# A Fragile multi-CPR Game††thanks: Research was supported by the Hellenic
Foundation for Research and Innovation (H.F.R.I.) under the ?First Call for
H.F.R.I. Research Projects to support Faculty members and Researchers and the
procurement of high-cost research equipment grant? (Project Number: HFRI-
FM17-2436).

Christos Pelekis School of Electrical and Computer Engineering, National
Technical University of Athens, Zografou, Greece, 15780, e-mail:
<EMAIL_ADDRESS>Panagiotis Promponas School of Electrical and Computer
Engineering, National Technical University of Athens, Zografou, Greece, 15780,
e-mail<EMAIL_ADDRESS>Juan Alvarado KU Leuven, Department of
Computer Sciences, Celestijnenlaan 200A, 3001, Belgium, e-mail:
<EMAIL_ADDRESS>Eirini Eleni Tsiropoulou Department of Electrical
and Computer Engineering, University of New Mexico, New Mexico, USA, 87131,
e-mail<EMAIL_ADDRESS>Symeon Papavassiliou School of Electrical and Computer
Engineering, National Technical University of Athens, Zografou, Greece, 15780,
e-mail<EMAIL_ADDRESS>
###### Abstract

A Fragile CPR Game is an instance of a resource sharing game where a common-
pool resource, which is prone to failure due to overuse, is shared among
several players. Each player has a fixed initial endowment and is faced with
the task of investing in the common-pool resource without forcing it to fail.
The return from the common-pool resource is subject to uncertainty and is
perceived by the players in a prospect-theoretic manner. It is shown in Hota
et al. [13] that, under some mild assumptions, a Fragile CPR Game admits a
unique Nash equilibrium. In this article we investigate an extended version of
a Fragile CPR Game, in which players are allowed to share multiple common-pool
resources that are also prone to failure due to overuse. We refer to this game
as a Fragile multi-CPR Game. Our main result states that, under some mild
assumptions, a Fragile multi-CPR Game admits a Generalized Nash equilibrium.
Moreover, we show that, when there are more players than common-pool
resources, the set consisting of all Generalized Nash equilibria of a Fragile
multi-CPR Game is of Lebesgue measure zero.

_Keywords and phrases_ : CPR games; prospect theory; Generalized Nash
equilibrium

_MSC(2010)_ : 91A06; 90C25

##  1 Prologue, related work and main results

In this article we shall be concerned with a _resource sharing game_. Such
games model instances in which a common-pool resource (henceforth CPR), which
is prone to failure due to overuse, is shared among several users who are
addressing the problem of choosing how much to exploit from / invest in the
CPR without forcing it to fail. Resource sharing games arise in a variety of
problems ranging from economics to computer science. Examples of CPRs include
arable lands, forests, fisheries, groundwater basins, spectrum and computing
resources, the atmosphere, among many others. Such CPRs are, on the one hand,
usually regenerative but, on the other hand, subject to failure when several
agents exploit the resource in an unsustainable manner. Each agent exploits /
invests in the CPR in order to obtain an individual benefit. However, it has
been observed that actions which are individually rational (e.g. Nash
equilibria) may result in outcomes that are collectively irrational, thus
giving rise to a particular social dilemma known as “the tragedy of the
commons” (see [12]). It is thus of interest to investigate equilibrium points
of resource sharing games, in order to better understand situations where such
a social dilemma arises. This is a topic that has drawn considerable
attention, both from a theoretical and a practical perspective. We refer the
reader to [1, 5, 13, 14, 17, 21, 22, 25, 26, 27] for applications, variations,
and for further references on resource sharing games. Let us remark that most
results in the literature appear to focus on games in which players invest on
a single CPR. In this article we investigate a resource sharing game in which
players are allowed to invest in several CPRs, whose performances are mutually
independent. To the best of our knowledge our work appears to be among the
first to consider resource sharing games on more than one CPR.

We shall be interested in a multi-version of a particular resource sharing
game, which is referred to as a _Fragile CPR Game_. It is initially introduced
in [13] and is played by several players, each of whom has a fixed initial
endowment and must decide how much to invest in the CPR without forcing it to
fail. The return from the CPR is subject to uncertainty, and is perceived by
the players in a prospect-theoretic manner. It is shown in [13] that a Fragile
CPR Game admits a unique Nash equilibrium. In this article we focus on an
extended version of a Fragile CPR Game in which players are allowed to share
multiple CPRs. We refer to the corresponding game as a _Fragile multi-CPR
Game_ and investigate its _Generalized Nash equilibria_. Our main result
states that the set consisting of all Generalized Nash equilibria of a Fragile
multi-CPR Game is non-empty and, when there are more players than CPRs,
“small” in a measure-theoretic sense. In the next subsection we introduce the
Fragile CPR Game and state the main result from [13]. We then proceed, in
Subsection 1.2, with defining the Fragile multi-CPR Game, which is the main
target of this work, and stating our main results.

###  1.1 Fragile CPR game

Throughout the text, given a positive integer $n$, we denote by $[n]$ the set
$\\{1,\ldots,n\\}$. In this article we extend a particular resource sharing
game to the case where the players are allowed to share multiple resources, by
determining how to distribute/invest their initial fixed endowment in the
available CPRs. The resource sharing game under consideration is referred to
as a _Fragile CPR Game_ , and may be seen as a prospect-theoretic version of
the _Standard CPR Game_ , introduced in [22, p. 109].

The Fragile CPR Game is introduced in [13], and is played by $n$ players, who
are assumed to be indexed by the set $[n]$. It is also assumed that there is a
single CPR, and each player has to decide how much to invest in the CPR. Each
player has an available endowment, which, without loss of generality, is
assumed to be equal to $1$. Every player, say $i\in[n]$, invests an amount
$x_{i}\in[0,1]$ in the CPR. The total investment of all players in the CPR is
denoted $\mathbf{x}_{T}=\sum_{i\in[n]}x_{i}$. The return from the CPR is
subject to uncertainty, that is there is a probability $p(\mathbf{x}_{T})$
that the CPR will fail, and this probability depends on the total investment
of the players in the CPR. In case the CPR fails, the players lose their
investment in the CPR. In case the CPR does not fail, then there is a _rate of
return_ from the CPR which depends on the total investment of all players, and
is denoted by $\mathcal{R}(\mathbf{x}_{T})$. The rate of return is assumed to
satisfy $\mathcal{R}(\mathbf{x}_{T})>1$, for all $\mathbf{x}_{T}$.

In other words, player $i\in[n]$ gains
$x_{i}\cdot\mathcal{R}(\mathbf{x}_{T})-x_{i}$ with probability
$1-p(\mathbf{x}_{T})$, and gains $-x_{i}$ with probability
$p(\mathbf{x}_{T})$. The situation is modelled through a prospect-theoretic
perspective, in the spirit of [16]. More precisely, let
$x^{(i)}=\sum_{j\in[n]\setminus\\{i\\}}x_{j}$; hence it holds
$x_{i}+x^{(i)}=\mathbf{x}_{T}$. Then the utility of player $i\in[n]$ is given
by the following utility function:

$\mathcal{V}_{i}(x_{i},x^{(i)})=\begin{cases}(x_{i}\cdot(\mathcal{R}(\mathbf{x}_{T})-1))^{a_{i}},&\text{
with probability }1-p(\mathbf{x}_{T}),\\\ -k_{i}x_{i}^{a_{i}},&\text{ with
probability }p(\mathbf{x}_{T}).\end{cases}$ (1)

The parameters $k_{i}$ and $a_{i}$ are fixed and player-specific. Let us note
that the parameter $k_{i}$ may be thought of as capturing the “behaviour” of
each player. More precisely, when $k_{i}>1$ then a player weighs losses more
than gains, a behaviour which is referred to as “loss averse”. On the other
hand, when $k_{i}\in[0,1]$ then a player weighs gains more than losses, a
behaviour which is referred to as “gain seeking”. Capturing behaviours of this
type among players constitutes a central aspect of prospect theory (see, for
example, [28]). Notice that when $k_{i}=1$ and $a_{i}=1$ then player $i\in[n]$
is _risk neutral_.

Each player of the Fragile CPR game is an expected utility maximizer, and
therefore chooses $x_{i}\in[0,1]$ that maximizes the expectation of
$\mathcal{V}(x_{i},x^{(i)})$, i.e, that maximizes the _utility_ of player
$i\in[n]$ which is given by

$\mathbb{E}\left(\mathcal{V}_{i}(x_{i},x^{(i)})\right)=x_{i}^{a_{i}}\cdot\mathcal{F}_{i}(\mathbf{x}_{T})\,,$

where

$\mathcal{F}_{i}(\mathbf{x}_{T})=(\mathcal{R}(\mathbf{x}_{T})-1)^{a_{i}}\cdot(1-p(\mathbf{x}_{T}))-k_{i}\cdot
p(\mathbf{x}_{T})$ (2)

is the  _effective rate of return_ to payer $i\in[n]$.

The main result in [13] establishes, among other things, the existence of a
unique Nash equilibrium for the Fragile CPR game, provided the following hold
true.

######  Assumption 1.

Consider a Fragile CPR game that satisfies the following properties.

  1. 1.

It holds $p(0)=0$ and $p(\mathbf{x}_{T})=1$, whenever $\mathbf{x}_{T}\geq 1$.

  2. 2.

$a_{i}\in(0,1]$ and $k_{i}>0$, for all $i\in[n]$.

  3. 3.

