# Variational Analysis of Generalized Games over Banach spaces

Asrifa Sultana111 Corresponding author. e-mail-<EMAIL_ADDRESS>222Department of Mathematics, Indian Institute of Technology Bhilai (491001),
India.  , Shivani Valecha222Department of Mathematics, Indian Institute of
Technology Bhilai (491001), India.

###### Abstract

We study generalized games defined over Banach spaces using variational
analysis. To reformulate generalized games as quasi-variational inequality
problems, we will first form a suitable principal operator and study some
significant properties of this operator. Then, we deduce the sufficient
conditions under which an equilibrium for the generalized game can be obtained
by solving a quasi-variational inequality. Based on this variational
reformulation, we derive the existence of equilibrium for generalized games
with non-ordered (that is, associated weak preference relations need not be
complete and transitive) and mid-point continuous preference maps.

Keywords: Nash equilibrium problem; quasi-variational inequality; binary
relations; non-ordered preference; generalized game
Mathematics Subject Classification: 49J40, 49J53, 91B06, 91B42

##  1 Introduction

Arrow-Debreu [2] initiated the notion of generalized Nash equilibrium problems
(GNEP) consisting of finitely many players with their strategy sets in
Euclidean spaces. Recently, several authors have studied these problems on
infinite dimensional spaces motivated by dynamic games defined over a time
interval and stochastic games consisting of infinite number of states of
nature (see [11, 5] and references therein). Consider a set of players
$\Lambda=\\{1,2,\cdots N\\}$ and finite a collection of Banach spaces
$\\{X_{\nu}|\,\nu\in\Lambda\\}\\}$. Suppose each $\nu\in\Lambda$ regulates a
strategy variable $x_{\nu}\in C_{\nu}$ where $C_{\nu}\subseteq X_{\nu}$.
Assume that,

$C_{-\nu}=\prod_{\mu\in\Lambda\setminus\\{\nu\\}}C_{\mu},C=\prod_{\nu\in\Lambda}C_{\nu},X_{-\nu}=\prod_{\mu\in\Lambda\setminus\\{\nu\\}}X_{\mu}~{}\text{and}~{}X=\prod_{\nu\in\Lambda}X_{\nu}.$
(1)

Then, any vector $x_{-\nu}\in C_{-\nu}$ consists of strategies of players
other then $\nu$ and we have $x=(x_{\nu},x_{-\nu})\in C$. Suppose
$u_{\nu}\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow\mathbb{R}$ is
objective function (or numerical representation of preferences) of player
$\nu$. For a given strategy vector $x_{-\nu}$ of rival players, the player
$\nu$ intends to find $x_{\nu}$ in the feasible strategy set
$K_{\nu}(x_{\nu},x_{-\nu})\subseteq C_{\nu}$ such that,

$u_{\nu}(x_{\nu},x_{-\nu})=\max_{z_{\nu}\in
K_{\nu}(x)}u_{\nu}(z_{\nu},x_{-\nu}).$ (2)

Suppose $Sol_{\nu}(x_{-\nu})$ consists of all the vectors $x_{\nu}$ which
satisfy (2). Then, any vector $\bar{x}\in X$ is known as equilibrium for GNEP
$\Upsilon=(X_{\nu},K_{\nu},u_{\nu})_{\nu\in\Lambda}$ if $\bar{x}_{\nu}\in
Sol_{\nu}(\bar{x}_{-\nu})$ for each $\nu\in\Lambda$ [2, 10].

A preference relation of any individual is a binary relation which represents
his choice in the set of available options. The preference relation of any
individual satisfy the completeness and transitivity property if he is able to
compare any pair of available options. Any complete, transitive and continuous
preference relation defined on connected and separable topological space
admits a numerical representation (see [8, Theorem I]). Hence, one can obtain
the above well known version of GNEP with real-valued objective functions if
preferences of players satisfy these properties. It is worth mentioning that
for numerical representation of preference relations the completeness and
transitivity properties are necessary (see [14, Proposition 1.3]). The fact
that preference relations are not always complete and transitive in real-world
scenarios (see for e.g. [3]) motivated Shafer-Sonnenschein [17] to study
generalized games in which players’ preference maps are non-ordered, that is,
there are no restrictions like completeness and transitivity of players’
preference relations. Thereafter, Yannelis- Prabhakar [20] extended the
generalized games studied in [17] to the case where strategy spaces are
contained in topological vector spaces.

Let us recall the notion of generalized games studied in [17, 20] which
extends the GNEP $\Upsilon=(X_{\nu},K_{\nu},u_{\nu})_{\nu\in\Lambda}$ to the
case of non-ordered preferences. Suppose the strategy set $C_{\nu}$ of any
player $\nu\in\Lambda$ is contained in Banach space $X_{\nu}$ same as GNEP
$\Upsilon$ and $C_{-\nu},C,X_{-\nu},X$ are defined as (1). Let us denote
generalized game (or abstract economy) by
$\Gamma=(X_{\nu},K_{\nu},P_{\nu})_{\nu\in\Lambda}$ where
$P_{\nu}\mathrel{\mathop{\mathchar 58\relax}}X\rightrightarrows X_{\nu}$ and
$K_{\nu}\mathrel{\mathop{\mathchar 58\relax}}C\rightrightarrows C_{\nu}$ are
non-ordered preference and feasible strategy maps of player $\nu$,
respectively. Suppose $\mathbb{S}(\Gamma)$ indicates the set of equilibrium
points for generalized game $\Gamma$. Then by following [17, 20], we say
$\bar{x}\in\mathbb{S}(\Gamma)$ if,

$\bar{x}_{\nu}\in K_{\nu}(\bar{x})~{}\text{and}~{}P_{\nu}(\bar{x})\cap
K_{\nu}(\bar{x})=\emptyset~{}\text{for every}~{}\nu\in\Lambda.$ (3)

Recently, the authors studied economic equilibrium problems with complete,
transitive and continuous preference relations in [15] using tools of
variational analysis. Thereafter, the maximization problem for incomplete and
non-transitive preferences has been studied in [16, 9] by reformulating it as
a variational inequality problem. In these articles, the principal operator of
associated variational inequality is formed by using normal cones to preferred
sets. This motivated Beuno et. al. [6] to study existence of equilibrium for
generalized games by employing variational approach. Further, Sultana-Valecha
deduced the existence of equilibrium and projected solutions for generalized
games with not necessarily bounded strategy maps [18] and non-self constraint
maps [19], respectively, by reformulating these games into appropriate quasi-
variational inequality problems. However, the mentioned articles [16, 9, 6,
18, 19] are restricted to finite dimensional strategy spaces. We observe that
the variational approach used in all these articles can not be administered on
the generalized games $\Gamma=(X_{\nu},K_{\nu},P_{\nu})_{\nu\in\Lambda}$
defined over infinite dimensional Banach spaces $X_{\nu}$ (see Remark 4.1).

Our aim is to study the generalized game
$\Gamma=(X_{\nu},K_{\nu},P_{\nu})_{\nu\in\Lambda}$ defined over Banach spaces
with non-ordered (that is, associated weak preference relations need not be
complete and transitive) and mid-point continuous preference maps, by
employing variational inequality theory. To reformulate generalized games as
quasi-variational inequality problems, we will first form a suitable principal
operator and study some significant properties of this operator. Then, we
deduce the sufficient conditions under which an equilibrium for the
generalized game can be obtained by solving an associated quasi-variational
inequality. Based on this variational reformulation, we derive the existence
of equilibrium for the proposed generalized games.

##  2 Notations and Definitions

Suppose $X$ is a Banach space with topological dual $X^{*}$ and duality
pairing $\langle\cdot,\cdot\rangle$. We denote unit ball in $X$ by
$B_{X}=\\{y\in X|\,\mathinner{\\!\left\lVert y\right\rVert}\leq 1\\}$ and unit
ball in $X^{*}$ by $B_{X}^{*}=\\{y\in X^{*}|\,\mathinner{\\!\left\lVert
y\right\rVert}_{*}\leq 1\\}$, where
$\mathinner{\\!\left\lVert\cdot\right\rVert}_{*}$ is usually defined as
$\mathinner{\\!\left\lVert y\right\rVert}_{*}=\sup_{\mathinner{\\!\left\lVert
u\right\rVert}\leq 1}\langle y^{*},u\rangle$ (see [1]). For any $C\subseteq
X$, we assume $co(C)$ and $\overline{C}$ denote convex hull and closure of a
set $C$, respectively. A point $x\in C$ is known as internal point of $C$ if
for any $u$ there exists $t_{\circ}>0$ such that $x+tu\in C$ for any
$|t|<t_{\circ}$ (see [1, Definition 5.58]). It is easy to notice that any
interior point is internal point. Suppose $P\mathrel{\mathop{\mathchar
58\relax}}C\rightrightarrows X$ is a multi-valued map, that is, $P(x)\subseteq
X$ for any $x\in C$. The graph of map $P$ is given as $Gr(P)=\\{(x,z)\in
C\times X\,|\,z\in P(x)\\}$. The reader may refer [1] for some important
concepts of upper semi-continuous (u.s.c.), lower semi-continuous (l.s.c.) and
closed multi-valued maps.

Let us recall the classical definitions of variational and quasi-variational
inequality problems [13]. Suppose $T\mathrel{\mathop{\mathchar
58\relax}}X\rightrightarrows X^{*}$ is a multi-valued map and $C\subseteq X$
is non-empty. Let $K\mathrel{\mathop{\mathchar 58\relax}}C\rightrightarrows C$
be a multi-valued map. Then, the quasi-variational inequality problem
$QVI(T,K)$ corresponds to find a $\bar{y}\in K(\bar{y})$ so that,

$\text{there exists}~{}\bar{y}^{*}\in
T(\bar{y})~{}\text{satisfying}~{}\langle\bar{y}^{*},y-\bar{y}\rangle\geq
0,~{}\text{for all}~{}y\in K(\bar{y}).$

If the constraint map $K$ is constant, that is, $K(y)=C$ for all $y\in C$ then
$QVI(T,K)$ reduces to a variational inequality problem $VI(T,C)$.

