# Extending the symmetry of the massless Klein-Gordon equation under the
general disformal transformation

Allan L. Alinea111corresponding author and Mark Ruel D. Chua  Institute of
Mathematical Sciences and Physics, University of the Philippines Los Baños,
4031 College, Los Baños, Laguna, Philippines<EMAIL_ADDRESS>
###### Abstract

The Klein-Gordon equation, one of the most fundamental equations in field
theory, is known to be not invariant under conformal transformation. However,
its massless limit exhibits symmetry under Bekenstein’s disformal
transformation, subject to some conditions on the disformal part of the metric
variation. In this study, we explore the symmetry of the Klein-Gordon equation
under the general disformal transformation encompassing that of Bekenstein and
a hierarchy of ‘sub-generalisations’ explored in the literature (within the
context of inflationary cosmology and scalar-tensor theories). We find that
the symmetry in the massless limit can be extended under this generalisation
provided that the disformal factors takes a special form in relation to the
conformal factor. Upon settling the effective extension of symmetry, we
investigate the invertibility of the general disformal transformation to avoid
propagating non-physical degrees of freedom upon changing the metric. We
derive the inverse transformation and the accompanying restrictions that make
this inverse possible.

  * August 2022

Keywords: disformal transformation, Klein-Gordon equation, symmetry,
invertibility

##  1 Introduction

Disformal transformation [1] is a metric transformation introduced by
Bekenstein in 1992 as a generalisation of the more commonly known conformal
transformation [2, 3]. The aim was to relate Finsler and Riemann geometries in
a single gravitational theory. Since then, disformal transformation found its
utility in many cutting-edge research areas in Physics beyond the original
scope of Bekenstein’s work. In the study of scalar-tensor theories, disformal
transformation plays an important role in relating one action to another. This
is with an insight to solve pressing problems about the Universe such as dark
energy and dark matter, and investigate the physical consistency of the
resulting action upon transformation; see Refs. [4, 5, 6, 7, 8, 9, 10]. In the
area of inflationary cosmology, disformal transformation is used in the study
of G-inflation and beyond [11, 12], Higgs inflation [13], and disformal
inflation [14], to name a few. The investigation of disformal transformation
also extends to hairy black holes [15], D-brane cosmology [16], particle
motion in semi-classical gravity [17], etc.

As in any other areas of physics involving the study of transformation,
symmetry is an ‘omnipresent’ idea that has to be taken into account. Of
particular interest are transformations that lend physical observables and
equations of motion describing physical systems, invariant. For disformal
transformation, it is found that both the gauge-invariant scalar and tensor
primordial cosmological perturbations [18, 19, 20, 21, 22]—the seeds that gave
rise to the current universe—are invariant under a hierarchy of this
transformation [23, 24, 25, 26, 27]. Furthermore, the ‘trinity’ of
relativistic field equations, namely, Klein-Gordon equation [28], Maxwell’s
equations [29], and Dirac equation (under the Inomata condition) [30] are
symmetric with respect to disformal transformation, subject to some
constraints on the conformal and disformal factors in the variation.

The focus of this work is the disformal transformation of the Klein-Gordon
equation. Falciano and Goulart [28] showed that subject to some conditions,
the massless Klein-Gordon equation is invariant under the disformal
transformation of the metric $g_{\mu\nu}$ that we rewrite as222Ref. [28]
started out with what we identify as the disformal transformation of the
inverse metric. Here, we follow a slightly different notation; i.e.,
$A_{FG}=1/A$ and $B_{FG}=-B/[A(A-2BX)]$, where the subscript ‘FG’ pertains to
Ref. [28].

$\displaystyle
g_{\mu\nu}\rightarrow\widehat{g}_{\mu\nu}=A(\phi,X)g_{\mu\nu}+B(\phi,X)\phi_{;\mu}\phi_{;\nu},$
(1)

where $A$ and $B$ are the conformal and disformal factors, respectively. They
depend on the scalar field $\phi$ and the kinetic term $X$ defined as
$X\equiv-\tfrac{1}{2}g^{\mu\nu}\phi_{;\mu}\phi_{;\nu}$. Subject to the
condition given by333The constant $c$ is a minor correction in the result in
Ref. [28] where $c^{2}$ is simply unity.

$\displaystyle B=\frac{A-c^{2}A^{n-1}}{2X},\quad(c=\text{const.})$ (2)

where $n$ is the dimension of the background spacetime, the Klein-Gordon
equation for the scalar field with mass $m$, transforms444We are using the
metric signature $(-,+,+,+)$. as
$\square\phi-m^{2}\phi=0\rightarrow(\square\phi)/(c^{2}A^{n-1})-m^{2}\phi=0$.
Consequently, it is invariant under the disformal transformation in the
massless limit. For the massive case, the mass may be seen as transforming
from $m^{2}\rightarrow m^{2}/(c^{2}A^{n-1})$.

The transformation given by (1) is not the only form of disformal
transformation. It can be generalised to a form involving higher-order
derivatives of $\phi$ and/or derivatives of the kinetic term. It is then
interesting to see if the Klein-Gordon equation would remain invariant under
such generalised transformation, and identify the needed conditions, if any,
for the preservation of symmetry. In this work, we consider a general
disformal transformation given by555As in Bekenstein’s disformal
transformation given by (1), this generalisation does not necessarily preserve
the metric signature. Interested readers may refer to Ref. [6] and references
therein for the discussion on restrictions involving Lorentzian signature and
causality.  [23, 31]

$\displaystyle
g_{\mu\nu}\rightarrow\widehat{g}_{\mu\nu}=Ag_{\mu\nu}+\Phi_{\mu}\Phi_{\nu},\quad\text{where}\quad\Phi_{\mu}\equiv
C\phi_{;\mu}+DX_{;\mu},$ (3)

with $A,C,$ and $D$ being functionals of $(\phi,X,Y,Z)$, in general, and the
quantities $Y$ and $Z$ are defined as $Y\equiv g^{\mu\nu}\phi_{;\mu}X_{;\nu}$
and $Z\equiv g^{\mu\nu}X_{;\mu}X_{;\nu}$. Note that this transformation
involves a derivative of the kinetic term in addition to that of the scalar
field; effectively, the second-order derivative of $\phi$ is encoded in
$X_{;\mu}$. It reduces to the special case of Bekenstein given by (1) when
$D=0$ and the dependencies on $(Y,Z)$ are removed; in this case, we can
identify $C^{2}(\phi,X)=B(\phi,X)$ in (1).

The form of the disformal transformation given by (3) is a generalisation of
the hierarchy of transformations investigated in Ref. [24, 25, 26, 27] within
the context of inflationary cosmology (see Refs. [32, 33] for other forms
covering dual scalars and vector disformal transformation.) Moreover, it is
inspired by Refs. [23] and [31]. In Ref. [23], it is shown that both the
scalar and tensor primordial cosmological cosmological perturbations are
invariant under (3) in the superhorizon limit [18, 19]. On the other hand, it
is explored in Ref. [31] within the scope of invertibility and physical
degrees of freedom in scalar-tensor theories.

The first study finds its relevance in connecting general actions through
disformal transformation and showing the physical quantities that are
symmetric. This is much like moving to and from the Einstein and Jordan frames
in conformal transformation while retaining the character of primordial
cosmological perturbations. The second study complements the first one in view
of the fact that metric transformation may lead to Ostrogradsky instability
[34, 35] and propagating non-physical degrees of freedom [36, 37, 38, 39, 40,
41] in scalar-tensor actions in which the Klein-Gordon action belongs;
invertibility ensures that the number of propagating physical degrees of
freedom remain the same under a metric transformation. Having been explored in
Refs. [23] and [31] the ‘umbrella’ disformal transformation encompassing that
of Bekenstein and those of other works [24, 25, 26, 27], it is time that we
revisit one of the most, if not the most fundamental equation in field theory,
namely, the Klein-Gordon equation, under the general disformal transformation.

To be clear, our aim in this study is two-fold. The first one is to determine
the form of the disformal factors $C$ and $D$ such that the massless Klein-
Gordon equation is invariant under the general disformal transformation given
by (3); this is in the same spirit as in Ref. [29]. The second one is to
demonstrate the invertibility of the general disformal transformation subject
to the result of the first one and other auxiliary conditions. In particular,
we wish to write $g_{\mu\nu}$ in terms of disformally transformed quantities
to the extent permitted by the form of the conformal and disformal factors in
the transformation.

To meet these ends, this paper is organised as follows. In Sec. 2, we derive
the general disformal transformation of the d’Alembertian of the scalar field,
$\square\phi$, following a pathway to symmetry of the Klein-Gordon equation.
Along the way, we determine the conditions on the functionals $C$ and $D$,in
relation to the conformal factor $A$, that render the massless Klein-Gordon
equation invariant. In addition to this, we also look at the transformation of
the mass in the massive case. In Sec. 3, we provide an alternative way to
derive the conditions in the previous section for the symmetry; this serves to
confirm our results. In Sec. 4, we comment on and derive the inverse general
disformal transformation covered in this work. Lastly, we state our concluding
remarks and future prospects in Sec. 5.

##  2 Symmetry and the General Disformal Transformation of the Klein-Gordon
Equation

In this section, we wish to derive the disformal transformation (3) of the
d’Alembertian of the scalar field contained in the Klein Gordon equation. The
original background $n$-dimensional spacetime is described by the metric,
$g_{\mu\nu}$, with signature $(-,+,+,+)$. We assume, as in Ref. [6], that the
transformation leading to $\widehat{g}_{\mu\nu}$, preserves the Lorentzian
signature and causality structure of spacetime. Cases where the disformal and
conformal factors lead to flipping of Lorentzian signature and acausal
behaviour are left for future studies.

In so finding the general disformal transformation of the Klein-Gordon
equation, our pathway is bent in search of symmetry for the massless case. In
other words, we do not simply derive the full transformation of the equation
under (3) but take opportunities along the way whenever they present
themselves as conditions that we may impose to have an invariant Klein-Gordon
equation.

The d’Alembertian of the scalar field, $\square\phi$, changes under the
disformal transformation to
$\widehat{\square}\phi=\widehat{g}^{\mu\nu}\widehat{\nabla}_{\mu}\widehat{\nabla}_{\nu}\phi$,
where $\widehat{g}^{\mu\nu}$ is the inverse of the disformal metric given by
(3) above. The hatted covariant derivative operators (as in all other hatted
quantities in this work) are disformally transformed quantities. From the
Sherman-Morrison formula for matrix inversion, we find

$\displaystyle\widehat{g}^{\mu\nu}$
$\displaystyle=\frac{g^{\mu\nu}}{A}-\frac{\Phi^{\mu}\Phi^{\nu}}{A(A-2\chi)},\quad\text{where}\quad\chi\equiv-\tfrac{1}{2}g^{\mu\nu}\Phi_{\mu}\Phi_{\nu}.$
(4)

It is straightforward to show that
$\widehat{g}^{\mu\alpha}\widehat{g}_{\alpha\nu}=\delta^{\mu}_{\nu}$, verifying
the validity of the expression for the inverse disformal metric above.