For all $i\in[n]$ and all $\mathbf{x}_{T}\in(0,1)$ it holds
$\frac{\partial}{\partial\mathbf{x}_{T}}\mathcal{F}_{i}(\mathbf{x}_{T}),\frac{\partial^{2}}{\partial\mathbf{x}_{T}^{2}}\mathcal{F}_{i}(\mathbf{x}_{T})<0$,
where $\mathcal{F}_{i}$ is given by (2).

In other words, the third condition in Assumption 1 states that the effective
rate of return of all players is a strictly decreasing and concave function.
An example of an effective rate of return $\mathcal{F}_{i}$ satisfying the
conditions of Assumption 1 is obtained by choosing $a_{i}<1/2$,
$p(\mathbf{x}_{T})=\mathbf{x}_{T}^{2}$, and
$\mathcal{R}(\mathbf{x}_{T})=2-e^{\mathbf{x}_{T}-1}$, as can be easily
verified.

Before proceeding with the main result from [13], let us recall here the
notion of Nash equilibrium, adjusted to the setting of the Fragile CPR Game.

######  Definition 1.

(Nash Equilibrium) A _Nash equilibrium_ for a Fragile CPR Game is a strategy
profile $(x_{1},\ldots,x_{n})\in[0,1]^{n}$ such that for all $i\in[n]$ it
holds:

$\mathbb{E}\left(\mathcal{V}(x_{i},x^{(i)})\right)\geq\mathbb{E}\left(\mathcal{V}(z_{i},x^{(i)})\right)\,,\text{
for all }\,z_{i}\in[0,1]\,.$

In other words, $(x_{1},\ldots,x_{n})\in[0,1]^{n}$ is a Nash equilibrium for a
Fragile CPR Game if no player can increase her utility by unilaterally
changing strategy. The main result in Hota et al. [13] reads as follows.

######  Theorem 1 ([13]).

Consider a Fragile CPR Game that satisfies Assumption 1. Then the game admits
a _unique_ Nash equilibrium.

We now proceed with defining the _Fragile multi–CPR Game_ , whose equilibria
are the main target of the present article.

###  1.2 Fragile multi-CPR game

In this article we introduce and investigate a multi-version of the Fragile
CPR game. In order to be more precise, we need some extra piece of notation.
If $m$ is a positive integer, let $C_{m}$ denote the set:

$C_{m}=\left\\{(x_{1},\ldots,x_{m})\in[0,1]^{m}:\sum_{i\in[m]}x_{i}\leq
1\right\\}\,.$ (3)

Moreover, let $\mathcal{C}_{n}$ denote the Cartesian product
$\prod_{i\in[n]}C_{m}$ and let
$\mathcal{C}_{-i}=\prod_{[n]\setminus\\{i\\}}C_{m}$ denote the Cartesian
product obtained from $\mathcal{C}_{n}$ by deleting its $i$-th component.
Elements in $\mathcal{C}_{-i}$ are denoted by $\mathbf{x}_{-i}$, as is
customary, and an element
$\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathcal{C}_{n}$ is
occasionally written $\mathbf{x}=(\mathbf{x}_{i},\mathbf{x}_{-i})$, for
$i\in[n]$, $\mathbf{x}_{i}\in C_{m}$ and $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$.
We now proceed with defining the _Fragile multi-CPR Game_.

Suppose that there are $n$ players, indexed by the set $[n]$, each having an
_initial endowment_ equal to $1$. Assume further that there are $m$ available
CPRs, where $m\geq 1$ is an integer. Every player has to decide how much to
invest in each CPR. More precisely, every player, say $i\in[n]$, chooses an
element $\mathbf{x}_{i}=(x_{i1},\ldots,x_{im})\in C_{m}$ and invests $x_{ij}$
in the $j$-th CPR. Given strategies $\mathbf{x}_{i}=(x_{i1},\ldots,x_{im})\in
C_{m},i\in[n]$, of the players and an integer $j\in[m]$, set

$\mathbf{x}_{T}^{(j)}=\sum_{i\in[n]}x_{ij}\quad\text{ and
}\quad\mathbf{x}_{T}^{j|i}=\sum_{\ell\in[n]\setminus\\{i\\}}x_{\ell j}\,.$ (4)

Hence it holds $\mathbf{x}_{T}^{(j)}=x_{ij}+\mathbf{x}_{T}^{j|i}$, for all
$i\in[n]$. In other words, $\mathbf{x}_{T}^{(j)}$ equals the total investment
of the players in the $j$-th CPR and $\mathbf{x}_{T}^{j|i}$ equals the total
investment of all players except player $i$ in the $j$-th CPR. As in the case
of the Fragile CPR Game, we assume that the performance of each CPR is subject
to uncertainty, and that each CPR has a corresponding rate of return, both
depending on the total investment of the players in each CPR. More precisely,
for $j\in[m]$, let $\mathcal{R}_{j}(\mathbf{x}_{T}^{(j)})$ denote the _return
rate_ of the $j$-th CPR and let $p_{j}(\mathbf{x}_{T}^{(j)})$ denote the
_probability that the $j$-th CPR fails_. We assume that
$\mathcal{R}_{j}(\mathbf{x}_{T}^{(j)})>1$ holds true, for all
$\mathbf{x}_{T}^{(j)}$.

The _utility_ of player $i\in[n]$ from the $j$-th CPR is given, as in the case
of the Fragile CPR game, via the following prospect-theoretic utility
function:

$\mathcal{V}_{ij}(x_{ij},\mathbf{x}_{T}^{j|i})=\begin{cases}(x_{ij}\cdot(\mathcal{R}_{j}(\mathbf{x}_{T}^{(j)})-1))^{a_{i}},&\text{
with probability }1-p_{j}(\mathbf{x}_{T}^{(j)}),\\\
-k_{i}x_{ij}^{a_{i}},&\text{ with probability
}p_{j}(\mathbf{x}_{T}^{(j)}).\end{cases}$ (5)

We assume that the performance of each CPR is _independent_ of the
performances of all remaining CPRs. Players in the Fragile multi-CPR Game are
expected utility maximizers. If player $i\in[n]$ plays the vector
$\mathbf{x}_{i}=(x_{i1},\ldots,x_{im})\in C_{m}$, and the rest of the players
play $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$ then her expected utility from the
$j$-th CPR is equal to

$\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j|i}):=\mathbb{E}\left(\mathcal{V}_{ij}(x_{ij},\mathbf{x}_{T}^{j|i})\right)=x_{ij}^{a_{i}}\cdot\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})\,,$
(6)

where

$\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)}):=(\mathcal{R}_{j}(\mathbf{x}_{T}^{(j)})-1)^{a_{i}}(1-p_{j}(\mathbf{x}_{T}^{(j)}))-k_{i}p_{j}(\mathbf{x}_{T}^{(j)})$
(7)

is the _effective rate of return_ to the $i$-th player from the $j$-th CPR.
Notice that, since we assume that the performance of each CPR is independent
of the performances of the remaining CPRs, $\mathcal{E}_{ij}$ depends only on
the values of $x_{ij},\mathbf{x}_{T}^{j|i}$ and does not depend on the values
of $x_{ik},\mathbf{x}_{T}^{k|i}$, for $k\neq j$. In other words, the (total)
prospect-theoretic _utility_ of player $i\in[n]$ in the Fragile multi-CPR Game
is given by:

$\mathcal{V}_{i}(\mathbf{x}_{i};\mathbf{x}_{-i})=\sum_{j\in[m]}\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})\,.$
(8)

In this article we establish the existence of a Generalized Nash equilibrium
for the Fragile multi-CPR game, provided the following holds true.

######  Assumption 2.

Consider a Fragile multi-CPR Game that satisfies the following properties:

  1. 1.

For every $j\in[m]$ it holds $p_{j}(0)=0$ and $p_{j}(\mathbf{x}_{T}^{(j)})=1$,
whenever $\mathbf{x}_{T}^{(j)}\geq 1$.

  2. 2.

It holds $a_{i}\in(0,1]$ and $k_{i}>0$, for all $i\in[n]$.

  3. 3.

For all $i\in[n]$ and all $j\in[m]$ it holds
$\frac{\partial}{\partial\mathbf{x}_{T}^{(j)}}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)}),\frac{\partial^{2}}{\partial(\mathbf{x}_{T}^{(j)})^{2}}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})<0$,
where $\mathcal{F}_{ij}$ is given by (7).

Notice that, similarly to the Fragile CPR Game, the third condition in
Assumption 2 states that the effective rate of return of every player from any
CPR is a _strictly decreasing and concave_ function. An example of an
effective rate of return satisfying Assumption 2 is obtained by choosing, for
$j\in[m]$, the return rate of the $j$-th CPR to be equal to
$\mathcal{R}_{j}(\mathbf{x}_{T}^{(j)})=c_{j}+1$, where $c_{j}>0$ is a
constant, and the probability that the $j$-th CPR fails to be a strictly
increasing and convex, on the interval $[0,1]$, function such that
$p_{j}(\mathbf{x}_{T}^{(j)})=1$, when $\mathbf{x}_{T}^{(j)}\geq 1$.

Before stating our main result, let us proceed with recalling the notion of
Generalized Nash equilibrium (see [11]).

