11institutetext: University of Bucharest 11email<EMAIL_ADDRESS>
# Existence of the equilibrium in choice

Monica Patriche  University of Bucharest, Faculty of Mathematics and Computer
Science, 14 Academiei Street, 010014 Bucharest, Romania

###### Abstract

In this paper, we prove the existence of the equilibrium in choice for games
in choice form. These games have recently been introduced by A. Stefanescu, M.
Ferrara and M. V. Stefanescu. Our results link the recent research to the
older approaches, regarding games in normal form or qualitative games.

###### Keywords:

game in choice form, equilibrium in choice , selection theorem, fixed point
theorem

##  1 Introduction

It has been widely agreed upon the fact that Nash’s concept of equilibrium
reflects the possibility of challenging the choices of the unilateral acts of
the players involved in noncooperative games. Having the aim of gaining the
most they can do, the players are in the situation of making choices in a
process described, mathematically, by the notion of ”game” (this was proposed
by Nash, and it was initially called ”game in normal form”). The original
definitions of Nash 80 ,79  have been extended, but the derived notions of
qualitative game or abstract economy and their corresponding concepts of
equilibrium reflect the initial meaning of flexible elections in any given
context to permit the players not to deviate, once they agreed on the best
solution for them.

The new extension of a game, called ”the game in choice form”, is due to
Stefanescu, Ferrara and Stefanescu SS . The game in choice form is a family of
the sets of individual strategies and choice profiles. The authors also
defined the concept of equilibrium in choice for this type of game. Their
interpretations evolved along with the problem of noncooperative solutions of
the voting operators. Firstly, Stefanescu and Ferrara proposed the concept of
Nash equilibrium in choice in st  and they renamed it in SS , founding
conditions under which the equilibrium in choice exists. The new adopted
definitions are coherent with the old underlying formalism. For instance, when
the utility functions represent the players’ options, a choice profile can be
seen as the family of players’ graphs best reply mappings. In this case, the
set of equilibria in choice coincides with the set of Nash equilibria. The
generality of the new concepts raise the interest of the formalist theorist of
games, which explicitly can show their significance. The authors themselves
developed these ideas in their work. They referred to the fact that the
players’ preferences need not be explicitly represented, at the same time
considering the possibility of recuperating the known solutions as particular
cases. The second problem raised is the one of the nonexistence of a best
reply. This interest is obviously at the core of our original research. In
this paper, we are looking for new conditions, in order to obtain the
existence of the equilibrium in choice. Our assumptions are different from the
ones proposed when the new theory was framed. They concern the properties of
the sets of choice profiles. We are now exploring a method of proof based on
continuous selection and fixed point theorems for correspondences defined by
using the upper sections of the sets which form the game.

The rest of the paper is organised as follows: In Section 2, some notation,
terminological convention, basic definition and results about correspondences
and games in choice form are given. Section 3 contains existence results for
equlibrium in choice.

##  2 Preliminaries and notation

Throughout this paper, we shall use the following notations and definitions:

Let $A$ be a subset of a vector space $X$, $2^{A}$ denotes the family of all
subsets of $A$ and co$A$ denotes the convex hull of $A$. If $A$ is a subset of
a topological space $X,$ cl$A$ denotes the closure of $A$ in $X$. If $T$,
$G:X\rightarrow 2^{Y}$ are correspondences, then co$G$ and $G\cap
T:X\rightarrow 2^{Y}$ are correspondences defined by $($co$G)(x):=$co$G(x)$
and $(G\cap T)(x):=G(x)\cap T(x),$ for each $x\in X$, respectively.

Given a correspondence $T:X\rightarrow 2^{Y},$ for each $x\in X,$ the set
$T(x)$ is called the upper section of $T$ at $x.$ For each $y\in Y,$ the set
$T^{-1}(y):=\\{x\in X:y\in T(x)\\}$ is called the lower section of $T$ at $y$.
The correspondence $T^{-1}:Y\rightarrow 2^{X},$ defined by $T^{-1}(y)=\\{x\in
X:y\in T(x)\\}$ for $y\in Y$, is called the (lower) inverse of $T.$

Let $X$, $Y$ be topological spaces. The correspondence $T:X\rightarrow 2^{Y}$
is called lower semicontinuous if for each x$\in X$ and for each open set $V$
in $Y$ with $T(x)\cap V\neq\emptyset$, there exists an open neighborhood $U$
of $x$ in $X$ so that $T(y)\cap V\neq\emptyset$ for each $y\in U$.

The following lemma will be useful to our study of existence of equilibrium in
choice.

###### Lemma 1

(see Yuan 134 ). Let $X$ and $Y$ be two topological spaces and let $W$ be an
open (resp. closed) subset of $X.$ Suppose $T_{1}:X\rightarrow 2^{Y}$ ,
$T_{2}:X\rightarrow 2^{Y}$ are upper semicontinuous (resp. lower
semicontinuous) correspondences such that $T_{2}(x)\subset T_{1}(x)$ for all
$x\in W.$ Then the correspondence $T:X\rightarrow 2^{Y}$ defined by

$T\mathit{(x)=}\left\\{\begin{array}[]{c}T_{1}(x)\text{, \ \ \ \ \ \ \ if
}x\notin W\text{, }\\\ T_{2}(x)\text{, \ \ \ \ \ \ if }x\in
W\end{array}\right.$

is also upper semicontinuous (resp. lower semicontinuous).

Further, we present the main models of noncooperative games we will deal with
in this paper. The corresponding notions of equilibrium are also recalled.

Let $(X_{i})_{i\in N}$ be the family of the individual sets of strategies and
let $X=\prod\nolimits_{i\in I}X_{i}$.

The normal form of an $n$-person game is $(X_{i},r_{i})_{i\in N}$, where, for
each $i\in N$, $X_{i}$ is a nonempty set (the set of individual strategies of
player $i$) and $r_{i}$ is the preference relation on $X$ of player $i$. The
individual preferences $r_{i}$ are often represented by utility functions,
i.e. for each $i\in N$ there exists a real valued function
$u_{i}:X\rightarrow\mathbb{R}$ (called the utility function of $i$), such
that: $xr_{i}y\Leftrightarrow u_{i}(x)\geq u_{i}(y),$ $\forall x,y\in X.$ Then
the normal form of n-person game is $(X_{i},u_{i})_{i\in N}$.

We denote $x_{-i}=(x_{1},...,x_{i-1},x_{i+1},...,x_{n})$,
$X_{-i}=\prod\nolimits_{i\neq j}X_{i}$ and
$(x_{-i},X_{i})=\\{(x_{-i},x_{i}):x_{i}\in X_{i}\\}.$

The Nash equilibrium for the game $(X_{i},u_{i})_{i\in N}$ is a point
$x^{\ast}\in$ $X$ which satisfies for each $i\in N:$ $u_{i}(x^{\ast})\geq
u_{i}(x_{-i}^{\ast},x_{i})$ for each $x_{i}\in X_{i}.$

For each $i\in N,$ the player’s i’s best reply mapping is the correspondence
$B_{i}:X_{-i}\rightarrow 2^{X_{i}},$ defined by $B_{i}(x_{-i})=\\{x_{i}\in
X_{i}:u_{i}(x_{-i},x_{i})\geq u_{i}(x_{-i},y_{i})$ for each $y_{i}\in
X_{i}\\}.$ Then, $x^{\ast}\in$ $X$ is a Nash equilibrium if only if
$x^{\ast}\in\bigcap\nolimits_{i\in N}$Gr$(B_{i}).$

The element $x^{\ast}\in$ $X$ is called weak Nash equilibrium (Stefanescu,
Ferrara, Stefanescu SS ):

$u_{i}(x^{\ast})\geq u_{i}(x_{-i}^{\ast},x_{i})$ for each $x_{i}\in X_{i}$ and
$i\in N$ such that $B_{i}(x_{-i}^{\ast})\neq\emptyset.\vskip 6.0pt plus 2.0pt
minus 2.0pt$

A qualitative game $\Gamma=(X_{i},P_{i})_{i\in N}$ is defined as a family of
$n$ ordered triplets $(X_{i},P_{i}),$ where for each $i\in N$:
$P_{i}:X\rightarrow 2^{X_{i}}$ is a preference correspondence. An equilibrium
for $\Gamma$ is a point $x^{\ast}\in$ $X$ which satisfies for each $i\in N:$
$P_{i}(x^{\ast})=\emptyset.$ A weak equilibrium (Stefanescu, Ferrara,
Stefanescu SS ) of $\Gamma=(X_{i},P_{i})_{i\in N}$ is a point $x^{\ast}\in$
$X$ which satisfies $P_{i}(x^{\ast})=\emptyset$ for each $i\in N$ such that
$\\{x_{i}\in X_{i}:P_{i}(x_{-i}^{\ast},x_{i})=\emptyset\\}\neq\emptyset.$

A choice profile (Stefanescu, Ferrara, Stefanescu SS ) is any collection
$C:=(C_{i})_{i\in N}$ of nonempty subsets of $X.$ A game in choice form is a
double family $((X_{i})_{i\in N},(C_{i})_{i\in N})$, where $C=(C_{i})_{i\in
N\text{ }}$is a choice profile.