For the covariant derivatives of the scalar field, we have
$\widehat{\nabla}_{\mu}\widehat{\nabla}_{\nu}\phi=\phi_{;\nu,\mu}-\widehat{\Gamma}^{\alpha}_{\mu\nu}\phi_{;\alpha},$
where the symbols comma ‘,’ and semicolon ‘;’ mean partial differentiation and
covariant differentiation, respectively666Needless to say, for the scalar
field, $\phi_{;\alpha}=\phi_{,\alpha}$, but
$\phi_{;\alpha\beta}\neq\phi_{,\alpha\beta}$.. The hatted Christoffel symbol
on the right hand side can be expressed in terms of the disformal metric,
derivative thereof, and its inverse, as
$\widehat{\Gamma}^{\alpha}_{\mu\nu}=\tfrac{1}{2}\widehat{g}^{\alpha\beta}\left(\widehat{g}_{\beta\mu,\nu}+\widehat{g}_{\nu\beta,\mu}-\widehat{g}_{\mu\nu,\beta}\right)$.
Using the equations for the disformal metric and its inverse given by (3) and
(4), respectively, the hatted Christoffel symbol can be decomposed as a sum of
the original Christoffel symbol and a tensor term. In particular,

$\displaystyle\widehat{\Gamma}^{\alpha}_{\mu\nu}$
$\displaystyle=\Gamma^{\alpha}_{\mu\nu}+C^{\alpha}_{\mu\nu},$ (5)

where $C^{\alpha}_{\mu\nu}$, the contribution due to the disformal part of the
transformation, is given by

$\displaystyle C^{\alpha}_{\mu\nu}$
$\displaystyle=\frac{A_{;\nu}\delta^{\alpha}_{\mu}+A_{;\mu}\delta^{\alpha}_{\nu}-A^{;\alpha}g_{\mu\nu}}{2A}-\frac{\Phi^{\alpha}(A_{;\nu}\Phi_{\mu}+A_{;\mu}\Phi_{\nu}-A_{;\beta}\Phi^{\beta}g_{\mu\nu})}{2A(A-2\chi)}$
$\displaystyle\qquad+\,\left[\frac{g^{\alpha\beta}}{A}-\frac{\Phi^{\alpha}\Phi^{\beta}}{A(A-2\chi)}\right](\Phi_{\mu}\Phi_{[\beta;\nu]}+\Phi_{\nu}\Phi_{[\beta;\mu]})+\frac{\Phi^{\alpha}\Phi_{(\mu;\nu)}}{A-2\chi}.$
(6)

Here, $\Phi_{[\mu;\nu]}\equiv\tfrac{1}{2}(\Phi_{\mu;\nu}-\Phi_{\nu;\mu})$ and
$\Phi_{(\mu;\nu)}\equiv\tfrac{1}{2}(\Phi_{\mu;\nu}+\Phi_{\nu;\mu})$. Observe
that $C^{\alpha}_{\mu\nu}$ is symmetric with respect to the lower indices. It
follows that $\widehat{\Gamma}^{\alpha}_{\mu\nu}$ retains the symmetry of the
original $\Gamma^{\alpha}_{\mu\nu}$. In the limit where $C=D=0$, it reduces to
that of the familiar conformal transformation [2]; i.e.,
$\widehat{\Gamma}^{\alpha}_{\mu\nu}-\Gamma^{\alpha}_{\mu\nu}$
$=C^{\alpha}_{\mu\nu}\rightarrow(A_{;\nu}\delta^{\alpha}_{\mu}+A_{;\mu}\delta^{\alpha}_{\nu}-A^{;\alpha}g_{\mu\nu})/2A$.

All the needed expressions are now in place to solve for
$\widehat{\square}\phi$. We have upon using (5) for the hatted Christoffel
symbol in the expression for the d’Alembertian,

$\displaystyle\widehat{\square}\phi$
$\displaystyle=\frac{\square\phi}{A}-\frac{g^{\mu\nu}C^{\alpha}_{\mu\nu}\phi_{;\alpha}}{A}-\frac{\Phi^{\mu}\Phi^{\nu}\phi_{;\mu\nu}}{A(A-2\chi)}+\frac{\Phi^{\mu}\Phi^{\nu}C^{\alpha}_{\mu\nu}\phi_{;\alpha}}{A(A-2\chi)}.$
(7)

We see that the transformed d’Alembertian of $\phi$ is a sum of the original
d’Alembertian of $\phi$, scaled by the conformal factor, $A$, and terms due to
the disformal part of the transformation. While these terms are
straightforward to compute using the equation for $C^{\alpha}_{\mu\nu}$ given
above, the result is quite long and the calculation is laborious. With an eye
for symmetry, as alluded above, we opt to cancel terms early on by imposing
‘symmetry’ condition(s) on the disformal factors $C$ and $D$, leading to the
invariance of the massless Klein-Gordon equation.

This cancellation starts with the second term on the right hand side of the
equation above for $\widehat{\square}\phi$. It can be expressed as

$\displaystyle g^{\mu\nu}C^{\alpha}_{\mu\nu}\phi_{;\alpha}$
$\displaystyle=\frac{(2-n)\phi^{;\alpha}A_{;\alpha}}{2A}-\frac{(2-n)\phi_{;\alpha}\Phi^{\alpha}\,A_{;\beta}\Phi^{\beta}}{2A(A-2\chi)}+\frac{2\phi^{;\alpha}\Phi^{\beta}\Phi_{[\alpha;\beta]}}{A}+\frac{\phi_{;\alpha}\Phi^{\alpha}\Phi^{\beta}{}_{;\beta}}{A-2\chi}.$

Observe that of all the terms in the equation for $\widehat{\square}\phi$
above, only this term generates $\square X$. Furthermore, only the term
involving ${\Phi^{\beta}}_{;\beta}$ on the right hand side of the equation
above for $\phi_{;\alpha}g^{\mu\nu}C^{\alpha}_{\mu\nu}$ generates $\square X$;
to wit,
${\Phi^{\beta}}_{;\beta}=C_{;\beta}\phi^{;\beta}+C\square\phi+D_{;\beta}X^{;\beta}+D\square
X$. If $\widehat{\square}\phi$ is then to be proportional to $\square\phi$,
leading to the equality between the two in the massless limit, the term
involving $\phi_{;\alpha}\Phi^{\alpha}\Phi^{\beta}{}_{;\beta}$ above should
vanish. For this to happen, we may impose the “orthogonality” condition777With
$\Phi^{\alpha}=C\phi^{;\alpha}+DX^{;\alpha}$, they are not orthogonal in the
usual sense of the word orthogonal in Linear Algebra and Hilbert spaces;
hence, the quotation marks. between $\Phi^{\alpha}$ and $\phi_{;\alpha}$; that
is,

$\displaystyle\Phi^{\alpha}\phi_{;\alpha}=0.$ (8)

As a consequence of this condition, we have the functional relation given by
$DY=2CX$. At this point, this relation can be satisfied by infinitely many
combinations of $D(\phi,X,Y,Z)$ and $C(\phi,X,Y,Z)$; e.g., $C=Y$ and $D=2X$.
However, as we shall see below, $C$ and $D$ should take special forms in
relation to $A$ and $(X,Y,Z)$ for the symmetry of the massless Klein-Gordon
equation.

Continuing, we find that given the “orthogonality” condition, the substitution
of the equation for $C^{\alpha}_{\mu\nu}$ given by (2) in the equation for
$\widehat{\square}\phi$ given by (7) leads us to a much simpler expression for
the latter.

$\displaystyle\widehat{\square}\phi$
$\displaystyle=\frac{\square\phi}{A}+\frac{[n(A-2\chi)-2A+6\chi]\phi^{;\alpha}A_{;\alpha}}{2A^{2}(A-2\chi)}-\frac{\Phi^{\alpha}\phi_{;\alpha\beta}\Phi^{\beta}+2\phi^{;\alpha}\Phi^{\beta}\Phi_{[\alpha;\beta]}}{A(A-2\chi)}.$
(9)

Needless to say, if the conformal factor is unity and $C=D=0$, then
$\widehat{\square}\phi=\square\phi$. On the other hand, when the
transformation is purely conformal, we find both $\chi$ and $\Phi_{\mu}$
vanish, and gain the expected result given by888We cannot set $D=0$
independent of $C$ as a consistency check because the condition
$\phi^{;\alpha}\Phi_{;\alpha}=0$ imposes $D\propto C$. Setting $D=0$ also
means $C=0$ or $B=0$ (since $C^{2}=B$).

$\displaystyle\widehat{\square}\phi$
$\displaystyle=\frac{\square\phi}{A}+\frac{(n-2)\phi^{;\alpha}A_{;\alpha}}{2A^{2}}.$
(10)

Interestingly, the Klein-Gordon equation is not symmetric with respect to
conformal transformation unless the dimension of the embedding manifold is
$n=2$. Being able to push it to symmetry [28] elevates the value of
Bekenstein’s disformal transformation. In this work, we push the symmetry
further under the general disformal transformation (3) and uncover the
conditions for this symmetry.

Beyond the conformal limit, the factors $C$ and $D$ are not necessarily
vanishing but are bound by the “orthogonality” condition given by (8). As a
consequence, for the numerator of the last term on the right hand side of (9),
we have

$\displaystyle\Phi^{\mu}\phi_{;\mu\nu}\Phi^{\nu}+2\phi^{;\alpha}\Phi^{\beta}\Phi_{[\alpha;\beta]}$
$\displaystyle=-\frac{B}{Y}(Y^{2}+4XZ)-\frac{X}{Y^{2}}(Y^{2}+2XZ)\phi^{;\alpha}B_{;\alpha}$
$\displaystyle\qquad-\,\frac{2BX^{2}\phi^{;\alpha}Z_{;\alpha}}{Y^{2}}+\frac{4BX^{2}Z\phi^{;\alpha}Y_{;\alpha}}{Y^{3}},$
(11)

where we identify $C^{2}=B$ from the correspondence of the usual disformal
transformation (1) and the general disformal transformation (3).

If $\widehat{\square}\phi$ is to be proportional to $\square\phi$ in (9), then
the two terms following the first term involving $\square\phi$ on the right
hand side should vanish. For this to happen, $B$ may be related to $A$. In an
attempt to find this relationship, we take the ansatz $B=q(X,Y,Z)h(X)f(A)$,
where the functionals $q,h,$ and $f$ are to be determined. Note that in this
ansatz, the dependence on $\phi$ is encoded in $A$ alone while $X$ is
distributed amongst $q,h,$ and $A$, with $q$ (and $A$) being also dependent on
$(Y,Z)$. Using this ansatz and (8) we find

$\displaystyle q$ $\displaystyle=\frac{Y^{2}}{Y^{2}+2XZ},\quad
h=\frac{1}{X},\quad\text{and}\quad f=-\chi.$ (12)

We conclude that under the disformal transformation given by
$\widehat{g}_{\mu\nu}=Ag_{\mu\nu}+\Phi_{\mu}\Phi_{\nu}$, the Klein-Gordon
equation for scalar field $\phi$ transforms as

$\displaystyle\boxed{\square\phi-m^{2}\phi=0\quad\rightarrow\quad\frac{\square\phi}{A}-m^{2}\phi=0,}$
(13)

provided that the disformal factors $C$ and $D$ are related to the conformal
factor and $(X,Y,Z)$ as

$\displaystyle\boxed{C^{2}=\frac{Y^{2}(bA^{3-n}-A)}{2X(Y^{2}+2XZ)}\quad\text{and}\quad
D^{2}=\frac{2X(bA^{3-n}-A)}{Y^{2}+2XZ}.}$ (14)

In the massless limit, the Klein-Gordon equation is invariant under the
transformation. On the other hand, for the massive case, the mass may be seen
as transforming by conformal rescaling (i.e., $m^{2}\rightarrow m^{2}/A$) from
the manifold described by $g_{\mu\nu}$ to the manifold described by
$\widehat{g}_{\mu\nu}$.