Consider the, above-mentioned, Fragile multi-CPR Game, denoted $G$. Assume
further that, for each player $i\in[n]$, there exists a correspondence
$\vartheta_{i}:\mathcal{C}_{-i}\to 2^{C_{m}}$ mapping every element
$\mathbf{x}_{-i}\in\mathcal{C}_{-i}$ to a set
$\vartheta_{i}(\mathbf{x}_{-i})\subset C_{m}$. The set-valued correspondence
$\vartheta_{i}$ is referred to as a _constraint policy_ and may be thought of
as determining the set of strategies that are feasible for player $i\in[n]$,
given $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$. We refer to the tuple
$(G,\\{\vartheta_{i}\\}_{i\in[n]})$ as the _Constrained Fragile multi-CPR
Game_ with constraint policies $\\{\vartheta_{i}\\}_{i\in[n]}$. Corresponding
to a constrained game is the following notion of _Constrained Nash
equilibrium_ (or _Generalized Nash equilibrium_):

######  Definition 2 (GNE).

A _Generalized Nash equilibrium_ for a Constrained Fragile multi-CPR Game
$(G,\\{\vartheta_{i}\\}_{i\in[n]})$ is a strategy profile
$\mathbf{x}^{\ast}=(\mathbf{x}_{1}^{\ast},\ldots,\mathbf{x}_{n}^{\ast})\in\mathcal{C}_{n}$
such that

  1. 1.

For all $i\in[n]$, it holds
$\mathbf{x}_{i}^{\ast}\in\vartheta_{i}(\mathbf{x}_{-i}^{\ast})$, for all
$i\in[n]$, and

  2. 2.

For all $i\in[n]$, it holds
$\mathcal{V}_{i}(\mathbf{x}_{i}^{\ast};\mathbf{x}_{-i}^{\ast})\geq\mathcal{V}_{i}(\mathbf{x}_{i};\mathbf{x}_{-i}^{\ast})$,
for all $\mathbf{x}_{i}\in\vartheta_{i}(\mathbf{x}_{-i}^{\ast})$, where
$\mathcal{V}_{i}(\,\cdot\,;\,\cdot\,)$ is the utility function of the $i$-th
player in a Fragile multi-CPR Game, given in (8).

In other words,
$\mathbf{x}^{\ast}=(\mathbf{x}_{1}^{\ast},\ldots,\mathbf{x}_{n}^{\ast})\in\mathcal{C}_{n}$
is a GNE if no player can increase her utility by unilaterally changing her
strategy to any other element of the set
$\vartheta_{i}(\mathbf{x}_{-i}^{\ast})$. We may now proceed with stating our
main results.

######  Theorem 2.

Consider a Fragile multi-CPR game, $G$, with $n\geq 1$ players and $m\geq 1$
CPRs, which satisfies Assumption 2. Then there exist constraint policies
$\\{\vartheta_{i}\\}_{i\in[n]}$ such that the Constraint Fragile multi-CPR
Game $(G,\\{\vartheta_{i}\\}_{i\in[n]})$ admits a Generalized Nash
equilibrium.

Given Theorem 2, it is natural to ask about the “size” of the set consisting
of all GNEs of a Fragile multi-CPR Game. Let us note that it is a well known
fact that Generalized Nash equilibrium problems tend to possess infinitely
many GNEs (see [11, p. 192]). In the case of a single CPR, i.e., when $m=1$,
the corresponding Constrained Fragile CPR Game admits a unique GNE.

######  Theorem 3.

Consider a Fragile multi-CPR Game with $n\geq 1$ players and $m=1$ CPR
satisfying Assumption 2. Then the game admits a unique GNE.

The proof of Theorem 3 is based upon a “first order condition” which is
satisfied by the best response correspondence in a Fragile multi-CPR Game. It
turns out that the aforementioned “first order condition” gives rise to _two
types_ of best responses for the players (see Theorem 8 below). In fact, we
show that Theorem 3 is a consequence of a more general statement (i.e.,
Theorem 9 below) which provides an upper bound on the numbers of GNEs in a
Fragile multi-CPR Game, subject to the assumption that best response of every
player is of the first type.

For general $m$ we are unable to determine the exact “size” of the set of
GNEs. We conjecture its size is always finite. Our main result, which is valid
when there are more players than CPRs, states that the set of GNEs is small in
a measure-theoretic sense.

######  Theorem 4.

Consider a Fragile multi-CPR game, $G^{(2)}$, with $n\geq 1$ players and
$m\geq 1$ CPRs, which satisfies Assumption 2. Assume further that $m\leq n$,
and let $\mathcal{N}(G^{(2)})$ be the set consisting of all Generalized Nash
equilibria of $G^{(2)}$. Then the $(n\cdot m)$-dimensional Lebesgue measure of
$\mathcal{N}(G^{(2)})$ is equal to zero.

As mentioned already, and despite the fact that GNE problems tend to possess
infinitely many solutions, we speculate that the “size” of the set
$\mathcal{N}(G^{(2)})$ in Theorem 4 can be reduced significantly.

######  Conjecture 1.

The set $\mathcal{N}(G^{(2)})$ is finite.

###  1.3 Brief outline of the proofs of main results

The proofs of our main results are inspired from the proof of Theorem 1, given
in [13]. Having said that, it should also be mentioned that in a Fragile
multi-CPR Game certain additional technicalities arise that are substantially
different from those addressed in the proof of Theorem 1 in [13]. First and
foremost, in a Fragile multi-CPR Game the strategy space of each player
consists of $m$-dimensional vectors, a setting which requires concepts and
ideas from multi-variable calculus.

In [13] the existence of a Nash equilibrium in a Fragile CPR Game is
established in two ways: the first approach employs Brouwer’s fixed point
theorem, and the second approach employs ideas from a particular class of
games known as _Weak Strategic Substitute Games_ (see [7]). The first approach
requires, among other things, the best response correspondence to be single-
valued. The second approach requires the best-response correspondence to be
decreasing. Both requirements may fail to hold true in a Fragile multi-CPR
Game. Instead, we establish the existence of a Generalized Nash equilibrium
for the Fragile multi-CPR Game by showing that it belongs to a particular
class of “convex constrained games” which are known to possess Generalized
Nash equilibria.

In [13] the uniqueness of the Nash equilibrium for a Fragile CPR Game is
established by showing that a particular auxiliary function, corresponding to
the fact that the best response correspondence satisfies a particular “first
order condition” (see [13, Eq. (6), p. 142] for the precise formulation of the
condition), is decreasing. Similar auxiliary functions are employed in the
proofs of Theorems 3 and 4. However, the corresponding “first order
conditions” are more delicate to characterise, and we do so by employing the
KKT conditions to the optimization program corresponding to the best response
correspondence (i.e., Problem (17) below). This allows to describe the best
responses via a system of equations, having unique solution, and results in
two types of “first order conditions” (see Theorem 8 below). Having
established the first order conditions in a Fragile multi-CPR Game, we
complete the proofs of our main results by employing monotonicity properties
of certain auxiliary functions, in a way which may be seen as an extension of
the approach taken in the proof of Theorem 1 in [13].

###  1.4 Organization

The remaining part of our article is organised as follows. In Section 2 we
show that the utility function of each player in a Fragile multi-CPR Game is
concave on a particular subset of the strategy space. In Section 3 we prove
Theorem 2, namely, we show that a Fragile multi-CPR Game admits a GNE. In
Section 4 we show that the best response of each player in a Fragile multi-CPR
Game satisfies certain “first order conditions”, which are then used, in
Section 5, in order to define suitable auxiliary functions whose monotonicity
properties play a key role in the proofs of Theorems 3 and 4. Theorem 3 is
proven in Section 6 and Theorem 4 is proven in Section 7. In Section 8 we show
that a “restricted” version of a Fragile multi-CPR Game admits finitely many
GNEs, a result which is then employed in order to formulate a conjecture which
is equivalent to Conjecture 1. Our paper ends with Section 9 which includes
some concluding remarks and conjectures.

##  2 Concavity of utility function

In this section we show that the utility function, given by (8), of each
player in a Fragile multi-CPR Games is concave in some particular subset of
$C_{m}$. Before proceeding with the details let us mention that this
particular subset will be used to define the constraint policies in the
corresponding Constrained Fragile multi-CPR Game.

We begin with the following result, which readily follows from [13, Lemma 1].
Recall the definition of $\mathbf{x}_{T}^{(j)}$ and $\mathbf{x}_{T}^{j|i}$,
given in (4), and the definition of the effective rate of return,
$\mathcal{F}_{ij}$, given in (7).

######  Lemma 1 (see [13], Lemma 1).

Let $i\in[n]$ and $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$ be fixed. Then, for
every $j\in[m]$, there exists a real number $\omega_{ij}\in(0,1)$ such that
$\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})>0$, whenever
$\mathbf{x}_{T}^{(j)}\in(0,\omega_{ij})$, and
$\mathcal{F}_{ij}(\omega_{ij})=0$. Furthermore, provided that
$\mathbf{x}_{T}^{j|i}<\omega_{ij}$, the function
$\mathcal{E}_{ij}(\,\cdot\,;\mathbf{x}_{T}^{j|i})$ is concave in the interval
$(0,\omega_{ij}-\mathbf{x}_{T}^{j|i})$.

