}+\omega_{2,\infty}\left(f;\frac{1}{2\sqrt{n}}\right)\right],$

which is completely different and essentially better than the estimate based
on the Shisha and Mond’s idea [17] in the paper Gal and Niculescu [13], namely

$\|f-T_{n}(f)\|_{\infty}\leq 2\omega_{1,\infty}(f;1/(2\sqrt{n})).$

For $p=1$, by Theorem 3, (ii), we get

$\|f-T_{n}(f)\|_{1}\leq
C\left[\|f\|_{1}\cdot\frac{1}{4n}+\omega_{3,1}\left(f;\frac{1}{(4n)^{1/3}}\right)\right].$

For $1<p<\infty$, since $T_{n}((e_{1}-x)^{2})(x)=\frac{x(1-x)}{n}$,
$\|T_{n}((e_{1}-x)^{2})(x)\|_{\infty}\leq\frac{1}{4n}$ and
$\|[T_{n}((e_{1}-x)^{2})(x)]^{1/2}\|_{\infty}\leq\frac{1}{2\sqrt{n}}$, by
Corollary 1 and by Remark 4 we obtain

$\|T_{n}(f)-f\|_{p}\leq
M_{p}\left[\|f\|_{p}\cdot\frac{1}{2\sqrt{n}}+\omega_{2,p}\left(f;\frac{1}{\sqrt{2}\cdot
n^{1/4}}\right)\right].$

######  Example 2.

Now, let us define the sequence of nonlinear operators
$T_{n}:C\left([0,1]\right)\rightarrow C\left([0,1]\right)$ defined by the
formulas

$T_{n}(f)(x)=\sum_{k=0}^{n}p_{n,k}(x)\sup_{[k/(n+1)\leq
t\leq,(k+1)/(n+1)]}f(t),$

which are monotone sublinear and strongly translatable. We have
$T_{n}(e_{0})(x)=1$, $T_{n}(e_{1})(x)=\frac{n}{n+1}e_{1}(x)$,
$T_{n}(-e_{1})(x)=-\frac{n}{n+1}e_{1}(x)$,

$\|T_{n}(e_{1})-e_{1}\|_{p}\leq\frac{1}{n+1},\|T_{n}(-e_{1})+e_{1}\|_{p}\leq\frac{1}{n+1},$
$T_{n}(e_{2})(x)=\sum_{k=0}^{n}p_{n,k}(x)\cdot\frac{(k+1)^{2}}{(n+1)^{2}}$
$=\left(\frac{n}{n+1}\right)^{2}\left(x^{2}+\frac{x(1-x)}{n}\right)+\frac{2n}{(n+1)^{2}}x+\frac{1}{(n+1)^{2}},$
$\|T_{n}(e_{2})-e_{2}\|_{p}=\left\|e_{2}\frac{-3n-1}{(n+1)^{2}}+\frac{3nx+1}{(n+1)^{2}}\right\|_{p}\leq\frac{6n+2}{(n+1)^{2}}\leq\frac{6}{n+1},$

and then

$\lambda_{n,p}\leq\frac{6}{n+1}.$

By Theorem 3, (i), we get

$\|f-T_{n}(f)\|_{\infty}\leq
C\left[\|f\|_{\infty}\cdot\frac{6}{n+1}+\omega_{2,\infty}\left(f;\frac{\sqrt{6}}{\sqrt{n+1}}\right)\right],$

and by Theorem 3, (ii), it follows

$\|f-T_{n}(f)\|_{1}\leq
C\left[\|f\|_{1}\cdot\frac{6}{n+1}+\omega_{3,1}\left(f;\frac{6}{(n+1)^{1/3}}\right)\right].$

For $1<p<\infty$, since by Remark 4 we have

$0\leq T_{n}((e_{1}-x)^{2})(x)\leq
T_{n}(e_{2})(x)+2xT_{n}(-e_{1})(x)+x^{2}T_{n}(e_{0})(x)$
$=\left(\frac{n}{n+1}\right)^{2}\left(x^{2}+\frac{x(1-x)}{n}\right)+\frac{2n}{(n+1)^{2}}x+\frac{1}{(n+1)^{2}}+2x^{2}\left(-\frac{n}{n+1}\right)+x^{2}$

which by simple calculation finally leads to

$0\leq T_{n}((e_{1}-x)^{2})(x)\leq\frac{1}{n}\cdot\frac{9}{4},$

by Corollary 1 we obtain

$\|T_{n}(f)-f\|_{p}\leq
M_{p}\left[\|f\|_{p}\cdot\frac{3}{2\sqrt{n}}+\omega_{2,p}\left(f;\frac{\sqrt{3}}{\sqrt{2}\cdot
n^{1/4}}\right)\right].$

######  Example 3.

Another example can be the Bernstein-Kantorovich-Choquet operators, given by
the formula

$K_{n,\mu}(f)(x)=\sum_{k=0}^{n}p_{n,k}(x)\cdot\frac{(C)\int_{k/(m+1)}^{(k+1)/(n+1)}f(t)\mathrm{d}\mu(t)}{\mu([k/(n+1),(k+1)/(n+1)])},$

where $(C)\int d\mu$ means the Choquet integral with respect to
$\mu=\sqrt{m}$, with $m$ the Lebesgue measure. We omit here the calculations.
For details concerning the properties of the Choquet integral and of
Bernstein-Kantorovich-Choquet operators, see, e.g., Gal-Niculescu [8], [10].

##  6\. Final remarks

######  Remark 5.

The estimate in Theorem 4 is worse than that for linear and positive operators
in Theorem 2, (i) in [18], where the quantity $\|T_{n}(|e_{1}-x|)(x)\|_{p}$ is
replaced by the smaller one $\|T_{n}(e_{1}-x)(x)\|_{p}$, since obviously
$\|T_{n}(e_{1}-x)(x)\|_{p}\leq\|T_{n}(|e_{1}-x|)(x)\|_{p}$. This seems to be
the price paid due to the more general hypothesis on the operators $L_{n}$ in
the above Theorem 4.

However, in the case of Example 1, since it is easy to show that

$\max\\{B_{n}(f),B_{n+1}(f)\\}-f=\max\\{B_{n}(f)-f,B_{n+1}(f)-f\\},$

and

$|\max\\{B_{n}(f)-f,B_{n+1}(f)-f\\}|\leq\max\\{|B_{n}(f)-f|,|B_{n+1}(f)-f|\\},$

we immediately get

$\|\max\\{B_{n}(f),B_{n+1}(f)\\}-f\|_{p}\leq\max\\{\|B_{n}(f)-f\|_{p},\|B_{n+1}(f)-f\|_{p}\\}$
$\leq\|B_{n}(f)-f\|_{p}+\|B_{n+1}(f)-f\|_{p}.$

Therefore, applying here the estimate in Swetits-Wood for $1<p<\infty$ and for
the linear and positive operators $B_{n}$, clearly that we get essentially
better estimate than that in Theorem 4 and Corollary 1, calculated in the
above Example 1.

In the case of Example 2, we can write

$T_{n}(f)(x)=\sum_{k=0}^{n}p_{n,k}(x)f(\xi_{n,k}),$

with $k/(n+1)\leq\xi_{n,k}\leq(k+1)/(n+1)$, which immediately leads to

$\|T_{n}(f)(x)-B_{n}(f)(x)|\leq\sum_{k=0}^{n}p_{n,k}(x)|f(\xi_{n,k})-f(k/n)|\leq\omega_{1}(f;1/(n+))$

and

$|T_{n}(f)(x)-f(x)|\leq|T_{n}(f)(x)-B_{n}(f)(x)|+|B_{n}(f)(x)-f(x)|.$

By the result in Swetits-Wood applied to the positive linear operators
$B_{n}(f)(x)$, for $1<p<\infty$, it immediately follows

$\|T_{n}(f)-f\|_{p}\leq
M_{p}\left(\frac{1}{n}\|f\|_{p}+\omega_{2,p}(f;1/\sqrt{n})_{p}\right)+\omega(f;1/(n+1)),$

which clearly it is essentially better than the estimate in the previous
section.

These considerations suggest that possibly the shortcoming in Remark 5 is due
to the method of proof in Swetits-Wood [18] and raise the following.

Open Question. Extend the classical interpolation technique from linear
continuous operators to sublinear, monotone and strongly translatable
operators.

A positive answer would allow to essentially improve the estimate in Theorem 4
by following now the method of proof in Theorem 3 of Berens-DeVore [7].