It is worth noting that our result here for the massive field is a little
different from that of the usual disformal transformation,
$\widehat{g}_{\mu\nu}=Ag_{\mu\nu}+B\phi_{;\mu}\phi_{;\nu}$, wherein
$\square\phi-m^{2}\phi=0\rightarrow(\square\phi)/(c^{2}A^{n-1})-m^{2}\phi=0$,
leading to $m^{2}\rightarrow m^{2}/(c^{2}A^{n-1})$ [28]. The dependence on the
dimension $n$ is apparent in the usual one, while for our transformation, it
is not present. In fact, for the massive field, the transformation of the
Klein-Gordon equation is not simply a special case of ours. The condition that
relates $B$ to $A$ given by (2) for the usual disformal transformation is not
a special case of the second of (14) due to the “orthogonality” condition
binding $C$ and $D$. This leads to different mass transformations with one
depending on $n$ while the other one does not. However, the massless limit
yields the same symmetry for both cases. It is in this sense that we are
extending the symmetry of the massless Klein-Gordon equation under the general
disformal transformation (3).

Before we leave this section, it is good to clarify one point leading to (14)
and consequently, to the invariance of the Klein-Gordon equation. Along the
way of our calculation, we encountered the ‘orthogonality’ condition, namely,
$\Phi^{\alpha}\phi_{;\alpha}=0$. Taken at face value, this seems to imply a
relation for $\phi$ in addition to the Klein-Gordon equation, making it
possibly over determined. However, a better and correct way to look at the
‘orthogonality’ condition is as part of the intermediate steps leading to
(14). In view of the idea that this result should not be taken as an equation
for $\phi$ but the forms $C$ and $D$ should take in order for the equation
given by $\square\phi=0$, to be invariant, the ‘orthogonality’ relation,
$\Phi^{\alpha}\phi_{;\alpha}=0\,\Leftrightarrow\,2CX=DY$, should be seen as a
functional relationship between $C$ and $D$ and a stepping stone leading to
their desired forms given by (14). Taking our derivation in this section as a
whole, it may not be taken as an auxiliary equation for $\phi$.

##  3 Alternative Derivation Leading to the Conditions for Symmetry

The immediately preceding section saw the calculation of
$\widehat{\square}\phi$ using a direct ‘expansion’ involving the Christoffel
symbol. In this section, we present a slightly different pathway, used in Ref.
[28] for the usual disformal transformation, to the conditions leading for
symmetry of the massless Klein-Gordon equation under the general disformal
transformation given by (3). The process of calculation is mainly for
verification of the previous section’s results.

Following this pathway, we start from the relation involving double covariant
differentiation, namely,
$\square\phi=g^{\mu\nu}\phi_{,\nu\mu}-g^{\mu\nu}\Gamma^{\alpha}_{\mu\nu}\phi_{;\alpha}$.
Knowing, however, that the contraction of the metric and the Christoffel
symbol can be written as
$g^{\mu\nu}\Gamma^{\alpha}_{\mu\nu}=-(g^{\alpha\beta}\sqrt{-g})_{,\beta}/\sqrt{-g}$,
leads us to a compact form given by

$\displaystyle\square\phi$
$\displaystyle=\frac{(g^{\mu\nu}\sqrt{-g}\phi_{,\mu})_{,\nu}}{\sqrt{-g}}.$
(15)

Because the derivative operators involved on the right hand side are not
covariant derivative operators but only partial derivative operators, the
disformal transformation of $\square\phi$ can simply be written as

$\displaystyle\widehat{\square}\phi$
$\displaystyle=\frac{(\widehat{g}^{\mu\nu}\,\sqrt{-\widehat{g}}\,\phi_{,\mu})_{,\nu}}{\sqrt{-\widehat{g}}}.$
(16)

With the inverse disformal metric already given by (4), we only need to find
the hatted determinant. Given the form of the disformal metric provided by
(3), we find from the matrix determinant lemma,
$\widehat{g}=A^{n-1}g(A-2\chi)$, where $\chi$ is defined by the second of (4).
Using this equation for $\widehat{g}$ and (4) for $\widehat{g}^{\mu\nu}$, in
the sub-expression $\widehat{g}^{\mu\nu}\,\sqrt{-\widehat{g}}\,\phi_{,\mu}$
above contained in the equation for $\widehat{\square}\phi$, yields

$\displaystyle\widehat{g}^{\mu\nu}\sqrt{-\widehat{g}}\,\phi_{,\mu}$
$\displaystyle=A^{\frac{n-3}{2}}(A-2\chi)^{\frac{1}{2}}g^{\mu\nu}\sqrt{-g}\,\phi_{;\mu}-\frac{A^{\frac{n-3}{2}}\sqrt{-g}}{(A-2\chi)^{\frac{1}{2}}}\phi^{;\mu}\Phi_{\mu}\Phi^{\nu}.$
(17)

Looking at (16), we see that $\widehat{\square}\phi$ may be made proportional
to $\square\phi$ if the second term on the right hand side of (17) vanishes.
As before, this leads to the “orthogonality” condition given by
$\phi^{;\mu}\Phi_{\mu}=0$.

While this condition is a significant stepping stone leading to the desired
final form of $C$ and $D$, it is not sufficient for the proportionality
between $\widehat{\square}\phi$ and $\square\phi$; the disformal factor $B$
should be related to $A$. We have upon performing the second differentiation
and dividing the result by $\sqrt{-\widehat{g}}$ to form
$\widehat{\square}\phi$, gain

$\displaystyle\widehat{\square}\phi=\frac{\square\phi}{A}+\frac{[A^{\frac{n-3}{2}}(A-2\chi)^{\frac{1}{2}}]_{,\nu}g^{\mu\nu}\sqrt{-g}\,\phi_{;\mu}}{A^{\frac{n-1}{2}}(A-2\chi)^{\frac{1}{2}}\sqrt{-g}}.$
(18)

The second term vanishes for
$A^{\frac{n-3}{2}}(A-2\chi)^{\frac{1}{2}}=\bar{c}$, where $\bar{c}$ is some
constant. This condition allows us to find the relationship between $B$ and
$A$. Indeed, noting our “orthogonality” condition for $\phi^{;\mu}$ and
$\Phi_{\mu}$, we get $\chi=-BX(Y^{2}+2XZ)/Y^{2}$. In effect,

$\displaystyle
B=-\frac{Y^{2}(A-\bar{c}^{2}A^{3-n})}{2X(Y^{2}+2XZ)}\quad\text{and}\quad
D^{2}=\frac{2X(bA^{3-n}-A)}{Y^{2}+2XZ}$ (19)

which matches the pair of equations given by (14) in the immediately preceding
section with the identification $\bar{c}^{2}=b$.

The “orthogonality” condition binds $D$ and $C^{2}=B$ in the general disformal
transformation given by
$\widehat{g}_{\mu\nu}=Ag_{\mu\nu}+(C\phi_{;\mu}+DX_{;\mu})(C\phi_{;\nu}+DX_{;\nu})$.
On the other hand, the condition given by (19) above binds $B$ and $A$. In
short, both $C$ and $D$ in the transformation are functionals of $A$ leaving
only the conformal factor free. Provided that these conditions hold, the
massless Klein-Gordon equation exhibits symmetry under the general disformal
transformation—restating the same conclusion in the previous section. For the
massive case, the mass may be seen as transforming999This is quite an
intriguing transformation. But as to its physical justification, making a
‘real-world’ example of this process remains a part of our future studies.
from $m^{2}\rightarrow\widehat{m}^{2}=m^{2}/A$, in going from the manifold
described by $g_{\mu\nu}$ to that with $\widehat{g}_{\mu\nu}$.

##  4 Invertibility of the Constrained General Disformal Transformation

The usual disformal transformation can be inverted in the sense that
$g_{\mu\nu}$ can be solved using the hatted quantities. Indeed, given
$\widehat{g}_{\mu\nu}=A(\phi,X)g_{\mu\nu}+B(\phi,X)\phi_{;\mu}\phi_{;\nu}$,
one finds that $\widehat{X}=X/(A-2BX)$, indicating that $X$ can be considered
as a functional of $(\phi,\widehat{X})$ provided101010Given
$\widehat{X}=X/(A-2BX)$, the conformal and disformal factors, $A$ and $B$,
respectively, must have restrictions on their functional forms in order to
find the analytical equation for $X$ in terms of $\widehat{X}$. However, even
if such an analytical equation does not exist, the inverse can still
mathematically exist under the conditions stated in Refs. [4, 6].
$\partial\widehat{X}/\partial X\neq 0$; more precisely [4, 6],
$B(A-A_{X}X+2B_{X}X^{2})\neq 0$ and $A-2BX>0$. As a consequence of this, we
may simply rewrite the conformal and disformal factors $A$ and $B$ as a
functional of $(\phi,\widehat{X})$. In effect, the inverse disformal
transformation may be written as

$\displaystyle
g_{\mu\nu}=\frac{\widehat{g}_{\mu\nu}}{A(\phi,\widehat{X})}-\frac{B(\phi,\widehat{X})}{A(\phi,\widehat{X})}\phi_{;\mu}\phi_{;\nu};$
(20)

or, in the same form as the usual disformal transformation,
$g_{\mu\nu}=\widehat{A}(\phi,\widehat{X})\widehat{g}_{\mu\nu}+\widehat{B}(\phi,\widehat{X})\phi_{;\mu}\phi_{;\nu}$,
with the identifications given by $\widehat{A}=1/A$ and $\widehat{B}=-B/A$.

The invertibility of disformal transformation is crucial in the study of
actions describing scalar-tensor theories; i.e., Horndeski theory and beyond
[12, 38, 39, 40, 41]. In the simplest scenario wherein both $A$ and $B$ are
functionals of $\phi$ alone as in Ref. [6], it effectively maps the Horndeski
theory back to itself; thus, preventing the existence of higher order
derivative terms (in the action or equation of motion) signaling Ostrogradsky
instability [34, 35]. Generalisations beyond this lead to beyond-Horndeski
terms whose corresponding equations of motion go beyond second order.
Nevertheless, if the transformation is invertible, the number of propagating
physical degrees of freedom generally remains the same as in the original
theory [36, 37, 42].

The Klein-Gordon action is a ‘small’ part but an important special case of the
much more complicated Horndeski and beyond-Horndeski actions. However, it is
important that we gain at least an insight about the invertibility of our
general disformal transformation if the transformed action is to be physically
sensible. To be honest, in the most general context, the transformation given
by
$\widehat{g}_{\mu\nu}=Ag_{\mu\nu}+(C\phi_{;\mu}+DX_{;\mu})(C\phi_{;\nu}+DX_{;\nu})$,
with $(A,C,D)$ all functionals of $(\phi,X,Y,Z)$, is not invertible;
invertibility needs to be supplemented by some requirements on functional
dependencies [31]. In this section, our task is to find the invertibility
conditions for this transformation subject to the forms of $C$ and $D$ given
by (14), ensuring the invariance of the massless Klein-Gordon equation.
Furthermore, we wish to derive the inverse general disformal transformation to
the extent permitted by this pair of equations.

As a requirement for the invertibility of the general disformal transformation
(3), we impose that all the hatted terms $\widehat{X},\widehat{Y},$ and
$\widehat{Z}$ be functionals of the unhatted quantities, namely,
$(\phi,X,Y,Z)$, alone. With this restriction, we may invert the relationship
between the hatted and unhatted quantities provided that the Jacobian
determinant is non-vanishing; i.e.,

$\displaystyle\left|\begin{array}[]{ccc}\partial_{X}\widehat{X}&\partial_{Y}\widehat{X}&\partial_{Z}\widehat{X}\\\\[3.50006pt]
\partial_{X}\widehat{Y}&\partial_{Y}\widehat{Y}&\partial_{Z}\widehat{Y}\\\\[3.50006pt]
\partial_{X}\widehat{Z}&\partial_{Y}\widehat{Z}&\partial_{Z}\widehat{Z}\end{array}\right|\neq
0.$ (24)

For the case in hand, the restrictions among the functional factors in the
general disformal transformation given by (14) allow us to ‘spell out’ the map
$(\phi,X,Y,Z)\rightarrow(\widehat{\phi},\widehat{X},\widehat{Y},\widehat{Z})$;
to a good extent, we can also determine the inverse map
$(\widehat{\phi},\widehat{X},\widehat{Y},\widehat{Z})\rightarrow(\phi,X,Y,Z)$.