###### Proof.

We repeat the proof for the sake of completeness. Notice that
$\mathcal{F}_{ij}(0)>0$. Moreover, Assumption 2 implies that
$\mathcal{F}_{ij}(1)<0$. Since $\mathcal{F}_{ij}$ is continuous, the
intermediate value theorem implies that there exists $\omega_{ij}\in(0,1)$
such that $\mathcal{F}_{ij}(\omega_{ij})=0$. Since $\mathcal{F}_{ij}$ is
assumed to be decreasing, the first statement follows, and we proceed with the
proof of the second statement. To this end, notice that (6) yields

$\frac{\partial^{2}}{\partial
x_{ij}^{2}}\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})=a_{i}(a_{i}-1)x_{ij}^{a_{i}-2}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})+2a_{i}x_{ij}^{a_{i}-1}\frac{\partial}{\partial
x_{ij}}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})+x_{ij}^{a_{i}}\frac{\partial^{2}}{\partial
x_{ij}^{2}}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})\,.$

Notice also that $\frac{\partial}{\partial
x_{ij}}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})=\frac{\partial}{\partial\mathbf{x}_{T}^{(j)}}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})$
as well as $\frac{\partial^{2}}{\partial
x_{ij}^{2}}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})=\frac{\partial^{2}}{\partial(\mathbf{x}_{T}^{(j)})^{2}}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})$.
Moreover, Assumption 2 guarantees that $\frac{\partial^{2}}{\partial
x_{ij}^{2}}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)}),\frac{\partial}{\partial
x_{ij}}\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})<0$ as well as that $a_{i}-1\leq
0$. Since $\mathcal{F}_{ij}(\mathbf{x}_{T}^{(j)})>0$ when
$\mathbf{x}_{T}^{(j)}\in(0,\omega_{ij})$, we conclude that
$\frac{\partial^{2}}{\partial
x_{ij}^{2}}\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})<0$ and therefore
$\mathcal{E}_{ij}(\,\cdot\,;\mathbf{x}_{T}^{j|i})$ is concave in the interval
$(0,\omega_{ij}-\mathbf{x}_{T}^{j|i})$, as desired. ∎

In other words, given the choices of all players except player $i$, the
utility of the $i$-th player from the $j$-th CPR is a concave function, when
restricted on a particular interval. The next result shows that an analogous
statement holds true for the total utility of each player in the Fragile
multi-CPR Game, namely, $\mathcal{V}_{i}(\mathbf{x}_{i};\mathbf{x}_{-i})$,
given by (8).

Given $i\in[n]$ and $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$, let

$A(\mathbf{x}_{-i}):=\\{j\in[m]:\mathbf{x}_{T}^{j|i}<\omega_{ij}\\}\,,$ (9)

where $\omega_{ij},j\in[m]$, is provided by Lemma 1. We refer to
$A(\mathbf{x}_{-i})$ as the set of _active CPRs_ corresponding to $i$ and
$\mathbf{x}_{-i}$.

######  Theorem 5.

Fix $i\in[n]$ and $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$. Let
$A(\mathbf{x}_{-i})$ be the set of active CPRs corresponding to $i$ and
$\mathbf{x}_{-i}$, and consider the set
$\mathcal{R}_{A(\mathbf{x}_{-i})}=\prod_{j\in
A(\mathbf{x}_{-i})}(0,\omega_{ij}-\mathbf{x}_{T}^{j|i})$. Then the function
$\mathcal{V}_{A(\mathbf{x}_{-i})}:=\sum_{j\in
A(\mathbf{x}_{-i})}\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j\setminus i})$ is
concave in $\mathcal{R}_{A(\mathbf{x}_{-i})}$.

###### Proof.

If $|A(\mathbf{x}_{-i})|=1$, then the result follows from Lemma 1 so we may
assume that $|A(\mathbf{x}_{-i})|\geq 2$. The set
$\mathcal{R}_{A(\mathbf{x}_{-i})}$ is clearly convex. Let $j,k\in
A(\mathbf{x}_{-i})$ be such that $j\neq k$ and notice that

$\frac{\partial^{2}\mathcal{V}_{A(\mathbf{x}_{-i})}}{\partial x_{ij}\;\partial
x_{ik}}=0\,.$ (10)

Moreover, by Lemma 1, we also have

$\frac{\partial^{2}\mathcal{V}_{A(\mathbf{x}_{-i})}}{\partial
x_{ij}^{2}}=\frac{\partial^{2}\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})}{\partial
x_{ij}^{2}}<0\,,\text{ for all
}x_{ij}\in(0,\omega_{ij}-\mathbf{x}_{T}^{j|i})\,.$ (11)

Given $\mathbf{x}\in\mathcal{R}_{A(\mathbf{x}_{-i})}$, denote by
$H(\mathbf{x})=\left(\frac{\partial^{2}\mathcal{V}_{A(\mathbf{x}_{-i})}(\mathbf{x})}{\partial
x_{ij}\;\partial x_{ik}}\right)_{j,k\in A(\mathbf{x}_{-i})}$ the Hessian
matrix of $\mathcal{V}_{A(\mathbf{x}_{-i})}$ evaluated at $\mathbf{x}$, and
let $\Delta_{k}(\mathbf{x})$, for $k\in A(\mathbf{x}_{-i})$, be the principal
minors of $H(\mathbf{x})$ (see [4, p. 111]). Notice that (10) implies that
$H(\mathbf{x})$ is a diagonal matrix. Therefore, using (11), it follows that
$(-1)^{k}\cdot\Delta_{k}(\mathbf{x})>0$, when
$\mathbf{x}\in\mathcal{R}_{A(\mathbf{x}_{-i})}$. In other words,
$H(\,\cdot\,)$ is negative definite on the convex set
$\mathcal{R}_{A(\mathbf{x}_{-i})}$ and we conclude (see [4, Theorem 3.3, p.
110]) that $\mathcal{V}_{A(\mathbf{x}_{-i})}$ is concave in
$\mathcal{R}_{A(\mathbf{x}_{-i})}$, as desired. ∎

##  3 Proof of Theorem 2: existence of GNE

In this section we show that the Fragile multi-CPR Game possesses a
Generalized Nash equilibrium. Recall that the notion of Generalized Nash
equilibrium depends upon the choice of constraint policies. Thus, before
presenting the details of the proof, we first define the constrained policies
under consideration.

Let $i\in[n]$ and
$\mathbf{x}_{-i}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{i-1},\mathbf{x}_{i+1},\ldots,\mathbf{x}_{n})\in\mathcal{C}_{-i}$,
where $\mathbf{x}_{j}=(x_{j1},\ldots,x_{jm})\in C_{m}$, for
$j\in[n]\setminus\\{i\\}$, be fixed. Recall that
$\mathbf{x}_{T}^{j|i}=\sum_{\ell\in[n]\setminus\\{i\\}}x_{\ell j}$ and
consider the set of active indices corresponding to $i$ and $\mathbf{x}_{-i}$,
i.e., consider the set $A(\mathbf{x}_{-i})$, defined in (9).

Define the constraint policy $\vartheta_{i}(\cdot)$ that maps each element
$\mathbf{x}_{-i}\in\mathcal{C}_{-i}$ to the set

$\vartheta_{i}(\mathbf{x}_{-i})=C_{m}\bigcap\left\\{\prod_{j\in
A(\mathbf{x}_{-i})}[0,\omega_{ij}-\mathbf{x}_{T}^{j|i}]\,\,\times\prod_{j\in[m]\setminus
A(\mathbf{x}_{-i})}\\{0\\}\right\\}\,,$ (12)

where $\\{\omega_{ij}\\}_{j\in A(\mathbf{x}_{-i})}$ is given by Lemma 1.
Notice that, for every $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$, the set
$\vartheta_{i}(\mathbf{x}_{-i})$ is _non-empty, compact and convex_. Fig. 1
provides a visualization of the aforementioned constraint policy, in the case
of $m=2$.

$\quad\quad\hat{\omega}_{2}$$\quad\quad\hat{\omega}_{1}$$\quad\quad\mathcal{C}_{2}$$\vartheta_{i}(\bm{x_{-i}})$$CPR_{1}$$CPR_{2}$
Figure 1: Visualization of an instance of the constraint policy
$\vartheta_{i}(\cdot)$ (blue shaded region) of player $i$ in the case of
$m=2$, where we denote $\hat{\omega}_{1}=\omega_{i1}-\mathbf{x}_{T}^{1|i}$ and
$\hat{\omega}_{2}=\omega_{i2}-\mathbf{x}_{T}^{2|i}$.

We aim to show that the Constrained Fragile multi-CPR Game, with constraint
policies given by (12), admits a Generalized Nash equilibrium. In order to do
so, we employ the following theorem. Recall (see [15, p. 32–33]) that a set-
valued correspondence $\phi:X\to 2^{Y}$ is _upper semicontinuous_ if for every
open set $G\subset Y$, it holds that $\\{x\in X:\phi(x)\subset G\\}$ is an
open set in $X$. A set-valued correspondence $\phi:X\to 2^{Y}$ is _lower
semicontinuous_ if every open set $G\subset Y$, it holds that $\\{x\in
X:\phi(x)\cap G\neq\emptyset\\}$ is an open set in $X$. Recall also that,
given $S\subset\mathbb{R}^{s}$, a function $f:S\to\mathbb{R}$ is quasi-concave
if
$f(\lambda\mathbf{x}+(1-\lambda)\mathbf{y})\geq\min\\{f(\mathbf{x}),f(\mathbf{y})\\}$,
for all $\mathbf{x}\neq\mathbf{y}$ in $S$ and $\lambda\in(0,1)$. Clearly, a
concave function is also quasi-concave.

######  Theorem 6.

Let $n$ players be characterized by strategy spaces $X_{i},i\in[n]$,
constraint policies $\phi_{i},i\in[n]$, and utility functions
$\mathcal{V}_{i}:\prod_{i}X_{i}\to\mathbb{R},i\in[n]$. Suppose further that
the following hold true for every $i\in[n]$:

  1. 1.