## References

  * [1] Abramovich, Y.A., Aliprantis, C.D.: Positive Operators. Handbook of the Geometry of Banach spaces, I, pp. 85–122. North-Holland, Amsterdam (2001)
  * [2] Altomare, F.: Korovkin-type theorems and positive operators. Surveys in Approximation Theory. 6, 92-164 (2010)
  * [3] Altomare, F.: On positive linear functionals and operators associated with generalized means. J. Math. Anal. Appl. 502, paper no. 125278 (2021)
  * [4] Altomare, F.: On the convergence of sequences of positive linear operators and functionals on bounded function spaces. Proc. Amer. Math. Soc. 149, 3837–3848 (2021)
  * [5] Altomare, F.: Korovkin-type theorems and local approximation problems. Expositiones Mathematicae (2022), doi: https://doi.org/10.1016/j.exmath.2022.06.001.
  * [6] Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications. de Gruyter Studies in Mathematics, 17, Berlin (1994, reprinted 2011)
  * [7] Berens, H., DeVore R.: Quantitative Korovkin theorems for positive linear operators on $L_{p}$-spaces. Trans. Amer. Math. Soc. 245, 349–361 (1978)
  * [8] Gal, S.G., Niculescu, C.P.: A nonlinear extension of Korovkin’s theorem. Mediterr. J. Math. 17, Article no. 145 (2020).
  * [9] Gal, S.G., Niculescu, C.P.: Choquet operators associated to vector capacities. J. Math. Anal. Appl. 500, article no. 125153 (2021)
  * [10] Gal, S.G., Niculescu, C.P.: A note on the Choquet type operators. Aequationes Math. 95, 433–447 (2021)
  * [11] Gal, S.G., Niculescu, C.P.: Nonlinear versions of Korovkin’s abstract theorems. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 116, Article number 68 (2022)
  * [12] Gal, S.G., Niculescu, C.P.: Korovkin type theorems for weakly nonlinear and monotone operators. Accepted at Mediterr. J. Math. Available on arXiv:2206.14102
  * [13] Gal, S.G., Niculescu, C.P.: Quantitative estimates in nonlinear operator extensions of Korovkin’s theorems. submitted.
  * [14] Goldberg, S., Meir, A.: Minimum moduli of ordinary differential operators. Proc. London Math. Soc. (3) 23, 1–15 (1971)
  * [15] Korovkin, P.P.: On convergence of linear positive operators in the space of continuous functions.(Russian). Doklady Akad. Nauk. SSSR. (NS) 90, 961–964 (1953).
  * [16] Korovkin, P.P.: Linear Operators and Approximation Theory. Fitzmatgiz, Moscow (1959) (in Russian) [English translation, Hindustan Publ. Corp., Delhi (1960)]
  * [17] Shisha, O., Mond,B.: The degree of convergence of sequences of linear positive operators. Proc. Nat. Acad. Sci. U.S.A. 60, 1196–1200 (1968)
  * [18] Swetits, J.J., Wood, B.: Quantitative estimates for $L_{p}$ approximation with positive linear operators. J. Approx. Theory 38, 81–89 (1983)


# Probabilistic Models of Profiles for Voting by Evaluation

Antoine Rolland  ERIC EA 3083, Université de Lyon, Université Lumière Lyon 2,
5 Pierre Mendès France, 69596 Bron Cedex, France  Jean-Baptiste Aubin  Univ
Lyon, INSA Lyon, UJM, UCBL, ECL, ICJ, UMR5208,
69621 Villeurbanne, France  Irène Gannaz  Univ. Grenoble Alpes, CNRS, Grenoble
INP111Institute of Engineering Univ. Grenoble Alpes, G-SCOP,
38000 Grenoble, France  Samuela Leoni  Univ Lyon, INSA Lyon, UJM, UCBL, ECL,
ICJ, UMR5208,
69621 Villeurbanne, France

###### Abstract

Considering voting rules based on evaluation inputs rather than preference
rankings modifies the paradigm of probabilistic studies of voting procedures.
This article proposes several simulation models for generating evaluation-
based voting inputs. These models can cope with dependent and non identical
marginal distributions of the evaluations received by the candidates. A last
part is devoted to fitting these models to real data sets.
Keywords: Voting rules, Evaluation-based voting rules, Simulation, IC model.

##  1 Introduction

Voting is a very general method to turn individual preferences on a finite set
of candidates into the choice of the best candidate, i.e. the winner of the
election. These individual preferences can be expressed as rankings or
evaluations on candidates. A review of both ranking-based and evaluation-based
voting rules can be found in [20]. Both social and mathematical approaches
consider positive or negative properties satisfied by a given voting rule as a
matter of interest, such as independence to irrelevant alternatives, Condorcet
paradox or manipulability. Studying the properties of voting rules can be done
either through an axiomatic approach or a probabilistic approach. The
axiomatic approach supposes to determine which properties characterize a
specific voting rule, _i.e._ which properties are to be observed, and which
are not, via formal theorems. The probabilistic approach aims at determining
whether these properties are likely to be observed, _i.e._ determining the
frequency of occurrence of such properties using a given voting rule.

Many works have been published on the probabilistic approach. For instance,
without claiming to be exhaustive, [46], [48], [41], [26] contain examples of
simulation-based studies of voting rules. Analogously, [23] looks at the
frequency of occurrence of Condorcet’s paradox. [20] studies the
manipulability rate of voting systems. [22] do the same focusing on
coalitional manipulability. Most of these articles consider only voting rules
based on rankings on candidates.

One can refer to [17] for a recent state-of-the-art of generative models for a
probabilistic approach of voting theory. Generative models for ranking-based
voting preferences (i.e. cardinal models) can be divided into two main
approaches. The first approach consists in directly generating random rankings
through a specific distribution on the space of ranking. It is difficult, if
not impossible, to generalize existing methods to evaluations because of the
very different nature of the information contained in a ranking and in
evaluations. The second approach entails working with utilities, incorporating
for instance spatial models. As observed by [14], one can generate any
distribution on rankings from continuous random variables. The Thurstone order
statistics models (see e.g. [1, chapter 8] and references therein) aim at
building probability distributions on rankings by ordering continuous random
variables, named utilities.

In continuous evaluation-based models, the grades can be viewed as bounded
utilities. Similarly to Thurstone models, a strict preference relation on the
candidates can be obtained from the evaluations for each voter. In case of
discrete grades, _ex-aequo_ may occur, and then a pre-order can be obtained
from evaluations, but not a complete order. The literature is mainly focused
on simulations for ranking-based voting rules, while our main matter of
interest is evaluation-based voting rules. Evaluation-based voting rules,
including the three main ones: range voting, see [44], majority judgment, see
[4], and approval voting, see [9], but also the new family of ”deepest
voting”, see [3], are based on the use of evaluations on a bounded scale
(which can be discrete or continuous) given by each voter to each candidate.
Generative models for evaluation-based voting rules did not get all the
attention they deserve in our opinion : filling this lack is our main
motivation in this paper.

Our objective here is therefore to propose generative models for evaluation-
based voting rules, in the framework of a probabilistic approach. We model
voting situations through the rationalization of probabilistic distributions
of evaluations upon candidates. We first propose in Section 2 some
distributions for univariate evaluations for one candidate. A social
interpretation of the univariate evaluations distributions will be given in
Section 2.3. Next, Section 3 is devoted to the distributions for multivariate
evaluations. The simplest approach is to consider independent and identically
distributed (i.i.d.) voters’ evaluations on each candidate. This assumption
can be relaxed to enlarge the scope of generating models. To this aim, we
introduce in Section 4 copula-based evaluations models and spatial models.
Section 5 provides an illustration on fitting on observed situations.

### Notations

In the following, we will consider situations with $n$ voters and $m$
candidates. Each voter associates an evaluation from a set $\mathcal{E}$ to
each candidate.

Evaluation of voter $v$ for candidate $c$ will be denoted $e_{vc}$, for
$v=1,\dots,n$, $c=1,\dots,m$. Observations
$\\{e_{vc},~{}{v=1,\dots,n,~{}c=1,\dots,m}\\}$ are $n$ independent
realizations of a random variable $E=(E_{1},\ldots,E_{m})$, which takes values
in $\mathcal{E}^{m}$. Note that we will not consider dependence among the
voters, as discussed later, in Section 2.4.

Under the assumption of independence between voters, defining a simulation
setting can be seen as defining a multivariate probability distribution on
$\mathcal{E}^{m}$. We will consider two cases with respect to the amount of
information contained in the set $\mathcal{E}$:

  * •

continuous grades: without loss of generality, $\mathcal{E}=[0,1]$ ;

  * •

discrete grades: without loss of generality, $\mathcal{E}=\\{0,\dots,K\\}$
with $K\in\mathbb{N}\setminus\\{0\\}$.

The easiest distributions one can consider for the $m$ random variables
$E_{1},\dots E_{m}$ are independent and identically distributed (i.i.d.)
distributions. When the evaluations are continuous, the preferences resulting
from the evaluations will satisfy almost surely an Impartial Culture (IC)
model, that is, a uniform distribution on rankings.

This model can be extended to more complex ones following two ways. The first
way is to remove the identical distribution assumption: two candidates may not
have the same evaluation distribution in the population of voters. The second
way is to introduce dependence between the evaluations of the candidates.
Indeed, two candidates viewed as almost similar (resp. in opposition) by the
voters may lead to positively (resp. negatively) correlated evaluations.