Let $X$ be a Banach space with topological dual $X^{*}$. A preference relation
on $X$ is a binary relation denoted by $\succeq$. As per [14, Chapter 1],
$\succeq$ is known as,

  * -

reflexive if $y\succeq y$ for any $y\in X$;

  * -

complete if for any $y,z\in X$ we have $y\succeq z$ or $z\succeq y$ or both;

  * -

transitive if for any $y,z,w\in X$ we have $y\succeq w$ and $w\succeq z$, then
$y\succeq z$;

  * -

continuous if the sets $\\{z\in X\,|\,z\succeq y\\}$ and $\\{z\in
X\,|\,y\succeq z\\}$ are closed for any $y\in X$;

Any preference relation $\succeq$ defined on $X$ induces a strict preference
relation $\succ$ as follows: For any pair $y,z\in X$, it is known that $y\succ
z$ iff $y\succeq z$ and $z\nsucceq y$. A vector $\bar{x}\in K$ is known as
maximal element of $K\subseteq X$ with respect to $\succeq$ if there no vector
$y\in K$ such that $y\succ\bar{x}$. A real-valued function
$u\mathrel{\mathop{\mathchar 58\relax}}X\rightarrow\mathbb{R}$ is known as
numerical representation of $\succeq$ if for any pair $y,z\in X,y\succeq z$
iff $u(y)\geq u(z)$. If $X$ is connected and separable topological space and
$\succeq$ is complete, transitive and continuous, then $\succeq$ admits a
numerical representation (see [8, Theorem I]) and problem of finding maximal
element reduces to classical optimization problem. On the other hand, a
preference relation $\succeq$ admits a numerical representation only if it is
complete and transitive [14, Proposition 1.3]. However, the completeness and
transitivity of $\succeq$ are not feasible in real world situations (refer
[3]). Hence, we consider a non-ordered preference map
$P\mathrel{\mathop{\mathchar 58\relax}}X\rightrightarrows X$ corresponding to
preference relation $\succeq$ which need not be complete and transitive,

$P(x)=\\{y\in X|\,y\succeq x~{}\text{and}~{}x\nsucceq y\\}=\\{y\in X|\,y\succ
x\\}.$ (4)

Now the notion of maximal elements can be given as (see [9, 20]),

######  Definition 2.1.

Suppose $K$ is non-empty subset of Banach space $X$. Any $\bar{x}\in K$ is
maximal element for the preference map $P$ over the set $K$ if $P(\bar{x})\cap
K\neq\emptyset$. We denote the set of maximal elements for preference $P$ over
set $K$ by $\mathbb{S}(P,K)$.

##  3 Mid-point continuity for Preference Maps

We define the concept of mid-point continuity for non-ordered preference maps
motivated by [12], which generalizes the concept of continuity for preference
relations studied in [16, Definition 2]:

######  Definition 3.1.

Let $X$ be a Banach space. A preference map $P\mathrel{\mathop{\mathchar
58\relax}}X\rightrightarrows X$ fulfils,

  * -

lower mid-point continuity at any $x\in X$ if for any $w\in{P}(x)$, there
exists $t\in[0,1)$ and open set $V$ in $X$ such that $tx+(1-t)w\in V$ and
$V\subset{P}(x)$;

  * -

upper mid-point continuity at any $x\in X$ if for any $w\in P(x)$, there
exists $t\in[0,1)$ and open set $W$ in $X$ such that
$tx+(1-t)w\in{P}(x^{\prime})$ for each $x^{\prime}\in W$;

  * -

mid-point continuity [12, Definition 1] at any $x\in X$ if it is upper and
lower mid-point continuous at $x$, that is, for any $w\in{P}(x)$, there exists
$t\in[0,1)$ and open sets $V$ and $W$ in $X$ such that

$\displaystyle tx+(1-t)w\in V,x\in
W~{}\text{and}~{}w^{\prime}\in{P}(x^{\prime})~{}\text{for
all}~{}(w^{\prime},x^{\prime})\in V\times W.$

Let $Y$ be a Banach space and $(x,y)\in X\times Y$ be arbitrary. We say
$P\mathrel{\mathop{\mathchar 58\relax}}X\times Y\rightrightarrows X$ is lower
mid-point continuous w.r.t. $x$ at $(x,y)$ if for any $w\in{P}(x,y)$, there
exists $t\in[0,1)$ and open set $V$ in $X$ such that $tx+(1-t)w\in V$ and
$V\subset{P}(x,y)$. Furthermore, we say $P$ is upper mid-point continuous
w.r.t. $x$ at $(x,y)$ if for any $w\in{P}(x,y)$ there exists $t\in[0,1)$ and
open sets $W$ in $X$ and $O$ in $Y$ such that
$tx+(1-t)w\in{P}(x^{\prime},y^{\prime})$ for each $(x^{\prime},y^{\prime})\in
W\times O$. Finally, we say $P$ is mid-point continuous w.r.t. $X$ if it is
mid-point continuous w.r.t. $x$ for all $(x,y)\in X\times Y$.

In [16, 15], the maximization of preference is studied through variational
inequalities under the continuity of preference relation $\succ$ (see [16,
Theorem 5]). The notion of mid-point continuity of preference maps is weaker
than existing concept of continuity of preference relations $\succ$ studied in
[15, Definition 2.3] and [16, Definition 2]. In fact, it is easy to observe
that if preference relation $\succ$ is lower (or upper) semi-continuous over
$X$ then $P\mathrel{\mathop{\mathchar 58\relax}}X\rightrightarrows X$ defined
as (4) is lower (or upper) mid-point continuous, respectively. But, the
converse need not be true as shown in following example.

######  Example 3.1.

Suppose $\succ$ is defined on $[0,1]$ as $x\succ y$ iff $u(x)>u(y)$ where,

$u(x)=\begin{cases}x,~{}&\text{if}~{}x<\frac{1}{2}\\\
2-x,~{}&\text{otherwise}.\end{cases}$ (5)

Then, clearly $\frac{1}{2}\succ 1$ but there no open set $V$ such that
$\frac{1}{2}\in V$ and $x^{\prime}\succ 1$ for each $x^{\prime}\in V$. Hence,
$\succ$ is not lower semi-continuous. But it is easy to observe that the
corresponding preference map $P\mathrel{\mathop{\mathchar
58\relax}}[0,1]\rightrightarrows[0,1]$ given as,

$\displaystyle P(x)=\begin{cases}(x,1],&~{}\text{if}~{}x\in[0,\frac{1}{2})\\\
\emptyset,&~{}\text{if}~{}x=\frac{1}{2}\\\
[\frac{1}{2},x),&~{}\text{if}~{}x\in(\frac{1}{2},1].\end{cases}$

is mid-point continuous at any $x\in[0,1]$.

The generalized games are studied in finite dimensional spaces through
variational reformulation in [6] under the assumption that preference maps are
lower semi-continuous with open convex values. Furthermore, the maximization
of preference maps is studied in finite dimensional spaces through variational
reformulation in [9] under the lower semi-continuity and openness like
assumptions on preference maps. Following result along with Example 3.1 and
3.2 shows that mid-point continuity is weaker than the assumptions in [6, 9].

######  Proposition 3.1.

Suppose $Z\subseteq\mathbb{R}^{m}$ is non-empty. A map
$P\mathrel{\mathop{\mathchar 58\relax}}Z\rightrightarrows Z$ is:

  * (a)

lower mid-point continuous at $x$ if $P(x)$ is open in $Z$;

  * (b)

lower mid-point continuous at $x$ if $P(x)$ is convex and any $y\in P(x)$ is
an internal point with respect to $Z$;

  * (c)

upper mid-point continuous at $x$ if it is l.s.c. with convex and open values
in $Z$;

  * (d)

upper mid-point continuous at $x$ if $P$ is l.s.c. at $x$, $P(x)$ is convex
and any $y\in P(x)$ is an internal point with respect to $Z$.

###### Proof.

To prove (a), suppose $P$ is open valued in $Z$. Then, for any $x\in Z$ the
lower mid-point continuity holds by assuming $t=0$. Further, we notice that
$P(x)$ is open if it satisfies the assumptions given in (b) (see [1, Lemma
5.60]). Hence, lower mid-point continuity of $P$ follows by (a). For (c), we
assume $P$ is l.s.c. with convex and open values in $Z$. Then, for any $x\in
Z$ the upper mid-point continuity holds by employing [21, Proposition 1] and
assuming $t=0$. Again, the proof of (d) follows from (c) and [1, Lemma 5.60].
∎

######  Remark 3.1.

We observe that the converse of the statements in Proposition 3.1 need not be
true. In fact, it is clear from Example 3.1 that $P$ is lower mid-point
continuous but it does not admit open values in $[0,1]$. Further, in following
example $P$ is nor lower semi-continuous neither it admits open values but it
is upper mid-point continuous for any $x\in[0,1]$.

######  Example 3.2.

Suppose $P\mathrel{\mathop{\mathchar 58\relax}}[0,1]\rightrightarrows[0,1]$ is
defined as,

$\displaystyle P(x)=\begin{cases}(x,1],&~{}\text{if}~{}x\in[0,\frac{1}{2}]\\\
[\frac{1}{2},\frac{3}{4}],&~{}\text{if}~{}x\in(\frac{1}{2},1].\end{cases}$

Clearly, $P$ is not l.s.c. at $x=\frac{1}{2}$ and it is not even open valued
but it is upper mid-point continuous for any $x\in[0,1]$.

##  4 Normal Cone Operators for Preference Maps

In this section, we derive some important properties of normal cone operators
corresponding to non-ordered preference map, which will be later employed to
study preference maximization problem and generalized games through
variational reformulation.