Starting with $\widehat{X}$ we find from its definition together with the
equations for $\widehat{g}^{\mu\nu}$ and $\Phi^{\mu}\phi_{;\mu}=0$,

$\displaystyle\widehat{X}=-\frac{1}{2A}\bigg{(}g^{\mu\nu}-\frac{\Phi^{\mu}\Phi^{\nu}}{A-2\chi}\bigg{)}\phi_{;\mu}\phi_{;\nu}=\frac{X}{A}.$
(25)

It is, in other words, a functional of $(\phi,X,Y,Z)$ only. For $\widehat{Y}$,
we find an additional dependence on $\phi^{;\alpha}Y_{;\alpha}$ and
$\phi^{;\alpha}Z_{;\alpha}$. Indeed,

$\displaystyle\widehat{Y}$
$\displaystyle=\widehat{g}^{\mu\nu}\phi_{;\mu}\widehat{X}_{;\nu}=\frac{Y}{A^{2}}+\frac{2A_{\phi}X^{2}}{A^{3}}-\frac{A_{X}XY}{A^{3}}-\frac{A_{Y}X}{A^{3}}\phi^{;\alpha}Y_{;\alpha}-\frac{A_{Z}X}{A^{3}}\phi^{;\alpha}Z_{;\alpha}.$
(26)

It follows that if $\widehat{Y}$ is to be dependent on $(\phi,X,Y,Z)$ alone,
then the partial derivatives of the conformal factor, namely, $A_{Y}$ and
$A_{Z}$ should both vanish leading to

$\displaystyle\widehat{Y}$
$\displaystyle=Y\frac{A-A_{X}X}{A^{3}}+\frac{2A_{\phi}X^{2}}{A^{3}}.$ (27)

Following the same logic for $\widehat{Z}$ we find

$\displaystyle\widehat{Z}$
$\displaystyle=\frac{(A-A_{X}X)^{2}(Y^{2}+2XZ)({{A}^{n}}-bA^{2})}{2bA^{7}X}$
$\displaystyle\qquad+\,\frac{(A-A_{X}X)^{2}Z-2A_{\phi}XY(A-{A_{X}}{X})-2A_{\phi}^{2}X^{3}}{A^{5}}.$
(28)

It is worth noting that although $A=A(\phi,X)$ only, the disformal factors
still depend on $(\phi,X,Y,Z)$ based on (14).

In summary, assuming $A=A(\phi,X)$, we have a ‘ladderised’ sequence of
functional dependencies as

$\displaystyle\widehat{X}=\widehat{X}(\phi,X),\quad\widehat{Y}=\widehat{Y}(\phi,X,Y),\quad\text{and}\quad\widehat{Z}=\widehat{Z}(\phi,X,Y,Z).$
(29)

Using (25), (27), and (4) we may invert these relations as

$\displaystyle X$ $\displaystyle=A\widehat{X},\quad
Y=\frac{A\widehat{Y}-2A_{\phi}\widehat{X}^{2}}{\widehat{X}_{X}},$
$\displaystyle Z$
$\displaystyle=\frac{b\widehat{Z}}{\widehat{X}_{X}^{2}A^{n-3}}-\frac{(A\widehat{Y}-2A_{\phi}\widehat{X}^{2})^{2}(1-bA^{2-n})}{2A\widehat{X}\widehat{X}_{X}^{2}}+\frac{2bA_{\phi}\widehat{X}(A\widehat{Y}-A_{\phi}\widehat{X}^{2})}{\widehat{X}_{X}^{2}A^{n-1}}.$
(30)

Note that the conformal factor, being unspecified, remains implicit in these
equations. The existence of the full analytical set of equations for the
inverse relations depends on the form of $A$. In the limit where $A=A(\phi)$
alone, the set of relations above completely solves for $(\phi,X,Y,Z)$ in
terms of $(\phi,\widehat{X},\widehat{Y},\widehat{Z})$. For general
$A=A(\phi,X)$, barring the divergences of the equations above, the inverse
relations exist, (although possibly, not analytically,) provided
$(A-A_{X}X)/A^{2}\neq 0$ in accord with Ref. [4].

Having found $(\phi,X,Y,Z)$ in terms of
$(\phi,\widehat{X},\widehat{Y},\widehat{Z})$, we seek to find the inverse of
the general disformal transformation; the aim is to write from

$\displaystyle\widehat{g}_{\mu\nu}$
$\displaystyle=Ag_{\mu\nu}+(C\phi_{;\mu}+DX_{;\mu})(C\phi_{;\nu}+DX_{;\nu}),$
$\displaystyle\text{to}\quad g_{\mu\nu}$
$\displaystyle=\widehat{A}\widehat{g}_{\mu\nu}+(\widehat{C}\phi_{;\mu}+\widehat{D}\widehat{X}_{;\mu})(\widehat{C}\phi_{;\nu}+\widehat{D}\widehat{X}_{;\nu}).$
(31)

In fulfilling this aim, we note from the first of (4) that $X_{;\mu}$ does not
depend on the derivative term $\widehat{X}_{;\mu}$ alone but also includes
$\phi_{;\mu}$. This leads to $\phi_{;\mu}X_{;\nu}$ producing
$\phi_{;\mu}\phi_{;\nu}$ and $\phi_{;\mu}\widehat{X}_{;\nu}$; in particular,
we have

$\displaystyle X_{;\mu}$
$\displaystyle=\frac{A_{\phi}\phi_{;\mu}\widehat{X}}{A\widehat{X}_{X}}+\frac{\widehat{X}_{;\mu}}{\widehat{X}_{X}}.$
(32)

At first sight, this might appear as a complication in deriving the second of
(4) due to additional hatted terms ‘generated’. The fear is that we might not
end up with exactly the right form matching the original disformal
transformation. However, as it turns out upon using (32),

$\displaystyle\boxed{g_{\mu\nu}=\frac{\widehat{g}_{\mu\nu}}{A}-\frac{AC^{2}}{(A\widehat{Y}-2A_{\phi}\widehat{X}^{2})^{2}}(\widehat{Y}\phi_{;\mu}+2\widehat{X}\widehat{X}_{;\mu})(\widehat{Y}\phi_{;\nu}+2\widehat{X}\widehat{X}_{;\nu}).}$
(33)

Our sought-for inverse of the general disformal transformation, subject to the
conditions given by (14), takes the form consistent with that of the second of
(4). Moreover, it is worth noting that with the identifications of
$\widehat{C}$ and $\widehat{D}$ in (33), we find
$\widehat{D}\widehat{Y}=2\widehat{C}\widehat{X}$, reflecting the same form as
the original “orthogonality” condition.

##  5 Concluding Remarks

Symmetry principles are ubiquitous in the fundamental laws of nature. In the
20th century, it dominated the exploration and formulation of theories in
modern physics. With the realisation of its significance in understanding our
Universe, we generalised a new class of symmetry related to the background
metrical structure in which a field propagates. This new symmetry is induced
by a broader class of metric transformations called the disformal
transformation generalising conformal transformation that is extensively used
in relativity and extensions thereof.

In this work, we extend the discovered symmetry under Bekenstein’s disformal
transformation of the massless Klein-Gordon equation. In particular, we show
that, subject to some conditions, the massless relativistic wave equation for
a scalar field in arbitrary dimensions is invariant under the general
disformal transformation of the form given by
$g_{\mu\nu}\rightarrow\widehat{g}_{\mu\nu}=Ag_{\mu\nu}+\Phi_{\mu}\Phi_{\nu}$,
where $\Phi_{\mu}=C\phi_{;\mu}+DX_{;\mu}$, with the functionals $A,C,$ and $D$
dependent on $X=-\tfrac{1}{2}\phi^{;\mu}\phi_{;\mu},\,Y=\phi^{;\mu}X_{;\mu}$,
and $Z=X^{;\mu}X_{;\mu}$. The conditions involve a functional relationship
between the disformal factors $(C,D)$ and the conformal factor $A$. In
addition to this, with an insight for the preservation of the number of
physical degrees of freedom upon transformation, we derive the inverse
disformal transformation. The inverse may exist if the conformal factor is
restricted to $A=A(\phi,X)$. Interestingly, while disformal factors $C$ and
$D$ are related to $A$ in view of the symmetry of the Klein-Gordon equation,
their functional dependencies include $(Y,Z)$ in addition to $(\phi,X)$ on
which the conformal factor depends.

For future studies, we have at least three tracks in sight. First, we can
revisit the other fundamental equations in physics, e.g., the Dirac equation
and Maxwell’s equations, that may exhibit an extended symmetry under a
hierarchy of disformal transformations. Second, we may ponder on the mass
transformation, together with the interaction terms in the action, induced by
disformal transformation. Lastly, it would be interesting to investigate
disformal transformation, at least of the scalar fields and derivatives
thereof, within the realm of quantum field theory in curved spacetime.

##  6 Acknowledgment

MRD Chua would like to thank the Department of Science and Technology Science
Education Institute (DOST-SEI) for providing financial assistance in support
of this study.

##  7 Data availability statement

No new data were created or analysed in this study.

##  8 ORCID iDs

Allan L. Alinea https://orcid.org/0000-0001-7511-1256
Mark Ruel D. Chua https://orcid.org/0000-0002-2451-0238

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# On egalitarian values for cooperative games with a priori unions

J.M. Alonso-Meijide1, J. Costa2,
I. García-Jurado3, J.C. Gonçalves-Dosantos3

###### Abstract

In this paper we extend the equal division and the equal surplus division
values for transferable utility cooperative games to the more general setup of
transferable utility cooperative games with a priori unions. In the case of
the equal surplus division value we propose three possible extensions. We
provide axiomatic characterizations of the new values. Furthermore, we apply
the proposed modifications to a particular cost sharing problem and compare
the numerical results with those obtained with the original values.

11footnotetext: Grupo MODESTYA, Departamento de Estatística, Análise
Matemática e Optimización, Universidade de Santiago de Compostela, Facultade
de Ciencias, Campus de Lugo, 27002 Lugo, Spain.22footnotetext: Grupo MODES,
Departamento de Matemáticas, Universidade da Coruña, Campus de Elviña, 15071 A
Coruña, Spain.33footnotetext: Grupo MODES, CITIC and Departamento de
Matemáticas, Universidade da Coruña, Campus de Elviña, 15071 A Coruña, Spain.

Keywords: cooperative games, a priori unions, equal division value, equal
surplus division value.

##  1 Introduction

Many economic problems deal with situations in which several agents cooperate
to generate benefits or to reduce costs. Cooperative game theory studies
procedures to allocate the resulting benefits (or costs) among the cooperating
agents in those situations.

One of the most commonly used allocating procedures is the Shapley value,
introduced in Shapley (1953) and analyzed more recently in Moretti and Patrone
(2008) or in Alonso-Meijide et al. (2019). Very often, however, agents
cooperate on the basis of a kind of egalitarian principle according to which
the benefits will be shared equitably. For instance, Selten (1972) indicates
that egalitarian considerations explain in a successful way observed outcomes
in experimental cooperative games.