##  2 Univariate distributions on evaluations

We propose in this section several distributions to model the evaluation
distribution on a single candidate by several independent voters, respectively
for continuous and discrete sets. A social interpretation of these models is
provided in Section 2.3. Independence between voters is justified in Section
2.4.

###  2.1 Continuous case

When the evaluations are continuous, that is, $\mathcal{E}=[0,1]$, we propose
to use Uniform, truncated Normal or Beta distributions, as illustrated in
Figure 1. Uniform distribution is the easiest model. Truncated Normal
distribution is useful to model uni-modal distributions when the proportions
of evaluations close to 0 or 1 are significant. Beta distribution is a very
versatile model that can model several situations, depending on the chosen
parameters; it is particularly adapted in the case of bi-modal distributions,
when most of the evaluations are around 0 or 1, but also in the case of uni-
modal distribution with a quick decreasing curve.

Figure 1: Probability distribution functions of evaluations on one candidate.
(a) Uniform distribution, (b) Truncated Normal distribution with $\mu=0.5$ and
$\sigma=0.35$, (c) Beta distribution with $\alpha=5$ and $\beta=2$ and (d)
Beta distribution with $\alpha=0.7$ and $\beta=0.5$.

It can be seen that these three families of distributions cover a large scope
of distributions. The initial distribution proposed by [45] is the standard
Gaussian distribution. Our truncated Normal distribution is, hence, a
generalization to bounded-support evaluations. Other distributions were
introduced in Thurstone’s like approach. The most famous is the Gumbel
distribution proposed by [33], which has the advantage of providing a closed
form distribution of rankings. Yet, Gumbel distribution does not have a
bounded support and, hence, cannot be used in our context.

###  2.2 Discrete case

When the evaluations are discrete, i.e. $\mathcal{E}=\\{0,\dots,K\\}$, we
propose to use discrete Uniform, Binomial or Beta-Binomial distributions,
which can be seen as the discrete counterpart of the Uniform, truncated Normal
and Beta continuous distributions. Examples are presented in Figure 2.

Figure 2: Examples of probability distributions on 7 levels ($K=6$). (a)
Uniform distribution, (b) Binomial distribution with $p=0.5$, (c) Beta-
Binomial distribution with $\alpha=5$ and $\beta=2$, and (d) Beta-Binomial
distribution with $\alpha=\beta=0.5$.

###  2.3 General remarks

Parameters for each model are summed up in Table 1.

Marginal continuous distributions | Parameters
---|---
Continuous Uniform $\mathcal{U}[0,1]$ | $\cdot$
Truncated Normal $\mathcal{N}_{T}(\mu,\sigma^{2})$ | $(\mu,\sigma)$
Beta $\mathcal{B}(\alpha,\beta)$ | $(\alpha,\beta)$

Marginal discrete distributions | Parameters
---|---
Discrete Uniform $\mathcal{U}\\{0,\dots,K\\}$ | $\cdot$
Binomial $\mathcal{B}(K,p)$ | $p$
Beta-Binomial $\mathcal{B}(K,\alpha,\beta)$ | $(\alpha,\beta)$

Table 1: Marginal distributions proposed for the distribution of the
evaluations. $K$ is not a flexible parameter since it corresponds to the known
scale of the evaluations.

In a similar way to traditional statistical studies, the distribution of
evaluations given to candidates can be chosen according to two main
parameters: a location parameter and a dispersion parameter. The location
parameter (e.g. the mean or median of evaluations) will be proportional to a
candidate’s general popularity. For example, a candidate whose evaluations
distribution is that of Figures 1(c) or 2(c) will be more popular than a
candidate whose distribution is that of case Figures 1(b) or 2(b). The
dispersion parameter indicates a candidate’s ability to divide opinions. In
the extreme case of a zero dispersion, all voters would associate the same
score to the candidate. Conversely, a high dispersion, like in Figures 1(d) or
2(d), is associated with a candidate that divides opinion. It is therefore
possible to associate one distribution rather than another a priori, depending
on the information we have on the candidates, with the uniform distribution
always being preferred if we have no information on the candidate’s
evaluations.

Of course, these distributions are not exhaustive and many choices are
possible. We focus here on the most natural, which already seem to cover
various shapes. An extension can be proposed by the use of a mixture model of
distributions. For example, [36] proposed mixture models for rankings. This is
especially interesting when fitting models on real voting elections, see [19]
or [25]. As our main objective is to propose generating models, we do not
explore this direction here. But note that none of the distributions presented
here is multimodal without modes in the extremities. This is a limit for the
fitting of real data, that mixtures can overcome.

The earlier methods require certain assumptions about the distribution of the
evaluations, which must be validated. These assumptions can be assessed using
goodness-of-fit tests such as the Chi-Square test, Kolmogorov-Smirnov test,
Anderson test, and others. In instances where the theoretical distribution
does not align with the observed data, more versatile techniques become
pertinent. Specifically, non-parametric approaches like kernel density
estimation or projection-based estimation of evaluation densities can be
employed. This spectrum of methods ensures both adaptability and verifiability
of the approach.

###  2.4 Dependence between voters

Some simulation approaches are based on introducing dependencies between the
voters, see e.g. [7, 29, 6, 10]. This corresponds to the so-called multi-stage
approach, see [1, Chapter 8]. This approach enables to generate non Uniform
distribution on rankings. It can be used for example to obtain Mallow’s Phi
distribution. Its main interest is the ease of the simulation process. The
objective of such models, rather than controlling the dependence between the
voters, is to obtain a given distribution on the rankings.

Introducing the dependencies between the voters does not bring relevant
information for the generation of evaluations. The dependence between voters
yields a different distribution of the evaluations, and this information is
lost when considering only the resulting values since the voters are
considered anonymous. An example is displayed in Figure 3. Hence, from a
generating point of view, it is more interesting to model the structure of the
voters through the distribution (with mixing of distributions to model
communities - see Figure 3(d)). From a fitting point of view, it is not
possible to estimate dependence between voters. Indeed, estimating a
dependence needs to have repeated observations of a variable.

Figure 3: Influence of the dependence between voters on evaluation
distributions. Evaluations are defined by $e_{v}=\Phi(x_{v})$ with
$(x_{1},\dots,x_{m})\sim\mathcal{N}(0,\Sigma)$, with $m=1000$, and $\Phi$ the
cumulative distribution function of the standard Gaussian distribution. In
subfigures (a) to (c), for all $(v,v^{\prime})\in\\{1,\dots,m\\}^{2}$, $v\neq
v^{\prime}$, $\Sigma_{v,v}=1$, $\Sigma_{v,v^{\prime}}=\rho$. From left to
right, (a) $\rho=0$, (b) $\rho=0.5$, (c) $\rho=0.5$ with a different seed in
simulation. In subfigure (d), $\Sigma$ is a block matrix, corresponding to 2
communities : $\Sigma_{v,v}=1$ for all $v=1,\dots,1000$, and if $v\neq
v^{\prime}$, $\Sigma_{v,v^{\prime}}=0.9$ if $1\leq v,v^{\prime}\leq 250$ or
$251\leq v,v^{\prime}\leq 1000$, and $\Sigma_{v,v^{\prime}}=0$ else.

######  Remark 1.

Evaluations $\\{{e_{v}\\}}$ in Figure 3 are dependent observations from a
Uniform distribution on [0,1]. Indeed, if a random variable $X$ follows a
standard Gaussian distribution, and $\Phi$ is its cumulative distribution
function, than $\Phi(X)$ follows a Uniform distribution on $[0,1]$. Repeating
simulations with different seeds, the resulting distribution would indeed be
uniform. As shown in the figure, if only one sample is observed, retrieving
the uniform distribution (as well as the dependence structure) is impossible.

##  3 Multivariate evaluation models

Generating evaluations on several candidates for each voter need to define a
multivariate distribution of evaluations. This section introduces
distributions for the random variables $E_{1},\dots,E_{m}$ corresponding to
voters’ evaluations for each of the $m$ candidates. First, i.i.d distributions
can be considered. Subsequently, the assumption of identical distributions
across candidates and the assumption of independence can be progressively
relaxed.

The simplest case of evaluation-based voting process is to consider that
voters’ evaluations of each candidate are i.i.d., which we will call an
Independent and Identically Distributed (IID) evaluations model.

######  Definition 1.

IID model
The Independent Identically Distributed (IID) evaluations model is such that
random variables $E_{1},\ldots,E_{m}$ are independent and identically
distributed.

As far as the evaluations are continuous and i.i.d., IID models almost surely
yield IC models on rankings. We propose hereafter alternatives to IID models.
The first possibility is to deal with independent and not identically
distributed distributions. Such models will be called Independent and
Differently Distributed (IDD) evaluations Models.