According to [4], the normal cone of the set $C$ at some point $x\in X$ is

$\displaystyle N_{C}(x)=\begin{cases}\\{x^{*}\in X^{*}|\,\langle
x^{*},y-x\rangle\leq 0~{}\text{for all}~{}y\in
C\\},&\text{if}~{}C\neq\emptyset\\\ X^{*},&\text{otherwise}.\end{cases}$ (6)

For Banach spaces $X$ and $Y$, consider a set-valued map
$P\mathrel{\mathop{\mathchar 58\relax}}X\times Y\rightrightarrows X$. We
define a normal cone operator $N_{P}\mathrel{\mathop{\mathchar
58\relax}}X\times Y\rightrightarrows X^{*}$ as
$N_{P}(x,y)=N_{P(x,y)}(x)=(P(x,y)-\\{x\\})^{\circ},$ that is,

$\displaystyle N_{P}(x,y)=\begin{cases}\\{x^{*}|\,\langle x^{*},z-x\rangle\leq
0~{}\forall\,z\in P(x,y)\\},&\text{if}~{}{P}(x,y)\neq\emptyset\\\
X^{*},&\text{otherwise}.\end{cases}$ (7)

We derive some properties related to normal cone operators defined as (7).

######  Proposition 4.1.

Suppose $N_{P}\mathrel{\mathop{\mathchar 58\relax}}X\times Y\rightrightarrows
X^{*}$ is defined as (7). If the map $P$ is convex valued and lower mid-point
continuous w.r.t. $x$ at $(x,y)$ and $x\notin P(x,y)$ then
$(N_{P}(x,y)\setminus\\{0\\})\neq\emptyset$ for any $(x,y)\in X\times Y$.

###### Proof.

In the case $P(x,y)=\emptyset$ then we have $N_{P}(x,y)=X^{*}$ and
$N_{P}(x,y)\setminus\\{0\\}\neq\emptyset$. Suppose $P(x,y)\neq\emptyset$ and
$w\in P(x,y)$. Then, by lower mid-point continuity of $P$ there exists
$t\in[0,1)$ and open set $V$ such that $tx+(1-t)w\in V\subset P(x,y)$. This
shows $P(x,y)$ contains an interior point $tx+(1-t)w$. As $x\notin P(x,y)$, we
obtain $0\neq x^{*}\in X^{*}$ by virtue of [1, Theorem 5.67] such that

$\sup_{w\in P(x)}\langle x^{*},w\rangle\leq\langle x^{*},x\rangle.$ (8)

This proves $N_{P}(x,y)\setminus\\{0\\}\neq\emptyset$. ∎

######  Proposition 4.2.

Suppose $N_{P}\mathrel{\mathop{\mathchar 58\relax}}X\times Y\rightrightarrows
X^{*}$ is defined as (7). If $P$ is lower mid-point continuous w.r.t. $x$ at
$(x,y)$ and $P(x,y)\neq\emptyset$ then,

$N_{P}(x,y)\cap-N_{P}(x,y)=\\{0\\}.$ (9)

###### Proof.

We observe that,

${N}_{P}(x,y)\cap-{N}_{P}(x,y)=(P(x,y)-\\{x\\})^{\perp}=\\{x^{*}|\,\langle
x^{*},w-x\rangle=0,~{}\forall~{}w\in P(x,y)\\}.$

Suppose $w\in P(x,y)$, then we have $t\in[0,1)$ and open set $V$ such that
$tx+(1-t)w\in V\subset P(x,y)$ by lower mid-point continuity. This implies
$int(P(x,y)-\\{x\\})\neq\emptyset$. Thus, $(P(x,y)-\\{x\\})^{\perp}=\\{0\\}$.
∎

######  Proposition 4.3.

Suppose $P\mathrel{\mathop{\mathchar 58\relax}}X\times Y\rightrightarrows X$
is upper mid-point continuous map with respect to $x$ at $(x,y)$. For
$C\subset X\times Y$, let $T\mathrel{\mathop{\mathchar
58\relax}}C\rightrightarrows X^{*}$ be defined as $T(x,y)=N_{P}(x,y)\cap H$,
where $H$ is weak∗ closed set in $X^{*}$. Then, the map $T$ is norm-weak∗
closed at $(x,y)$ if $T(C)\subset B_{X}^{*}$.

###### Proof.

Consider a net $(x_{\beta},y_{\beta},x_{\beta}^{*})_{\beta}\subseteq Gr(T)$
such that $(x_{\beta},y_{\beta})_{\beta}$ converges to $(x,y)$ with respect to
norm topology and $x_{\beta}^{*}$ converges to $x^{*}$ with respect to weak∗
topology. We aim to show that $x^{*}\in T(x,y)$. Since
$(x_{\beta}^{*})_{\beta}\subseteq H$ weak∗ converges to $x^{*}$ and $H$ is
weak∗ closed set, we have $x^{*}\in H$. It remains to show that $x^{*}\in
N_{P}(x,y)$, that is,

$\langle x^{*},z-x\rangle\leq 0,~{}\text{for each}~{}z\in P(x,y).$ (10)

As we know $x_{\beta}^{*}\in N_{P}(x_{\beta})$, it appears

$\langle x_{\beta}^{*},z-x_{\beta}\rangle\leq 0~{}\text{for all}~{}z\in
P(x_{\beta},y_{\beta}).$ (11)

To prove (10), we assume $z\in P(x,y)$ is arbitrary. Since
$P\mathrel{\mathop{\mathchar 58\relax}}X\times Y\rightrightarrows X$ is upper
mid-point continuous w.r.t. $x$ at $(x,y)$, we obtain $t\in[0,1)$ and a subnet
$(x_{\gamma},y_{\gamma})_{\gamma}\subseteq(x_{\beta},y_{\beta})_{\beta}$ such
that $tx+(1-t)z\in P(x_{\gamma},y_{\gamma})$ for each $\gamma$. As per (11),
we have

$\langle x_{\gamma}^{*},tx+(1-t)z-x_{\gamma}\rangle\leq 0.$ (12)

Since the evaluation $\langle\cdot,\cdot\rangle$ restricted to
$B_{X}^{*}\times X$ is jointly continuous and
$(x^{*}_{\gamma})_{\gamma}\subseteq B_{X}^{*}$, we observe (10) holds by
taking limits in (12). ∎

In [7, Theorem 2.5] Castellani-Giuli derived the presence of norm-weak∗ u.s.c.
compact convex valued sub-map for an adjusted normal operator corresponding to
lower semi-continuous real valued function. In following result, we show that
a sub-map with similar properties exists for normal cone operators
corresponding to non-ordered preference maps motivated by the approach in [7].

######  Proposition 4.4.

Suppose $P\mathrel{\mathop{\mathchar 58\relax}}X\times Y\rightrightarrows X$
is convex valued mid-point continuous map w.r.t. to $X$ and $x\notin P(x,y)$
for any $(x,y)\in X\times Y$. Then, there exists a norm-weak∗ upper semi-
continuous map $F\mathrel{\mathop{\mathchar 58\relax}}X\times
Y\rightrightarrows X^{*}$ with non-empty convex weak∗ compact values such that
$F(x,y)\subseteq N_{P}(x,y)\setminus\\{0\\}$ whenever $P(x,y)\neq\emptyset$.

###### Proof.

Suppose $(\bar{x},\bar{y})\in(X\times Y)\setminus E$ is arbitrary, where
$E=\\{(x,y)\in X\times Y|\,P(x,y)=\emptyset\\}$. We aim to find an open set
$O_{(\bar{x},\bar{y})}$ in $(X\times Y)\setminus E$ containing
$(\bar{x},\bar{y})$ and a map $T_{(\bar{x},\bar{y})}\mathrel{\mathop{\mathchar
58\relax}}O_{(\bar{x},\bar{y})}\rightrightarrows X^{*}$ such that
$T_{(\bar{x},\bar{y})}(x,y)\subseteq N_{P}(x,y)\setminus\\{0\\}$ for all
$(x,y)\in O_{(\bar{x},\bar{y})}$. Suppose $\bar{w}\in P(\bar{x},\bar{y})$ then
by mid-point continuity of $P$ w.r.t. $\bar{x}$ there exists
$\epsilon_{1},\epsilon_{2}>0$ and $t=t_{(\bar{x},\bar{y})}\in[0,1)$, such that

$t\bar{x}+(1-t)\bar{w}+\epsilon_{1}B_{X}\subset P(x,y)~{}\text{for
each}~{}(x,y)\in(\bar{x},\bar{y})+\epsilon_{2}(B_{X}\times B_{Y}).$ (13)

Suppose $2\epsilon=\min\\{\epsilon_{1},\epsilon_{2}\\}$ then we have

$t\bar{x}+(1-t)\bar{w}+2\epsilon B_{X}\subset P(x,y)~{}\text{for
each}~{}(x,y)\in(\bar{x},\bar{y})+\epsilon(B_{X}\times B_{Y}).$ (14)

For any $(x,y)\in(\bar{x},\bar{y})+\epsilon(B_{X}\times B_{Y})$ and $x^{*}\in
N_{P}(x,y)=\\{x^{*}|\,\langle x^{*},w-x\rangle\leq 0,~{}\text{for any}~{}w\in
P(x,y)\\}$ we have,

$\langle x^{*},t\bar{x}+(1-t)\bar{w}+2\epsilon u-x\rangle\leq 0~{}\text{for
all}~{}u\in B_{X}.$ (15)

This further implies,

$\displaystyle 2\epsilon\mathinner{\\!\left\lVert x^{*}\right\rVert}_{*}$
$\displaystyle=2\epsilon\sup_{u\in B_{X}}\langle x^{*},u\rangle$
$\displaystyle\leq\langle x^{*},x-(t_{z}\bar{x}+(1-t_{z})w_{z})\rangle$
$\displaystyle=\langle x^{*},x-\bar{x}\rangle+(1-t_{z})\langle
x^{*},\bar{x}-\bar{w}\rangle$
$\displaystyle\leq\epsilon\mathinner{\\!\left\lVert
x^{*}\right\rVert}+(1-t_{z})\langle x^{*},\bar{x}-\bar{w}\rangle.$