$X_{i}$ is non-empty, compact, convex subset of a Euclidean space.

  2. 2.

$\phi_{i}(\cdot)$ is both upper semicontinuous and lower semicontinuous in
$X_{-i}$.

  3. 3.

For all $\mathbf{x}_{-i}\in X_{-i}$, $\phi_{i}(\mathbf{x}_{-i})$ is nonempty,
closed and convex.

  4. 4.

$\mathcal{V}_{i}$ is continuous in $\prod_{i}X_{i}$.

  5. 5.

For every $\mathbf{x}_{-i}\in X_{-i}$, the map
$x_{i}\mapsto\mathcal{V}_{i}(x_{i},\mathbf{x}_{-i})$ is quasi-concave on
$\phi_{i}(\mathbf{x}_{-i})$.

Then there exists a Generalized Nash equilibrium.

###### Proof.

This is a folklore result that can be found in various places. See, for
example, [2], [11, Theorem 6], [15, Theorem 4.3.1], [3, Theorem 12.3], or [8,
Theorem 3.1]. ∎

We are now ready to establish the existence of a GNE in the Constrained
Fragile multi-CPR Game. In the following proof, $\|\cdot\|_{d}$ denotes
$d$-dimensional Euclidean distance, and
$B_{d}(\varepsilon):=\\{\mathbf{x}\in\mathbb{R}^{d}:\|\mathbf{x}\|_{d}\leq\varepsilon\\}$
is the closed ball of radius $\varepsilon$ centered at the origin. Moreover,
given $A\subset\mathbb{R}^{d}$ and $\varepsilon>0$, we denote by
$\\{A\\}_{\varepsilon}$ the set $A+B_{d}(\varepsilon):=\\{a+b:a\in A\text{ and
}b\in B_{d}(\varepsilon)\\}$ and by $(1-\varepsilon)\cdot A$ the set
$\\{(1-\varepsilon)\cdot a:a\in A\\}$.

###### Proof of Theorem 2.

We apply Theorem 6. The strategy space of each player is equal to $C_{m}$,
which is non-empty, compact and convex. Hence the first condition of Theorem 6
holds true. The third condition also holds true, by (12). Moreover, the fourth
condition of Theorem 6 is immediate from the definition of utility, given in
(8), while the fifth condition follows from Theorem 5.

It remains to show that the second condition of Theorem 6 holds true, i.e.,
that for each $i\in[n]$ the constrained policy $\vartheta_{i}(\cdot)$, given
by (12), is both upper and lower semicontinuous. Towards this end, fix
$i\in[n]$ and let $G\subset C_{m}$ be an open set. Consider the sets

$G^{+}:=\\{\mathbf{x}_{-i}\in\mathcal{C}_{-i}:\vartheta_{i}(\mathbf{x}_{-i})\subset
G\\}\quad\text{and}\quad
G^{-}:=\\{\mathbf{x}_{-i}\in\mathcal{C}_{-i}:\vartheta_{i}(\mathbf{x}_{-i})\cap
G\neq\emptyset\\}\,.$

We have to show that both $G^{+}$ and $G^{-}$ are open subsets of
$\mathcal{C}_{-i}$. We first show that $G^{+}$ is open.

If $G^{+}$ is empty then the result is clearly true, so we may assume that
$G^{+}\neq\emptyset$. Let
$\mathbf{y}=(\mathbf{y}_{1},\ldots,\mathbf{y}_{i-1},\mathbf{y}_{i+1},\ldots,\mathbf{y}_{n})\in
G^{+}$; hence $\vartheta_{i}(\mathbf{y})\subset G$. We have to show that there
exists $\varepsilon>0$ such that for every $\mathbf{x}\in\mathcal{C}_{-i}$
with $\|\mathbf{x}-\mathbf{y}\|_{(n-1)m}<\varepsilon$, we have
$\vartheta_{i}(\mathbf{x})\subset G$. Since $\vartheta_{i}(\mathbf{y})$ is a
compact subset of the open set $G$, it follows that there exists
$\varepsilon_{0}>0$ such that
$\\{\vartheta_{i}(\mathbf{y})\\}_{\varepsilon_{0}}\subset G$. Since summation
is continuous, there exists $\varepsilon_{1}>0$ such that for every
$\mathbf{x}\in\mathcal{C}_{-i}$ with
$\|\mathbf{x}-\mathbf{y}\|_{(n-1)m}<\varepsilon_{1}$ it holds
$\mathbf{x}\in\\{\vartheta_{i}(\mathbf{y})\\}_{\varepsilon_{0}}$. The desired
$\varepsilon$ is given by $\varepsilon_{1}$. Hence $G^{+}$ is an open set, and
we proceed with showing that $G^{-}$ is open as well.

We may assume that $G^{-}$ is non-empty. For each $i\in[n]$, let
$g_{i}:\mathcal{C}_{-i}\to\mathbb{R}^{m}_{\geq 0}$ be the continuous function
whose $j$-th coordinate, for $j\in[m]$, is given by

$g_{ij}(\mathbf{x}_{-i})=\begin{cases}\omega_{ij}-\mathbf{x}_{T}^{j|i}\,,&\mbox{
if }\omega_{ij}-\mathbf{x}_{T}^{j|i}>0\\\ 0\,,&\mbox{ if
}\omega_{ij}-\mathbf{x}_{T}^{j|i}\leq 0\,\,,\end{cases}$

where $\omega_{ij}$ is given by Lemma 1. Let $h:\mathbb{R}^{m}_{\geq 0}\to
2^{C_{m}}$ be the set-valued function defined by
$h(z_{1},\ldots,z_{m})=\prod_{j\in[m]}[0,z_{j}]$, with the convention
$[0,0]:=\\{0\\}$. Clearly, it holds that $\vartheta_{i}=h\circ g_{i}$, for all
$i\in[n]$.

We claim that $h$ is lower semicontinuous. If the claim holds true then it
follows that the set $H:=\\{\mathbf{z}\in\mathbb{R}^{m}_{\geq
0}:h(\mathbf{z})\cap G\neq\emptyset\\}$ is open. Notice that
$G^{-}\neq\emptyset$ implies that $H\neq\emptyset$. Since $g_{i}$ is
continuous, it follows that the preimage of $H$ under $g_{i}$, i.e.,
$g^{-1}_{i}(H)$, is open. In other words, the set
$\\{\mathbf{x}\in\mathcal{C}_{-i}:h\circ g_{i}(\mathbf{x})\cap
G\neq\emptyset\\}=\\{\mathbf{x}\in\mathcal{C}_{-i}:\vartheta_{i}(\mathbf{x})\cap
G\neq\emptyset\\}$ is open and the proof of the theorem is complete.

It remains to prove the claim, i.e., that $h$ is lower semicontinuous. To this
end, let $G\subset C_{m}$ be an open set, and let
$G^{\ast}:=\\{\mathbf{z}\in\mathbb{R}^{m}_{\geq 0}:h(\mathbf{z})\cap
G\neq\emptyset\\}$. We have to show that $G^{\ast}$ is open; that is, we have
to show that for every $\mathbf{z}\in G^{\ast}$ there exists $\varepsilon>0$
such that $\mathbf{w}\in G^{\ast}$, for all $\mathbf{w}$ with
$\|\mathbf{z}-\mathbf{w}\|_{m}<\varepsilon$. Fix $\mathbf{z}\in G^{\ast}$.
Since $h(\mathbf{z})$ is compact and $G$ is open, it follows that there exists
$\varepsilon_{0}>0$ such that $(1-\varepsilon_{0})\cdot h(\mathbf{z})\cap
G\neq\emptyset$. Now choose $\varepsilon>0$ such that for every $\mathbf{w}\in
C_{m}$ for which $\|\mathbf{z}-\mathbf{w}\|_{m}<\varepsilon$ it holds
$(1-\varepsilon_{0})\cdot h(\mathbf{z})\subset h(\mathbf{w})$. In other words,
for this particular choice of $\varepsilon>0$ it holds $h(\mathbf{w})\cap
G\neq\emptyset$, for every $\mathbf{w}$ with
$\|\mathbf{z}-\mathbf{w}\|_{m}<\varepsilon$. The claim follows. ∎

##  4 Best response correspondence

Having established the existence of a GNE for a Fragile multi-CPR Game, we now
proceed with the proofs of Theorems 3 and 4. The proofs will be obtained in
two steps. In the first step we deduce certain “first order conditions” which
are satisfied by the best response correspondence of each player in the game.
In the second step we employ the first order conditions in order to define
certain auxiliary functions, whose monotonicity will be employed in the proofs
of the aforementioned theorems. In this section we collect some results
pertaining to the first step. We begin with recalling the notion of the best
response correspondence (see [19]).