######  Definition 2.

IDD model
The Independent and Differently Distributed (IDD) evaluations model is such
that the random variables $E_{1},\ldots,E_{m}$ are independent and non
identically distributed.

The marginal distributions of $(E_{1},\ldots,E_{m})$ can be different by
changing parameters of a given distribution family, or by changing the
distribution family. As noticed previously, Beta (continuous) distributions
and Beta-Binomial (discrete) distributions enable to consider various shapes
of distributions.

A possibility to extend IID models and IDD models is to remove the
independence hypothesis between the evaluations of each candidate. In such a
case, the marginal distributions can be either identical or different. Hence,
these models can be considered as Dependent and Identically Distributed
evaluations models (DID).

######  Definition 3.

DID model
The Dependent and Identically Distributed (DID) evaluations model is such that
random variables $E_{1},\ldots,E_{m}$ are dependent and identically
distributed.

Considering non identical distributions provide Dependent and Differently
Distributed evaluations models (DDD).

######  Definition 4.

DDD model
The Dependent and Differently Distributed (DDD) evaluations model is such that
the random variables $E_{1},\ldots,E_{m}$ are dependent and non identically
distributed.

Many DDD models can be proposed. Some are detailed in the next section.

##  4 Construction of dependent models

Several approaches can lead to a DDD model. We propose first to use copula-
based models to introduce dependence between candidates. This approach enables
to define separately the univariate distributions of each candidate. Next, we
introduce multinomial and Dirichlet models to describe the special case of
cumulative voting. Finally, spatial models offer a large scope of multivariate
distributions.

###  4.1 Copula-based models

Copulas are useful tools to represent dependencies between variables. If the
distributions of the evaluations of each candidate are not independent, a
multivariate copula can be used to take into account their dependencies. In a
nutshell, a copula is a multivariate cumulative distribution function which
has all its margins uniformly distributed on the unit interval. It can also be
applied on transform of random variables to generate dependence with non
uniform marginals. See [24, 38] for a formal presentation of the subject.

######  Definition 5.

Copula DDD models
The Copula DDD models are defined by

$E=(E_{1},\ldots,E_{m})\sim C(\mathcal{D}_{1},\ldots\mathcal{D}_{m}),$

where $C$ is a multivariate copula and
$\mathcal{D}_{1},\ldots,\mathcal{D}_{m}$ are distributions on $\mathcal{E}$.
That is, $C$ is a function from $[0,1]^{m}$ to $[0,1]$ such that the
cumulative distribution function (c.d.f.) of $E$ writes as
$C(\Delta_{1}(E_{1}),\dots,\Delta_{m}(E_{m}))$ with $\Delta_{c}$ c.d.f. of
marginal distributions $\mathcal{D}_{c}$ of $E_{c}$, for $c=1,\dots,m$.

A strength of copulas is that they allow any marginal distributions.
Therefore, the model should specify both the marginal distribution for each
candidate, and the copula used to model the dependencies between variables. In
this sense, Copula DDD models include also IID, IDD and DID models, which
appear as particular cases of copula DDD models (see below).

Note that in Thurstone order statistics modeling, with continuous variables,
copulas were introduced by [34] and studied e.g. by [28]. In the following, we
distinguish the cases of a continuous evaluations set and a discrete
evaluations set $\mathcal{E}$.

#### Choice of a copula in a continuous case

In the continuous case, Gaussian copulas offer a simple way to model
dependencies between each pairs of candidates, through the correlation
coefficients. For a given correlation matrix $R\in\mathcal{R}^{m\times m}$,
the associated Gaussian copula writes as
$C_{R}(\mathcal{D}_{1},\ldots\mathcal{D}_{m})=\Phi_{R}(\Phi^{-1}(\mathcal{D}_{1}),\dots,\Phi^{-1}(\mathcal{D}_{m})),$
where $\Phi^{-1}$ is the inverse c.d.f. of a real standard normal and
$\Phi_{R}$ is the joint c.d.f. of a multivariate centered normal distribution
with covariance matrix $R$. The dependencies of the copula between two
variables are exactly characterized by the correlation coefficients. This
model has therefore the advantage to be easy to simulate and to enable to
define in a very comprehensive way the dependence between the evaluations.
Another usual family of copulas for the continuous case is Archimedean
copulas. We decide not to detail this family of copulas for the sake of
simplicity.

Another interesting copula class is the checkerboard copula class, see [15],
which represents a good compromise between the richness of the expression and
the complexity of the model. This copula is based on a partition of
$[0,1]^{m}$, $(B_{b})_{b\in\mathcal{B}}$,
$\mathcal{B}\subset\\{0,\dots,B-1\\}^{m}$, $B\in\mathbb{N}\setminus\\{0\\}$,
with $B_{b}=(b_{1}/B,(b_{1}+1)/B)\times\dots\times(b_{m}/B,(b_{m}+1)/B)$. On
each $B_{b}$, $b\in\mathcal{B}$, the copula is constant, equal to the value of
the empirical copula at the left corner, that is, the c.d.f.
$C(x_{1},\dots,x_{m})$ is proportional to the cardinal of
$\\{j,\;(e_{1j},e_{2j},\dots,e_{mj})\in B_{b},\;\forall
c\in{1,\dots,m},e_{cj}\leq b_{j}/B\\}$. This copula is, hence, a piecewise
constant version of the empirical copula. It reduces the overfitting compared
with the empirical copula. We refer _e.g._ to [21] for a discussion on the
choice of a copula.

#### Choice of the copula in a discrete case

Evaluations on discrete scales need the use of specific discrete copulas for
simulation. The discrete copula function $C$ has to be defined only on a
finite subset of $[0,1]^{m}$, which is in our context
$\\{0,\frac{1}{K},\dots,\frac{K-1}{K},1\\}^{m}$. Among others, Vines pair-
copulas, see [40], and Gaussian copulas, see [5], have been proposed to
simulate dependent discrete data, and therefore can be used also to model
discrete evaluations in a social choice framework. In particular, the Gaussian
copula is defined as in the continuous case.

#### Marginal distributions

Additionally, marginal distributions must be chosen. One can consider the same
distribution for each candidate and obtain DID models, or different
distributions following DDD models. Except for the Copula DID Uniform
continuous models, which belong almost surely to the IC model since all
candidates have the same distributions, each of these models allows for
different marginal distributions for the evaluations of each candidate.
Additionally to this non identical setting, the dependence modeling, through
the copula, yields a large scope of models. These copula-based models appear
very rich and adapted for covering much framework of simulations of
evaluations.

Figure 4 presents a simulation based on the use of two different marginal
distributions for two candidates (Beta distribution with parameters (0.7,0.5)
for the first candidate and (0.5,0.7) for the second one), using a Gaussian
copula of parameter 0.8.

Figure 4: Simulation examples of DDD Copula evaluations for $m=2$ candidates
and $n=200$ voters.

###  4.2 Multinomial and Dirichlet models for cumulative voting

DDD models suppose that there are dependencies between evaluations given by a
voter to the candidates. Among these models, a specific case is related to
cumulative voting, which consists in dividing a total score on the candidate
evaluations, such that the evaluations’ sum on the set of candidates is the
same for each voter. We refer for example to [12, 11] and references therein.
In such a context, the evaluations are not independent as there is a link
between the evaluations for all the candidates given by a single voter.

Both discrete and continuous models are available, using multinomial
distribution (a multivariate version of the Binomial distribution) in the
discrete case and Dirichlet distribution (a multivariate generalization of the
Beta distribution) for the continuous case.

When the evaluations are discrete, i.e. $\mathcal{E}=\\{0,\dots,K\\}$, the DDD
multinomial model is the following.

######  Definition 6.

DDD multinomial model
The DDD multinomial model is defined by
$(E_{1},\ldots,E_{m})\sim\mathcal{M}\\{K,p_{1},\ldots,p_{m}\\}$, where
$\mathcal{M}\\{K,p_{1},\ldots,p_{m}\\}$ is the multinomial distribution of
parameters $K$ and $p_{1},\ldots,p_{m}$, with for all $c=1,\ldots,m$,
$p_{c}\geq 0$ and $\sum_{c=1}^{m}p_{c}=1$.

The marginal distributions of a multinomial distribution are Binomial
distributions; the higher $p_{c}$, the higher the mean score of candidate $c$,
$c=1,\dots,m$. The multinomial model introduces dependence between the
evaluations of the candidates. More precisely, if parameters
$(p_{c})_{c=1,\dots,m}$ do not have the same value, then the evaluations are
not identically distributed and the associated multinomial model is then an
DDD model. If for all $c=1,\ldots,m$, $p_{c}=1/m$, then the associated
multinomial model is an DID model. Note that $K$ is the maximal authorized
grade.

Figure 5 shows an example of a DDD multinomial model for 3 candidates and
$K=6$, with the probability vector (0.5, 0.3, 0.2).