Finally, we have $\epsilon\mathinner{\\!\left\lVert
x^{*}\right\rVert}_{*}\leq(1-t)\langle x^{*},\bar{x}-\bar{w}\rangle$. Let us
assume,

$H_{(\bar{x},\bar{y})}=\\{x^{*}\in X^{*}|\,(1-t)\langle
x^{*},\bar{x}-\bar{w}\rangle=\epsilon\\}.$

Clearly, $N_{P}(x,y)\cap H_{z}\subset N_{P}(x,y)\setminus\\{0\\}$ is weak∗
compact as it is contained in $B_{X}^{*}$ and non-empty as per Proposition
4.1. Suppose $O_{(\bar{x},\bar{y})}\mathrel{\mathop{\mathchar
58\relax}}=(\bar{x},\bar{y})+\epsilon(B_{X}\times B_{Y})$ and define
$T_{(\bar{x},\bar{y})}\mathrel{\mathop{\mathchar
58\relax}}O_{(\bar{x},\bar{y})}\rightrightarrows X^{*}$ by,

$T_{(\bar{x},\bar{y})}(x,y)=N_{P}(x,y)\cap H_{(\bar{x},\bar{y})}~{}\text{for
any}~{}(x,y)\in O_{(\bar{x},\bar{y})}.$ (16)

Now, we construct a map $T\mathrel{\mathop{\mathchar 58\relax}}X\times
Y\rightrightarrows X^{*}$ as convex combination of the maps like
$T_{(\bar{x},\bar{y})}$ by employing a partition of unity technique. We
observe that,

$\mathcal{O}=\\{O_{(\bar{x},\bar{y})}\mathrel{\mathop{\mathchar
58\relax}}={(\bar{x},\bar{y})}+\epsilon(B_{X}\times
B_{Y})|\,{(\bar{x},\bar{y})}\in(X\times Y)\setminus E\\}$

forms an open cover for $(X\times Y)\setminus E$. Since $X\times Y$ is
paracompact, there exists a locally finite open refinement
$\mathcal{U}=\\{U_{\nu}\\}_{\nu\in\Lambda}$ for $\mathcal{O}$ such that for
each $\nu\in\Lambda$ there is $(\bar{x},\bar{y})\in X\times Y$ satisfying
$U_{\nu}\subset O_{(\bar{x},\bar{y})}$. Further, we obtain a family of
continuous functions $\\{f_{\nu}\\}_{\nu\in\Lambda}$ which forms a partition
of unity corresponding to $\mathcal{U}$ as per [1, Theorem 2.90]. Then,
$f_{\nu}(x,y)=0$ if $(x,y)\notin U_{\nu}$ and
$\sum_{\nu\in\Lambda}f_{\nu}(x,y)=1$ for any $(x,y)\in X\times Y$. Further,
the set $\lambda(x,y)=\\{\nu\in\Lambda|\,f_{\nu}(x,y)>0\\}$ is non-empty
finite. Since for each $\nu\in\Lambda$ there exists $(\bar{x},\bar{y})\in X$
satisfying $U_{\nu}\subset O_{(\bar{x},\bar{y})}$, we define
$T_{\nu}\mathrel{\mathop{\mathchar 58\relax}}U_{\nu}\rightrightarrows X^{*}$
by $T_{\nu}(x,y)=T_{(\bar{x},\bar{y})}(x,y)$.

Define $T\mathrel{\mathop{\mathchar 58\relax}}(X\times Y)\setminus
E\rightrightarrows X^{*}$ as
$T(x,y)=\sum_{\nu\in\Lambda(x,y)}f_{\nu}(x,y)T_{\nu}(x,y)$. We claim that
$T(x,y)\subset N_{P}(x,y)\setminus\\{0\\}$.On contrary, suppose $0\in T(x,y)$.
Then, there exists $x_{\nu}^{*}\in T_{\nu}(x,y)\subset N_{P}(x,y)$ for all
$\nu\in\Lambda(x,y)$ such that
$0=\sum_{\nu\in\Lambda(x,y)}f_{\nu}(x,y)x_{\nu}^{*}$. Since $f_{\nu}(x,y)>0$
for all $\nu\in\Lambda(x,y)$, we have

$-x_{\nu}^{*}=\sum_{\mu\in\Lambda(x,y)\setminus\\{\nu\\}}\frac{f_{\mu}(x,y)}{f_{\nu}(x,y)}x_{\mu}^{*}\in
N_{P}(x,y).$

In the view of Proposition 4.2, we have $x_{\nu}^{*}=\\{0\\}$. But, this
contradicts the fact that $x_{\nu}^{*}\in H_{z}$. This proves $T(x,y)\subset
N_{P}(x,y)\setminus\\{0\\}$.

We claim that $F\mathrel{\mathop{\mathchar 58\relax}}X\times
Y\rightrightarrows X^{*}$ constructed as,

$F(x,y)=\begin{cases}B_{X}^{*},&~{}\text{if}~{}P(x,y)=\emptyset\\\
T(x,y),&~{}\text{otherwise}\end{cases}$ (17)

is norm-weak∗ upper semi-continuous. Since $F(x,y)\subseteq B_{X}^{*}$, it is
sufficient to prove that $F$ is norm-weak∗ closed. Consider a net
$(x_{\beta},y_{\beta},x_{\beta}^{*})_{\beta}\subseteq Gr(F)$ such that
$(x_{\beta},y_{\beta})_{\beta}$ converges to $(x,y)$ with respect to norm
topology and $x_{\beta}^{*}$ converges to $x^{*}$ with respect to weak∗
topology. We aim to show that $x^{*}\in F(x,y)$. If $(x,y)\in E$, we observe
that $x^{*}\in F(x,y)=B_{X}^{*}$ as $x_{\beta}^{*}\in
F(x_{\beta},y_{\beta})\subset B_{X}^{*}$ and $B_{X}^{*}$ is weak∗ closed.
Suppose $(x,y)\in(X\times Y)\setminus E$. We notice that $E$ is closed set due
to mid-point continuity of $P$. Hence, it is not restrictive to consider
$(x_{\beta},y_{\beta})_{\beta}\subset(X\times Y)\setminus E$. Since
$(x_{\beta},y_{\beta})_{\beta}$ converges to $(x,y)$, we can assume without
loss of generality that $\Lambda(x,y)\subseteq\Lambda(x_{\beta},y_{\beta})$
for each $\beta$. Hence, it appears,

$T(x_{\beta},y_{\beta})=\sum_{\nu\in\Lambda(x_{\beta},y_{\beta})\setminus\Lambda(x,y)}f_{\nu}(x_{\beta},y_{\beta})T_{\nu}(x_{\beta},y_{\beta})+\sum_{\nu\in\Lambda(x,y)}f_{\nu}(x_{\beta},y_{\beta})T_{\nu}(x_{\beta},y_{\beta}).$

Since $(x^{*}_{\beta})_{\beta}$ weak∗ converges to $x^{*}$, we have nets
$(x^{*}_{\nu,\beta})_{\beta}$ weak∗ converging to $x^{*}_{\nu}$ for each
$\nu\in\Lambda(x_{\beta},y_{\beta})$ such that,

$x^{*}_{\beta}=\sum_{\nu\in\Lambda(x_{\beta},y_{\beta})\setminus\Lambda(x,y)}f_{\nu}(x_{\beta},y_{\beta})x^{*}_{\nu,\beta}+\sum_{\nu\in\Lambda(x,y)}f_{\nu}(x_{\beta},y_{\beta})x^{*}_{\nu,\beta}.$
(18)

Since $\\{f_{\nu}\\}_{\nu\in\Lambda}$ forms a continuous family of functions
we have,

$\lim_{\beta}\sum_{\nu\in\Lambda(x_{\beta},y_{\beta})\setminus\Lambda(x,y)}f_{\nu}(x_{\beta},y_{\beta})=1-\lim_{\beta}\sum_{\nu\in\Lambda(x,y)}f_{\nu}(x_{\beta},y_{\beta})=0.$

This further implies first term in (18) weak∗ converges to $0$ as,

$\mathinner{\\!\left\lVert\sum_{\nu\in\Lambda(x_{\beta},y_{\beta})\setminus\Lambda(x,y)}f_{\nu}(x_{\beta},y_{\beta})x^{*}_{\nu,\beta}\right\rVert}_{*}\leq\sum_{\nu\in\Lambda(x_{\beta},y_{\beta})\setminus\Lambda(x,y)}f_{\nu}(x_{\beta},y_{\beta}).$

In the view of Proposition 4.3, we observe that each $T_{\nu}$ is norm-weak∗
closed map and $x^{*}_{\nu}\in T_{\nu}(x,y)$. Hence, by taking limits in (18)
we have,

$x^{*}=\sum_{\nu\in\Lambda(x,y)}f_{\nu}(x,y)x^{*}_{\nu}\in T(x,y).$

This proves the map $F$ defined as (17) is norm-weak∗ u.s.c. with non-empty
convex weak∗ compact values such that $F(x,y)\subseteq N_{P}(x,y)$. ∎

######  Remark 4.1.

In the case $X$ and $Y$ are finite dimensional spaces, the map $F$ defined as
$F(x,y)=co(N_{P}(x,y)\cap S^{*}_{X})$, where $S^{*}_{X}=\\{y\in
X^{*}\,|\,\mathinner{\\!\left\lVert y\right\rVert}_{*}=1\\}$, is u.s.c. with
non-empty, convex and compact values as per [9]. However, the similar
properties does not hold for $F$ if $X$ is infinite dimensional space, as it
is well known that the sphere $S^{*}_{X}$ is not weak∗ compact in $X^{*}$.