We denote $C(x_{-i})$ the upper section through $x_{-i}$ of the set $C_{i},$
i.e., $C(x_{-i})=\\{y_{i}\in X_{i}:(x_{-i},y_{i})\in C_{i}\\}.$

We will make the following assumption:

(A) assume that for each $x\in X,$ there exists $i\in N$ such that
$C_{i}(x_{-i})\neq\emptyset.$

The game strategy $x^{\ast}\in X$ is an equilibrium in choice (denoted EC) (A.
Stefanescu, M. Ferrara, V. Stefanescu SS ) if $\forall i\in N,$
$(x_{-i}^{\ast},X_{i})\cap C_{i}\neq\emptyset\Rightarrow x^{\ast}\in C_{i},$
equivalently, $x_{i}^{\ast}\in C_{i}(x_{-i}^{\ast}),$ for every $i\in N$ for
which $C_{i}(x_{-i}^{\ast})\neq\emptyset.$ The strategy $x^{\ast}\in X$ is a
strong equilibrium in choice (denoted SEC) if
$x_{i}^{\ast}\in\bigcap\nolimits_{i\in N}C_{i}.$

##  3 Equilibrium results

This section provides a summary of different theorems concerning the existence
of equilibria in choice for games in choice form. In order to underline the
novelty and the importance of our work, we also must discuss here the
additional benefit of obtaining corollaries which state, under new conditions,
the existence of the weak Nash equilibria for games in normal form, or the
existence of the weak equilibria of qualitative games. To prove our point, we
will use continuous selection theorems or fixed point theorems for the
correspondences that we will form, considering upper sections of the sets
defining the game of the choice form. The advantage we have by doing this,
deserves a great prominence in the assessment of the new assumptions which
characterize the new statements. These results differ very much from the ones
obtained by Stefanescu, Ferrara and Stefanescu in SS  and they link the recent
research to the older approaches, regarding games in normal form or
qualitative games. In order to suggest priorities to the reader, we keep the
relationship between the main theorems and their consequences on particular
games. For a better understanding of the paper, we recall all properties of
the correspondences which will be used and the tools of proofs. Our study
gives a new perspective of unifying of different approaches and results on the
equilibrium concepts and the existence of noncooperative theory of games.
Finally, we note that we obtain the existence of the strong equilibrium in
choice for all situations considered in this section, if we suppose, in
addition, that $C_{i}(x_{-i})\neq\emptyset$ for each $x_{-i}\in X_{-i}.\vskip
6.0pt plus 2.0pt minus 2.0pt$

Let $X$, $Y$ be topological spaces. We recall that the correspondence
$T:X\rightarrow 2^{Y}$ has the local intersection property if $x\in X$ with
$T(x)\neq\emptyset$ implies the existence of an open neighborhood $V(x)$ of
$x$ such that $\cap_{z\in V(x)}T(z)\neq\emptyset.$

To prove Theorem 1, we need the following lemma.

###### Lemma 2

(Wu, Shen, shen ). Let $X$ be a nonempty paracompact subset of a Hausdorff
topological space $E$ and $Y$ be a nonempty subset of a Hausdorff topological
vector space. Let $S$,$T:X\rightarrow 2^{Y}$ be correspondence which verify:

a) for each $x\in X,$ $S(x)\neq\emptyset$ and co$S(x)\subset T\left(x\right);$

b) $S$ has the local intersection property.

Then, $T$ has a continuous selection.

Our main result cites conditions which ensure the existence of equilibria in
choice for a game in choice form in the lack of convexity of the upper
sections of the sets $C_{i}$. The framework for our general next theorem
consists of Hausdorff topological vector spaces. The proof is based on an
argument that implies the above lemma.

###### Theorem 3.1

Let $((X_{i})_{i\in N},(C_{i})_{i\in N})$ be a game in choice form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) there exists a nonempty subset $D_{i}$ of $C_{i}$ such that
$W_{i}=\\{x_{-i}\in X_{-i}:D_{i}(x_{-i})\neq\emptyset\\}$ is closed and
$D_{i}(x_{-i})\neq\emptyset$ if only if $C_{i}(x_{-i})\neq\emptyset;$

c) if $D_{i}(x_{-i})\neq\emptyset,$ there exists an open neighborhood
$V(x_{-i})$ of $x_{-i}$ such that $\cap_{z_{-i}\in
V(x_{-i})}D_{i}(z_{-i})\neq\emptyset;$

d) $D_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits equilibria in choice.

###### Proof

For each $i\in N,$ let us define the correspondences
$S_{i},T_{i}:X_{-i}\rightarrow 2^{X_{i}},$ by

$S_{i}(x_{-i})=\left\\{\begin{array}[]{c}\text{co}(\bigcup\nolimits_{\\{y_{-i}:D_{i}(y_{-i})\neq\emptyset\\}}D_{i}(y_{-i}))\text{
if }x_{-i}\notin W_{i};\\\ D_{i}(x_{-i})\text{ if }x_{-i}\in
W_{i}\end{array}\right.$ and

$T_{i}(x_{-i})=\left\\{\begin{array}[]{c}\text{co}(\bigcup\nolimits_{\\{y_{-i}:C_{i}(y_{-i})\neq\emptyset\\}}C_{i}(y_{-i}))\text{
if }x_{-i}\notin W_{i};\\\ C_{i}(x_{-i})\text{ if }x_{-i}\in
W_{i}.\end{array}\right.$

We call $T_{i}$ the correspondence of the upper sections of the sets $C_{i}.$

The correspondence $S_{i}$ has nonempty and convex values and
$S_{i}(x_{-i})\subset T_{i}(x_{-i})$ for each $x_{-i}\in X_{-i}.$

Assumption c) implies that $S_{i\mid(X_{-i}\cap W_{i})}$ has the local
intersection property. If $x_{-i}\in^{C}W_{i},$ then,
$S_{i}(x_{-i})=$co$(\bigcup\nolimits_{\\{y_{-i}:D_{i}(y_{-i})\neq\emptyset\\}}D_{i}(x_{-i}))\neq\emptyset$.
According to b), there exists an open neighborhood $V_{i}(x_{-i})$ of $x_{-i}$
such that $D_{i}(z_{-i})=\emptyset$ for each $z_{-i}\in V_{i}(x_{-i}).$ Then,
$S_{i}(z_{-i})=$co$(\bigcup\nolimits_{\\{y_{-i}:D_{i}(y_{-i})\neq\emptyset\\}}D_{i}(y_{-i}))$
for each $z_{-i}\in V_{i}(x_{-i})$ and, $\cap_{z_{-i}\in
V_{i}(x_{-i})}S_{i}(z_{-i})=$co$(\bigcup\nolimits_{\\{y_{-i}:D_{i}(y_{-i})\neq\emptyset\\}}D_{i}(y_{-i}))\neq\emptyset.$
It follows that $S_{i}:X_{-i}\rightarrow 2^{X_{i}}$ has the local intersection
property.

The Wu-Shen Lemma implies that $T_{i}$ has a continuous selection
$f_{i}:X_{-i}\rightarrow X_{i}.$

Let $f:X\rightarrow X$ be defined by $f(x)=\prod\nolimits_{i\in
N}f_{i}(x_{-i})$ for each $x\in X.$ The function $f$ is continuous, and,
according to the Brouwer fixed point Theorem, there exists $x^{\ast}\in X$
such that $f(x^{\ast})=x^{\ast}.$ Hence, $x^{\ast}\in\prod\nolimits_{i\in
N}T_{i}(x_{-i}^{\ast})$ and obviously, $x_{i}^{\ast}\in T_{i}(x_{-i}^{\ast})$
for each $i\in N.$ Suppose that $(x_{-i}^{\ast},X_{i})\cap
C_{i}\neq\emptyset,$ for some $i\in N.$ Then,
$C_{i}(x_{-i}^{\ast})\neq\emptyset$ and $x_{i}^{\ast}\in
C_{i}(x_{-i}^{\ast}),$ which implies $x^{\ast}=(x_{-i}^{\ast},x_{i}^{\ast})\in
C_{i}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

As a corollary, we obtain sufficient conditions for a game in normal form to
admit weak Nash equilibria. The main assumption is new in literature and it
refers to the existence of a best reply for each player $i$ to the common
strategy of the other players, which lies in open intervals of the product
space $X_{-i}.$ The meaning is that each player $i$ can remain stable in the
choice of his own best strategy in the situation that the decisions of the
opponents can vary slightly in any manner profitable to themselves.

###### Corollary 1

Let $((X_{i})_{i\in N},(u_{i})_{i\in N})$ be a game in normal form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) the set $\\{x\in X:u_{i}(x)\geq u_{i}(x_{-i},y_{i})$ for each $y_{i}\in
X_{i}\\}$ is nonempty;

c) $W_{i}=\\{x_{-i}\in X_{-i}:B_{i}(x_{-i})\neq\emptyset\\}$ is closed;

d) if $B_{i}(x_{-i})\neq\emptyset,$ there exists an open neighborhood
$V(x_{-i})$ of $x_{-i}$ so that $\cap_{z_{-i}\in
V(x_{-i})}B_{i}(z_{-i})\neq\emptyset;$

e) $B_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits weak Nash equilibria.

The following corollary concerns the existence of the weak equilibria for
qualitative games.

###### Corollary 2

Let $((X_{i})_{i\in N},(P_{i})_{i\in N})$ be a qualitative game. Assume that,
for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) the set $\\{x\in X:P_{i}(x)=\emptyset\\}$ is nonempty and
$W_{i}=\\{x_{-i}\in X_{-i}:\exists x_{i}\in X_{i}$ such that
$P_{i}(x_{-i},x_{i})=\emptyset\\}$ is closed;

c) for each $x_{-i}\in X_{-i},$ if there exists $x_{i}\in X_{i}$ such that
$P_{i}(x)=\emptyset,$ then, there exist an open neighborhood $V(x_{-i})$ of
$x_{-i}$ and $z_{i}\in X_{i}$ such that $P_{i}(z_{-i},z_{i})=\emptyset$ for
each $z_{-i}\in V_{i}(x_{-i});$

d) $\\{x_{i}\in X_{i}:P_{i}(x_{-i},x_{i})=\emptyset\\}$ is convex or empty for
each $x_{-i}\in X_{-i}.$

Then, the game admits weak equilibria.