Figure 5: Barplots of evaluations for each of the $m=3$ candidates in a DDD
multinomial model with $K=6$ and a probability vector (0.5, 0.3, 0.2).

The continuous counterpart of the multinomial distribution is obtained through
the use of a Dirichlet distribution on $[0,1]^{m}$ as follows (see [39] for
details about Dirichlet distribution).

######  Definition 7.

DDD Dirichlet model
The DDD Dirichlet model is defined by
$(E_{1},\ldots,E_{m})\sim\mathcal{D}ir\\{\alpha_{1},\ldots,\alpha_{m}\\}$,
where $\mathcal{D}ir\\{\alpha_{1},\ldots,\alpha_{m}\\}$ is the Dirichlet
distribution of parameters $\alpha_{1},\ldots,\alpha_{m}$, with for all
$c=1,\ldots,m$, $\alpha_{c}>0$.

The Dirichlet distribution is an extension of the Beta distribution to the
multivariate case. The marginal distributions are Beta distributions and
Dirichlet modeling introduces dependence between the evaluations of the
candidates; the higher $\alpha_{c}$, the higher the mean score of candidate
$c$, $c=1,\dots,m$. If parameters $(\alpha_{c})_{c=1,\ldots,m}$ do not have
all the same value, then the Dirichlet model is an DDD model. If for all
$c=1,\ldots,m$, $\alpha_{c}=1$, then the Dirichlet model is an DID model. We
refer to [39, Chapter 1 and Chapter 2] and [31] for an overview on Dirichlet
distribution.

An example of simulation with 3 candidates using different probabilities for
the candidates is shown in Figure 6. The links between the evaluations of the
three candidates is shown in Figure 7.

Figure 6: Histograms of evaluations for each of the $m=3$ candidates in a DDD
Dirichlet model with parameters vector (5, 3, 2). Figure 7: Simulations of
evaluations for each of the $m=3$ candidates ($n=1000$ voters) in a DDD
Dirichlet model with (a) $(\alpha_{1},\alpha_{2},\alpha_{3})=(0.5,0.3,0.2)$,
(b) $(\alpha_{1},\alpha_{2},\alpha_{3})=(2,0.5,0.5)$, (c)
$(\alpha_{1},\alpha_{2},\alpha_{3})=(5,3,2)$.

###  4.3 Spatial models

Spatial voting simulations have been developed following the early work of
[18] for votes based on rankings. The model is based on the use of an
euclidean distance between the candidates and the voters, living in the same
uni- or multi-dimensional space: the smaller the distance, the better the
rank. [47] conclude that a spatial model “describes the observations in data
sets much more accurately” than other models.

We propose an adaptation of the spatial model in the framework of evaluation-
based voting. Let $d$ be a given dimension parameter. Parameter $d$ should be
seen as the number of latent characteristics which are used to build an
opinion on the candidates. Typically, $d=2$ or $3$, see [2] for a discussion
on the choice of $d$. Voters and candidates are then randomly generated as
points inside the hyperspace $\mathbb{R}^{d}$.

Spatial voting is next based on the distances between the generated points.
The closer a voter to a candidate, the higher their evaluation of this
candidate.

######  Definition 8.

DDD Spatial models
Let $x_{v}$, $v=1,\dots,n$, and $y_{c}$, $c=1,\dots,m$ be independent
realizations from a distribution on $\mathbb{R}^{d}$,
$d\in\mathbb{N}\setminus\\{0\\}$. The DDD spatial model for the evaluation
$e_{vc}$ of candidate $c$ by voter $v$ is defined as

$\forall v=1,\ldots,n,\forall c=1,\ldots,m,\quad
e_{vc}=f(\delta(x_{v},y_{c})),$

where $\delta$ is a distance between $x_{v}$ and $y_{c}$ and $f$ a non-
increasing function mapping $\mathbb{R}^{+}$ to $[0,1]$.

Typically, an intuitive spatial simulation model is given by the choices of

  * •

a spatial distribution for the voters and the candidates, $x_{v}$,
$v=1,\dots,n$ and $y_{c}$, $c=1,\dots,m$. A classical choice is the Uniform
distribution without any additive information about the voters, see Figure 9.
More specific distributions like a Gaussian distribution (resp. a mixing of
Gaussian distributions) allow to obtain a bigger concentration of voters in a
specific area of the hyperspace (resp. different areas), see e.g. [37].

  * •

a distance A usual distance is the Euclidean distance $\delta_{e}$. Numerous
distances exist, each with specific properties.

  * •

a link function, $f$. For instance, for $v=1,\ldots,n$ and $c=1,\ldots,m$,
$e_{vc}=\max\\{0,(1-\ell\times\delta_{e}(x_{v},y_{c}))\\}$ with $\delta_{e}$
the Euclidean distance and $\ell>0$. The parameter $\ell$ defines the
decreasing rate of the evaluations with respect to the distance. For example,
$\ell$ greater than $2$ ensures that a voter being anywhere on the frontier of
the unit cube will give a null score to a candidate who is on the center of
the unit cube or, equivalently, that a voter who is on the center of the cube
will give a null score to an extreme candidate on the frontier of the cube.

Other link functions $f$ are also possible, as for example the sigmoïd which
is defined as follows: for $v=1,\ldots,n$, $c=1,\ldots,m$,
$e_{vc}=\bigl{(}1+e^{\lambda(\beta\delta_{e}(x_{v},y_{c})-1)}\bigr{)}^{-1}$,
with $\lambda>0$ and $\beta>0$. Figure 9 presents an example of such a
function for $\lambda=5$ and $\beta=2$.

Figure 8: Simulation example of spatial model, with $m=2$ candidates
($\blacktriangle$ is candidate 1 and $\bullet$ is candidate 2) and $n=100$
voters in a 2-dimensional space ($d=2$), obtained through an Uniform
distribution on $[0,1]^{2}$.

Figure 9: Example of sigmoïd transformation,
$\delta\mapsto\bigl{(}1+e^{\lambda(\beta\delta-1)}\bigr{)}^{-1}$, with
$\lambda=5$ and $\beta=2$.

For a given position of candidates in $[0,1]^{d}$, the spatial model is
clearly an DDD model, since the marginal distributions are different (see for
example Figure 10) and potentially correlated (see for example Figure 11).

An example of evaluations obtained through a spatial model is presented in
Figure 9. Two candidates and $n=100$ voters have been randomly generated in
$[0,1]^{2}$, that is, with $d=2$. Evaluations are then obtained from the
euclidean distance of each voter to each candidate. We propose in Figure 10
histograms of three different ways to obtain evaluations, based on the same
spatial situation (the one described in Figure 9), for each candidate:

  * •

evaluations in model 1 are obtained using
$e_{vc}=max\\{0,(1-2\times\delta_{e}(x_{v},y_{c}))\\}$,

  * •

evaluations in model 2 are obtained using
$e_{vc}=(1+e^{\lambda(\beta\delta_{e}(x_{v},y_{c})-1)})^{-1}$ with $\lambda=5$
and $\beta=2$,

  * •

evaluations in model 3 are obtained using
$e_{vc}=(1+e^{\lambda(\beta\delta_{e}(x_{v},y_{c})-1)})^{-1}$ with $\lambda=2$
and $\beta=2$.

Figure 10: Histograms of evaluations obtained through the spatial model of
Figure 9 and calculation models 1, 2 and 3 defined above.

Figure 11: Plot of evaluations of candidate 2 vs candidate 1 given by each
voter, with calculation models 1, 2 and 3 defined above.

Spatial models with discrete evaluations on $\\{0,\dots,K\\}$ can easily be
obtained from continuous models by dividing the $[0,1]$ interval onto the
$K+1$ intervals. Then if the continuous grade obtained with the spatial model
belongs to the $l^{\text{th}}$ interval, the discrete evaluation is set to
$l-1$. That is, for a continuous evaluation $e_{vc}$ on $[0,1]$, we consider
$\lfloor(K+1)e_{vc}\rfloor$ as the resulting discrete evaluation, where
$\lfloor u\rfloor$ denotes the greatest integer lower than $u$. The spatial
interpretation of such a process is to determine $K$ spheres centered on the
candidate and to give the evaluation of $K$ if the voter is into the smallest
sphere, $K-1$ for the second smallest sphere and so on, until 0.

Note that the spatial model is interesting for generating evaluations since it
offers various dependence structures between voters. Moreover, it is easily
interpretable, with the latent space $\mathbb{R}^{d}$ seen as the space of the
criteria used for discriminating the candidates. Considering the fitting power
of spacial model, its non-parametric nature makes it a very versatile tool,
but with a high risk of overfitting. Yet, a drawback of this modeling is the
fact that we cannot explicit the resulting distribution on evaluations, unlike
when using parametric copulas models.

##  5 Fitting real data

We determine which models defined in the previous sections fit better real
data sets.