######  Proposition 4.5.

Suppose $P$ is convex valued lower mid-point continuous map w.r.t. $x$ at
$(x,y)$. If $x^{*}\in N_{P}(x,y)$ and $w\in P(x,y)$ satisfy $\langle
x^{*},w-x\rangle\geq 0$ then $x^{*}=0$.

###### Proof.

Let $C$ be non-empty subset of $X$. We claim that if $y\in int(C)$ and
$y^{*}\in C^{\circ}$ satisfy $\langle y^{*},y\rangle\geq 0$ then $y^{*}=0$.
Since $y\in int(C)$, we have $\epsilon>0$ such that $y+\epsilon B_{X}\subset
C$. Hence,

$\langle y^{*},y+\epsilon u\rangle\leq 0~{}\text{for all}~{}u\in B_{X}.$ (19)

Since $y^{*}\in C^{\circ}$, we have $\langle y^{*},y\rangle=0$. From (19), we
obtain $\langle y^{*},u\rangle\leq 0~{}\text{for all}~{}u\in B_{X}.$ Thus,
$\langle y^{*},u\rangle=0~{}\text{for all}~{}u\in B_{X},$ which proves
$y^{*}=0$.

Suppose $x^{*}\in N_{P}(x,y)$ and $w\in P(x,y)$ satisfy $\langle
x^{*},w-x\rangle\geq 0$. Considering $w_{t}=tx+(1-t)w$, we have

$\langle x^{*},w_{t}-x\rangle=(1-t)\langle x^{*},w-x\rangle\geq 0~{}\text{for
any}~{}t\in[0,1).$ (20)

By lower mid-point continuity, we get $t\in[0,1)$ such that $w_{t}\in
int(P(x,y))$. In the view of (20) and the fact that $w_{t}-x\in
int(P(x,y)-\\{x\\})$ and $x^{*}\in(P(x,y)-\\{x\\})^{\circ}$ we get $x^{*}=0$.
∎

##  5 Existence Results

###  5.1 Variational Reformulation of Preference Maximization Problems

Let us reconsider the problem of finding maximal elements for non-ordered
preference map $P\mathrel{\mathop{\mathchar 58\relax}}X\rightrightarrows X$
over $K\subset X$, by following the Definition 2.1. We show that any solution
of variational inequality with principal operator $F\mathrel{\mathop{\mathchar
58\relax}}X\rightrightarrows X^{*}$ defined as (17) is maximal element for $P$
over $K$. Note that the results derived in the previous section for normal
cone operators can be easily adapted for preference maps $P$ which have no
dependence on $y\in Y$.

######  Theorem 5.1.

Let $K$ be non-empty closed convex subset of Banach space $X$. Suppose
$P\mathrel{\mathop{\mathchar 58\relax}}X\rightrightarrows X$ is convex valued
mid-point continuous map and $x\notin P(x)$ for any $x\in X$. Then,
$\bar{x}\in\mathbb{S}(P,K)$ if $\bar{x}\in K$ solves $VI(F,K)$ where $F$ is
defined as (17).

###### Proof.

Suppose $\bar{x}\in K$ solves $VI(F,K)$, that is

$\exists\,\bar{x}^{*}\in F(\bar{x}),\langle\bar{x}^{*},y-\bar{x}\rangle\geq
0~{}\text{for all}~{}y\in K.$ (21)

If $P(\bar{x})=\emptyset$ then $\bar{x}$ is maximal element. Suppose
$P(\bar{x})\neq\emptyset$, then we aim to show that $P(\bar{x})\cap
K=\emptyset$. On contrary, suppose $z\in P(\bar{x})\cap K$. From (21), it
appears that $\bar{x}^{*}\in F(\bar{x})\subseteq
N_{P}(\bar{x})\setminus\\{0\\}$ satisfies,

$\langle\bar{x}^{*},z-\bar{x}\rangle\geq 0.$ (22)

According to Proposition 4.5, we have $\bar{x}^{*}=0$. But, we know that
$P(\bar{x})\neq\emptyset$ and $x^{*}\in F(\bar{x})\subset
N_{P}(\bar{x})\setminus\\{0\\}$ by following the arguments in proof of
Proposition 4.4. This shows our assumption $P(\bar{x})\cap K\neq\emptyset$ is
false and $\bar{x}\in\mathbb{S}(P,K)$. ∎

Based on the variational reformulation of preference maximization problems
obtained in the above result Theorem 5.1, we now deduce the sufficient
conditions for the existence of maximal elements.

######  Theorem 5.2.

Suppose $P\mathrel{\mathop{\mathchar 58\relax}}X\rightrightarrows X$ is convex
valued mid-point continuous map satisfying $x\notin P(x)$ for any $x\in X$ and
$K\subset X$ is non-empty compact and convex. Then, the set of maximal
elements $\mathbb{S}(P,K)\neq\emptyset$.

###### Proof.

In the view of Theorem 5.1, it is sufficient to show that $VI(F,K)$ admits a
solution. According to Proposition 4.4, the map $F$ is norm-weak∗ upper semi-
continuous with non-empty convex and weak∗ compact values. Hence, we see that
variational inequality $VI(F,K)$ admits a solution $\bar{x}\in K$ by assuming
the constraint map $K$ is a constant for all $x$ in [7, Theorem 3.3]. Finally,
$\bar{x}\in K$ is required maximal element of $P$ over $K$. ∎

###  5.2 Variational Reformulation of Generalized Games

In this section, we aim to show that any solution of quasi-variational
inequality with principal operator $\mathcal{F}$ (defined as (23)) and
constraint map $K=\prod_{\nu\in\Lambda}K_{\nu}$ is an equilibrium for game
$\Gamma=(X_{\nu},K_{\nu},P_{\nu})_{\nu\in\Lambda}$ (refer (3) in Section 1).

By following (7), we define ${N}_{P_{\nu}}(x)\mathrel{\mathop{\mathchar
58\relax}}X_{\nu}\times X_{-\nu}\rightrightarrows X^{*}_{\nu}$ corresponding
to preference $P_{\nu}$ as
$N_{P_{\nu}}(x_{\nu},x_{-\nu})=N_{P_{\nu}(x)}(x_{\nu})$, that is,

$N_{P_{\nu}}(x_{\nu},x_{-\nu})=\begin{cases}\\{x_{\nu}^{*}\in
X_{\nu}^{*}|\,\langle x_{\nu}^{*},y_{\nu}-x_{\nu}\rangle\leq 0~{}\text{for
all}~{}y_{\nu}\in P_{\nu}(x)\\},&\text{if}~{}P_{\nu}(x)\neq\emptyset\\\
X_{\nu}^{*},&\text{otherwise}.\end{cases}$

In addition to notations in (1), let $X^{*}=\prod_{\nu\in\Lambda}X_{\nu}^{*}$.
Suppose $\mathcal{F}\mathrel{\mathop{\mathchar 58\relax}}X\rightrightarrows
X^{*}$ is defined as,

$\mathcal{F}(x)=\prod_{\nu\in\Lambda}F_{\nu}(x),$ (23)

where $F_{\nu}\mathrel{\mathop{\mathchar 58\relax}}X_{\nu}\times
X_{-\nu}\rightrightarrows X_{\nu}^{*}$ is considered as (17).

Following result shows that any solution of $QVI(\mathcal{F},K)$ is
equilibrium for the generalized game
$\Gamma=(X_{\nu},K_{\nu},P_{\nu})_{\nu\in\Lambda}$ considered in Section 1.

######  Theorem 5.3.

For any $\nu\in\Lambda$, assume that,

  * (a)

$P_{\nu}\mathrel{\mathop{\mathchar 58\relax}}X\rightrightarrows X_{\nu}$ is
convex valued mid-point continuous map with respect to $X_{\nu}$ such that
$x_{\nu}\notin P(x)$ for any $x\in X$;

  * (b)

$K_{\nu}\mathrel{\mathop{\mathchar 58\relax}}C\rightrightarrows C_{\nu}$
admits non-empty closed convex values.

Then, $\bar{x}\in\mathbb{S}(\Gamma)$ if $\bar{x}$ solves $QVI(\mathcal{F},K)$
where $\mathcal{F}$ is defined as (23).

###### Proof.

If $\bar{x}$ solves $QVI(\mathcal{F},K)$, then

$\exists\,\bar{x}^{*}\in\mathcal{F}(\bar{x}),\langle\bar{x}^{*},y-\bar{x}\rangle\geq
0,~{}\text{for all}~{}y\in K(\bar{x}).$ (24)

We claim that $\bar{x}\in\mathbb{S}(\Gamma).$ Suppose $\nu\in\Lambda$ is
arbitrary. If $P_{\nu}(\bar{x})=\emptyset$ then $P_{\nu}(\bar{x})\cap
K_{\nu}(\bar{x})=\emptyset$. In the case $P_{\nu}(\bar{x})\neq\emptyset$, we
aim to show that $P_{\nu}(\bar{x})\cap K_{\nu}(\bar{x})=\emptyset$. On
contrary, suppose $z_{\nu}\in P_{\nu}(\bar{x})\cap K_{\nu}(\bar{x})$. By
substituting $y=(z_{\nu},\bar{x}_{-\nu})$ in (24) it appears that
$\bar{x}_{\nu}^{*}\in F_{\nu}(\bar{x}_{\nu},\bar{x}_{-\nu})\subset
N_{P_{\nu}}(\bar{x}_{\nu},\bar{x}_{-\nu})$ satisfies,

$\langle\bar{x}_{\nu}^{*},z_{\nu}-\bar{x}_{\nu}\rangle\geq 0.$ (25)

According to from Proposition 4.5, we have $\bar{x}_{\nu}^{*}=0$. But, we know
that $P_{\nu}(\bar{x})\neq\emptyset$ and $\bar{x}_{\nu}^{*}\in
F_{\nu}(\bar{x})\subseteq N_{P_{\nu}}(\bar{x})\setminus\\{0\\}$ by following
the arguments given in proof of Proposition 4.4. This shows, our assumption
$P_{\nu}(\bar{x})\cap K_{\nu}(\bar{x})\neq\emptyset$ is false. Since $\nu$ is
chosen arbitrarily, we observe $P_{\nu}(\bar{x})\cap
K_{\nu}(\bar{x})=\emptyset$ for any $\nu\in\Lambda$ and
$\bar{x}\in\mathbb{S}(\Gamma)$. ∎

Based on the variational reformulation of generalized games obtained in the
above result Theorem 5.3, we now deduce the sufficient conditions for the
existence of equilibrium.