A new approach to the existence of equilibria in choice implies the concept of
weakly convex graph for a correspondence, proposed by X. Ding and Y He in 126
. Firstly, we recall the definition.

###### Definition 1

(see 126 ). Let $X$ be a nonempty convex subset of a topological vector space
$E$ and $Y$ be a nonempty subset of $E$. The correspondence
$T:X\longrightarrow 2^{Y}$ is said to have _weakly convex graph_ (in short, it
is a WCG correspondence) if for each $n\in\mathbb{N}$ and for each finite set
$\\{x_{1},x_{2},...,x_{n}\\}\subset X$, there exists $y_{i}\in T(x_{i})$,
$(i=1,2,...,n)$, such that

co$(\\{(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})\\})\subset$Gr$(T).$

We note that either the graph Gr$(T)$ is convex, or $\
\mathop{\textstyle\bigcap}\\{T(x):x\in X\\}\neq\emptyset$, then $T$ has a
weakly convex graph.

It is obvious that a WCG correspondence may have nonempty values and may not
be convex-valued.

###### Example 1

$T:[0,2]\rightarrow[0,2],$
$T(x)=\left\\{\begin{array}[]{c}[0,\frac{1}{2}]\cup[\frac{3}{2},2]\text{ \ \ \
\ \ if }x=1,\\\ [0,2-\frac{1}{2}x]\text{ \ if
}x\in[0,2]\backprime\\{1\\}\end{array}\right.$ is a WCG correspondence (since
$\cap\\{T(x):x\in[0,2]\\}=\\{0\\}\neq\emptyset$), but $T(1)$ is not convex and
Gr$(T)$ is not convex either.

The theorem below is a continuous selection theorem for correspondences having
weakly convex graph.

###### Lemma 3

(Patriche, 85 ). Let $Y$ be a nonempty subset of a topological vector space
$E$ and $K$ be a $(n-1)$\- dimensional simplex in a topological vector space
$F$. Let $T:K\rightarrow 2^{Y}$ be a WCG correspondence. Then, $T$ has a
continuous selection on $K$.

The following lemma guarantees the existence of a fixed point for a product of
lower semi-continuous correspondences. It will be useful for proving our
second result on equilibria in choice.

###### Lemma 4

(Wu, 120 ) Let $I$ be an index set. For each $i\in I,$ let $X_{i}$ be a
nonempty convex subset of a Hausdorff locally convex topological vector space
$E_{i}$, $D_{i}$ a nonempty compact metrizable subset of $X_{i}$ and
$S_{i},T_{i}:X\rightarrow 2^{D_{i}}$ two correspondences with the following
conditions:

(1) for each $x\in X$, clco$S_{i}(x)\subset T_{i}(x)$ and
$S_{i}(x)\neq\emptyset$.

(2) $S_{i}$ is lower semi-continuous.

Then, there exists $x^{\ast}=\underset{i\in
I}{\mathop{\textstyle\prod}}x_{i}^{\ast}\in D=\underset{i\in
I}{\mathop{\textstyle\prod}}D_{i}$ such that $x_{i}^{\ast}\in T_{i}(x^{\ast})$
for each $i\in I.\vskip 6.0pt plus 2.0pt minus 2.0pt$

The assumption that each correspondence of the upper sections of the sets
$C_{i}$ has a selection which is an WCG correspondence also assures the
existence of the equilibria in choice. The following theorem presents
precisely this.

###### Theorem 3.2

Let $((X_{i})_{i\in N},(C_{i})_{i\in N})$ be a game in choice form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty convex compact metrizable set in a locally convex
space $E_{i}$ and $C_{i}$ is nonempty;

b) for each $x_{i}\in X_{i},$ $W_{i}=\\{x_{-i}\in
X_{-i}:C_{i}(x_{-i})\neq\emptyset\\}$ is a $(n_{i}-1)$-dimensional simplex in
$X;$

c) there exists a WCG correspondence $S_{i}:W_{i}\rightarrow 2^{X_{i}}$ such
that $S_{i}(x_{-i})\subset C_{i}(x_{-i})$ for each $x_{-i}\in W_{i};$

d) $C_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits equilibria in choice.

Proof. Let be $i\in I.$ From the assumption (c) and the selection Lemma 3, it
follows that there exists a continuous function $f_{i}:W_{i}\rightarrow X_{i}$
so that for each $x_{-i}\in W_{i}$, $f_{i}(x_{-i})\in S_{i}(x_{-i})\subset
C_{i}(x_{-i}).$

Let us define the correspondence $T_{i}:X_{-i}\rightarrow 2^{X_{i}}$, by

$T_{i}(x_{-i}):=\left\\{\begin{array}[]{c}\\{f_{i}(x_{-i})\\}\text{ \ \ \ \ \
\ \ \ \ \ \ \ \ if \ \ \ \ \ \ \ \ \ \ \ }x_{-i}\in W_{i};\\\
\text{co}(\bigcup\nolimits_{\\{y_{-i}:C_{i}(y_{-i})\neq\emptyset\\}}C_{i}(y_{-i}))\text{
if }x_{-i}\notin W_{i}.\end{array}\right.$

We appeal to Lemma 1 to conclude that $T_{i}$ is lower semicontinuous on $X$.
Tychonoff’s Theorem implies that $X$ is compact.

According to Wu’s fixed-point Theorem, applied for the correspondences
$S_{i}=T_{i}$ and $T_{i}:X\rightarrow 2^{X_{i}},$ there exists
$\mathit{x}^{\ast}\in X$ such that for each $i\in I$,
$\mathit{x}_{i}^{\ast}\in T_{i}(\mathit{x}^{\ast})$. Suppose that
$(x_{-i}^{\ast},X_{i})\cap C_{i}\neq\emptyset,$ for some $i\in N.$ Then,
$C_{i}(x_{-i}^{\ast})\neq\emptyset$ and $x_{i}^{\ast}\in
C_{i}(x_{-i}^{\ast}),$ which implies $x^{\ast}=(x_{-i}^{\ast},x_{i}^{\ast})\in
C_{i}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

###### Remark 1

We can obtain two corollaries to the above theorem, if we replace assumption
c) with:

c’) there exists a correspondence $S_{i}:W_{i}\rightarrow 2^{X_{i}}$ such
that$\mathit{\ }S_{i}$ has a convex graph and $S_{i}(x_{-i})\subset
C_{i}(x_{-i})$ for each $x_{-i}\in W_{i};$

or

c”) there exists a correspondence $S_{i}:W_{i}\rightarrow 2^{X_{i}}$ with
closed values such that$\mathit{\ }S_{i}$ has the property that for any finite
set $\\{x_{-i}^{1},x_{-i}^{2},...x_{-i}^{m}\\}\subset X,$
$\mathop{\textstyle\bigcap}_{j=1}^{m}S_{i}(x_{-i}^{j})\neq\emptyset$ and
$S_{i}(x_{-i})\subset C_{i}(x_{-i})$ for each $x_{-i}\in W_{i};$

Proof. In the first case, since a correspondence with a convex graph is a WCG
one, it follows that $S_{i}$ verifies Assumption c) from Theorem 2, then we
can apply this theorem.

In the second case, $X$ is a compact space and for each $i\in I$ the closed
sets $S_{i}(x_{-i}),$ $x_{-i}\in X_{-i}$ have the finite intersection
property, then $\mathop{\textstyle\bigcap}\\{S_{i}(x_{-i}):x_{-i}\in
X_{-i}\\}\neq\emptyset$. It follows that $S_{i}$ is a WCG correspondence and
the conclusion comes from Theorem 2.

As in the first case, we obtain the following corollaries concerning the
existence of the weak Nash equilibria for games in normal form and,
respectively, of the weak equilibria for qualitative games.

###### Corollary 3

Let $((X_{i})_{i\in N},(u_{i})_{i\in N})$ be a game in normal form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) the set $\\{x\in X:u_{i}(x)\geq u_{i}(x_{-i},y_{i})$ for each $y_{i}\in
X_{i}\\}$ is nonempty;

c) for each $x_{i}\in X_{i},$ $W_{i}=\\{x_{-i}\in
X_{-i}:B_{i}(x_{-i})\neq\emptyset\\}$ is a $(n_{i}-1)$-dimensional simplex in
$X;$

d) there exists a WCG correspondence $S_{i}:W_{i}\rightarrow 2^{X_{i}}$ such
that $S_{i}(x_{-i})\subset B_{i}(x_{-i})$ for each $x_{-i}\in W_{i};$

e) $B_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits weak Nash equilibria.

###### Corollary 4

Let $((X_{i})_{i\in N},(P_{i})_{i\in N})$ be a qualitative game. Assume that,
for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) the set $\\{x\in X:P_{i}(x)=\emptyset\\}$ is nonempty;

c) for each $x_{i}\in X_{i},$ $W_{i}=\\{x_{-i}\in X_{-i}:\exists x_{i}\in
X_{i}$ such that $P_{i}(x_{-i},x_{i})=\emptyset\\}$ is a
$(n_{i}-1)$-dimensional simplex in $X;$

d) there exists a WCG correspondence $S_{i}:W_{i}\rightarrow 2^{X_{i}}$ such
that $S_{i}(x_{-i})\subset\\{x_{i}\in X_{i}$ such that
$P_{i}(x_{-i},x_{i})=\emptyset\\}$ for each $x_{-i}\in W_{i};$

e) $\\{x_{i}\in X_{i}:P_{i}(x_{-i},x_{i})=\emptyset\\}$ is convex or empty for
each $x_{-i}\in X_{-i}.$

Then, the game admits weak equilibria.