In the easiest cases, a parametric approach will fit the real data in a
satisfactory way. It will be very convenient for the interpretation of the
study. For example, mixture models can be a solution to deal with
heterogeneous situations and capture the complexity of real life situations.
But a danger is then to overfit the data with too much parameters. A non
parametric approach can also be considered. Yet, it can lead to more
complicated interpretation of the model.

######  Remark 2.

Note that a Bayesian procedure could also be used to fit the models (see e.g.
[51, 35, 27] for Bayesian estimation of Thurstone order statistics models). We
rather use a frequentist approach here.

Except for spatial modeling, a guideline for the choice of a specific model
could be the following:

  1. 1.

Choose a general parametric model for each distribution and fit the marginal
distributions of the evaluations of each candidate, $E_{1},\ldots,E_{m}$ to
the chosen model.

  2. 2.

Test if these marginal distributions can be considered as identical.

  3. 3.

Test the independence of $E_{1},\ldots,E_{m}$ and propose a dependence
modeling if the evaluations are not independent.

  4. 4.

Build the final model, using previous steps.

The results of the distributions equality and independence tests lead to
choose the appropriate model within IID, IDD, DID and DDD models.

This guideline is relevant only in case of small amount of data, as it is
well-known that statistical tests are inclined to systematically reject the
null hypothesis when the sample size is too large, see [30, 32]. So we will
not do any goodness-of-fit hypothesis test, but rather compare the quality of
fitting for different models by providing a distance statistic between the
observed distribution and the modeled one.

As a matter of illustration, we propose in the following two examples of
fitting real voting situations through the proposed models. The first one
deals with continuous evaluations, whereas the other focus on discrete
evaluations.

###  5.1 Continuous case

The first example, in a continuous framework, is based on the use of a survey
concerning the 2017 presidential election in France. Data are available in
[8], and deal with 13 candidates evaluations on $\\{0,\dots,100\\}$ by
$n=20210$ voters. We transform this 0-100 scale into a continuous scale,
adding to any value given by a voter to a candidate a random value uniformly
distributed between $0$ and $1$. These values included in the $[0,101]$
interval are then scaled to the $[0,1]$ interval.

We focus on $m=3$ candidates: François Fillon (FF), Benoit Hamon (BH) and
Emmanuel Macron (EM) (who finally won the election). Illustrations of the
observed distributions are presented in Figure 12, with the distributions of
Uniform, truncated Normal, Beta and checkerboard copula models.

Figure 12: Histograms of observed evaluations of $m=3$ candidates given by
$n=20210$ voters at the French 2017 election. Uniform model distribution is in
dashed black, Beta model in red, truncated Normal in blue and checkerboard
copula in solid black.

#####  Step 1. Fit a parametric model for each candidate’s evaluations.

When choosing a general parametric model for each marginal evaluations
distributions, considering empirical evaluations distributions is the first
thing to do. The marginal distributions are simply the sample distributions,
divided into $G$ classes. We tried $G=101$ to recover the initial discrete
values on the 0-100 scale, and, as suggested in [15], we also tried $G=40$. In
practice this approach is often efficient to fit a dataset but not appropriate
to generate a predictive modeling, as the risk of overfitting is strong.

We then propose to model the marginal distributions using either Uniform
distributions, Beta distributions or Truncated Normal distribution as a matter
of example. Obviously, truncated Normal distribution does not seem accurate
with these data, since, as illustrated in Figure 12, the shapes of the
distributions are far from a Gaussian curve.

The Uniform distribution does not need any parameter estimation. The Beta
distribution depends on two parameters $\alpha$ and $\beta$. These parameters
can be estimated using the method of moments. Let $\hat{\mu}$ denote the
sample mean and $s$ the sample standard deviation. Then parameters $\alpha$
and $\beta$ can be estimated respectively by
$\hat{\alpha}=\hat{\mu}\left({\frac{\hat{\mu}(1-\hat{\mu})}{s^{2}}}-1\right)$
and
$\hat{\beta}=(1-\hat{\mu})\left({\frac{\hat{\mu}(1-\hat{\mu})}{s^{2}}}-1\right)$.
The truncated Normal distribution depends on two parameters, the mean and
standard deviation, which are estimated via a maximum likelihood method (see
[13]), using the package developed by [50].

The first three lines of Table 2 gives the Kolmogorov-Smirnov statistic,
corresponding to the distance between the distribution of observed evaluations
and the distribution of simulated evaluations. The smaller the distance, the
more the model fits the sample distribution. Concerning the non parametric
approach, the statistics are presented in the last two lines of Table 2. It is
obvious that the empirical distribution with $G=101$ classes overfits the
data. The empirical distribution with $G=40$ classes is competitive with the
Beta distribution.

| $FF$ | $BH$ | $EM$
---|---|---|---
Uniform marginals dist. | $0.570$ | $0.354$ | $0.180$
Trunc. normal marginals dist. | $0.492$ | $0.344$ | $0.179$
Beta marginals dist. | $0.269$ | $0.099$ | $0.119$
Sample dist. divided into 40 classes | $0.267$ | $0.142$ | $0.105$
Sample dist. divided into 101 classes | $0.006$ | $0.007$ | $0.005$
Table 2: Kolmogorov-Smirnov statistic for 5 fitting models.

##### Step 2. Test if these marginal distributions can be considered as
identical.

It is straightforward from Figure 12 that the marginal distributions are
different, and, hence, that the modeling of the data is either DID or DDD.
Note that in the general case, such test can be done with a Kruskall-Wallis
test.

##### Step 3. Test the independence of $E_{1},\ldots,E_{m}$ and model the
dependence if necessary.

Let us now focus on the independence between the evaluations. As one can see
in Table 3, evaluations given to FF and BH are negatively correlated, whereas
evaluations given to FF and EM are slightly positively correlated, and
evaluations given to BH and EM are not correlated. Based on the observed
correlation values, the independence assumption is not realistic. Formally, a
Bartlett’s sphericity test on the correlation matrix can be applied to check
the dependence property. We, hence, consider a DDD model. We consider below,
as a matter of example, copulas models to capture the dependencies between the
evaluations given to the candidates.

######  Remark 3.

Dirichlet model deals with cumulative voting, where the evaluations provided
by each voter for all candidates collectively sum to a constant. Obviously
this is not the case here, as shown in Figure 13.

Figure 13: Histogram of the sum of the evaluations given by each voter to the
candidates

We consider two Copula-based models.

  * •

Parametric Copula model. We consider a Gaussian copula, with marginal
distributions fitted by Beta distributions as established above, and
correlation coefficients equal to the observed correlation coefficients shown
in Table 3. It has the advantage to be parametric and to provide a
reproducible modeling.

  * •

Non parametric Copula model. We consider a checkerboard Copula model, which
does not require any assumption on the marginal distributions. The
checkerboard copula model is simply the multivariate sample distribution,
divided into $B^{m}$ classes, where the marginal distributions are determined
as above. The value of $B$ in the following is 40. Note that $B=101$ leads to
the empirical copula which overfits the data.

Non-parametric Copula model exhibits a superior fit to the data compared to
the parametric Copula model, particularly when the number of classes is
sufficiently large. This outcome is unsurprising, given the inherent
parametric/non-parametric nature of the respective models.

Table 4 displays the correlation coefficients obtained with the two modelings.

| $FF$ | $BH$ | $EM$
---|---|---|---
FF | $1$ | $-0.41$ | $0.30$
BH | $-0.41$ | $1$ | $0.0008$
EM | $0.30$ | $0.0008$ | $1$
Table 3: Empirical correlations between three candidates at the French 2017
presidential election.

| $FF$ | $BH$ | $EM$
---|---|---|---
FF | $1$ | $-0.33$ | $0.23$
BH | $-0.33$ | $1$ | $0.002$
EM | $0.23$ | $0.002$ | $1$

| $FF$ | $BH$ | $EM$
---|---|---|---
FF | $1$ | $-0.41$ | $0.30$
BH | $-0.41$ | $1$ | $0.01$
EM | $0.30$ | $0.01$ | $1$

Table 4: Correlations between candidates obtained with copula DDD models at
the French 2017 presidential election. On the left, correlations obtained with
a Normal copula and Beta marginals, on the right correlations obtained through
the use of a checkerboard copula with $B=40$ classes associated with empirical
marginal distributions with $G=40$ classes.

##### Step 4. Build the final model, using previous steps.

As a conclusion, in this case Copula models seems more appropriate with Beta
marginal distributions. A non-parametric approach of course better fits the
data distributions, but a parametric approach seems more appropriate to model
the situation without overfitting.