Given $i\in[n]$ and $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$, let
$\vartheta_{i}(\cdot)$ denote the constraint policy given by (12), and
consider the _best response_ of the $i$-th player in the Fragile multi-CPR
Game defined as follows:

$B_{i}(\mathbf{x}_{-i})=\arg\max_{\mathbf{x}_{i}\in\vartheta_{i}(\mathbf{x}_{-i})}\,\mathcal{V}_{i}(\mathbf{x}_{i};\mathbf{x}_{-i})\,,$
(13)

where $\mathcal{V}_{i}$ is the utility of the $i$-th player, given by (8).
Notice that $B_{i}(\cdot)$ is a correspondence $B_{i}:\mathcal{C}_{-i}\to
2^{C_{m}}$, where $2^{C_{m}}$ denotes the class consisting of all subsets of
$C_{m}$. For $j\in[m]$, we denote by $B_{ij}(\mathbf{x}_{-i})$ the $j$-th
component of $B_{i}(\mathbf{x}_{-i})$; hence we have

$B_{i}(\mathbf{x}_{-i})=(B_{i1}(\mathbf{x}_{-i}),\ldots,B_{im}(\mathbf{x}_{-i}))\,.$

######  Remark 1.

Notice that Definition 2 implies that if
$\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathcal{C}_{n}$ is a GNE
of a Constrained Fragile multi-CPR Game, with constraint policies given by
(12), then for each $i\in[n]$ it holds $\mathbf{x}_{i}\in
B_{i}(\mathbf{x}_{-i})$.

Recall that $A(\mathbf{x}_{-i})$ denotes the set of active CPRs corresponding
to $\mathbf{x}_{-i}$, defined in (9), and notice that
$B_{ij}(\mathbf{x}_{-i})=0$, for all $j\in[m]\setminus A(\mathbf{x}_{-i})$.

For $x_{ij}\in[0,1]$, let $\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})$ be the
function defined via

$\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})=x_{ij}\cdot\frac{\partial}{\partial
x_{ij}}\mathcal{F}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i})+a_{i}\mathcal{F}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i})\,.$
(14)

######  Lemma 2.

Fix $i\in[n]$ and $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$ and let
$\mathcal{R}_{A(\mathbf{x}_{-i})}=\prod_{j\in
A(\mathbf{x}_{-i})}(0,\omega_{ij}-\mathbf{x}_{T}^{j|i})$, where $\omega_{ij}$
is provided by Lemma 1. Then a global maximum of the function
$\mathcal{V}_{\mathbf{x}_{-i}}:=\sum_{j\in
A(\mathbf{x}_{-i})}\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})$ defined on
the set $\mathcal{R}_{A(\mathbf{x}_{-i})}$ is given by the unique solution of
the following system of equations:

$\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})=0,\,\text{ for }\,j\in
A(\mathbf{x}_{-i})\,,$ (15)

where $\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})$ is defined in (14).

###### Proof.

To simplify notation, we write $\psi_{ij}(\cdot)$ instead of
$\psi_{ij}(\,\cdot\,;\mathbf{x}_{T}^{j|i})$. Using (6) and (8), it is
straightforward to verify that for every $j\in A(\mathbf{x}_{-i})$ it holds

$\frac{\partial\mathcal{V}_{\mathbf{x}_{-i}}}{\partial
x_{ij}}=\frac{\partial\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})}{\partial
x_{ij}}=x_{ij}^{a_{i}-1}\cdot\psi_{ij}(x_{ij})\,.$ (16)

Now notice that $\psi_{ij}(0)>0$ as well as
$\psi_{ij}(\omega_{ij}-\mathbf{x}_{T}^{j|i})<0$. Moreover, Assumption 2
readily implies that $\psi_{ij}(\cdot)$ is strictly decreasing on the interval
$(0,\omega_{ij}-\mathbf{x}_{T}^{j|i})$. The intermediate value theorem implies
that there exists unique $\lambda_{ij}\in(0,\omega_{ij}-\mathbf{x}_{T}^{j|i})$
such that $\psi_{ij}(\lambda_{ij})=0$. Hence, it follows from (16) that the
points $\lambda_{ij}$, for $j\in A(\mathbf{x}_{-i})$, are critical points of
the function $\mathcal{V}_{\mathbf{x}_{-i}}$, which is concave on the open and
convex set $\mathcal{R}_{A(\mathbf{x}_{-i})}$, by Theorem 5. It follows (see
[4, Theorem 2.4, p. 132]) that $\\{\lambda_{ij}\\}_{j\in A(\mathbf{x}_{-i})}$
is a global maximum of $\mathcal{V}_{\mathbf{x}_{-i}}$ on
$\mathcal{R}_{A(\mathbf{x}_{-i})}$. We conclude that
$\mathcal{V}_{\mathbf{x}_{-i}}$ is maximized when $x_{ij}=\lambda_{ij}$, for
$j\in A(\mathbf{x}_{-i})$, as desired. ∎

######  Remark 2.

Let us remark that the solution of the system of equations given by (15) may
not belong to the set $C_{m}$. More precisely, it could happen that the
solution of the system of equations (15), say $\\{\lambda_{ij}\\}_{j\in
A(\mathbf{x}_{-i})}$, satisfies $\sum_{j\in
A(\mathbf{x}_{-i})}\lambda_{ij}>1$. This is a crucial difference between the
Fragile CPR Game and the Fragile multi-CPR Game.

Now notice that, given $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$, the best response
of player $i$ is a local maximum of the following program:

$\displaystyle\underset{\\{x_{ij}\\}_{j\in
A(\mathbf{x}_{-i})}}{\text{maximize}}$
$\displaystyle\mathcal{V}_{\mathbf{x}_{-i}}:=\sum_{j\in
A(\mathbf{x}_{-i})}\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})$ subject to
$\displaystyle\sum_{j\in A(\mathbf{x}_{-i})}x_{ij}\leq 1$ $\displaystyle 0\leq
x_{ij}\leq\omega_{ij}-\mathbf{x}_{T}^{j|i},\text{ for all }\,j\in
A(\mathbf{x}_{-i})\,.$

Equivalently, the best response of player $i$ is a local minimum of the
following program:

$\displaystyle\underset{\\{x_{ij}\\}_{j\in
A(\mathbf{x}_{-i})}}{\text{minimize}}$ $\displaystyle-\sum_{j\in
A(\mathbf{x}_{-i})}\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})$ (17) subject
to $\displaystyle\sum_{j\in A(\mathbf{x}_{-i})}x_{ij}\leq 1$ $\displaystyle
0\leq x_{ij}\leq\omega_{ij}-\mathbf{x}_{T}^{j|i},\text{ for all }\,j\in
A(\mathbf{x}_{-i})\,.$

Notice that since $\mathcal{E}_{ij}(\,\cdot\,;\mathbf{x}_{T}^{j|i})$ is
concave on $(0,\omega_{ij}-\mathbf{x}_{T}^{j|i})$, by Lemma 1, it follows that
Problem (17) is a separable convex knapsack program (see [20, 24]). We are
going to describe the optima of Problem (17) using the KKT conditions. The KKT
conditions pertain to the Lagrangian corresponding to Problem (17), which is
defined as the following quantity:

$\mathcal{L}:=-\mathcal{V}_{\mathbf{x}_{-i}}+\kappa_{0}\cdot\left(\sum_{j\in
A(\mathbf{x}_{-i})}x_{ij}-1\right)+\sum_{j\in
A(\mathbf{x}_{-i})}\mu_{j}\cdot(x_{ij}+\mathbf{x}_{T}^{j|i}-\omega_{ij})+\sum_{j\in
A(\mathbf{x}_{-i})}\nu_{j}\cdot(-x_{ij}),$

where $\kappa_{0},\\{\mu_{j}\\}_{j},\\{\nu_{j}\\}_{j}$ are real numbers. The
KKT conditions corresponding to problem (17) read as follows (see [18, Theorem
3.8]).

######  Theorem 7 (KKT conditions for Problem (17)).

If $\\{x_{ij}\\}_{j\in A(\mathbf{x}_{-i})}$ is a local minimum of Problem
(17), then there exist _non-negative_ real numbers $\kappa_{0}$,
$\\{\mu_{j}\\}_{j\in A(\mathbf{x}_{-i})}$, and $\\{\nu_{j}\\}_{j\in
A(\mathbf{x}_{-i})}$ such that:

  1. 1.

For all $j\in A(\mathbf{x}_{-i})$ it holds
$-x_{ij}^{a_{i}-1}\cdot\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})+\kappa_{0}+\mu_{j}-\nu_{j}=0\,$,
where $\psi_{ij}$ is given by (14).

  2. 2.

$\kappa_{0}\cdot\left(\sum_{j\in A(\mathbf{x}_{-i})}x_{ij}-1\right)=0$.

  3. 3.

$\mu_{j}\cdot(x_{ij}+\mathbf{x}_{T}^{j|i}-\omega_{ij})=0\,$, for all $j\in
A(\mathbf{x}_{-i})$.

  4. 4.

$\nu_{j}\cdot x_{ij}=0\,$, for all $j\in A(\mathbf{x}_{-i})$.

  5. 5.

$0\leq x_{ij}\leq\omega_{ij}-\mathbf{x}_{T}^{j|i}\,$, for all $j\in
A(\mathbf{x}_{-i})$.

We aim to employ Theorem 7 in order to describe a local maximum of Problem
(17) via the solution of a system of equations. This will require the
following result, which is presumably reported somewhere in the literature
but, lacking a reference, we include a proof for the sake of completeness.

######  Lemma 3.

Fix a positive integer $s$ and, for each $j\in[s]$, let
$f_{j}:\mathbb{R}\to\mathbb{R}$ be a strictly decreasing function. Then there
exists at most one vector $(c,x_{1},\ldots,x_{s})\in\mathbb{R}^{s+1}$ such
that

$f_{j}(x_{j})=c,\,\text{ for all }\,j\in[s],\,\text{ and
}\,\sum_{j\in[s]}x_{j}=1\,.$

###### Proof.

Suppose that there exist two distinct vectors, say $(c,x_{1},\ldots,x_{s})$
and $(d,y_{1},\ldots,y_{s})$. If $c=d$, then there exists $j\in[s]$ such that
$x_{j}\neq y_{j}$ and

$f_{j}(x_{j})=c=d=f_{j}(y_{j})\,,$

contrariwise to the assumption that the function $f_{i}(\cdot)$ is strictly
decreasing. Hence $c\neq d$.