######  Theorem 5.4.

For any $\nu\in\Lambda$, assume that,

  * (a)

$P_{\nu}\mathrel{\mathop{\mathchar 58\relax}}X\rightrightarrows X_{\nu}$ is
convex valued mid-point continuous map with respect to $X_{\nu}$ such that
$x_{\nu}\notin P(x)$ for any $x\in X$;

  * (b)

$K_{\nu}\mathrel{\mathop{\mathchar 58\relax}}C\rightrightarrows C_{\nu}$ is
closed lower semi-continuous map with non-empty convex values and $K_{\nu}(C)$
relatively compact.

Then, the set of equilibrium points $\mathbb{S}(\Gamma)\neq\emptyset$.

###### Proof.

In the view of Theorem 5.3, it is sufficient to prove that
$QVI(\mathcal{F},K)$ admits a solution. As per Proposition 4.4, we know that
the map $\mathcal{F}=\prod_{\nu}F_{\nu}$ is norm-weak∗ upper semi-continuous
with non-empty convex and weak∗ compact values. Finally, we observe that
$QVI(\mathcal{F},K)$ admits a solution $\bar{x}\in K(\bar{x})$ as per [7,
Theorem 3.3]. Finally, $\bar{x}\in\mathbb{S}(\Gamma)$ is the required
equilibrium for generalized game $\Gamma$. ∎

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# GCDT: A Chinese RST Treebank for
Multigenre and Multilingual Discourse Parsing

Siyao Peng Yang Janet Liu Amir Zeldes
Department of Linguistics, Georgetown University
{sp1184, yl879<EMAIL_ADDRESS>
###### Abstract

A lack of large-scale human-annotated data has hampered the hierarchical
discourse parsing of Chinese. In this paper, we present GCDT, the largest
hierarchical discourse treebank for Mandarin Chinese in the framework of
Rhetorical Structure Theory (RST). GCDT covers over 60K tokens across five
genres of freely available text, using the same relation inventory as
contemporary RST treebanks for English. We also report on this dataset’s
parsing experiments, including state-of-the-art (SOTA) scores for Chinese RST
parsing and RST parsing on the English GUM dataset, using cross-lingual
training in Chinese and English with multilingual embeddings.

##  1 Introduction

Hierarchical discourse parsing has shown its importance in document-level
natural language understanding (NLU) tasks, such as text summarization Yoshida
et al. (2014); Goyal and Eisenstein (2016); Xu et al. (2020); Xiao et al.
(2020); Huang and Kurohashi (2021) and sentiment analysis Bhatia et al.
(2015); Markle-Hus et al. (2017); Kraus and Feuerriegel (2019); Huber and
Carenini (2020). Among discourse frameworks, Rhetorical Structure Theory (RST,
Mann and Thompson 1988) is a document-level discourse analysis formalism that
assumes a single-rooted, labeled constituent tree for each document. Unlike
the Penn Discourse Treebank (PDTB, Miltsakaki et al. 2004), which primarily
focuses on local discourse relations and for which more data exists in
Chinese, RST builds a document tree using nested relations within a sentence,
across sentences, and across paragraphs. RST is thus particularly significant
at the macro-level, which is more challenging for understanding discourse
organization than at the micro-level Jia et al. (2018); Hou et al. (2020);
Zhang et al. (2020).

Figure 1: A RST subtree with two relative clauses annotated as elaboration-
attribute and same-unit in GCDT_academic_dingzhen with automatic
$zh\rightarrow en$ translations appended after the source Chinese texts.

Despite the complexity of RST and the human labor required, many new datasets
have come out in the past decades Zeldes et al. (2019, 2021), including
English Carlson et al. (2001); Zeldes (2017), Basque Iruskieta et al. (2013),
Bangla Das and Stede (2018), Brazilian Portuguese Cardoso et al. (2011), Dutch
Redeker et al. (2012), German Stede and Neumann (2014), Persian Shahmohammadi
et al. (2021), Russian Toldova et al. (2017), Spanish da Cunha et al. (2011),
and the Spanish-Chinese parallel corpus Cao et al. (2018).

However, a substantial gap remains in the availability of document-level
hierarchical discourse datasets for non-European languages, particularly
Chinese, of sufficient magnitude for training contemporary neural parsers.
Aside from the small parallel Spanish-Chinese dataset by Cao et al. (2018, see
below) with only 400+ discourse relation instances, there are no available
Chinese treebanks in the RST framework. Thus, neither monolingual nor
multilingual RST constituent parsers are trained in Chinese and cannot benefit
downstream tasks.

In this paper, we present the Georgetown Chinese Discourse Treebank (GCDT)
corpus,111 The source texts, annotations, and guidelines are open-source (CC-
BY) and available at https://github.com/logan-siyao-peng/GCDT. The corpus is
also searchable in the ANNIS interface Krause and Zeldes (2014) at
https://gucorpling.org/annis/#_c=R0NEVA==. a new, freely available, multi-
genre RST corpus of 50 medium to long documents for Mandarin Chinese, as the
sample subtree shown in Figure 1. The corpus covers over 60K tokens and 9K
Elementary Discourse Units (EDUs). In addition to presenting the SOTA parsing
results in monolingual settings for the dataset, we jointly train a model with
both English and Chinese datasets, testing finetuning and automatic
translation-based approaches to improve performance in both Chinese and
English as the target language. Experimenting with different monolingual and
multilingual embeddings, we find that joint training and translation improve
performance on the smaller Chinese and larger English datasets. However,
finetuning only helps with the smaller Chinese data. Finally, we show that
monolingual RoBERTa embeddings outperform multilingual embeddings in
applicable settings. Still, the best overall performance is achieved using
Chinese and English data in a multilingual training regime.

##  2 Previous Work

### RST Datasets in English and Chinese

The English RST-DT corpus Carlson et al. (2001) is the primary benchmark in
the RST framework. The large corpus (205K tokens) includes only news articles
from the Penn Treebank (Marcus et al., 1993). Another English RST corpus is
GUM Zeldes (2017), a multi-genre corpus growing in size yearly and currently
(V8.0.0) contains 180K tokens from 12 written or spoken genres. GUM is thus
slightly smaller in the token count but has a larger number of discourse
relation instances due to a shorter average unit length in tokens. Moreover,
the dynamic aspect of GUM makes it different to set up benchmark scores
compared to other RST corpora. To our knowledge, this paper publishes the
first set of RST parsing performances on GUM V8.0.0.

The Spanish-Chinese parallel corpus Cao et al. (2018) is a small Chinese RST
corpus (15K tokens) constructed for translation studies. To support this goal,
its EDUs are adjusted to align between Spanish and Chinese rather than staying
faithful to the syntax of the individual languages. Its relation inventory is
also distinct from inventories used for English corpora, as are the
segmentation criteria used in the corpus, limiting its compatibility with
other datasets. Another older Chinese RST corpus was reported in Yue (2008)
with 97 news commentaries annotated. However, to our knowledge, the dataset is
no longer accessible or used in RST parsing or other tasks Cao (2018).

### Other Hierarchical Chinese Discourse Datasets

There are a few other hierarchical discourse corpora in Mandarin Chinese, but
none of them annotate single-rooted RST trees for longer documents. The CDT-
CDTB corpus Li et al. (2014b) uses connectives to build up discourse trees
only within paragraphs, for 500 news documents from the Chinese Penn Treebank
Xue et al. (2005). Not only are many of the connectives ambiguous in Chinese
Li et al. (2014a); Lu et al. (2018), discourse trees in CDT-CDTB are also
small (only 4.5 EDUs/tree). This dataset, therefore, differs substantially
from the expected structure of an RST treebank, in which EDUs are expected to
be all clauses in the text with functionally motivated relation labels, such
as cause or background.

MCDTB Jiang et al. (2018) further utilizes a set of discourse relations to
connect between paragraphs within 720 documents. The design choice to use
specific inter-paragraph-only annotations creates an interesting distinction
between micro-level versus macro-level relations Sporleder and Lascarides
(2004); Wang et al. (2017), but also deviates from RST’s fundamental idea of
constructing a single tree for an entire document, in which the same inventory
of labels is used for all nodes.

Moving beyond constituent-based discourse trees, Cheng and Li (2019) annotated
108 scientific abstracts in their Sci-CDTB corpus using Discourse Dependency
Structure (DDS; Hirao et al., 2013; Morey et al., 2018). Cheng et al. (2021)
further converted other Chinese discourse corpora into the DDS representation.
Even though DDS simplifies parsing and is more similar to other linguistic
annotation schemes, such as Universal Dependencies Nivre et al. (2016) for
syntax, the dependency-style discourse annotation loses significant
information on the ordering or scope of satellite attachments. For example,
whether a unit with cause and attribution satellites means that both the cause
and the result are attributed to someone, as in Appendix 0.C, or that
something caused an attributed statement. In other words, when multiple
discourse units modify the same nucleus, the relative importance of the
satellites and their scopes are ignored.

### Multilingual RST Parsers

RST parsing is a task that merges a sequence of gold or predicted EDUs and
forms a labeled tree structure for the entire document. Since RST datasets
share the same unlabeled constituent tree structure, based on the principle
that more prominent units should serve as nuclei to less prominent satellite
units, multilingual joint training has achieved SOTA results in multilingual
RST parsing in several languages. Translating EDUs across languages Cheng and
Li (2019); Liu et al. (2020, 2021) and mapping word embeddings into the same
space Braud et al. (2017); Iruskieta and Braud (2019); Liu et al. (2020, 2021)
are two common approaches to encoding EDUs across languages in joint training.
Among this line of work, Liu et al. (2021) presented a SOTA multilingual RST
parser with a pointer-network decoder for top-down depth-first span splitting.
The model uses the multilingual xlm-roberta-base (Conneau et al., 2020) and
trains jointly with six languages: English, Portuguese, Spanish, German,
Dutch, and Basque. The current work uses the parser from Liu et al. (2021) for
training between the Chinese GCDT corpus and the English GUM corpus.