##### Spatial representation

The spatial model introduced in Section 4.3 can also be used as a
representation of candidates and voters in the same space. However, the
experiments show that it is difficult to simulate new data from a spatial
representation: even if the spatial representation of the voting situation is
accurate, it is not an easy task to identify the latent space (its dimension
$d$ and its metric) nor the distribution of the voters in the voting space.
Suppose the dimension $d$ known, that the metric is given, and let us focus on
the distributions of voters in the latent space. We propose a two steps
process, to simulate new data:

  1. 1.

estimate both candidates and voters positions into a $d$-dimensions space, for
example by the use of the SMACOF method, see [16],

  2. 2.

estimate the distribution of voters into the $d$-dimensions space, in order to
generate new voters with the same distribution.

The second step needs to fit a multidimensional distribution. The fitting is
not done directly on the data but on the latent positions obtained with the
first step. This can be done for example with a Copula model, as previously.

We choose hereafter to consider $d=1$, $d=2$ and $d=3$, and the euclidean
distance. We refer to [42] for the choice of $d$. As stated therein, the
choice is not easy. The higher $d$, the higher the complexity of the model.
Actually, choosing $d\leq 3$ allows an easy visualization of the latent space.
The spatial representations obtained by the SMACOF algorithm are shown in
Figure 14. As shown by the figure, the distribution with $d=3$ does not differ
much to the distribution obtained with $d=2$. Considering $d=2$ seems
sufficient to capture the information on the evaluations. This supports the
conclusion of [2] that small values of $d$ are often sufficient.

Figure 14: Spatial representation of candidates (in red) and voters of the
French 2017 election with by the SMACOF algorithm, in dimension $d=1$, $d=2$
and $d=3$, from left to right.

Hence, from real data, we have calibrated the parameters of a spatial model,
which allows generating new data. Note that the intrinsic dimension and the
used distance are to be set by the user. Some post-hoc measures of the quality
of the adjustment are available for comparing different choices, see [16,
chapter 4].

###  5.2 Discrete case

We study in this section voting data collected in the general assembly of a
french scout association where 7 options have been presented to 93 voters for
a specific resolution222Data are available in the supplementary material. The
association is the Eclaireuses and Eclaireurs Unionistes de France and the
general assembly stood in January 2022 in Bordeaux, France. Data are from
private communication. . The voting procedure used in this assembly was the
majority judgment with evaluations on a 7 levels discrete scale.

#####  Step 1. Fit a parametric model for each candidate’s evaluations.

We first propose to model the marginal distributions on the candidates using
Uniform distributions, Binomial distributions and Beta-Binomial distributions.

The Binomial distribution needs the estimation of a parameter $p$, which can
be estimated by $\hat{p}=\frac{\hat{\mu}}{K+1}$, where $\hat{\mu}$ is the
sample mean and $K+1$ is the number of scales in the evaluations. The Beta-
Binomial distribution needs the estimation of parameters $\alpha$ and $\beta$,
obtained as follows:
$\hat{\alpha}=\frac{K\hat{\mu}-\hat{\mu}^{2}-s^{2}}{K(s^{2}/\hat{\mu}-1)+\hat{\mu}}$
and
$\hat{\beta}=\frac{(K-\hat{\mu})(K-\hat{\mu}-s^{2}/\hat{\mu})}{K(s^{2}/\hat{\mu}-1)+\hat{\mu}}$,
with $\hat{\mu}$ the sample mean and $s$ the sample standard deviation, see
[49].

For each candidate, we compute the $\chi^{2}$ statistics of the distances
between the distribution of observed evaluations and the theoretical Uniform,
Binomial and Beta-Binomial distributions. The smaller the distance, the better
the model fits the observed distribution.

Table 5 shows the summary statistics of the $\chi^{2}$ distances between the
observed distributions of the evaluations and the simulated distributions for
the 7 candidates. The Beta-Binomial distribution has a better fit than the
Binomial or the Uniform distributions. Bar plots for observed distributions
and models for three propositions are displayed in Figure 15.

Proposition | $A$ | $B$ | $C$ | $D$ | $E$ | $F$ | $G$
---|---|---|---|---|---|---|---
binomial | $225.98$ | $58.11$ | $41.19$ | $24.81$ | $163.46$ | $121.74$ | $2042.11$
uniform | $79.14$ | $16.82$ | $23.90$ | $61.38$ | $21.33$ | $18.62$ | $100.21$
beta-binomial | $8.23$ | $18.91$ | $18.30$ | $48.94$ | $27.10$ | $16.88$ | $9.60$
Table 5: Summary of $\chi^{2}$ distances between the observed distributions of
the evaluations and the simulated distributions for the 7 candidates. Figure
15: Bar plots of observed values for propositions C, F, and G. The dashed
black line represents the uniform distribution. The distribution in red
corresponds to the binomial model, and the distribution in blue corresponds to
the beta-binomial model.

##### Step 2. Test if these marginal distributions can be considered as
identical.

Figure 15 shows that the hypothesis of identical distributions cannot be
assumed. We will hence consider DID or DDD models.

##### Step 3. Test the independence of $E_{1},\ldots,E_{m}$ and model the
dependence if necessary.

On the 21 pairs of candidates 9 have a negative correlation, and 12 have a
positive correlation. The correlation coefficient is significantly different
to zero for 7 pairs of candidates. This shows the importance of taking into
account dependence in modeling. Therefore, as in the continuous case, the
evaluations in the discrete case are not independent and we will consider a
DDD model.

As a matter of example, Table 6 shows the correlations between the three
candidates C, F and G. First, the marginal distributions are fitted as
described above by a Beta-Binomial distribution. Then, a Gaussian discrete
copula is used to model the correlation between candidates. A checkerboard
copula based on $B=40$ classes has also been used, considering empirical
distributions on the same $G=40$ classes of the copula. The obtained
correlations are presented in Table 6, and bar plots are presented in Figure
16.

##### Step 4. Build the final model, using previous steps.

The discrete copula captures the dependence structure, but may introduce
overfitting of the dependence, when generating simulated observations. No
parametric modeling, among the ones proposed, are well adapted in such
context. In this case, a non parametric approach seems more convenient.

| $C$ | $F$ | $G$
---|---|---|---
C | $1$ | $-0.18$ | $-0.37$
F | $-0.18$ | $1$ | $0.47$
G | $-0.37$ | $0.48$ | $1$

| $C$ | $F$ | $G$
---|---|---|---
C | $1$ | $-0.15$ | $-0.18$
F | $-0.15$ | $1$ | $0.48$
G | $-0.18$ | $0.48$ | $1$

Table 6: Spearmann correlations between the evaluations of three candidates.
On the left the observed correlations, on the right the correlations obtained
with a Copula DDD Beta-Binomial model. Figure 16: Correlation plots of
observed evaluations between candidates C and G (left) and candidates F and G
(right).

######  Remark 4.

Note that a multinomial model is not accurate in the observed situation as the
sum of scores for each voters is neither fixed nor equal to the maximum score.

##  6 Conclusion

As explained in the preamble, simulations can be done in two different
settings.

  * •

On the one hand, simulations can be done without any specific context, and the
tuning of the distribution of the evaluations is let free or determined by
external considerations. One has therefore to choose a model and set the
parameters to arbitrary values. Examples of such simulation settings have been
proposed with the description of each model above.

  * •

On the other hand, one can wish to simulate observations in harmony with real
data. In that case, an adjustment of the model to the observed data is
necessary. The aim is therefore 1) to choose the appropriate model 2) to infer
the model parameters from the available data. This situation is detailed in
Section 5 for both discrete and continuous cases.

We introduced in this paper several models to simulate evaluation-based voting
data. Three main families of distributions were proposed for the marginal
distributions of the evaluations, in a continuous setting and in a discrete
setting. On the contrary to preference rankings models, where the key notion
is the impartiality, a more refined discussion is needed for evaluation-based
processes.

The simplest model is when evaluations are independent and identically
distributed (IID models). We propose first to distinguish either the marginal
distributions are identical or not (IDD models). Such models do not imply
Impartial Culture on preferences. Next, introducing dependence (DDD models)
creates more complex models. We give examples of dependent distributions with
identical marginals (DID models) which provide Impartial Culture on
preferences. In particular, we introduce Copula DID and DDD models which allow
to model the dependence between the evaluations. The variety of modeling
described here offers the possibility of studying the properties of
evaluation-based voting processes with an extensive probabilistic approach. It
also provides new IC and non IC simulation approaches for preferences, since
preferences rankings can be deduced from evaluations. Finally, as some
proposed settings are parametric, the studied models can be fitted to real
dataset to deduce more realistic frameworks. We present examples of such an
approach on real data for continuous and discrete evaluations.

Finally, the R package VoteSim in [43], available on the CRAN repository, aims
at providing the models introduced in this paper for simulating distributions
of evaluations given by voters on candidates concerning a voting situation
within an evaluation-based perspective. The proposed simulation functions
cover Uniform, Truncated Normal, Beta, Dirichlet, multinomial, copula-based
and spatial models. Basically a function is proposed for each model which
produces a matrix with candidates in rows and voters in columns.