Since $f_{j}(\cdot),j\in[s]$, is strictly decreasing, it is injective and
therefore it follows that it is invertible. Let us denote its inverse by
$f_{j}^{-1}(\cdot)$. We then have

$x_{j}=f_{j}^{-1}(c)\,\text{ and }y_{j}=f_{j}^{-1}(d),\,\text{ for all
}\,j\in[s],$

which in turn implies that $x_{j}\neq y_{j}$, for all $j\in[s]$. Assume,
without loss of generality, that $c<d$. The assumption that $f_{j}$ is
strictly decreasing then implies $x_{j}>y_{j}$, for all $j\in[s]$, and
therefore $1=\sum_{j\in[s]}x_{j}>\sum_{j\in[s]}y_{j}=1$, a contradiction. The
result follows. ∎

We may now proceed with describing the best responses of each player in the
Fragile multi-CPR Game via a system of “first order conditions”.

######  Theorem 8.

Let $i\in[n]$ and $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$ be fixed. Suppose that
$\\{x_{ij}\\}_{j\in A(\mathbf{x}_{-i})}$ is a best response of player $i$ in
the Fragile multi-CPR Game. Then $\\{x_{ij}\\}_{j\in A(\mathbf{x}_{-i})}$ is
either of the following two types:

  * •

Type I:  There exists $J_{\mathbf{x}_{-i}}\subset A(\mathbf{x}_{-i})$ such
that $x_{ij}=0$, when $j\in A(\mathbf{x}_{-i})\setminus J_{\mathbf{x}_{-i}}$,
and $\\{x_{ij}\\}_{j\in J_{\mathbf{x}_{-i}}}$ satisfy the following
inequality, and are given by the unique solution of the following system of
equations:

$\sum_{j\in J_{\mathbf{x}_{-i}}}x_{ij}<1\quad\text{ and
}\quad\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})=0,\,\text{ for }\,j\in
J_{\mathbf{x}_{-i}}\,,$

where $\psi_{ij}(\,\cdot\,;\mathbf{x}_{T}^{j|i})$ is defined in (14).

  * •

Type II:  There exists $J_{\mathbf{x}_{-i}}\subset A(\mathbf{x}_{-i})$ and a
real number $\kappa_{0}\geq 0$ such that $x_{ij}=0$, when $j\in
A(\mathbf{x}_{-i})\setminus J_{\mathbf{x}_{-i}}$, and $\\{x_{ij}\\}_{j\in J}$
are given by the unique solution of the following system of equations:

$\sum_{j\in J_{\mathbf{x}_{-i}}}x_{ij}=1\quad\text{and}\quad
x_{ij}^{a_{i}-1}\cdot\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})=\kappa_{0},\,\text{
for }\,j\in J_{\mathbf{x}_{-i}}\,,$

where $\psi_{ij}(\,\cdot\,;\mathbf{x}_{T}^{j|i})$ is defined in (14).

###### Proof.

Let $\\{x_{ij}\\}_{j\in A(\mathbf{x}_{-i})}$ be a best response of player
$i\in[n]$. Then $\\{x_{ij}\\}_{j\in A(\mathbf{x}_{-i})}$ is a local minimum of
Problem (17); hence it satisfies the KKT Conditions of Theorem 7.

If $x_{ij}=\omega_{ij}-\mathbf{x}_{T}^{j|i}$, for some $j\in
A(\mathbf{x}_{-i})$, then Lemma 1 and (6) imply that
$\mathcal{E}_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})=0$. Hence player $i$ could
achieve the same utility from the $j$-th CPR by choosing $x_{ij}=0$. Thus we
may assume that $x_{ij}<\omega_{ij}-\mathbf{x}_{T}^{j|i}$, for all $j\in
A(\mathbf{x}_{-i})$ and therefore Theorem 7.(3) implies that $\mu_{j}=0$, for
all $j\in A(\mathbf{x}_{-i})$. Now let

$J_{\mathbf{x}_{-i}}=\\{j\in A(\mathbf{x}_{-i}):x_{ij}\neq 0\\}\,,$ (18)

and notice that Theorem 7.(4) implies that $\nu_{j}=0$ for $j\in
J_{\mathbf{x}_{-i}}$. We distinguish two cases.

Suppose first that $\sum_{j\in J_{\mathbf{x}_{-i}}}x_{ij}<1$. Then Theorem
7.(2) yields $\kappa_{0}=0$, and therefore Theorem 7.(1) implies that
$\\{x_{ij}\\}_{i\in J_{\mathbf{x}_{-i}}}$ is given by the unique solution of
the following system of equations:

$\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})=0,\,\text{ for }\,j\in
J_{\mathbf{x}_{-i}}\,.$

In other words, if $\sum_{j\in J_{\mathbf{x}_{-i}}}x_{ij}<1$ then
$\\{x_{ij}\\}_{j\in A(\mathbf{x}_{-i})}$ is of Type I.

Now assume that $\sum_{j\in J_{\mathbf{x}_{-i}}}x_{ij}=1$. Then Theorems 7.(1)
and 7 .(2) imply that there exists $\kappa_{0}\geq 0$ such that
$-x_{ij}^{a_{i}-1}\cdot\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})=-\kappa_{0}$,
for all $j\in J_{\mathbf{x}_{-i}}$. In other words, $\\{x_{ij}\\}_{j\in
J_{\mathbf{x}_{-i}}}$ and $\kappa_{0}$ are given by the solution of the
following system of equations:

$\sum_{j\in J_{\mathbf{x}_{-i}}}x_{ij}=1\,\text{ and
}\,x_{ij}^{a_{i}-1}\cdot\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})=\kappa_{0},\,\text{
for all }\,j\in J_{\mathbf{x}_{-i}}\,.$ (19)

Since the functions
$f_{ij}(x_{ij}):=x_{ij}^{a_{i}-1}\cdot\psi_{ij}(x_{ij};\mathbf{x}_{T}^{j|i})$,
for $j\in J_{\mathbf{x}_{-i}}$, are strictly decreasing, Lemma 3 implies that
the system of equations in (19) has a unique solution. Hence
$\\{x_{ij}\\}_{j\in A(\mathbf{x}_{-i})}$ is of Type II and the result follows.
∎

We refer to the set $J_{\mathbf{x}_{-i}}$ provided by Theorem 8, defined in
(18), as the set of _effective_ CPRs corresponding to $i\in[n]$ and
$\mathbf{x}_{-i}\in\mathcal{C}_{-i}$. In the next section we employ Theorem 8
in order to define auxiliary functions (i.e., (24) and (25) below) whose
monotonicity will play a key role in the proof of Theorem 4.

##  5 Auxiliary functions

In this section we define and state basic properties of certain auxiliary
functions, whose monotonicity will be used in the proofs of Theorems 3 and 4,
and whose definition depends upon the “first order conditions” provided by
Theorem 8.

Let us begin with some notation and remarks. Fix $i\in[n]$ and
$\mathbf{x}_{-i}\in\mathcal{C}_{-i}$, and recall from (13) that
$B_{i}(\mathbf{x}_{-i})$ denotes a best response of player $i$ and that
$B_{ij}(\mathbf{x}_{-i})$ is its $j$-th component. To simplify notation, let
us denote $b_{ij}:=B_{ij}(\mathbf{x}_{-i})$. From Theorem 8 we know that there
exists $J_{\mathbf{x}_{-i}}\subset A(\mathbf{x}_{-i})$ such that $b_{ij}=0$,
for $j\in A(\mathbf{x}_{-i})\setminus J_{\mathbf{x}_{-i}}$, and either

$\sum_{j\in
J_{\mathbf{x}_{-i}}}b_{ij}<1\quad\text{and}\quad\psi_{ij}(b_{ij};\mathbf{x}_{-i})=0,\,\text{
for all }\,j\in J_{\mathbf{x}_{-i}},$ (20)

or

$\sum_{j\in J_{\mathbf{x}_{-i}}}b_{ij}=1\quad\text{ and }\quad
b_{ij}^{a_{i}-1}\cdot\psi_{ij}(b_{ij};\mathbf{x}_{-i})=\kappa_{0},\,\text{ for
all }\,j\in J_{\mathbf{x}_{-i}}\,\text{ and some }\,\kappa_{0}\geq 0.$ (21)

In particular, it holds $b_{ij}>0$, for all $j\in J_{\mathbf{x}_{-i}}$. Using
(14), it follows that the second statement of (20) is equivalent to

$b_{ij}\cdot\frac{\partial}{\partial
x_{ij}}\mathcal{F}_{ij}(b_{ij}+\mathbf{x}_{T}^{j|i})+a_{i}\mathcal{F}_{ij}(b_{ij}+\mathbf{x}_{T}^{j|i})=0,\,\text{
for all }\,j\in J_{\mathbf{x}_{-i}},$ (22)

and that the second statement of (21) is equivalent to

$b_{ij}^{a_{i}-1}\cdot\left(b_{ij}\cdot\frac{\partial}{\partial
x_{ij}}\mathcal{F}_{ij}(b_{ij}+\mathbf{x}_{T}^{j|i})+a_{i}\mathcal{F}_{ij}(b_{ij}+\mathbf{x}_{T}^{j|i})\right)=\kappa_{0},\,\text{
for all }\,j\in J_{\mathbf{x}_{-i}}\,.$ (23)