Now, we present a continuous selection lemma on Banach spaces which was
proposed by Yuan 134 .

###### Lemma 5

(see 134 ). Let $X$ be a paracompact space, $Y$ be a Banach space and
$T:X\rightarrow 2^{Y}$ be a lower semicontinuous correspondences with closed
convex values. Let $S:X\rightarrow 2^{Y}$ be a correspondence whose graph is
open in $X\times Y$ such that $T(x)\cap S(x)\neq\emptyset$ for each $x\in X.$
Then, there exists a continuous function $f:X\rightarrow Y$ such that
$f(x)\in$co$(T(x)\cap S(x))$ for each $x\in X$.

The previous lemma leads us to the enunciation of Theorem 3, which gives new
conditions under which the equilibria in choice exist.

###### Theorem 3.3

Let $((X_{i})_{i\in N},(C_{i})_{i\in N})$ be a game in choice form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Banach space $E_{i};$

b) $C_{i}$ is nonempty and open;

c) the set $W_{i}=\\{x_{-i}\in X_{-i}:C_{i}(x_{-i})\neq\emptyset\\}$ is
nonempty and open;

d) there exists a lower semicontinuous correspondence $S_{i}:W_{i}\rightarrow
2^{X_{i}}$ with closed convex values such that $C_{i}(x_{-i})\cap
S_{i}(x_{-i})\neq\emptyset$ for each $x_{-i}\in X_{-i}$;

e) $C_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits equilibria in choice.

###### Proof

Let be $i\in I.$ From the assumption d) and the above lemma, it follows that
there exists a continuous function $f_{i}:W_{i}\rightarrow X_{i}$ such that
$f_{i}(x_{-i})\in C_{i}(x_{-i})\cap S_{i}(x_{-i})$ for each $x_{-i}\in W_{i}$.

Let us define the correspondence $T_{i}:X_{-i}\rightarrow 2^{X_{i}}$, by

$T_{i}(x_{-i}):=\left\\{\begin{array}[]{c}\\{f_{i}(x_{-i})\\}\text{ \ \ \ \ \
\ \ \ \ \ \ \ if \ \ \ \ \ \ \ \ \ \ \ \ }x_{-i}\in W_{i};\\\
\text{co}(\bigcup\nolimits_{\\{y_{-i}:C_{i}(y_{-i})\neq\emptyset\\}}C_{i}(y_{-i}))\text{
if }x_{-i}\notin W_{i}.\end{array}\right.$

We appeal to Lemma 1 to conclude that $T_{i}$ is upper semicontinuous on $X$.

According to Kakutani’s fixed-point Theorem, there exists
$\mathit{x}^{\ast}\in X$ such that for each $i\in I$,
$\mathit{x}_{i}^{\ast}\in T_{i}(\mathit{x}_{-i}^{\ast})$. Suppose that
$(x_{-i}^{\ast},X_{i})\cap C_{i}\neq\emptyset,$ for some $i\in N.$ Then,
$C_{i}(x_{-i}^{\ast})\neq\emptyset$ and $x_{i}^{\ast}=f_{i}(x_{-i}^{\ast})\in
C_{i}(x_{-i}^{\ast}),$ which implies $x^{\ast}=(x_{-i}^{\ast},x_{i}^{\ast})\in
C_{i}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

As corollaries of Theorem 3, we obtain the following results which assume the
lower semicontinuity of the involved correspondences. We note that these
results are different from the old ones obtained in literature so far.

###### Corollary 5

Let $((X_{i})_{i\in N},(u_{i})_{i\in N})$ be a game in normal form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Banach space $E_{i}$;

b) the set $\\{x\in X:u_{i}(x)\geq u_{i}(x_{-i},y_{i})$ for each $y_{i}\in
X_{i}\\}$ is nonempty and open;

c) the set $W_{i}=\\{x_{-i}\in X_{-i}:B_{i}(x_{-i})\neq\emptyset\\}$ is
nonempty and open;

d) there exists a lower semicontinuous correspondence $S_{i}:X_{-i}\rightarrow
2^{X_{i}}$ with closed convex values such that $S_{i}(x_{-i})\cap
B_{i}(x_{-i})\neq\emptyset$ for each $x_{-i}\in X_{-i}$;

e) $B_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits weak Nash equilibria.

###### Corollary 6

Let $((X_{i})_{i\in N},(P_{i})_{i\in N})$ be a qualitative game. Assume that,
for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Banach space $E_{i}$;

b) the set $\\{x\in X:P_{i}(x)=\emptyset\\}$ is nonempty and open;

c) the set $W_{i}=\\{x_{-i}\in X_{-i}:$ $\mathit{\exists}$ $x_{i}\in X_{i}$
such that $P_{i}(x_{-i},x_{i})=\emptyset\\}$ is nonempty and open;

d) there exists a lower semicontinuous correspondence $S_{i}:W_{i}\rightarrow
2^{X_{i}}$ with closed convex values such that $S_{i}(x_{-i})\cap\\{x_{i}\in
X_{i}:P_{i}(x_{-i},x_{i})=\emptyset\\}\neq\emptyset$ for each $x_{-i}\in
X_{-i}$;

e) $\\{x_{i}\in X_{i}:P_{i}(x_{-i},x_{i})=\emptyset\\}$ is convex or empty for
each $x_{-i}\in X_{-i}.$

Then, the game admits weak equilibria.

Further, we will prove the existence of the equilibrium in choice under new
conditions. The proof we will provide explicitly relies on lemmas concerning
the fixed points for the correspondences we will construct based on the upper
sections of the sets $(C_{i})_{i\in N}.$

First, we recall the following definition.

If $X$ is a nonempty set and $Y$ is a topological space, the correspondence
$T:X\rightarrow 2^{Y}$ is said to be transfer open-valued tian  if for any
$(x,y)\in X\times Y$ with $y\in T(x),$ there exists an $x^{\prime}\in X$ such
that $y\in$int$T(x^{\prime}).\vskip 6.0pt plus 2.0pt minus 2.0pt$

Further, we present the following useful statement about the transfer open-
valued correspondences (Proposition 1 in lin2001 ).

###### Lemma 6

Let $Y$ be a nonempty set, $X$ be a topological space and $S:X\rightarrow
2^{Y}$ be a correspondence. The following assertions are equivalent:

a) $S^{-1}:Y\rightarrow 2^{X}$ is transfer open-valued and $S$ has nonempty
values;

b) $X=\bigcup\nolimits_{y\in Y}$int$S^{-1}(y).\vskip 6.0pt plus 2.0pt minus
2.0pt$

In this context, Ansari and Yao proved in an  a fixed point result.

###### Lemma 7

(Ansari, Yao an ). Let $X$ be a compact convex subset of a Hausdorff
topological vector space. Let $S:X\rightarrow 2^{X}$ be a correspondence with
nonempty convex values. If $X=\cup\\{$int${}_{X}S^{-1}(y):y\in X\\}$ (or,
$S^{-1}:Y\rightarrow 2^{X}$ is transfer open-valued), then, $S$ has fixed
points.

We will apply the previous lemma in order to prove the existence of the
equilibria in choice for games in choice form.

###### Theorem 3.4

Let $((X_{i})_{i\in N},(C_{i})_{i\in N})$ be a game in choice form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$ and $C_{i}$ is nonempty;

b) $X=\bigcup\nolimits_{y\in X}\\{$int${}_{X}\bigcap\nolimits_{i\in
N}(^{C}W_{i}\cup\\{x_{-i}\in X_{-i}:x_{-i}\in C_{i}(y_{i})\\})\\}$, where
$W_{i}=\\{x_{-i}\in X_{-i}:C_{i}(x_{-i})\neq\emptyset\\};$

c) $C_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits equilibria in choice.

###### Proof

For each $i\in N,$ let us define the correspondence $T_{i}:X_{-i}\rightarrow
2^{X_{i}},$ by

$T_{i}(x_{-i})=\left\\{\begin{array}[]{c}\text{co}(\bigcup\nolimits_{\\{y_{-i}:C_{i}(y_{-i})\neq\emptyset\\}}C_{i}(y_{-i}))\text{
if }x_{i};\notin W_{i}\\\ C_{i}(x_{-i})\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if \
\ \ \ \ \ \ \ \ \ \ \ \ \ }x_{-i}\in W_{i}.\end{array}\right.$

The correspondence $S_{i}$ has nonempty and convex values.

Let $T:X\rightarrow 2^{X}$ be defined by $T(x)=\prod\nolimits_{i\in
N}T_{i}(x_{-i})$ for each $x\in X.$

The correspondence $S$ also has nonempty and convex values.

If $y\in X,$ then $T^{-1}(y)=\bigcap\nolimits_{i\in N}\\{x\in X:y_{i}\in
T_{i}(x_{-i})\\}=\bigcap\nolimits_{i\in N}(^{C}W_{i}\cup\\{x_{-i}\in
X_{-i}:x_{-i}\in C_{i}(y_{i})\\}).$

$X=\bigcup\nolimits_{y\in X}$int${}_{X}T^{-1}(y)$, according to assumption b).