In recent years, the game theoretical literature has dealt with several
egalitarian solutions in cooperative games. For instance, van den Brink (2007)
provides a comparison of the equal division value and the Shapley value, and
Casajus and Hüttner (2014) compare those two solutions with the equal surplus
division value (studied first in Driessen and Funaki, 1991). In van den Brink
and Funaki (2009), Chun and Park (2012), van den Brink et al. (2016),
Ferrières (2017) and Béal et al. (2019) several axiomatic characterizations of
the equal division and equal surplus division values are provided. Ju et al.
(2007) introduce and characterize the consensus value, a new solution that
somewhat combines the Shapley value and the equal division rule. Dutta and Ray
(1989) introduce the egalitarian solution for cooperative games, closely
related to Lorenz dominance, that considers cooperating agents who believe in
equality as a desirable social goal and negotiate accordingly; this solution
was later characterized by Dutta (1990), Klijn et al. (2000) and Arín et al.
(2003), and modified by Dietzenbacher et al. (2017).

Another stream of literature in cooperative game theory started in Owen
(1977), where a variant of the Shapley value for games with a priori unions is
introduced and characterized. In a game with a priori unions there exists a
partition of the set of players, whose classes are called unions, that is
interpreted as an a priori coalition structure that conditions the negotiation
among the players and, consequently, modifies the fair outcome of the
negotiation. There is a large literature concerning the Owen value and its
applications; just to cite some recent papers, Lorenzo-Freire (2016) provides
new axiomatic characterizations of the Owen value, Costa (2016) deals with an
application in a cost allocation problem, and Saavedra-Nieves et al. (2018)
propose a sampling procedure to approximate it. Not only the Shapley value but
also other values have been modified for the case with a priori unions. For
instance, Alonso-Meijide and Fiestras-Janeiro (2002) deal with the Banzhaf
value for games with a priori unions, Casas-Méndez et al. (2003) introduce the
$\tau$-value for games with a priori unions, Alonso-Meijide et al. (2011)
study the Deegan-Packel index for simple games with a priori unions, and Hu et
al. (2019) introduce an egalitarian efficient extension of the Aumann-Drèze
value (Aumann and Drèze, 1974).

In this paper we modify the equal division value and the equal surplus
division value for games with a priori unions. In Section 2 we illustrate the
interest of our study describing a cost allocation problem that arises in the
installation of an elevator in an apartment building. In Section 3 we define
and characterize the equal division rule for games with a priori unions. In
Section 4 we introduce and characterize three alternative extensions of the
equal surplus division rule for games with a priori unions. In Section 5 we
include some final remarks.

##  2 An example

In this section we consider an example where the owners of apartments in a
building have agreed to install an elevator and share the corresponding costs.
This example is inspired by a problem analyzed in Crettez and Deloche (2018)
from the point of view of French legislation. The French Law on Apartment
Ownership of Buildings does not provide a precise method for sharing the cost
of an improvement but indicates that the co-owners must pay “in proportion to
the advantages” they will receive. In the case of elevators in France, Crettez
and Deloche (2018) indicate that there is a de facto sharing method that they
call the _elevator rule_. In their paper they study the elevator rule and
other proposals in the spirit of the French legislation.

However, Crettez and Deloche (2018) explain that in other European countries
the legislation is based on principles of egalitarian character. For example,
in The Netherlands each of the owners of the apartments must “participate for
an equal part in the debts and costs which are for account of all apartments
owned pursuant to law or the internal arrangements, unless the internal
arrangements provide for another proportion of participation.”

The Spanish Horizontal Property Law 49/1960 (modified by the Act 8/2013)
indicates that “to each apartment or local will be attributed a quota of
participation in relation to the total of the value of the building (…). This
quota will serve as a module to determine the participation in the burdens and
benefits due to the community.” These quotas generally depend on the surface
area of each apartment but can take into account other aspects.

In a particular example, let us see how the Dutch and Spanish rules would
share the costs of installing an elevator. Consider the following three-storey
building with no apartments or offices on the ground floor: on the first floor
there is a single apartment of 180 square meters, on the second floor there
are two apartments, one of 100 and other one of 90 square meters, and on the
third floor there are three apartments of 60 square meters each. The second
floor has a slightly larger area because one of the two apartments on the
floor has an additional gallery. Suppose now that the cost of installing the
elevator is 120 (in thousands of euros), 50 of which correspond to the
machine, 40 to the works to make the hollow of the elevator, and 30 to the
works to be done on each floor to allow access to the elevator (10 in each of
them). Table 1 below shows the distribution of costs for each of the
apartments according to the Dutch and Spanish rules (the latter with quotas
for each apartment given by its surface). Notice that both rules are based on
egalitarian principles and can be interpreted as the equal division rule; the
difference is that in the case of the Dutch rule the subjects that receive the
equitable distribution are the apartments, whereas in the case of the Spanish
rule the equitable distribution subjects are the quota units.111In this
example the quota units are the square meters of the apartments. For the
approach we adopt to be meaningful, the quota unit numbers must be integers.
Notice that the same egalitarian spirit of these rules can be maintained
despite changing the equitable distribution subjects. For instance, it would
be natural to consider a kind of two-step equitable distribution subjects,
where the subjects in the first step are the floors and the subjects in the
second step are the apartments (in the case of the Dutch rule) or the quota
units (in the case of the Spanish rule). This would result in the distribution
of costs shown in Table 2 below. Observe that this variation arises from
considering that the floors of the building naturally give rise to a structure
of a priori unions in the sense of Owen (1977) and, thus, the convenience of
extending the equal division value for games with a priori unions emerges
spontaneously in this example. We do it formally in Section 3.

There are other possible variations of these Dutch and Spanish rules with and
without the structure of a priori unions when using the equal surplus division
value instead of the equal division value; thus, the convenience of extending
the equal surplus division value for games with a priori unions can also be
motivated on the basis of this example. We do it in Section 4, where we also
analyse in more depth how the equal surplus division value for games with a
priori unions can be applied in the example we have discussed in this section.

| Dutch rule | Spanish rule
---|---|---
3rd floor | 20 20 20 | 13.0909 13.0909 13.0909
2nd floor | 20 20 | 21.8182 19.6364
1st floor | 20 | 39.2727
Table 1: Distribution according to the Dutch and Spanish rules | Dutch rule | Spanish rule
---|---|---
3rd floor | 13.3333 13.3333 13.3333 | 13.3333 13.3333 13.3333
2nd floor | 20 20 | 21.0526 18.9474
1st floor | 40 | 40
Table 2: Distribution according to the two-step Dutch and Spanish rules

##  3 The equal division value for TU-games with a priori unions

In this section we extend the equal division value for TU-games to the more
general setup of TU-games with a priori unions. To start with, we introduce
the basic concepts and notations we use in this paper.

A transferable utility cooperative game (from now on a TU-game) is a pair
$(N,v)$ where $N$ is a finite set of $n$ players, and $v$ is a map from
$2^{N}$ to $\mathbb{R}$ with $v(\emptyset)=0$, that is called the
characteristic function of the game. In the sequel, $\mathcal{G}_{N}$ will
denote the family of all TU-games with player set $N$ and $\mathcal{G}$ the
family of all TU-games. A value for TU-games is a map $f$ that assigns to
every TU-game $(N,v)\in\mathcal{G}$ a vector $f(N,v)=(f_{i}(N,v))_{i\in
N}\in\mathbb{R}^{N}$ with $\sum_{i\in N}f_{i}(N,v)=v(N)$.

As it was remarked in the introduction, sometimes agents cooperate on the
basis of a kind of egalitarian principle according to which the benefits will
be shared equitably. This gives rise to the equal division value $ED$ that
distributes $v(N)$ equally among the players in $N$. Formally, the equal
division value $ED$ is defined for every $(N,v)\in\mathcal{G}$ and for all
$i\in N$ by

$ED_{i}(N,v)=\frac{v(N)}{n}.$

Now denote by $P(N)$ the set of all partitions of $N$. A TU-game with a priori
unions is a triplet $(N,v,P)$ where $(N,v)\in\mathcal{G}$ and
$P=\\{P_{1},\dots,P_{m}\\}\in P(N)$. The set of TU-games with a priori unions
and with player set $N$ will be denoted by $\mathcal{G}_{N}^{U}$, and the set
of all TU-games with a priori unions by $\mathcal{G}^{U}$. A value for TU-
games with a priori unions is a map $g$ that assigns to every
$(N,v,P)\in\mathcal{G}^{U}$ a vector $g(N,v,P)=(g_{i}(N,v,P))_{i\in
N}\in\mathbb{R}^{N}$ with $\sum_{i\in N}g_{i}(N,v,P)=v(N)$. The next
definition provides the natural extension of the equal division value to TU-
games with a priori unions.

###### Definition 3.1

The equal division value for TU-games with a priori unions $ED^{U}$ is defined
by

$ED_{i}^{U}\left(N,v,P\right)=\frac{v(N)}{mp_{k}}$

for all $i\in N$ and all $(N,v,P)\in\mathcal{G}^{U}$ with
$P=\\{P_{1},\dots,P_{m}\\}$ and $i\in P_{k}$; $p_{k}$ denotes the cardinal of
$P_{k}$.

Notice that the equal division value for TU-games with a priori unions has
been used in the motivating example in Section 2 (see Table 2 and the
corresponding comments). Next we provide an axiomatic characterization of this
value. We start giving some properties of a value $g$ for TU-games with a
priori unions.

Additivity (ADD). A value $g$ for TU-games with a priori unions satisfies
additivity if, for all $(N,v,P),(N,w,P)\in\mathcal{G}^{U}$, it holds that

$g(N,v+w,P)=g(N,v,P)+g(N,w,P).$

Take a TU-game $(N,v)\in\mathcal{G}^{N}$ and $i,j\in N$. We say that $i,j$ are
indistinguishable in $v$ if $v(S\cup i)=v(S\cup j)$ for all $S\subseteq
N\setminus\\{i,j\\}$.

Symmetry within unions (SWU). A value $g$ for TU-games with a priori unions
satisfies symmetry within unions if, for all $(N,v,P)\in\mathcal{G}^{U}$, all
$P_{k}\in P$, and all $i,j\in P_{k}$ indistinguishable in $v$, it holds that
$g_{i}(N,v,P)=g_{j}(N,v,P)$.

Take $(N,v,P)\in\mathcal{G}^{U}$ with $P=\\{P_{1},\dots,P_{m}\\}$ and denote
$M=\\{1,\dots,m\\}$. The quotient game of $(N,v,P)$ is the TU-game $(M,v/P)$
where

$(v/P)(R)=v(\cup_{r\in R}P_{r})\quad\mbox{for all}\;R\subseteq M.$

Symmetry among unions (SAU). A value $g$ for TU-games with a priori unions
satisfies symmetry among unions if, for all $(N,v,P)\in\mathcal{G}^{U}$ and
all $k,l\in M$ indistinguishable in $v/P$, it holds that $\sum_{i\in
P_{k}}g_{i}(N,v,P)=\sum_{i\in P_{l}}g_{i}(N,v,P)$.

Take a TU-game $(N,v)\in\mathcal{G}^{N}$ and $i\in N$. We say that $i$ is a
nullifying player in $v$ if $v(S\cup i)=0$ for all $S\subseteq N$.

Nullifying player property (NPP). A value $g$ for TU-games with a priori
unions satisfies the nullifying player property if, for all
$(N,v,P)\in\mathcal{G}^{U}$ and all $i\in N$ nullifying player in $v,$ it
holds that $g_{i}(N,v,P)=0$.

An analogous to NPP above is used in van den Brink (2007) to characterize the
equal division value for TU-games. In the next theorem, we extend van den
Brink’s result to TU-games with a priori unions.

###### Theorem 3.2

$ED^{U}$ is the unique value for TU-games with a priori unions that satisfies
ADD, SWU, SAU and NPP.