##  3 GCDT: Georgetown Chinese Discourse Treebank

GCDT is an open-source multi-genre RST dataset in Mandarin Chinese. Following
the design of GUM Zeldes (2017), GCDT contains 50 documents, 10 from each of 5
genres which also appear in GUM: academic articles, biographies (bio),
interview conversations, news, and how-to guides (whow), as shown in Table 1.
Unlike existing Chinese discourse corpora, GCDT focuses on building larger
discourse trees for medium-to-long documents. We select documents with an
average of 1K+ tokens to provide more training data for learning higher-level
discourse structures.

Genre | #Docs | #Toks | # EDUs | Source
---|---|---|---|---
academic | 10 | 14,168 | 2,033 | hanspub.org/
bio | 10 | 13,485 | 2,018 | zh.wikipedia.org/
interview | 10 | 11,464 | 1,810 | zh.wikinews.org/
news | 10 | 11,249 | 1,652 | zh.wikinews.org/
whow | 10 | 12,539 | 2,197 | zh.wikihow.com/
Total | 50 | 62,905 | 9,710 |

Table 1: GCDT Corpus Statistics.

### EDU Segmentation

Elementary Discourse Unit (EDU) segmentation is fundamental to RST. We deviate
from previous corpora that predominately use potentially ambiguous punctuation
Li et al. (2014a) to segment EDUs, regardless of the surrounding structures.
Instead, our Chinese EDU segmentation mirrors the syntactic criteria
established in the English RST-DT and GUM corpora Carlson and Marcu (2001);
Carlson et al. (2001); Zeldes (2017), largely equating EDUs with the
propositional structure of clauses. We use the Penn Chinese Treebank Xue et
al. (2005) as our syntactic guidelines. We first manually tokenize according
to Xia (2000b) and conduct EDU segmentation based on parts-of-speech defined
in Xia (2000a).

Most notably, we segment relative clauses in GCDT, following the practice in
English and other corpora Carlson et al. (2001); Zeldes (2017); Das and Stede
(2018); Cardoso et al. (2011); Redeker et al. (2012); Toldova et al. (2017).
Chinese relative clauses present a unique feature in the existing RST
treebanks. To our knowledge, GCDT is the first RST corpus in any language in
which prenominal relative clauses are annotated for discourse relations.
Cross-referencing Dryer (2013a, b) with languages of existing RST corpora
suggests that only Basque also exhibits the Relative-Noun order found in
Chinese. Yet, relative clauses are not segmented in the Basque RST dataset
Iruskieta et al. (2015). Moreover, since relative clauses intervene between
Verb-Object in Chinese, the pseudo-relation same-unit is used to express
discontinuous EDUs, as shown in Figure 1. Segmenting and annotating discourse
relations for relative clauses is one of the reasons that GCDT has relatively
short EDUs, on average 6.5 tokens/EDU.

### Relation Annotation

GCDT builds up constituent discourse trees based on gold EDUs using rstWeb
Zeldes (2016). We use the enhanced two-level relation labels from GUM V8.0.0
with 15 coarse and 32 fine-grained relations (see Appendix 0.A for relation
distributions in GCDT and GUM).

### Data Split

We provide an 8-1-1 train-dev-test split per genre to facilitate future RST
parsing experiments. Both human inter-annotator agreements and parsing results
are assessed on the five test documents, with one from each of the five
genres.

### Inter-Annotator Agreement (IAA)

We evaluate agreement on the five test documents to obtain human ceiling
scores for parser performance. One Chinese native-speaker linguist annotated
the entire corpus, and another read the guidelines and conducted independent
EDU segmentation. We measured segmentation agreement, adjudicated segmentation
between the two annotators, and then separately annotated relation trees on
gold EDUs to measure relation agreement. We also release the double
annotations in GCDT for future experiments on annotation disagreements. On
segmentation, we obtained a token-wise agreement of 97.4% and Cohen’s
$\kappa=$0.89. The agreements on micro-averaged original Parseval F1 of Span,
Nuclearity, and Relation are 84.27, 66.15, and 57.77 respectively. The IAA of
GCDT is similar to that of the English RST-DT benchmark – 78.7, 66.8, and 57.1
– when evaluated using the original Parseval Morey et al. (2017). The results
show that the GCDT annotation agreement is highly satisfactory even though the
documents are much longer and exhibit more genre diversity than RST-DT.

corpus | monolingual embedding | Span | Nuc | Rel | multilingual embedding | Span | Nuc | Rel
---|---|---|---|---|---|---|---|---
GCDT | bert-base-chinese | 73.15±0.53 | 55.71±0.66 | 50.81±0.65 | bert-base-multilingual-cased | 67.34±1.32 | 47.66±0.73 | 43.97±0.93
hfl/chinese-roberta-wwm-ext | 75.51±0.68 | 57.08±0.81 | 51.76±0.97 | xlm-roberta-base | 74.35±0.54 | 54.17±1.20 | 50.45±1.09
GUM-5 | bert-base-cased | 64.61±1.42 | 49.58±1.51 | 40.43±1.56 | bert-base-multilingual-cased | 64.52±2.68 | 51.63±2.07 | 44.96±1.46
roberta-base | 73.85±0.70 | 58.95±0.79 | 50.35±1.18 | xlm-roberta-base | 72.45±0.97 | 56.78±0.80 | 47.69±0.88
GUM-12 | bert-base-cased | 60.93±0.63 | 47.92±0.62 | 40.20±0.40 | bert-base-multilingual-cased | 64.47±0.50 | 50.69±0.32 | 43.25±0.35
roberta-base | 68.59±0.58 | 55.32±0.27 | 46.29±0.46 | xlm-roberta-base | 66.12±0.59 | 52.58±0.52 | 45.06±0.45

Table 2: Monolingual parsing results on the test sets of GCDT, GUM-5, and
GUM-12 with Chinese, English, and multilingual BERT and RoBERTa embeddings
(mean±std over five runs).

Experiment | Span | Nuc | Rel | Experiment | Span | Nuc | Rel
---|---|---|---|---|---|---|---
Train on GCDT+GUM-5 and Dev/Test on GCDT | Train on GUM-5+GCDT and Dev/Test on GUM-5
joint training w/ XLM RoBERTa | 74.24±0.48 | 56.68±0.86 | 52.21±0.83 | joint training w/ XLM RoBERTa | 72.56±0.71 | 60.63±0.43 | 52.57±0.77
+finetuning w/ XLM RoBERTa | 76.97±0.32 | 57.94±0.82 | 53.38±0.51 | +finetuning w/ XLM RoBERTa | 73.44±0.36 | 59.40±0.56 | 50.57±0.97
+enzh trans. w/ XLM RoBERTa | 74.80±0.78 | 56.58±0.98 | 51.18±1.15 | +zhen trans. w/ XLM RoBERTa | 72.21±1.11 | 60.07±1.25 | 52.32±1.05
+enzh trans. w/ ZH RoBERTa | 77.66±0.42 | 59.29±0.59 | 54.66±0.76 | +zhen trans. w/ EN RoBERTa | 74.73±0.40 | 62.65±0.72 | 54.32±0.82
Train on GCDT+GUM-12 and Dev/Test on GCDT | Train on GUM-12+GCDT and Dev/Test on GUM-12
joint training w/ XLM RoBERTa | 74.33±0.49 | 57.24±0.99 | 52.61±1.13 | joint training w/ XLM RoBERTa | 70.32±0.37 | 57.49±0.73 | 49.14±0.34
+finetuning w/ XLM RoBERTa | 76.95±0.65 | 59.40±0.64 | 55.28±0.23 | +finetuning w/ XLM RoBERTa | 66.00±0.24 | 53.13±0.22 | 45.47±0.42
+enzh trans. w/ XLM RoBERTa | 73.99±0.79 | 56.31±1.43 | 51.51±1.34 | +zhen trans. w/ XLM RoBERTa | 70.28±0.55 | 57.63±0.55 | 49.26±0.39
+enzh trans. w/ ZH RoBERTa | 78.11±0.39 | 59.42±0.90 | 54.41±1.23 | +zhen trans. w/ EN RoBERTa | 71.41±0.47 | 59.17±0.35 | 50.63±0.48

Table 3: Multilingual parsing results with finetuning and automatic
translation on the test sets of GCDT+GUM combinations with highest-performing
Chinese (ZH), English (EN), and multilingual (XLM) RoBERTa embeddings.

Genre | Trained on GCDT |  | Trained on GCDT+GUM-5
---
w/ zhen trans.
| Trained on GCDT+GUM-12
---
w/ zhen trans.
| Human
---
Agreement
Span | Nuc | Rel | Span | Nuc | Rel | Span | Nuc | Rel | Span | Nuc | Rel
academic | 74.64 | 54.07 | 48.33 | 72.25 | 47.37 | 43.54 | 75.12 | 51.20 | 44.98 | 80.38 | 59.33 | 49.76
bio | 72.87 | 54.26 | 52.71 | 74.81 | 57.75 | 53.49 | 77.52 | 59.69 | 55.43 | 81.57 | 63.92 | 55.69
interview | 74.68 | 56.33 | 52.53 | 80.38 | 61.39 | 55.70 | 77.85 | 56.96 | 48.73 | 83.55 | 62.50 | 54.61
news | 76.63 | 56.52 | 50.54 | 83.15 | 64.13 | 57.07 | 78.80 | 60.33 | 54.35 | 80.98 | 61.96 | 54.35
whow | 77.89 | 57.76 | 54.79 | 80.20 | 66.34 | 62.71 | 80.20 | 65.68 | 61.06 | 91.99 | 77.70 | 69.34
Overall | 75.45 | 55.85 | 52.07 | 77.97 | 59.71 | 55.04 | 78.06 | 59.44 | 53.87 | 84.27 | 66.15 | 57.77

Table 4: GCDT genre-wise performances on sample models trained on GCDT, as
well as translation-augmented GCDT+GUM-5 and GCDT+GUM-12 combinations using
hfl/chinese-roberta-wwm-ext.