We can apply the Ansari and Yao Lemma and we obtain that there exists
$x^{\ast}\in X$ such that $x^{\ast}\in T(x^{\ast}).$ Obviously,
$x_{i}^{\ast}\in T_{i}(x_{-i}^{\ast})$ for each $i\in N.$ Suppose that
$(x_{-i}^{\ast},X_{i})\cap C_{i}\neq\emptyset,$ for some $i\in N.$ Then,
$C_{i}(x_{-i}^{\ast})\neq\emptyset$ and $x_{i}^{\ast}\in
C_{i}(x_{-i}^{\ast}),$ which implies $x^{\ast}=(x_{-i}^{\ast},x_{i}^{\ast})\in
C_{i}.$

###### Remark 2

According to Lemma 7, we can replace condition b) in Theorem 4 with

b’) if $x_{i}\in C_{i}(y_{-i}),$ then, there exists $z_{i}\in X_{i}$ such that
$y_{-i}\in$int${}_{X_{-i}}\\{x_{-i}\in X_{-i}:z_{i}\in C_{i}(x_{-i})\\}$ and
the set $W_{i}=\\{x_{-i}\in X_{-i}:C_{i}(x_{-i})\neq\emptyset\\}$ is closed.

A new result involving the equilibria in choice will naturally follow directly
from Theorem 4.

###### Theorem 3.5

Let $((X_{i})_{i\in N},(C_{i})_{i\in N})$ be a game in choice form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$ and $C_{i}$ is nonempty;

b) for each $x_{i}\in X_{i},$ $\\{x_{-i}\in X_{-i}:x_{i}\in
C_{i}(x_{-i})\\}\cup^{C}W_{i}$ is open, where $W_{i}=\\{x_{-i}\in
X_{-i}:C_{i}(x_{-i})\neq\emptyset\\};$

c) $C_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits equilibria in choice.

Another proof of the above theorem appeals to Yannelis and Prabhakar’
continuous selection lemma 127  applied for the correspondences
$T_{i}:X_{-i}\rightarrow 2^{X_{i}}$ defined by

$T_{i}(x_{-i})=\left\\{\begin{array}[]{c}\text{co}(\bigcup\nolimits_{\\{y_{-i}:C_{i}(y_{-i})\neq\emptyset\\}}C_{i}(y_{-i}))\text{
if }x_{-i}\notin W_{i};\\\ C_{i}(x_{-i})\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
\ \ \ \ \ \ \ \ \ \ \ \ \ \ }x_{-i}\in W_{i}.\end{array}\right.$ for each
$i\in N.$

We present below the lemma.

###### Lemma 8

(Yannelis and Prabhakar, 127 ). Let $X$ be a paracompact Hausdorff topological
space and $Y$ be a Hausdorff topological vector space. Let $T:X\rightarrow
2^{Y}$ be a correspondence with nonempty convex values and for each $y\in Y$,
$T^{-1}(y)$ is open in $X$. Then, $T$ has a continuous selection that is,
there exists a continuous function $f:X\rightarrow Y$ so that $f(x)\in T(x)$
for each $x\in X$.

For the proof, we note that for each $i\in N,$ the correspondence $T_{i}$ has
nonempty and convex values and if $x_{i}\in X_{i},$ then
$T_{i}^{-1}(x_{i})=^{C}W_{i}\cup\\{x_{-i}\in X_{-i}:x_{i}\in C_{i}(x_{-i})\\}$
is an open set, according to assumption b).

We can apply the Yannelis and Prabhakar Lemma and we obtain that there exists
$f_{i}:X_{-i}\rightarrow X_{i}$, a continuous selection of $T_{i}.$ Let
$f:X\rightarrow X$ be defined by $f(x)=\prod\nolimits_{i\in N}f_{i}(x_{-i})$
for each $x\in X.$ The function $f$ is continuous, and, according to the
Brouwer fixed point Theorem, there exists $x^{\ast}\in X$ such that
$f(x^{\ast})=x^{\ast}.$ Hence, $x^{\ast}\in\prod\nolimits_{i\in
N}T_{i}(x_{-i}^{\ast})$ and obviously, $x_{i}^{\ast}\in T_{i}(x_{-i}^{\ast})$
for each $i\in N.$ Suppose that $(x_{-i}^{\ast},X_{i})\cap
C_{i}\neq\emptyset,$ for some $i\in N.$ Then,
$C_{i}(x_{-i}^{\ast})\neq\emptyset$ and $x_{i}^{\ast}\in
C_{i}(x_{-i}^{\ast}),$ which implies $x^{\ast}=(x_{-i}^{\ast},x_{i}^{\ast})\in
C_{i}.$

###### Remark 3

If for each $x_{i}\in X_{i},$ $\\{x_{-i}\in X_{-i}:x_{i}\in C_{i}(x_{-i})\\}$
is open, and the set $W_{i}=\\{x_{-i}\in X_{-i}:C_{i}(x_{-i})\neq\emptyset\\}$
is closed, then the above Theorem holds.

Corollary 7 is mainly obtained by verifying an assumption concerning the union
of all lower sections of the best reply correspondences for a game in normal
form.

###### Corollary 7

Let $((X_{i})_{i\in N},(u_{i})_{i\in N})$ be a game in normal form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) the set $\\{x\in X:u_{i}(x)\geq u_{i}(x_{-i},y_{i})$ for each $y_{i}\in
X_{i}\\}$ is nonempty;

c) $X=\bigcup\nolimits_{y\in X}\\{$int${}_{X}\bigcap\nolimits_{i\in
N}(^{C}W_{i}\cup B_{i}^{-1}(y_{i}))\\}$, where $W_{i}=\\{x_{-i}\in
X_{-i}:B_{i}(x_{-i})\neq\emptyset\\};$

d) $B_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits weak Nash equilibria.

###### Remark 4

In Corollary 7, condition c) can also be replaces with

c’) the best reply correspondence $B_{i}:X_{-i}\rightarrow 2^{X_{i}}$ is
transfer open valued and the set $W_{i}=\\{x_{-i}\in
X_{-i}:B_{i}(x_{-i})\neq\emptyset\\}$ is closed.

A new statement can be deduced explicitly from Corollary 7.

###### Corollary 8

Let $((X_{i})_{i\in N},(u_{i})_{i\in N})$ be a game in normal form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) the set $\\{x\in X:u_{i}(x)\geq u_{i}(x_{-i},y_{i})$ for each $y_{i}\in
X_{i}\\}$ is nonempty;

c) $B_{i}^{-1}(x_{i})\cup\\{x_{-i}\in X_{-i}:B_{i}(x_{-i})=\emptyset\\}$ is
open for each $x_{i}\in X_{i};$

d) $B_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits weak Nash equilibria.

The following results refer to the existence of weak equilibria for the
qualitative games. They are consequences of Theorem 5.

###### Corollary 9

Let $((X_{i})_{i\in N},(P_{i})_{i\in N})$ be a qualitative game. Assume that,
for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) the set $\\{x\in X:P_{i}(x)=\emptyset\\}$ is nonempty;

c) $X=\bigcup\nolimits_{y\in X}\\{$int${}_{X}\bigcap\nolimits_{i\in
N}(^{C}W_{i}\cup\\{x_{-i}\in X_{-i}:$ $P_{i}(x_{-i},y_{i})=\emptyset\\})\\},$
where $W_{i}=\\{x_{-i}\in X_{-i}:\exists x_{i}\in X_{i}$ such that
$P_{i}(x_{-i},x_{i})=\emptyset\\};$

d) $\\{x_{i}\in X_{i}:P_{i}(x_{-i},x_{i})=\emptyset\\}$ is convex or empty for
each $x_{-i}\in X_{-i}.$

Then, the game admits weak equilibria.

###### Remark 5

In Corollary 9, condition c) can also be replaced with

c”)if $P_{i}(y_{-i},x_{i})=\emptyset,$ then, there exists $z_{i}\in X_{i}$
such that $y_{-i}\in$int${}_{Y_{-i}}\\{x_{-i}\in
X_{-i}:P_{i}(x_{-i},z_{i})=\emptyset\\}$ and the set $W_{i}=\\{x_{-i}\in
X_{-i}:\exists x_{i}\in X_{i}$ such that $P_{i}(x_{-i},x_{i})=\emptyset\\}$ is
open.

###### Corollary 10

Let $((X_{i})_{i\in N},(P_{i})_{i\in N})$ be a qualitative game. Assume that,
for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) the set $\\{x\in X:P_{i}(x)=\emptyset\\}$ is nonempty;

c) $\\{x_{-i}\in X_{-i}:\nexists x_{i}\in X_{i}$ such that
$P_{i}(x_{-i},x_{i})=\emptyset\\}\cup\\{x_{-i}\in X_{-i}:$
$P_{i}(x_{-i},x_{i})=\emptyset\\}$ is open for each $x_{i}\in X_{i};$

d) $\\{x_{i}\in X_{i}:P_{i}(x_{-i},x_{i})=\emptyset\\}$ is convex or empty for
each $x_{-i}\in X_{-i}.$

Then, the game admits weak equilibria.

A very great importance in the fixed point theory has Tarafdar’s fixed point
Theorem, which we present below.

###### Lemma 9

(Tarafdar, 114 ). Let $\\{X_{i}\\}_{i\in I}$ be a family of nonempty compact
convex sets, each in a topological vector space $E_{i},$ where $\mathit{I}$ is
an index set. Let $X=\prod\limits_{i\in I}X_{i}.$ For each $i\in I,$ let
$S_{i}:X\rightarrow 2^{X_{i}}$ be a correspondence such that

a) for each $x\in X,$ $S_{i}(x)$ is a nonempty, convex subset of $X_{i};$

b) for each $x_{i}\in X_{i},$ $S_{i}^{-1}(x_{i})$ contains a relatively open
subset $O_{x_{i}}$ of $X$ such that

$\cup_{x_{i}\in X_{i}}O_{x_{i}}=X$ $(O_{x_{i}}$ may be empty for some
$x_{i}).$

Then, there exists a point $x\in X$ such that $x\in S(x)=\prod\limits_{i\in
I}S_{i}(x)$, that is, $x_{i}\in S_{i}(x)$ for each $i\in I,$ where $x_{i}$ is
the projection of $x$ onto $X_{i}$ for each $i\in I.$

By using Lemma 9, we establish Theorem 6, which is a slightly different
version of Theorem 4.