###### Proof.

It is immediate to check that $ED^{U}$ satisfies ADD, SWU, SAU and NPP. To
prove the unicity, consider a value $g$ for TU-games with a priori unions that
satisfies ADD, SWU, SAU and NPP. Fix $N$ and define for all
$\alpha\in\mathbb{R}$ and all non-empty $T\subseteq N$ the TU-game
$(N,e_{T}^{\alpha})$ given by $e_{T}^{\alpha}(S)=\alpha$ if $S=T$ and
$e_{T}^{\alpha}(S)=0$ if $S\neq T$. If $T=N$, since $g$ satisfies SWU and SAU,
it is clear that $g_{i}(N,e_{N}^{\alpha},P)=\frac{\alpha}{mp_{k}}$ for any
$P=\\{P_{1},\dots,P_{m}\\}$ and all $i\in P_{k}\subseteq N$, because all
players in $N$ are indistinguishable in $e_{N}^{\alpha}$ and all players in
$M$ are indistinguishable in $e_{N}^{\alpha}/P$. If $T\subset N$ notice that
all players in $N\setminus T$ are nullifying players in $e_{T}^{\alpha}$ and
then, since $g$ satisfies NPP,

$\sum_{i\in T}g_{i}(N,e_{T}^{\alpha},P)=\sum_{i\in
N}g_{i}(N,e_{T}^{\alpha},P)=e_{T}^{\alpha}(N)=0$

for any $P$. Then, since $g$ satisfies SWU and SAU it is not difficult to
check that $g(N,e_{T}^{\alpha},P)=0$. Finally, the additivity of $g$ and the
fact that $v=\sum_{T\subseteq N}e_{T}^{v(T)}$ imply that

$g_{i}(N,v,P)=\sum_{T\subseteq
N}g_{i}(N,e_{T}^{v(T)},P)=g_{i}(N,e_{N}^{v(N)},P)=\frac{v(N)}{mp_{k}}=ED_{i}^{U}(N,v,P)$

for any $P$ and all $i\in P_{k}\subseteq N$. ∎

##  4 The equal surplus division value for TU-games with a priori unions

In this section we extend the equal surplus division value for TU-games to the
more general setup of TU-games with a priori unions. To start with, remember
that the equal surplus division value $ESD$ is defined for every
$(N,v)\in\mathcal{G}$ and for all $i\in N$ by

$ESD_{i}(N,v)=v(i)+\frac{v^{0}(N)}{n},$

where $v^{0}(S)=v(S)-\sum_{i\in S}v(i)$ for all $S\subseteq N$. Notice that
$ESD$ is a variant of $ED$ in which we first allocate $v(i)$ to each player
$i\in N$, and then distribute $v^{0}(N)$ among the players using $ED$. $ESD$
is a reasonable alternative to $ED$ for situations where individual benefits
and joint benefits are neatly separable. Let us illustrate this with the
example of Section 2 (notice that it deals with costs instead of with
benefits).

Consider again the three-storey building of Section 2 and the cost of
installing the elevator. Clearly, the cost of the machine is a joint cost,
whereas the cost due to the works to be done on each floor should be paid by
the owners of each floor. With respect to the costs of the hollow, assume that
there is a fixed cost of 10 and an individual cost of 10 for the owners of the
first floor that is incremented by 10 for the owners of the second floor and
by an additional 10 for the owners of the third floor. According to this, the
cost $c(i)$ in which each player is involved is:

  * •

50 (machine) + 10 (floor) + 40 (hollow) = 100, for the players of the third
floor,

  * •

50 (machine) + 10 (floor) + 30 (hollow) = 90, for the players of the second
floor,

  * •

50 (machine) + 10 (floor) + 20 (hollow) = 80, for the players of the first
floor.

Now we can compute the equal surplus division value for the game in which the
players are the apartments and $c(N)=120$ (this is what we call the ES-Dutch
rule) and the equal surplus division value for the game in which the players
are the quota units and $c(N)=120$ (this is what we call the ES-Spanish rule).
Table 3 below displays the distributions of the cost among the apartments
using both rules. Notice that these distributions are not satisfactory because
they seem to penalize too much the apartments on the third floor, specially
the ES-Spanish rule that even proposes that the apartment on the first floor
is recompensed if the elevator is installed. The reason for this seems to be
that the individual costs in this example actually belong to the floors
instead of to the players; consequently it would be more reasonable to use a
kind of two-step rule for the equal surplus division value analogous to the
two-step rule for the equal division value introduced in Section 2. In other
words, this example suggests that we should consider the structure of a priori
unions given by the floors and distribute the costs using an extension of the
equal surplus division value to TU-games with a priori unions.

| Dutch rule | Spanish rule
---|---|---
3rd floor | 26.6666 26.6666 26.6666 | 613.0860 613.0860 613.0860
2nd floor | 16.6666 16.6666 | 21.8100 19.6290
1st floor | 6.6666 | -1760.7240
Table 3: Distribution according to the ES-Dutch and ES-Spanish rules

Next we propose three alternative ways for extending the equal surplus
division value to TU-games with a priori unions. The first one divides the
value of the grand coalition in the quotient game using the equal surplus
division value and then divides the amount assigned to each union equally
among its members.

###### Definition 4.1

The equal surplus division value (one) for TU-games with a priori unions
$ESD1^{U}$ is defined by

$ESD1^{U}_{i}\left(N,v,P\right)=\frac{(v/P)(k)}{p_{k}}+\frac{(v/P)^{0}(M)}{mp_{k}}=\frac{v(P_{k})}{p_{k}}+\frac{v(N)-\sum_{l\in
M}v(P_{l})}{mp_{k}}$

for all $i\in N$ and all $(N,v,P)\in\mathcal{G}^{U}$ with
$P=\\{P_{1},\dots,P_{m}\\}$ and with $i\in P_{k}$.

The second extension divides again the value of the grand coalition in the
quotient game using the equal surplus division value; then it distributes the
amount $\frac{v(N)-\sum_{l\in M}v(P_{l})}{m}$ equally among the players in
each union, and the amount $v(P_{k})$ giving $v(i)$ to each player $i\in
P_{k}$ and dividing $v(P_{k})-\sum_{j\in P_{k}}v(j)$ equally among the players
in $P_{k}$.

###### Definition 4.2

The equal surplus division value (two) for TU-games with a priori unions
$ESD2^{U}$ is defined by

$ESD2_{i}^{U}\left(N,v,P\right)=v(i)+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(j)}{p_{k}}+\frac{v(N)-\sum_{l\in M}v(P_{l})}{mp_{k}}$

for all $i\in N$ and all $(N,v,P)\in\mathcal{G}^{U}$ with
$P=\\{P_{1},\dots,P_{m}\\}$ and with $i\in P_{k}$.

Finally, the third extension assigns $v(i)$ to each player $i$ and then
divides $v^{0}(N)$ among the players using $ED^{U}$.

###### Definition 4.3

The equal surplus division value (three) for TU-games with a priori unions
$ESD3^{U}$ is defined by

$ESD3^{U}_{i}\left(N,v,P\right)=v(i)+ED^{U}(N,v^{0},P)=v(i)+\frac{v(N)-\sum_{j\in
N}v(j)}{mp_{k}}$

for all $i\in N$ and all $(N,v,P)\in\mathcal{G}^{U}$ with
$P=\\{P_{1},\dots,P_{m}\\}$ and with $i\in P_{k}$.

Now we can compute the equal surplus division values one, two and three for
the game with a priori unions in which the players are the apartments, the
unions are the floors and $c(N)=120$ (they are what we call the $ESD1^{U}$,
$ESD2^{U}$ and $ESD3^{U}$-Dutch rules) and the equal surplus division values
one, two and three for the game with a priori unions in which the players are
the quota units, the unions are the floors and $c(N)=120$ (they are what we
call the $ESD1^{U}$, $ESD2^{U}$ and $ESD3^{U}$-Spanish rules). Tables 4, 5 and
6 below display the distributions of the cost among the apartments using these
rules.222Notice that Tables 4 and 5 are identical. This is because $ESD1$ and
$ESD2$ coincide when, as in this example, for each union $P_{k}$ and each
$i,j\in P_{k}$, it is satisfied that $v(i)=v(j)$. The results in Tables 4 and
5 seem to be more reasonable than those in Table 3; notice that they slightly
penalize the higher floors in comparison with the results in Table 2. The
result in Table 6 is not satisfactory since it penalizes too much the
apartments on the third floor. It shows that $ESD3^{U}$ is not an appropriate
extension of $ESD$, at least for this example; we informally discuss why in
the section of concluding remarks.

| Dutch rule | Spanish rule
---|---|---
3rd floor | 16.6666 16.6666 16.6666 | 16.6666 16.6666 16.6666
2nd floor | 20 20 | 21.0584 18.9525
1st floor | 30 | 30
Table 4: Distribution according to $ESD1^{U}$ | Dutch rule | Spanish rule
---|---|---
3rd floor | 16.6666 16.6666 16.6666 | 16.6666 16.6666 16.6666
2nd floor | 20 20 | 21.0584 18.9525
1st floor | 30 | 30
Table 5: Distribution according to $ESD2^{U}$ | Dutch rule | Spanish rule
---|---|---
3rd floor | 51.1111 51.1111 51.1111 | 513.3333 513.3333 513.3333
2nd floor | 16.6666 16.6666 | 336.8421 303.1579
1st floor | -66.6666 | -2060
Table 6: Distribution according to $ESD3^{U}$

In the remainder of this section we study $ESD1^{U}$, $ESD2^{U}$ and
$ESD3^{U}$ from the point of view of their properties; in particular, we
provide axiomatic characterizations of these values. We start by introducing
new properties of a value $g$ for TU-games with a priori unions. Take
$(N,v)\in\mathcal{G}$ and $i\in N$. We say that $i$ is a dummifying player in
$v$ if $v(S\cup i)=\sum_{j\in S\cup i}v(j)$ for all $S\subseteq N$. Take now a
TU-game with a priori unions $(N,v,P)\in\mathcal{G}^{U}$ where
$P=\\{P_{1},\dots,P_{m}\\}$. We say that $P_{k}$ is a dummifying union in
$(v,P)$ if $k$ is a dummifying player in $v/P$. Dummifying players and
dummifying unions should play a relevant role in the characterizations of
$ESD1^{U}$, $ESD2^{U}$ and $ESD3^{U}$ since a property on dummifying players
is used in Casajus and Hüttner (2014) for characterizing $ESD$. In fact they
use the following property (for $\mathcal{G}$ instead of $\mathcal{G}^{U}$).

Dummifying player property (DPP). A value $g$ for TU-games with a priori
unions satisfies the dummifying player property if, for all
$(N,v,P)\in\mathcal{G}^{U}$ and all $i\in N$ dummifying player in $v$, it
holds that $g_{i}(N,v,P)=v(i)$.

Notice that $ESD3^{U}$ satisfies DPP, but neither $ESD1^{U}$ nor $ESD2^{U}$
satisfy it. In the search of properties that $ESD1^{U}$ or $ESD2^{U}$ might
satisfy, we propose the following variations of DPP and NPP.

Dummifying union/player property (DUPP). A value $g$ for TU-games with a
priori unions satisfies the dummifying union/player property if, for all
$(N,v,P)\in\mathcal{G}^{U}$ and all $P_{k}\in P$ dummifying union in $(v,P)$
with $i\in P_{k}$ being a dummifying player in $v_{P_{k}},$333$v_{P_{k}}$
denotes the characteristic function of the TU-game $(P_{k},v_{P_{k}})$, where
$v_{P_{k}}(S)=v(S)$ for all $S\subseteq P_{k}$. it holds that
$g_{i}(N,v,P)=v(i)$.