##  4 Experiments

We present benchmark results on GCDT using the SOTA multilingual parser, DMRST
Liu et al. (2021). Results are shown in two experimental settings: monolingual
training using only one dataset (either Chinese GCDT or English GUM V8.0.0)
and multilingual training using data from both corpora (GCDT+GUM). Besides
directly combining corpora from the two languages, we also experiment with
finetuning and automatic EDU-wise translation. We use the same set of
hyperparameters as reported in Liu et al. (2021). Similarly, we also report
monolingual and multilingual parsing performance on GUM V8.0.0.

### Datasets

Cross-genre adaptability remains a bottleneck in RST parsing Nishida and
Matsumoto (2022); Atwell et al. (2021). To isolate cross-lingual versus cross-
genre influences, we conduct monolingual and multilingual experiments using
the following data compositions: 1) GCDT: 50 Chinese documents from 5 genres;
2) GUM-12: 193 English documents from 12 genres; 3) GUM-5: 99 GUM documents
from the same 5 genres in GCDT.

### Language Models

We test different monolingual and multilingual BERT and RoBERTa embeddings
(see Appendix 0.B for details).

### Metrics

We use the 15 coarse relation classes shared between GCDT and GUM and follow
the recommendation of Morey et al. (2017) to use the original Parseval micro-
averaged F1 for Span, Nuclearity (Nuc), and Relation (Rel).

### Multilingual Training Setups

In addition to training with combinations of the original GCDT and GUM
datasets using multilingual embeddings (see Appendix 0.D for specific data
partitions used in the GCDT+GUM-combined experiments), we also experiment with
two techniques to improve performances on both target datasets. Specifically,
to improve on Chinese GCDT:

1) Finetuning: we first train models with both English and Chinese data and
then continue training only on the training partition of the target dataset
(i.e., GCDT).

2) Automatic EDU-wise Translation: we use GoogleTranslator222
https://github.com/nidhaloff/deep-translator to automatically translate EDUs
from the other dataset to the target language (i.e., EDU-wise enzh
translations of GUM) and train on the original GCDT and translated GUM data.
The advantage of the translation approach is that we can replace the
multilingual embeddings with higher-performing monolingual embeddings.

##  5 Results

We present monolingual and multilingual results on GCDT and GUM in Tables 2
and 3, as well as genre-wise performance on GCDT in Table 4.

### Monolingual Parsing

Similar to previous observations Staliūnaitė and Iacobacci (2020); Naseer et
al. (2021); Tarunesh et al. (2021), Table 2 shows that RoBERTa outperforms
BERT in both languages. Monolingual RoBERTa embeddings achieve the best
performance when training with monolingual data, e.g., hfl/chinese-roberta-
wwm-ext obtained 51.76 on the relation level on GCDT.

### Multilingual Parsing

Our multilingual parsing experiments include joint training, finetuning, and
automatic EDU-wise translation. Based on the monolingual results, we use the
best-performing multilingual embedding xlm-roberta-base Conneau et al. (2020)
with the GCDT+GUM-combined multilingual data. Different aspects of the
multilingual parsing results are shown in Table 3.

Firstly, joint training outperformed monolingual results in all three test
scenarios: GCDT, GUM-5, and GUM-12. For example, training on GCDT+GUM-12 using
XLM RoBERTa achieved an F_Rel of 52.61 on GCDT, higher than the 50.45 trained
with only GCDT, and the same embedding.

Secondly, more genres from GUM (GCDT+GUM-12) achieved better performance than
training only using the same genres (GCDT+GUM-5) when tested on GCDT.

Thirdly, pretraining on the GCDT+GUM-combined training sets and training on
the training set of the target corpus improves performance on Chinese GCDT but
deteriorates on the English GUM. We hypothesize that with more English
training data available, there is less headroom for improvement. In contrast,
finetuning for the smaller Chinese dataset added to the comparatively little
information available to the parser.

Lastly, results show that augmenting with automatic translation and using
monolingual embeddings achieved the best performance on three of the four test
scenarios, while the best result on GCDT was achieved by training together
with GUM-12 and finetuning on GCDT.

### Genre-wise Analysis

We further select three models trained in the monolingual GCDT and
translation-augmented scenarios, GCDT+GUM-5 and GCDT+GUM-12, using the Chinese
RoBERTa embedding Cui et al. (2021). Table 4 provides per-genre parsing
results of the models on the five test genres. On the one hand, the average
performance on how-to guides (whow) is much higher than academic articles for
both models and humans. This demonstrates a good human-model alignment
regarding which genre is the hardest or easiest (cf. Zeldes and Simonson
2016). On the other, model results are the farthest from the human ceiling
scores on the highest performing whow genre. We hypothesize that
characteristics of genres triggered the different performances. Future multi-
genre experiments could be conducted across datasets to study out-of-domain
effects in multilingual RST parsing scenarios.

##  6 Conclusion

This paper presents GCDT, the largest RST dataset for Mandarin Chinese, which
closely follows established RST guidelines and is highly comparable to
existing English RST corpora. Besides evaluating annotation quality and
establishing SOTA results on this dataset in monolingual settings, we also
jointly train on GCDT and a similar English corpus—GUM—and demonstrate that
multilingual training and automatic EDU translation boost parser performance.
However, finetuning is only helpful when targeting the smaller Chinese
dataset. We further conduct per-genre analyses and show that parsing
performance varies widely between some genres but less between others. We hope
that this dataset can alleviate the lack of training resources for
hierarchical discourse parsing in Chinese and facilitate multilingual and
multi-genre RST parsing, as well as other downstream NLP tasks.

## Acknowledgements

We thank Nianwen Xue for providing insights on Chinese syntax, which helped
refine the EDU segmentation guidelines. We also thank the anonymous reviewers
and Nathan Schneider for their insightful comments.

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##  Appendix 0.A Label Distributions

Table 5 gives descriptive statistics of the distribution of relations in GCDT
and numbers for comparison from the GUM corpus, which uses the same inventory
of relations and covers all and more genres in the GCDT dataset.

Relation Name | GCDT% | GUM%
---|---|---
Nucleus-Satellite Relations
elaboration-attribute | 7.71% | 4.60%
attribution-positive | 4.37% | 3.08%
elaboration-additional | 4.25% | 4.81%
explanation-evidence | 4.15% | 2.08%
context-background | 2.89% | 2.66%
context-circumstance | 2.69% | 2.40%
organization-preparation | 2.36% | 1.83%
causal-cause | 1.98% | 1.63%
organization-heading | 1.78% | 1.49%
contingency-condition | 1.77% | 1.67%
adversative-concession | 1.68% | 2.04%
purpose-goal | 1.54% | 1.63%
restatement-partial | 1.32% | 1.13%
evaluation-comment | 1.15% | 2.29%
mode-means | 1.09% | 0.55%
explanation-justify | 0.87% | 1.60%
causal-result | 0.87% | 1.54%
adversative-antithesis | 0.58% | 1.47%
mode-manner | 0.52% | 0.89%
topic-question | 0.44% | 1.10%
organization-phatic | 0.27% | 1.37%
attribution-negative | 0.23% | 0.30%
purpose-attribute | 0.21% | 0.87%
explanation-motivation | 0.2% | 0.71%
topic-solutionhood | 0.01% | 0.20%
Multi-Nucleus Relations
joint-list | 22.28% | 12.90%
same-unit | 18.69% | 4.71%
joint-sequence | 4.99% | 4.41%
joint-other | 4.83% | 4.48% | adversative-contrast | 3.32% | 2.40%
joint-disjunction | 0.64% | 1.13%
restatement-repetition | 0.32% | 1.82%

Table 5: Distribution of 32 relations (15 classes, including same-unit) in
GCDT and GUM V8.0.0.

##  Appendix 0.B Specific PLMs Used in the Experiments

Table 6 shows the Chinese, English, and multilingual BERT and RoBERTa
pretrained language models used in the experiments described in §4.

Type | Details
---|---
BERT | Chinese: bert-base-chinese Devlin et al. (2019)
English: bert-base-cased Devlin et al. (2019)
Multilingual: bert-base-multilingual-cased Devlin et al. (2019)
RoBERTa | Chinese: hfl/chinese-roberta-wwm-ext Cui et al. (2021)
English: roberta-base Liu et al. (2019)
Multilingual: xlm-roberta-base Conneau et al. (2020)

Table 6: An overview of pretrained BERT and RoBERTa language models used in
the experiments.

##  Appendix 0.C A Fragment of RST Annotation in GCDT

Figure 2 presents a relation hierarchy of attribution-positive scoping over
causal-cause.

Figure 2: A RST subtree with attribution-positive scoping over causal-cause
from GCDT_academic_dingzhen with automatic $zh\rightarrow en$ translation.

##  Appendix 0.D Data Splits for Multilingual Experiments

Table 7 presents the train/dev/test splits when jointly training with GCDT and
GUM in multilingual experiments.

|  |  train: GCDT+GUM
---
dev/test: GUM
|  train: GCDT+GUM
---
dev/test: GCDT
train |  | GUM-train
---
\+ GCDT-train
\+ GCDT-dev
| GCDT-train
---
\+ GUM-train
\+ GUM-dev
dev | GUM-dev | GCDT-dev
test | GUM-test | GCDT-test

Table 7: An overview of the train/dev/test splits of GCDT and GUM used for
training in the multilingual experiments.