###### Theorem 3.6

Let $((X_{i})_{i\in N},(C_{i})_{i\in N})$ be a game in choice form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$ and $C_{i}$ is nonempty;

b) for each $x_{i}\in X_{i},$ $\\{x_{-i}\in X_{-i}:x_{i}\in C_{i}(x_{-i})\\}$
contains a relatively open subset $O_{x_{i}}$ of $X_{-i}$, such that

$\cup_{x_{i}\in X_{i}}O_{x_{i}}=X_{-i}$ $(O_{x_{i}}$ may be empty for some
$x_{i});$

c) $C_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits equilibria in choice.

###### Proof

For each $i\in N,$ let us define the correspondence $T_{i}:X_{-i}\rightarrow
2^{X_{i}},$ by

$T_{i}(x_{-i})=\left\\{\begin{array}[]{c}\text{co}(\bigcup\nolimits_{\\{y_{-i}:C_{i}(y_{-i})\neq\emptyset\\}}C_{i}(y_{-i}))\text{
if }x_{-i}\notin W_{i};\\\ C_{i}(x_{-i})\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
\ \ \ \ \ \ \ \ \ \ \ \ \ \ }x_{-i}\in W_{i},\end{array}\right.,$ where
$W_{i}=\\{x_{-i}\in X_{-i}:C_{i}(x_{-i})\neq\emptyset\\}.$

The correspondence $T_{i}$ has nonempty and convex values.

If $x_{i}\in X_{i},$ then $T_{i}^{-1}(x_{i})=^{C}W_{i}\cup\\{x_{-i}\in
X_{-i}:x_{i}\in C_{i}(x_{-i})\\}.$

According to assumption b), for each $x_{i}\in X_{i},$ $T_{i}^{-1}(x_{i})$
contains a relatively open subset $O_{x_{i}}$ of $X$ such that

$\cup_{x_{i}\in X_{i}}O_{x_{i}}=X_{-i}$ $(O_{x_{i}}$ may be empty for some
$x_{i}).$ Then, $X=\cup_{x_{i}\in X_{i}}(O_{x_{i}}\times X_{i}).$ For each
$i\in N,$ let us define the correspondence $S_{i}:X\rightarrow 2^{X_{i}},$ by
$S_{i}(x)=T_{i}(x_{-i}).$ If $x_{i}\in X_{i},$ then
$S_{i}^{-1}(x_{i})=T_{i}^{-1}(x_{i})\times X_{i}.$ We denote
$U_{x_{i}}=O_{x_{i}}\times X_{i}$ and we obtain that for each $x_{i}\in
X_{i},$ $S_{i}^{-1}(x_{i})$ contains a relatively open subset $U_{x_{i}}$ of
$X$ such that $\cup_{x_{i}\in X_{i}}U_{x_{i}}=X$ $(U_{x_{i}}$ may be empty for
some $x_{i}).$

We can apply the previous lemma and we obtain that there exists $x^{\ast}\in
X$ such that $x_{i}^{\ast}\in S_{i}(x^{\ast})=T_{i}(x_{-i}^{\ast})$ for each
$i\in N.$ Suppose that $(x_{-i}^{\ast},X_{i})\cap C_{i}\neq\emptyset,$ for
some $i\in N.$ Then, $C_{i}(x_{-i}^{\ast})\neq\emptyset$ and $x_{i}^{\ast}\in
C_{i}(x_{-i}^{\ast}),$ which implies $x^{\ast}=(x_{-i}^{\ast},x_{i}^{\ast})\in
C_{i}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Now, we get the following corollaries from the previous result.

###### Corollary 11

Let $((X_{i})_{i\in N},(u_{i})_{i\in N})$ be a game in normal form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) the set $\\{x\in X:u_{i}(x)\geq u_{i}(x_{-i},y_{i})$ for each $y_{i}\in
X_{i}\\}$ is nonempty;

c) for each $x_{i}\in X_{i},$ $B_{i}^{-1}(x_{i})$ contains a relatively open
subset $O_{x_{i}}$ of $X$ such that

$\cup_{x_{i}\in X_{i}}O_{x_{i}}=X_{-i}$ $(O_{x_{i}}$ may be empty for some
$x_{i});$

d) $B_{i}(x_{-i})=\\{x_{i}\in X_{i}:u_{i}(x)\geq u_{i}(x_{-i},y_{i})$ for each
$y_{i}\in X_{i}\\}$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits weak Nash equilibria.

###### Corollary 12

Let $((X_{i})_{i\in N},(P_{i})_{i\in N})$ be a qualitative game. Assume that,
for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$;

b) the set $\\{x\in X:P_{i}(x)=\emptyset\\}$ is nonempty;

c) for each $x_{i}\in X_{i},$ $\\{x_{-i}\in
X_{-i}:P_{i}(x_{-i},x_{i})=\emptyset\\}$ contains a relatively open subset
$O_{x_{i}}$ of $X$ such that

$\cup_{x_{i}\in X_{i}}O_{x_{i}}=X_{-i}$ $(O_{x_{i}}$ may be empty for some
$x_{i});$

d) $\\{x_{i}\in X_{i}:P_{i}(x_{-i},x_{i})=\emptyset\\}$ is convex or empty for
each $x_{-i}\in X_{-i}.$

Then, the game admits weak equilibria.

###### Remark 6

In a particular case, we can weaken condition b) of Theorem 6 by condition
b’):

b’) for each $x_{i}\in X_{i},$ $\\{x_{-i}\in X_{-i}:x_{i}\in
C_{i}(x_{-i})\\}=O_{x_{i}}$ is an open subset of $X$ such that

$\cup_{x_{i}\in X_{i}}O_{x_{i}}=X_{-i}$ $(O_{x_{i}}$ may be empty for some
$x_{i}).$ According to Lemma 7, this condition is equivalent with the fact
that the correspondence $T_{i}^{-1}:X_{i}\rightarrow 2^{X_{-i}}$ is transfer
open-valued and $T_{i}$ has nonempty values, where $T_{i}:X_{-i}\rightarrow
2^{X_{i}}$ is defined by $T_{i}(x_{-i})=C_{i}(x_{-i})$ for each $x_{-i}\in
X_{-i}$.

In this case, we obtain the following theorem concerning the existence of the
strong equilibrium in choice.

###### Theorem 3.7

Let $((X_{i})_{i\in N},(C_{i})_{i\in N})$ be a game in choice form. Assume
that, for each $i\in N,$ the following conditions are fulfilled:

a) $X_{i}$ is a nonempty, convex and compact set in a Hausdorff topological
vector space $E_{i}$ and $C_{i}$ is nonempty;

b) for each $x_{i}\in X_{i},$ $\\{x_{-i}\in X_{-i}:x_{i}\in C_{i}(x_{-i})\\}$
$=O_{x_{i}}$ is an open subset of $X$ such that

$\cup_{x_{i}\in X_{i}}O_{x_{i}}=X_{-i}$ $(O_{x_{i}}$ may be empty for some
$x_{i});$

c) $C_{i}(x_{-i})$ is convex or empty for each $x_{-i}\in X_{-i}.$

Then, the game admits strong equilibria in choice.

##  4 Concluding remarks

Our study is a new perspective unifying different approaches and results on
the equilibrium concepts and the existence of noncooperative theory of games.
We have proposed to the reader a synthesis of theorems and consequences which
state, under new conditions, the existence of the equilibrium for games in
choice form, in normal form and also for qualitative games. Our approach
differs essentially from the one of Stefanescu, Ferrara and Stefanescu, who
proposed the new concept of game in choice form and the corresponding
equilibrium in choice (2012). A further research may consist of the
integration of all research instruments and perspectives. The advantage of
using these new ideas is that they are more systematic and can cover more
general situations. This paper reflects the integrity of this kind of thinking
and can reopen the problem of the equilibrium existence under new conditions.

## References

  * (1) Ansari, QH, Yao, J-C: A fixed point theorem and its applications to a system of variational inequalities. Bull. Aust. Math. Soc.. 59, 433–442 (1999)
  * (2) X. Ding, Y. He, Best Approximation Theorem for Set-valued Mappings without Convex Values and Continuity. Appl Math. and Mech. English Edition, 19, 9 (1998), 831-836.
  * (3) L. J. Lin, Applications of a fixed point theorem in $G$-convex spaces, Nonlinear Anal. 46 (2001) 601-608.
  * (4) J. F. Nash, Equilibrium points in n-person games. Proc. Nat. Acad. Sci. U. S. A., vol 36, no 1, (1950), 48-49.
  * (5) J. F. Nash, Non-cooperative games. Ann. Math. 54 (1951), 286-295.
  * (6) M. Patriche,Equilibrium in abstract economies with weakly convex graph correspondences. Math. Reports, 10(60), 4 (2008).
  * (7) S. P. Singh, E. Tarfdar and B. Watson, A generalized fixed point theorem and equilibrium point of an abstract economy. Journal of Computational and Applied Mathematics 113 (2000), 65-71.
  * (8) A. Stefanescu, M. Ferrara, Implementation of voting operators, J. Math. Econ. 42 (2006), 315-324.
  * (9) A. Stefanescu, M. Ferrara, M.V. Stefanescu, Equilibria of the Games in Choice Form, J. Optim. Theory Appl. 155 (2012), 160-1072
  * (10) E. Tarafdar - A fixed point theorem and equilibrium point of an abstract economy. J. Math. Econ. 20 (1991),211-218.
  * (11) C. G. Tian, Generalizations of the KKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity, J. Math. Anal. Appl. 170 (1992) 457-471.
  * (12) X. Wu, A new fixed point theorem and its applications. Proc. Amer. Math. Soc. 125 (1997), 1779-1783.
  * (13) X. Wu, S. Shen, A further generalisation of Yannelis-Prabhakar’s continuous selection theorem and its applications. J. Math. Anal. Appl. 197 (1996), 61-74.
  * (14) N. C. Yannelis, N. D. Prabhakar, Existence of maximal elements and equilibrium in linear topological spaces. J. Math. Econom. 12 (1983), 233-245.
  * (15) X. Z. Yuan, The Study of Minimax inequalities and Applications to Economies and Variational inequalities. Memoirs of the American Mathematical Society 132, 625, (1988).