Dummifying union/nullifying player property (DUNPP). A value $g$ for TU-games
with a priori unions satisfies the dummifying union/nullifying player property
if, for all $(N,v,P)\in\mathcal{G}^{U}$ and all $P_{k}\in P$ dummifying union
in $(v,P)$ with $i\in P_{k}$ being a nullifying player in $v_{P_{k}},$ it
holds that $g_{i}(N,v,P)=0$.

Now we give parallel characterizations of the three extensions of $ESD$ using
the properties we have introduced above.

###### Theorem 4.4

$ESD1^{U}$ is the unique value for TU-games with a priori unions that
satisfies ADD, SWU, SAU and DUNPP.

###### Proof.

It is immediate to check that $ESD1^{U}$ satisfies ADD, SWU, SAU and DUNPP. To
prove the unicity, consider a value $g$ for TU-games with a priori unions that
satisfies ADD, SWU, SAU and DUNPP. Take $(N,v,P)\in\mathcal{G}^{U}$ with
$P=\\{P_{1},\dots,P_{m}\\}$ and define the TU-game $(N,v^{1})$ given by

$v^{1}(S)=\sum_{P_{l}\subseteq S}v(P_{l})=\sum_{l=1}^{m}v^{P_{l}}(S)$

for all $S\subseteq N$, where $v^{P_{l}}(S)=v(P_{l})$ if $P_{l}\subseteq S$
and $v^{P_{l}}(S)=0$ otherwise.

Take $P_{k}\in P$. Since $g$ is a value for TU-games with a priori unions,
then

$\sum_{i\in N}g_{i}(N,v^{P_{k}},P)=v^{P_{k}}(N)=v(P_{k}).$

All unions $P_{l}\in P$ are dummifying unions in $(v^{P_{k}},P)$ and all
players $i\in P_{l}$, with $l\neq k$, are nullifying players in
$(v^{P_{k}})_{P_{l}}$. By DUNPP, $g_{i}(N,v^{P_{k}},P)=0$ for all $i\notin
P_{k}$. And since all players in $P_{k}$ are indistinguishable in $v^{P_{k}}$,
then SWU implies that, for all $i\in P_{k}$,
$g_{i}(N,v^{P_{k}},P)=\frac{v(P_{k})}{p_{k}}$. Using the additivity of $g$,
for all $i\in P_{k}$,

$g_{i}(N,v^{1},P)=\frac{v(P_{k})}{p_{k}}.$ (1)

Define now $v^{2}=v-v^{1}$ and, for all $\alpha\in\mathbb{R}$ and all non-
empty $T\subseteq N$, $e_{T}^{\alpha}$ by $e_{T}^{\alpha}(S)=\alpha$ if $S=T$
and $e_{T}^{\alpha}(S)=0$ if $S\neq T$. It is clear that
$v^{2}=\sum_{T\subseteq N}e^{v^{2}(T)}_{T}$. If $T=N$, since all players in
$N$ are indistinguishable in $e^{v^{2}(N)}_{N}$ and all players in $M$ are
indistinguishable in $e^{v^{2}(N)}_{N}/P$, SWU and SAU imply that, for all
$i\in P_{k}$,

$g_{i}(N,e^{v^{2}(N)}_{N},P)=\frac{{v^{2}(N)}}{mp_{k}}=\frac{v(N)-\sum_{l\in
M}v(P_{l})}{mp_{k}}.$

If $T\subset N$, consider two cases:

  * •

Take $T=\cup_{l\in L}P_{l}$, with $\emptyset\subset L\subset M$. For all
$P_{u}\in P$, if $T\neq P_{u}$ then $e_{T}^{v^{2}(T)}(P_{u})=0$ and if
$T=P_{u}$ then $e_{T}^{v^{2}(T)}(P_{u})=v^{2}(P_{u})=0$. Hence, it is easy to
see that all the unions in $M\setminus L$ are dummifying unions in
$(e_{T}^{v^{2}(T)},P)$. Also, since all players in $N\setminus T$ are
nullifying players in $e^{v^{2}(T)}_{T}$, DUNPP implies that
$g_{i}(N,e^{v^{2}(T)}_{T},P)=0$ for all $i\notin T$. Notice that since all
unions in $L$ are indistinguishable in $e^{v^{2}(T)}_{T}$, then by SAU
$\sum_{i\in P_{k}}g_{i}(N,e^{v^{2}(T)}_{T},P)=\sum_{i\in
P_{l}}g_{i}(N,e^{v^{2}(T)}_{T},P)$ for all $k,l\in L$; notice also that since

$\sum_{i\in T}g_{i}(N,e^{v^{2}(T)}_{T},P)=\sum_{i\in
N}g_{i}(N,e^{v^{2}(T)}_{T},P)=e^{v^{2}(T)}_{T}(N)=0$

then $\sum_{i\in P_{k}}g_{i}(N,e^{v^{2}(T)}_{T},P)=0$ for all $k\in L$. To
conclude, SWU implies that $g_{i}(N,e^{v^{2}(T)}_{T},P)=0$ for all $i\in
P_{k}$, with $k\in L$, and therefore for all $i\in N$.

  * •

For any other $T\subset N$ that is not in the previous case, the quotient game
$(M,e^{v^{2}(T)}_{T}/P)$ satisfies that $(e^{v^{2}(T)}_{T}/P)(R)=0$ for all
$R\subseteq M$ and, thus, all the unions in $P$ are indistinguishable and
dummifying unions in $(e^{v^{2}(T)}_{T},P)$. If $i\notin T$, then $i$ is a
nullifying player in $e^{v^{2}(T)}_{T}$ and DUNPP implies that
$g_{i}(N,e^{v^{2}(T)}_{T},P)=0$. Analogously as in the previous case, SAU and
SWU imply that $g_{i}(N,e^{v^{2}(T)}_{T},P)=0$ for all $i\in T$.

Now ADD implies that, for all $i\in P_{k}$ with $P_{k}\in P$,

$g_{i}(N,v^{2},P)=\sum_{T\subseteq
N}g_{i}(N,e_{T}^{v^{2}(T)},P)=\frac{{v^{2}(N)}}{mp_{k}}.$ (2)

Finally, from (1), (2), ADD and $v=v^{1}+v^{2}$ it is clear that

$g(N,v,P)=ESD1^{U}(N,v,P).$

∎

###### Theorem 4.5

$ESD2^{U}$ is the unique value for TU-games with a priori unions that
satisfies ADD, SWU, SAU and DUPP.

###### Proof.

It is immediate to check that $ESD2^{U}$ satisfies ADD, SWU, SAU and DUPP. To
prove the unicity, consider a value $g$ for TU-games with a priori unions that
satisfies ADD, SWU, SAU and DUPP. Take $(N,v,P)\in\mathcal{G}^{U}$ with
$P=\\{P_{1},\dots,P_{m}\\}$ and define $v^{a}$, $v^{01}$ and $v^{02}$ by:

  * •

$v^{a}(S)=\sum_{i\in S}v(i)$,

  * •

$v^{01}(S)=\sum_{P_{l}\subseteq S}v^{0}(P_{l})=\sum_{l=1}^{m}v^{0P_{l}}$(S),

  * •

$v^{02}(S)=v^{0}(S)-\sum_{P_{l}\subseteq S}v^{0}(P_{l})$,

for all $S\subseteq N$, where $v^{0P_{l}}(S)=v^{0}(P_{l})$ if $P_{l}\subseteq
S$ and $v^{0P_{l}}(S)=0$ otherwise.

Since all unions are dummifying in $(v^{a},P)$ and all players are dummifying
in $v^{a}$, then DUPP implies that, for all $i\in N$,

$g_{i}(N,v^{a},P)=v^{a}(i)=v(i).$ (3)

Take $P_{k}\in P$. Since $g$ is a value for TU-games with a priori unions,
then

$\sum_{i\in N}g_{i}(N,v^{0P_{k}},P)=v^{0P_{k}}(N)=v^{0}(P_{k}).$

All unions $P_{l}\in P$ are dummifying unions in $(v^{0P_{k}},P)$ and all
players $i\in P_{l}$, with $l\neq k$, are dummifying players in
$(v^{0P_{k}})_{P_{l}}$. By DUPP, $g_{i}(N,v^{0P_{k}},P)=v^{0P_{k}}(i)=0$ for
all $i\notin P_{k}$. And since all players in $P_{k}$ are indistinguishable in
$v^{0P_{k}}$, then SWU implies that, for all $i\in P_{k}$,
$g_{i}(N,v^{0P_{k}},P)=\frac{v^{0}(P_{k})}{p_{k}}$. Using ADD, for all $i\in
P_{k}$,

$g_{i}(N,v^{01},P)=\frac{v^{0}(P_{k})}{p_{k}}.$ (4)

Take now into account that $v^{02}=\sum_{T\subseteq N}e^{v^{02}(T)}_{T}$. If
$T=N$, since all players in $N$ are indistinguishable in $e^{v^{02}(N)}_{N}$
and all players in $M$ are indistinguishable in $e^{v^{02}(N)}_{N}/P$, SWU and
SAU imply that, for all $i\in P_{k}$,

$g_{i}(N,e^{v^{02}(N)}_{N},P)=\frac{{v^{02}(N)}}{mp_{k}}.$

If $T\subset N$, consider two cases:

  * •

Take $T=\cup_{l\in L}P_{l}$, with $\emptyset\subset L\subset M$. Since
$e_{T}^{v^{02}(T)}(P_{u})=0$ for all $P_{u}\in P$ and
$(e_{T}^{v^{02}(T)}/P)(R)=0$ for all $R\subseteq M$ with $R\cap(M\setminus
L)\neq\emptyset$, all the unions in $M\setminus L$ are dummifying unions in
$(e_{T}^{v^{02}(T)},P)$. Also, since all players in $N\setminus T$ are
dummifying players in $e^{v^{02}(T)}_{T}$, DUPP implies that
$g_{i}(N,e^{v^{02}(T)}_{T},P)=e^{v^{02}(T)}_{T}(i)=0$ for all $i\notin T$.
Notice that since all unions in $L$ are indistinguishable in
$e^{v^{02}(T)}_{T}$, then by SAU $\sum_{i\in
P_{k}}g_{i}(N,e^{v^{02}(T)}_{T},P)=\sum_{i\in
P_{l}}g_{i}(N,e^{v^{02}(T)}_{T},P)$ for all $k,l\in L$, and notice that since

$\sum_{i\in T}g_{i}(N,e^{v^{02}(T)}_{T},P)=\sum_{i\in
N}g_{i}(N,e^{v^{02}(T)}_{T},P)=e^{v^{02}(T)}_{T}(N)=0$

then $\sum_{i\in P_{k}}g_{i}(N,e^{v^{02}(T)}_{T},P)=0$ for all $k\in L$.
Hence, SWU implies that $g_{i}(N,e^{v^{02}(T)}_{T},P)=0$ for all $i\in T$.

  * •

For any other $T\subset N$ that is not in the previous case, the quotient game
$(M,e_{T}^{v^{02}(T)}/P)$ satisfies that $(e_{T}^{v^{02}(T)}/P)(R)=0$ for all
$R\subseteq M$ and, thus, all the unions in $P$ are indistinguishable and
dummifying unions in $(e_{T}^{v^{02}(T)},P)$. If $i\notin T$, then $i$ is a
dummifying player in $e^{v^{02}(T)}_{T}$ and DUPP implies that
$g_{i}(N,e^{v^{02}(T)}_{T},P)=e^{v^{02}(T)}_{T}(i)=0$. Analogously as in the
previous case, SAU and SWU imply that $g_{i}(N,e^{v^{02}(T)}_{T},P)=0$ for all
$i\in T$.