## References

  * [1] Mayer Alvo and LH Philip.  Statistical methods for ranking data, volume 1341.  Springer, New York, 2014.
  * [2] David A. Armstrong, Rayan Bakker, Roice Carroll, Christopher Hare, Keith T. Poole, and Howard Rosenthal.  Analyzing Spatial Models of Choice and Judgment (2nd ed.).  CRC Press, Boca Raton, 2021.
  * [3] Jean-Baptiste Aubin, Irène Gannaz, Samuela Leoni, and Antoine Rolland.  Deepest voting: A new way of electing.  Mathematical Social Sciences, 116:1–16, 2022.
  * [4] Michel Balinski and Rida Laraki.  A theory of measuring, electing and ranking.  Proceedings of the National Academy of Sciences, 104(21):8720–8725, 2007.
  * [5] Alessandro Barbiero and Pier Alda Ferrari.  An R package for the simulation of correlated discrete variables.  Communications in Statistics - Simulation and Computation, 46(7):5123–5140, 2017.
  * [6] Daniel Berend and Luba Sapir.  Monotonicity in condorcet’s jury theorem with dependent voters.  Social Choice and Welfare, 28:507–528, 2007.
  * [7] Sven Berg.  Evaluation of some weighted majority decision rules under dependent voting.  Mathematical Social Sciences, 28(2):71–83, 1994.
  * [8] Sylvain Bouveret, Renaud Blanch, Antoinette Baujard, François Durand, Herrade Igersheim, Jérôme Lang, Annick Laruelle, Jean-François Laslier, Isabelle Lebon, and Vincent Merlin.  Voter autrement 2017 - online experiment, July 2018.
  * [9] Steven J. Brams and Peter C. Fishburn.  Approval voting.  Springer, New York, second edition, 2007.
  * [10] Lorán Chollete, Victor de la Peña, and Michael Klass.  The price of independence in a model with unknown dependence.  Mathematical Social Sciences, 123:51–58, 2023.
  * [11] Duane Cooper and Arthur Zillante.  A comparison of cumulative voting and generalized plurality voting.  Public Choice, 150:363–383, 2012.
  * [12] Duane A Cooper.  The potential of cumulative voting to yield fair representation.  Journal of Theoretical Politics, 19(3):277–295, 2007.
  * [13] Bradford R. Crain.  Estimating the parameters of a truncated normal distribution.  Applied Mathematics and Computation, 5(2):149–156, 1979.
  * [14] Douglas E Critchlow, Michael A Fligner, and Joseph S Verducci.  Probability models on rankings.  Journal of mathematical psychology, 35(3):294–318, 1991.
  * [15] Andrés Cuberos, Esterina Masiello, and Véronique Maume-Deschamps.  Copulas checker-type approximations: Application to quantiles estimation of sums of dependent random variables.  Communications in Statistics - Theory and Methods, 49(12):3044–3062, 2020.
  * [16] Jan de Leeuw and Patrick Mair.  Multidimensional scaling using majorization: SMACOF in R.  Journal of Statistical Software, 31(3):1–30, 2009.
  * [17] Mostapha Diss and Eric Kamwa.  Simulations in models of preference aggregation.  Œconomia, 10(2):279–308, 2020.
  * [18] Anthony Downs.  An economic theory of political action in a democracy.  Journal of Political Economy, 65(2):135–150, 1957.
  * [19] Jeffrey A Dubin and Elisabeth R Gerber.  Patterns of voting on ballot propositions: A mixture model of voter types.  Technical report, California Institute of Technology, 1992.
  * [20] François Durand.  Towards less manipulable voting systems.  Theses, Université Pierre et Marie Curie - Paris VI, September 2015\.
  * [21] Valdo Durrleman, Ashkan Nikeghbali, and Thierry Roncalli.  Which copula is the right one?  SSRN Electronic Journal, 08 2000.
  * [22] Pierre Favardin and Dominique Lepelley.  Some further results on the manipulability of social choice rules.  Social Choice and Welfare, 26(3):485–509, 2006.
  * [23] W.V. Gehrlein.  Condorcet’s Paradox.  Theory and Decision Library C. Springer, Berlin Heidelberg, 2006.
  * [24] Christian Genest and Anne-Catherine Favre.  Everything you always wanted to know about copula modeling but were afraid to ask.  Journal of Hydrologic Engineering, 12(4):347–368, 2007.
  * [25] Isobel Claire Gormley and Thomas Brendan Murphy.  Exploring voting blocs within the Irish electorate: A mixture modeling approach.  Journal of the American Statistical Association, 103(483):1014–1027, 2008.
  * [26] James Green-Armytage, Nicolaus Tideman, and Rafael Cosman.  Statistical evaluation of voting rules.  Social Choice and Welfare, 46(1):183–212, 2016.
  * [27] John Guiver and Edward Snelson.  Bayesian inference for Plackett-Luce ranking models.  In proceedings of the 26th annual international conference on machine learning, pages 377–384, 2009.
  * [28] Harry Joe.  Multivariate extreme value distributions and coverage of ranking probabilities.  Journal of Mathematical Psychology, 45(1):180–188, 2001.
  * [29] Krishna K Ladha.  Information pooling through majority-rule voting: Condorcet’s jury theorem with correlated votes.  Journal of Economic Behavior & Organization, 26(3):353–372, 1995\.
  * [30] Björn Lantz.  The large sample size fallacy.  Scandinavian Journal of Caring Sciences, 27(2):487–492, 2013.
  * [31] Jiayu Lin.  On the Dirichlet distribution.  Technical report, Department of Mathematics and Statistics, Queens University, 2016.
  * [32] Mingfeng Lin, Henry C. Lucas, and Galit Shmueli.  Research commentary: Too big to fail: Large samples and the p-value problem.  Information Systems Research, 24(4):906–917, 2013.
  * [33] R Duncan Luce.  Individual choice behavior: A theoretical analysis.  Wiley, New York, 1959.
  * [34] D McFadden.  Modeling the choice of residential location.  Spatial Interaction Theory and Planning Models, pages 75–96, 1978\.
  * [35] Glenn B Milewski.  Using Bayesian methods to evaluate Thurstone’s simple structure concept.  Fordham University, New York, 2004.
  * [36] Thomas Brendan Murphy and Donal Martin.  Mixtures of distance-based models for ranking data.  Computational statistics & data analysis, 41(3-4):645–655, 2003\.
  * [37] Anghel Negriu and Cyrille Piatecki.  On the performance of voting systems in spatial voting simulations.  Journal of Economic Interaction and Coordination, 7(1):63–77, 2012\.
  * [38] Roger B. Nelsen.  An Introduction to Copulas.  Springer, New York, second edition, 2006.
  * [39] Kai Wang Ng, Guo-Liang Tian, and Man-Lai Tang.  Dirichlet and related distributions: Theory, methods and applications.  John Wiley & Sons, New York, 2011.
  * [40] Anastasios Panagiotelis, Claudia Czado, and Harry Joe.  Pair copula constructions for multivariate discrete data.  Journal of the American Statistical Association, 107(499):1063–72, 2012.
  * [41] Florenz Plassmann and Nicolaus Tideman.  How frequently do different voting rules encounter voting paradoxes in three-candidate elections?  Social Choice and Welfare, 42(1):31–75, January 2014.
  * [42] Keith T Poole, Fallaw B Sowell, and Stephen E Spear.  Evaluating dimensionality in spatial voting models.  Mathematical and Computer Modelling, 16(8-9):85–101, 1992.
  * [43] Antoine Rolland and Nathan Grimault.  voteSim: Generate Simulated Data for Voting Rules using Evaluations, 2023.  R package version 0.1.0.
  * [44] Warren D. Smith.  Range voting, 2000.
  * [45] Louis L Thurstone.  A law of comparative judgment.  Psychological review, 34(4):273, 1927.
  * [46] Nicolaus Tideman and Florenz Plassmann.  Developing the empirical side of computational social choice.  In Annals of Mathematics and Artificial Intelligence, volume 68, 01 2012.
  * [47] Nicolaus Tideman and Florenz Plassmann.  Modeling the outcomes of vote-casting in actual elections.  In Dan S. Felsenthal and Moshé Machover, editors, Electoral Systems: Paradoxes, Assumptions, and Procedures, pages 217–251, New York, 2012\. Springer.
  * [48] Nicolaus Tideman and Florenz Plassmann.  Which voting rule is most likely to choose the “best” candidate?  Public Choice, 158(3/4):331–357, 2014.
  * [49] Ram C. Tripathi, Ramesh C. Gupta, and John Gurland.  Estimation of parameters in the beta binomial model.  Annals of the Institute of Statistical Mathematics, 46(2):317–331, 1994.
  * [50] Stefan Wilhelm and Manjunath B G.  tmvtnorm: Truncated Multivariate Normal and Student t Distribution, 2023.  R package version 1.6.
  * [51] Philip LH Yu.  Bayesian analysis of order-statistics models for ranking data.  Psychometrika, 65:281–299, 2000.