# Fermion integrals for finite spectral triples

John W. Barrett

School of Mathematical Sciences
University of Nottingham
University Park
Nottingham NG7 2RD, UK

E-mail<EMAIL_ADDRESS>
(March 27th, 2024)

###### Abstract

Fermion functional integrals are calculated for the Dirac operator of a finite
real spectral triple. Complex, real and chiral functional integrals are
considered for each KO-dimension where they are non-trivial, and phase
ambiguities in the definition are noted.

##  1 Introduction

Spectral triples provide a precise mathematical framework for the Dirac
operator, allowing the generalisation to non-commutative spaces [1]. In the
traditional (commutative) case of a Dirac operator on a manifold, the
formulation of the fermion functional integral is well known, and results in
the computation of a regularised Pfaffian or determinant.

In this paper, the corresponding fermion integrals are calculated for finite
spectral triples, that is, ones in which the vector space of spinor fields is
finite-dimensional. Finite spectral triples are richer in the non-commutative
setting and include the fuzzy sphere [2] and the fuzzy torus [3]. Moreover,
random finite spectral triples have been used to model random geometries [4,
5, 6, 7, 8, 9, 10, 11], reviewed in [12]. These models have the potential to
model quantum gravity and bosonic matter fields in a finite setting. Thus it
is interesting to add fermion fields and the fermion functional integral to
these models.

Three different types of functional integral are considered:

  * •

The complex integral, over a set of fermionic variables and their conjugates.

  * •

The real integral, over a set of fermionic variables (without conjugates).

  * •

The chiral integral, a real integral where the spinors are restricted to a
definite chirality.

There are eight different types of spectral triple distinguished by the KO-
dimension $s$, which is an integer mod 8. The results of this paper are to
determine the non-trivial cases, giving a formula for fermion integral in
terms of a Pfaffian or a determinant. In some cases there is a sign or a phase
ambiguity. Finally, there are some remarks about the physical significance of
the results.

##  2 Finite real spectral triples

A finite spectral triple has a finite-dimensional Hilbert space $\mathcal{H}$,
on which acts a self-adjoint operator $D$, the Dirac operator, and an algebra
$\mathcal{A}$. The eight different types of spectral triple are determined by
the properties of a chirality operator $\Gamma$ on $\mathcal{H}$, which is
Hermitian and squares to $1$, and an antilinear unitary operator $J$ on
$\mathcal{H}$ called the real structure.

For even parameter $s$, $D$ anticommutes with $\Gamma$. The elements of the
$+1$ eigenspace of $\Gamma$ are called the left-handed fermions and the $-1$
eigenspace the right-handed fermions. For odd $s$, $\Gamma$ commutes with $D$
and one can decompose the spectral triple so that $\Gamma=1$ (or $-1$); the
chirality operator is therefore not important.

The properties of $J$ are $J^{2}=\epsilon=\pm 1$, $DJ=\epsilon^{\prime}JD=\pm
JD$ and $J\Gamma J^{-1}\Gamma=\epsilon^{\prime\prime}=\pm 1$. These three
signs depend only on the value of $s=q-p\mod 8$, as shown in Figure 1 [13].

$s$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
$\epsilon$ | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ | 1 | 1
$\epsilon^{\prime}$ | 1 | $-1$ | 1 | 1 | 1 | $-1$ | 1 | 1
$\epsilon^{\prime\prime}$ | 1 | 1 | $-1$ | 1 | 1 | 1 | $-1$ | 1
| ${\mathbb{R}}$ | ${\mathbb{C}}$ | ${\mathbb{H}}$ | ${\mathbb{H}}$ | ${\mathbb{H}}$ | ${\mathbb{C}}$ | ${\mathbb{R}}$ | ${\mathbb{R}}$
Figure 1: Table of signs for real spectral triples.

The properties of $J$ mirror those for a real structure in a Clifford module
and therefore arise also where the fermions are spinors of signature $(p,q)$,
with $s=q-p\mod 8$ [14].

There are some further axioms that relate the algebra to the Dirac operator,
chirality operator and the real structure. However, these axioms are not
needed in this paper, except to note that the Dirac operator is constrained by
the algebra action. There are no such constraints if one takes
$\mathcal{A}={\mathbb{C}}$. Note that since the Hilbert space has a positive
definite inner product, any symmetries are unitary and the spectral triples
are therefore an abstraction of Euclidean signature physics.

##  3 Complex fermion integral

The simplest fermionic action is

$\langle\psi,D\psi\rangle$ (1)

using the Hermitian inner product from $\mathcal{H}$. The fermion integral is

$Z=\int_{\psi\in\mathcal{H}}\int_{\overline{\psi}\in\mathcal{H}}e^{i\langle\psi,D\psi\rangle}\;{\mathrm{d}}\overline{\psi}\,{\mathrm{d}}\psi=\det
iD.$ (2)

which is the standard result, though it is worth writing the detail for later
use. It is called a complex integral because the field and its conjugate are
integrated independently.

The integral is defined by picking an orthonormal basis $e_{j}$ of
$\mathcal{H}$, writing $\psi=\sum_{j}\psi_{j}e_{j}$. The $\psi_{j}$ are the
component functions with respect to the basis and are assumed to be
anticommuting variables. Then evaluating

$Z=\int e^{i\sum_{jk}\langle
e_{j},De_{k}\rangle\overline{\psi}_{j}\psi_{k}}\;{\mathrm{d}}\overline{\psi}_{1}\,{\mathrm{d}}\psi_{1}\,{\mathrm{d}}\overline{\psi}_{2}\,{\mathrm{d}}\psi_{2}\ldots=\det[\langle
e_{j},iDe_{k}\rangle].$ (3)

This gives the determinant of the matrix on the right-hand side, which is the
determinant of the operator $iD$. In particular, it is independent of the
orthonormal basis chosen.

If the spectrum is symmetric about zero, as it is for $s=0,1,2,4,5\text{ or
}6$, then this determinant is the product of terms $(i\lambda)(-i\lambda)$,
and possibly zeroes, and so is a non-negative real number.

##  4 Real fermion integral

Define a new inner product on $\mathcal{H}$

$[\psi,\phi]=\langle J\psi,\phi\rangle.$ (4)

This is bilinear, and a calculation with commuting $\psi$ and $\phi$ shows
that $[\psi,\phi]=\langle J\psi,\phi\rangle=\langle
J\phi,J^{2}\psi\rangle=\epsilon[\phi,\psi]$. Including the Dirac operator
gives
$[\psi,D\phi]=\epsilon[D\phi,\psi]=\epsilon^{\prime}\epsilon[\phi,D\psi]$.

An action is defined for the anticommuting field $\psi$ as

$\frac{1}{2}[\psi,D\psi]=\frac{1}{2}\langle J\psi,D\psi\rangle$ (5)

which is only non-zero as an element of the Grassman algebra if
$\epsilon^{\prime}\epsilon=-1$, due to the anticommutation of fermions.

Since the action depends on $\psi$ but not its complex conjugate, the fermion
integration is symbolically

$F=\int_{\psi\in\mathcal{H}}e^{\frac{i}{2}[\psi,D\psi]}\;{\mathrm{d}}\psi.$
(6)

It is called a real integral because the conjugate variables are absent (like
a contour integral in complex analysis). Picking an ordered orthonormal basis
$e_{j}$, the precise definition of this integral is

$F=\int_{\psi\in\mathcal{H}}e^{\frac{i}{2}\sum_{jk}[e_{j},De_{k}]\psi_{j}\psi_{k}}\;{\mathrm{d}}\psi_{1}\,{\mathrm{d}}\psi_{2}\ldots=\operatorname{pf}(iM)$
(7)

with $M$ the matrix with components $M_{jk}=[e_{j},De_{k}]$. This is an
antisymmetric matrix if $\epsilon\epsilon^{\prime}=-1$, and so the KO-
dimension can only be $s=1,2,3,4$ if the fermion integral is non-zero.

The Pfaffian of $M$ depends on the choice of basis: in a new basis determined
by unitary matrix $U$ this is

$\operatorname{pf}(UMU^{T})=\det U\operatorname{pf}M.$ (8)

It follows that $|\operatorname{pf}M|$ is independent of the basis.

The complex conjugate of the matrix is determined by

$\overline{M_{jk}}=\overline{\langle Je_{j},De_{k}\rangle}=\langle
J^{2}e_{j},JDe_{k}\rangle=\epsilon^{\prime}[Je_{j},DJe_{k}].$ (9)

Further analysis depends on the sign $J^{2}=\epsilon$.