Now ADD implies that, for all $i\in P_{k}$ with $P_{k}\in P$,

$g_{i}(N,v^{02},P)=\sum_{T\subseteq
N}g_{i}(N,e_{T}^{v^{02}(T)},P)=\frac{{v^{02}(N)}}{mp_{k}}.$ (5)

Finally, from (3), (4), (5), ADD and $v=v^{a}+v^{01}+v^{02}$ it is clear that

$g(N,v,P)=ESD2^{U}(N,v,P).$

∎

Now we provide a characterization of $ESD3^{U}$. In order to do it we
introduce a new property that is a weaker version of SAU.

Weak symmetry among unions (WSAU). A value $g$ for TU-games with a priori
unions satisfies weak symmetry among unions if, for all
$(N,v,P)\in\mathcal{G}^{U}$ with $v(j)=0$ for all $j\in N$, and for all
$k,l\in M$ indistinguishable in $v/P$, it holds that $\sum_{i\in
P_{k}}g_{i}(N,v,P)=\sum_{i\in P_{l}}g_{i}(N,v,P)$.

###### Theorem 4.6

$ESD3^{U}$ is the unique value for TU-games with a priori unions that
satisfies ADD, SWU, WSAU and DPP.

###### Proof.

It is immediate to check that $ESD3^{U}$ satisfies ADD, SWU, WSAU and DPP. To
prove the unicity, consider a value $g$ for TU-games with a priori unions that
satisfies ADD, SWU, WSAU and DPP. Take now $(N,v,P)\in\mathcal{G}^{U}$ and
$i\in P_{k}$ with $P_{k}\in P$, and define $v^{a}=v-v^{0}$. ADD implies that

$g_{i}(N,v,P)=g_{i}(N,v^{a},P)+g_{i}(N,v^{0},P).$ (6)

Since all players are dummifying in $v^{a}$, then DPP implies that

$g_{i}(N,v^{a},P)=v^{a}(i)=v(i).$ (7)

Now, using for $(N,v^{0})$ analogous arguments as those used in the proof of
Theorem 3.2, it is clear that ADD, SWU, WSAU and DPP imply that

$g_{i}(N,v^{0},P)=ED_{i}(N,v^{0},P).$ (8)

Finally, from (6), (7) and (8) it is clear that

$g(N,v,P)=ESD3^{U}(N,v,P).$

∎

##  5 Concluding remarks

In this last section, we include some supplementary information.

a) It is immediate to prove that $ESD3^{U}$ does not satisfy SAU. Since WSAU
is a weaker version of SAU, and $ESD3^{U}$ is characterized with ADD, SWU,
WSAU and DPP, we conclude that there does not exist a value for TU-games with
a priori unions satisfying ADD, SWU, SAU and DPP.

b) Given a value $f$ for TU-games, a _coalitional_ $f$ _value_ is a value $g$
for TU-games with a priori unions that coincides with $f$ when the partition
$P$ is such that each union is a singleton. That is, if we denote by $P^{n}$
the partition
$\left\\{\left\\{1\right\\},\left\\{2\right\\},\ldots,\left\\{n\right\\}\right\\},$
it holds that $g(N,v,P^{n})=f(N,v)$. It is easy to check that $ED^{U}$ is a
_coalitional equal division value_ , and $ESD1^{U}$, $ESD2^{U}$ and $ESD3^{U}$
are _coalitional equal surplus division values_.

c) A value $g$ for TU-games with a priori unions satisfies the _quotient game
property_ (QGP) if, for all $(N,v,P)\in\mathcal{G}^{U}$ with
$P=\\{P_{1},\dots,P_{m}\\}$ and for its quotient game $(M,v/P)$, it holds that
$\sum_{i\in P_{k}}g_{i}\left(N,v,P\right)=g_{k}\left(M,v/P,P^{m}\right)$ for
all $P_{k}\in P$. It is easy to check that $ED^{U}$, $ESD1^{U}$ and $ESD2^{U}$
satisfy QGP. However, $ESD3^{U}$ does not satisfy QGP. Maybe that is the
reason why it does not behave in an appropriate way in the example we dealt
with in Section 4.

d) The properties in the theorems of this paper are independent. We prove it
in the Appendix.

## Acknowledgements

This work has been supported by the ERDF, the MINECO/AEI grants
MTM2017-87197-C3-1-P, MTM2017-87197-C3-3-P, and by the Xunta de Galicia
(Grupos de Referencia Competitiva ED431C-2016-015 and ED431C-2017/38 and
Centro Singular de Investigación de Galicia ED431G/01).

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## Appendix

a) Independence of the properties of Theorem 3.2:

  * •

$\varphi_{i}=v(i)$ satisfies ADD, SWU, SAU and NPP, but not EFF.

  * •

$\varphi_{i}=$ $\frac{v(P_{k})}{p_{k}}+\frac{v(N)-\sum_{l\in
M}v(P_{l})}{mp_{k}}$ satisfies EFF, ADD, SWU and SAU, but not NPP.

  * •

$\varphi_{i}=$ $\frac{v(N)}{n}$ satisfies EFF, ADD, SWU and NPP, but not SAU.

  * •

$\varphi_{i}=\frac{2v(N)}{mp_{k}}$ if $\displaystyle i=\min_{j\in P_{k}}j$ or
$\varphi_{i}=\frac{(p_{k}-2)v(N)}{mp_{k}(p_{k}-1)}$ if $i\in P_{k}$ and
$i\neq\min_{j\in P_{k}}j$, satisfies EFF, ADD, SAU and NPP, but not SWU.

  * •

$\varphi_{i}=\frac{2v(N)}{mp_{k}|Z_{k}|}$ if $\displaystyle i\in Z_{k}=\\{j\in
P_{k}/v(j)=\min_{z\in P_{k}}v(z)\\}$,
$\varphi_{i}=\frac{(p_{k}-2)v(N)}{mp_{k}(p_{k}-|Z_{k}|)}$ if $i\in
P_{k}\backslash Z_{k}$, satisfies EFF, SWU, SAU and NPP, but not ADD.

b) Independence of the properties of Theorem 4.4:

  * •

$\varphi_{i}=v(i)$ satisfies ADD, SWU, SAU and DUNPP, but not EFF.

  * •

$\varphi_{i}=\frac{v(P_{k})}{p_{k}}+\frac{v(N)-\sum_{l\in M}v(P_{l})}{n}$
satisfies EFF, ADD, SWU, DUNPP, but not SAU.

  * •

$\varphi_{i}=\frac{v(P_{k})}{p_{k}}+\frac{2(v(N)-\sum_{l\in
M}v(P_{l}))}{mp_{k}}$ if $\displaystyle i=\min_{j\in P_{k}}j$ or
$\varphi_{i}=\frac{v(P_{k})}{p_{k}}+\frac{(p_{k}-2)(v(N)-\sum_{l\in
M}v(P_{l}))}{mp_{k}(p_{k}-1)}$ if $i\in P_{k}$ and $i\neq\min_{j\in P_{k}}j$,
satisfies EFF, ADD, SAU and DUNPP, but not SWU.

  * •

$\varphi_{i}=\frac{v(P_{k})}{p_{k}}+\frac{2(v(N)-\sum_{l\in
M}v(P_{l}))}{mp_{k}|Z_{k}|}$ if $\displaystyle i\in Z_{k}=\\{j\in
P_{k}/v(j)=\min_{z\in P_{k}}v(z)\\}$,
$\varphi_{i}=\frac{v(P_{k})}{p_{k}}+\frac{(p_{k}-2)(v(N)-\sum_{l\in
M}v(P_{l}))}{mp_{k}(p_{k}-|Z_{k}|)}$ if $i\in P_{k}\backslash Z_{k}$,
satisfies EFF, SWU, SAU and DUNPP, but not ADD.

  * •

$\varphi_{i}=v(i)+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(j)}{p_{k}}+\frac{v(N)-\sum_{l\in M}v(P_{l})}{mp_{k}}$ satisfies EFF,
ADD, SWU and SAU, but not DUNPP.

c) Independence of the properties of Theorem 4.5:

  * •

$\varphi_{i}=v(i)$ satisfies ADD, SWU, SAU and DUPP, but not EFF.

  * •

$\varphi_{i}=v(i)+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(j)}{p_{k}}+\frac{v(N)-\sum_{l\in M}v(P_{l})}{n}$ satisfies EFF, ADD,
SWU and DUPP, but not SAU.

  * •

$\varphi_{i}=v(i)+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(j)}{p_{k}}+\frac{2(v(N)-\sum_{l\in M}v(P_{l}))}{mp_{k}}$ if
$\displaystyle i=\min_{j\in P_{k}}j$ or
$\varphi_{i}=v(i)+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(j)}{p_{k}}+\frac{(p_{k}-2)(v(N)-\sum_{l\in
M}v(P_{l}))}{mp_{k}(p_{k}-1)}$ if $i\in P_{k}$ and $i\neq\min_{j\in P_{k}}j$,
satisfies EFF, ADD, SAU and DUPP, but not SWU.

  * •

$\varphi_{i}=v(i)+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(j)}{p_{k}}+\frac{2(v(N)-\sum_{l\in M}v(P_{l}))}{mp_{k}|Z_{k}|}$ if
$\displaystyle i\in Z_{k}=\\{j\in P_{k}/v(j)=\min_{z\in P_{k}}v(z)\\}$,
$\varphi_{i}=v(i)+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(j)}{p_{k}}+\frac{(p_{k}-2)(v(N)-\sum_{l\in
M}v(P_{l}))}{mp_{k}(p_{k}-|Z_{k}|)}$ if $i\in P_{k}\backslash Z_{k}$,
satisfies EFF, SWU, SAU and DUNPP, but not ADD.

  * •

$\varphi_{i}=\frac{v(P_{k})}{p_{k}}+\frac{v(N)-\sum_{l\in M}v(P_{l})}{mp_{k}}$
satisfies EFF, ADD, SWU and SAU, but not DUPP.

d) Independence of the properties of Theorem 4.6:

  * •

$\varphi_{i}=v(i)$ satisfies ADD, SWU, WSAU and DPP, but not EFF.

  * •

$\varphi_{i}=v(i)+\frac{v(N)-\sum_{j\in N}v(j)}{n}$ satisfies EFF, ADD, SWU
and DPP, but not WSAU.

  * •

$\varphi_{i}=v(i)+\frac{2(v(N)-\sum_{j\in N}v(j))}{mp_{k}}$ if $\displaystyle
i=\min_{j\in P_{k}}j$ or $\varphi_{i}=v(i)+\frac{(p_{k}-2)(v(N)-\sum_{j\in
N}v(j))}{mp_{k}(p_{k}-1)}$ if $i\in P_{k}$ and $i\neq\min_{j\in P_{k}}j$,
satisfies EFF, ADD, WSAU and DPP, but not SWU.

  * •

$\varphi_{i}=v(i)+\frac{2(v(N)-\sum_{j\in N}v(j))}{mp_{k}|Z_{k}|}$ if
$\displaystyle i\in Z_{k}=\\{j\in P_{k}/v(j)=\min_{z\in P_{k}}v(z)\\}$,
$\varphi_{i}=v(i)+\frac{(p_{k}-2)(v(N)-\sum_{j\in
N}v(j))}{mp_{k}(p_{k}-|Z_{k}|)}$ if $i\in P_{k}\backslash Z_{k}$, satisfies
EFF, SWU, WSAU and DPP, but not ADD.

  * •

$\varphi_{i}=\frac{v(P_{k})}{p_{k}}+\frac{v(N)-\sum_{l\in M}v(P_{l})}{mp_{k}}$
satisfies EFF, ADD, SWU, WSAU but not DPP.