### Case $\epsilon=1$.

The only non-trivial case this applies to is $s=1$. The basis can be chosen to
be a set of real vectors $Je_{k}=e_{k}$. Since $\epsilon^{\prime}=-1$, (9)
shows that $M_{jk}$ is imaginary, and so $\operatorname{pf}iM$ is a real
number. A change of basis matrix $U$ is real, and so $\det U=\pm 1$. Thus the
sign of the Pfaffian depends on the ordering of the basis, up to even
permutation. Another way of seeing that there is only a sign ambiguity is that
$(\operatorname{pf}M)^{2}=\det[\langle e_{j},De_{k}\rangle]=\det D$, which is
independent of the basis.

### Case $\epsilon=-1$.

In this quaternionic case $J^{2}=-1$, nontrivial for $s=2,3,4$, an orthonormal
basis can be chosen such that $e_{2k+1}=Je_{2k}$ for $k=1,\ldots,n/2$. Then
$\epsilon^{\prime}=1$ and $\overline{M}=VMV^{T}$ with $V$ the direct sum of
the $2\times 2$ blocks

$\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},$ (10)

which has determinant $1$. Thus $\operatorname{pf}M$ is real (and
$\operatorname{pf}iM$ either imaginary or real depending on whether $n/2$ is
odd or even). If also $e^{\prime}_{2k+1}=Je^{\prime}_{2k}$, the change of
basis matrix $U$ preserves the antisymmetric form $[e_{j},e_{k}]$, and so
$\det U=1$. Thus $\operatorname{pf}M$ is independent of the choice of basis.

The Pfaffian can be calculated using an orthonormal basis of eigenvectors of
$D$. Let $e_{1}$ be any eigenvector of $D$, with eigenvalue $\lambda$. Then
$e_{2}=Je_{1}$ is a linearly independent eigenvector with the same eigenvalue
and so $M_{12}=-M_{21}=\lambda$, which is the contribution to the Pfaffian.
Repeating this with further eigenvectors shows that
$(\operatorname{pf}M)^{2}=\det(D)$.

In the cases $s=2$ or $4$, the chirality operator $\Gamma$ determines further
eigenvectors $e_{3}=\Gamma e_{1}$ and $e_{4}=Je_{3}$, both with eigenvalue
$-\lambda$. Thus (assuming all eigenvalues are non-zero) $n/2$ is even and
$\operatorname{pf}iM$ is a product of terms $(i\lambda)(-i\lambda)$, and is
therefore positive. In fact, $\operatorname{pf}iM=\sqrt{\det D}$.

If $s=3$, the chirality operator is trivial. A $2\times 2$ example shows that
$\operatorname{pf}M$ can have either sign, and so this case has a sign
ambiguity in general.

##  5 Chiral fermions

For $s$ even, a chiral model is achieved by splitting
$\mathcal{H}=\mathcal{H}_{+}\oplus\mathcal{H}_{-}$ according to the
eigenvalues of the chirality operator $\Gamma$ and using only vectors in
$\mathcal{H}_{+}$, or only vectors in $\mathcal{H}_{-}$.

For action (1) this does not work because $D$ maps $\mathcal{H}_{+}$ to
$\mathcal{H}_{-}$. Putting $P_{\pm}=\frac{1}{2}(\Gamma\pm 1)$, this problem is
explicitly that $\langle P_{+}\psi,DP_{+}\psi\rangle=0$.

The chirality condition can be imposed in the action (5) for the real integral
if the KO-dimension $s=2$. This is because $[P_{+}\psi,DP_{+}\psi]$ is non-
zero in this case, due to the fact that $\epsilon^{\prime\prime}=-1$. (The
only other case satisfying this condition is $s=6$, but the action for this
case is zero.)

The chiral fermionic integrals are

$F_{+}=\int_{\psi\in\mathcal{H}_{+}}e^{\frac{i}{2}[\psi,D\psi]}\;{\mathrm{d}}\psi=\operatorname{pf}iM_{+}$
(11)

and

$F_{-}=\int_{\psi\in\mathcal{H}_{-}}e^{\frac{i}{2}[\psi,D\psi]}\;{\mathrm{d}}\psi=\operatorname{pf}iM_{-}$
(12)

with the matrix $M_{\pm}$ the restriction of $M$ to a basis of
$\mathcal{H}_{\pm}$. Once again, this depends on the choice of basis and the
ambiguity is a phase factor $\det U_{+}$ determined by the analogue of (8).

The phase ambiguity can be eliminated by considering an integral that has two
chiral fields. The (non-chiral) fermion integral (6) for $s=2$ can be
considered a product of two chiral integrals

$F=F_{+}F_{-}$ (13)

using a basis of vectors each having definite chirality. The corresponding
phase factors $\det U_{+}$ and $\det U_{-}$ cancel for the left- and right-
handed bases because they are related with $J$. Explicitly,

$\psi=\begin{pmatrix}\psi_{L}\\\
\psi_{R}\end{pmatrix},\quad\Gamma\psi=\begin{pmatrix}\psi_{L}\\\
-\psi_{R}\end{pmatrix},\quad J\psi=\begin{pmatrix}J_{R}\psi_{R}\\\
J_{L}\psi_{L}\end{pmatrix},\quad D\psi=\begin{pmatrix}D_{R}\psi_{R}\\\
D_{L}\psi_{L}\end{pmatrix}$ (14)

and the action is

$\frac{1}{2}[\psi,D\psi]=\frac{1}{2}[\psi_{L},D_{L}\psi_{L}]+\frac{1}{2}[\psi_{R},D_{R}\psi_{R}]$
(15)

giving the splitting (13) on integration.

A different example is obtained by starting with a real spectral triple
$(\mathcal{A},\mathcal{H},D,J,\Gamma)$ for $s=4$. The Hilbert space is
doubled, setting $\mathcal{H}^{\prime}=\mathcal{H}\oplus\mathcal{H}$ and

$\Gamma^{\prime}=\begin{pmatrix}\Gamma&0\\\ 0&-\Gamma\end{pmatrix},\quad\quad
D^{\prime}=\begin{pmatrix}D&\overline{\mu}\Gamma\\\
\mu\,\Gamma&D\end{pmatrix},\quad\quad J^{\prime}=\begin{pmatrix}0&J\\\
J&0\end{pmatrix}$ (16)

with a constant $\mu\in{\mathbb{C}}$. One can check this gives a spectral
triple
$(\mathcal{A},\mathcal{H}^{\prime},D^{\prime},J^{\prime},\Gamma^{\prime})$ of
KO-dimension $s=2$. The chiral condition for this spectral triple is the
restriction to
$\psi=(\chi_{L},\phi_{R})\in\mathcal{H}^{\prime}_{+}\cong\mathcal{H}_{+}\oplus\mathcal{H}_{-}$,
giving the chiral action

$\frac{1}{2}[\psi,D^{\prime}\psi]^{\prime}=[\phi_{R},D\chi_{L}]-\frac{\overline{\mu}}{2}[\phi_{R},\phi_{R}]+\frac{\mu}{2}[\chi_{L},\chi_{L}]$
(17)

where the right-hand side is the expansion into two chiral fermions in the
original $s=4$ Hilbert space $\mathcal{H}$. There is no phase or sign
ambiguity for this chiral fermion functional integral because both
$\mathcal{H}_{+}$ and $\mathcal{H}_{-}$ have bases that are determined by $J$
up to unitary matrices $U_{+}$ and $U_{-}$ which satisfy $\det U_{+}=1$ and
$\det U_{-}=1$.

##  6 Discussion

Non-trivial fermion integrals exist for all $s$ in the complex case without
any ambiguity, but in the case of the real fermion functional integral, only
for $s=1$ or $3$ with a sign ambiguity, or $s=2$ or $4$ with no ambiguity. In
the chiral case, only $s=2$ is possible, and there is in general a phase
ambiguity. There are examples of the chiral construction where the ambiguity
disappears if the Hilbert space is constructed from two chiral fermions.

In a sense, the ambiguities are not fatal because one can just pick a basis
and obtain a definite value for the integral, despite the fact that the
problem is not uniquely specified in an abstract mathematical sense. However,
the issue may resurface when considering ensembles, because all of the
possible signs or phases are liable to be equally likely and give cancelling
contributions to a partition function when summing over Dirac operators.

Thus the models with no ambiguity are the most important ones. Although the
spectral triples considered here are all finite, the results that can be
formulated in terms of determinants lead to a definition in the infinite-
dimensional case, using regularised determinants. The regularisation can even
be with finite-dimensional approximations [15].

The physical interpretation of the KO dimension is subtle. In the simplest
cases, $s$ is determined by a Clifford module and so could be the dimension of
an analogous differentiable manifold. However, this is far from always the
case, for example, the simplest spectral triple for the fuzzy 2-sphere [2] has
$s=3$. In fact, in general one can tensor the spectral triple for a manifold
of dimension $d$ with an ‘internal space’ of KO dimension $s$ to get an
overall spectral triple with KO dimension $s^{\prime}=d+s\mod 8$. This
construction was used for the standard model of particle physics [16] with an
internal space of KO dimension 6 and $d=4$ to give the chiral theory with
$s^{\prime}=2$. This construction is simplified and abstracted here to give
the example (17). Physically, the parameter $\mu$ is a Majorana mass for the
left-handed fermion $\chi_{L}$, and $\phi_{R}$ is its conjugate field. This is
the Euclidean analogue of the Dirac conjugate of a chiral fermion in
Lorentzian field theory [17].

## References

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