Now, given $\mathbf{x}_{-i}\in\mathcal{C}_{-i}$, $j\in J_{\mathbf{x}_{-i}}$
and $\kappa_{0}\geq 0$, define for each $i\in[n]$ the functions

$\mathcal{G}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i}):=-\frac{a_{i}\mathcal{F}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i})}{\frac{\partial}{\partial
x_{ij}}\mathcal{F}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i})}\,,\text{ for
}\,x_{ij}\in(0,\omega_{ij}-\mathbf{x}_{T}^{j|i})$ (24)

and

$\mathcal{H}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i};\kappa_{0}):=-\frac{a_{i}\mathcal{F}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i})}{\frac{-\kappa_{0}}{x_{ij}^{a_{i}}}+\frac{\partial}{\partial
x_{ij}}\mathcal{F}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i})},\text{ for
}\,x_{ij}\in(0,\omega_{ij}-\mathbf{x}_{T}^{j|i})\,.$ (25)

Notice that (22) implies that when $b_{ij}$ is of Type I it holds

$\mathcal{G}_{ij}(b_{ij}+\mathbf{x}_{T}^{j|i})=b_{ij}\,,$ (26)

while (23) implies that when $b_{ij}$ is of Type II it holds

$\mathcal{H}_{ij}(b_{ij}+\mathbf{x}_{T}^{j|i};\kappa_{0})=b_{ij},\,.$ (27)

Observe also that it holds
$\mathcal{G}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i})\geq\mathcal{H}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i};\kappa_{0})$,
for all $x_{ij}\in[0,\omega_{ij}-\mathbf{x}_{T}^{j|i}]$. Let us, for future
reference, collect a couple of observations about the functions
$\mathcal{G}_{ij},\mathcal{H}_{ij}$.

######  Lemma 4.

Let $i\in[n]$ and $j\in[m]$ be fixed. Then the functions
$\mathcal{G}_{ij}(\cdot)$ and $\mathcal{H}_{ij}(\,\cdot\,;\kappa_{0})$,
defined in (24) and (25) respectively, are strictly decreasing in the interval
$[0,\omega_{ij}]$.

###### Proof.

To simplify notation, let
$\mathcal{F}:=\mathcal{F}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i})$,
$\mathcal{F}^{\prime}:=\frac{\partial}{\partial x_{ij}}\mathcal{F}$ and
$\mathcal{F}^{\prime\prime}:=\frac{\partial^{2}}{\partial
x_{ij}^{2}}\mathcal{F}$. For $x_{ij}\in(0,\omega_{ij}-\mathbf{x}_{T}^{(j)})$,
we compute

$\displaystyle\frac{\partial}{\partial
x_{ij}}\mathcal{H}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i};\kappa_{0})$
$\displaystyle=$
$\displaystyle\frac{-a_{i}\mathcal{F}^{\prime}\cdot(\frac{-\kappa_{0}}{x_{ij}^{a_{i}}}+\mathcal{F}^{\prime})+a_{i}\mathcal{F}\cdot(-a_{i}\frac{-\kappa_{0}}{x_{ij}^{a_{i}+1}}+\mathcal{F}^{\prime\prime})}{(\frac{-\kappa_{0}}{x_{ij}^{a_{i}}}+\mathcal{F}^{\prime})^{2}}$
$\displaystyle=$ $\displaystyle
a_{i}\cdot\frac{\kappa_{0}x_{ij}^{a_{i}-1}\left(x_{ij}\mathcal{F}^{\prime}+a_{i}\mathcal{F}\right)-x_{ij}^{2a_{i}}(\mathcal{F}^{\prime})^{2}+x_{ij}^{2a_{i}}\mathcal{F}\cdot\mathcal{F}^{\prime\prime}}{(-\kappa_{0}+x_{ij}^{a_{i}}\mathcal{F}^{\prime})^{2}}$
$\displaystyle<$ $\displaystyle
a_{i}\cdot\frac{\kappa_{0}x_{ij}^{a_{i}-1}\left(x_{ij}\mathcal{F}^{\prime}+a_{i}\mathcal{F}\right)-x_{ij}^{2a_{i}}(\mathcal{F}^{\prime})^{2}}{(-\kappa_{0}+x_{ij}^{a_{i}}\mathcal{F}^{\prime})^{2}}\,,$

where the last estimate follows from the fact that, by Assumption 2, it holds
$\mathcal{F}^{\prime\prime}<0$. If $\kappa_{0}=0$, then it readily follows
that $\frac{\partial}{\partial
x_{ij}}\mathcal{H}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i};\kappa_{0})<0$ and
therefore $\mathcal{H}_{ij}$ is strictly decreasing; thus $\mathcal{G}_{ij}$
is strictly decreasing as well. So we may assume that $\kappa_{0}>0$. If
$x_{ij}\mathcal{F}^{\prime}+a_{i}\mathcal{F}<0$, then it also follows that
$\mathcal{H}_{ij}$ is strictly decreasing; thus we may also assume that
$A:=x_{ij}\mathcal{F}^{\prime}+a_{i}\mathcal{F}\geq 0$. Now notice that
$\frac{\partial A}{\partial
x_{ij}}=\mathcal{F}^{\prime}+x_{ij}\mathcal{F}^{\prime\prime}+a_{i}\mathcal{F}^{\prime}<0$,
and define the function

$H(x_{ij}):=\kappa_{0}x_{ij}^{a_{i}-1}\cdot
A-x_{ij}^{2a_{i}}(\mathcal{F}^{\prime})^{2}\,;$

hence it holds $\frac{\partial}{\partial
x_{ij}}\mathcal{H}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i};\kappa_{0})<a_{i}\cdot\frac{H(x_{ij})}{(-\kappa_{0}+x_{ij}^{a_{i}}\mathcal{F}^{\prime})^{2}}$.
Moreover, it holds

$\frac{\partial}{\partial
x_{ij}}H(x_{ij})=(a_{i}-1)\kappa_{0}x_{ij}^{a_{i}-2}\cdot
A+\kappa_{0}x_{ij}^{a_{i}-1}\cdot\frac{\partial A}{\partial
x_{ij}}-2a_{i}x_{ij}^{2a_{i}-1}(\mathcal{F}^{\prime})^{2}-2x_{ij}^{2a_{i}}\mathcal{F}^{\prime}\mathcal{F}^{\prime\prime}\,.$

Since $a_{i}\leq 1$, $A\geq 0$ and
$\mathcal{F}^{\prime},\mathcal{F}^{\prime\prime},\frac{\partial A}{\partial
x_{ij}}<0$, it readily follows that all addends in the previous equation are
negative, and therefore $\frac{\partial}{\partial x_{ij}}H(x_{ij})<0$. In
other words, $H(\cdot)$ is strictly decreasing in
$[0,\omega_{ij}-\mathbf{x}_{T}^{j|i}]$ and, since $H(0)=0$,
$H(\omega_{ij}-\mathbf{x}_{T}^{j|i})<0$, we conclude that $H(x_{ij})\leq 0$,
for all $x_{ij}\in[0,\omega_{ij}-\mathbf{x}_{T}^{j|i}]$. This implies that
$\frac{\partial}{\partial
x_{ij}}\mathcal{H}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i};\kappa_{0})<0$ for
$x_{ij}\in[0,\omega_{ij}-\mathbf{x}_{T}^{j|i}]$. Since
$\frac{\partial}{\partial\mathbf{x}_{T}^{(j)}}\mathcal{H}_{ij}(\mathbf{x}_{T}^{(j)};\kappa_{0})=\frac{\partial}{\partial
x_{ij}}\mathcal{H}_{ij}(x_{ij}+\mathbf{x}_{T}^{j|i};\kappa_{0})$, and
similarly for $\mathcal{G}_{ij}$, we conclude that both $\mathcal{G}_{ij}$ and
$\mathcal{H}_{ij}$ are strictly decreasing in the interval $[0,\omega_{ij}]$,
as desired. ∎

##  6 Proof of Theorem 3

In this section we prove Theorem 3. We first introduce some notation. Consider
a GNE, say
$\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathcal{C}_{n}$, where
$\mathbf{x}_{i}=(x_{i1},\ldots,x_{im})\in C_{m}$, of a Fragile multi-CPR Game
satisfying Assumption 2. Given $j\in[m]$, let

$\mathcal{S}(\mathbf{x}_{T}^{(j)})=\\{i\in[n]:\mathbf{x}_{T}^{(j)}<\omega_{ij}\text{
and }x_{ij}>0\\}$ (28)

be the _support_ of the $j$-th CPR and let

$\mathcal{S}_{I}(\mathbf{x}_{T}^{(j)})=\\{i\in\mathcal{S}(\mathbf{x}_{T}^{(j)}):\mathbf{x}_{i}\text{
is of Type\leavevmode\nobreak\ I}\\}$ (29)

be the _support of Type I_ , consisting of those players in the support of the
$j$-th CPR whose best response is of Type I, and

$\mathcal{S}_{II}(\mathbf{x}_{T}^{(j)})=\\{i\in\mathcal{S}(\mathbf{x}_{T}^{(j)}):\mathbf{x}_{i}\text{
is of Type\leavevmode\nobreak\ II}\\}$ (30)

be the _support of Type II_ , consisting of those players in the support of
the $j$-th CPR whose best response is of Type II. Clearly, in view of Theorem
8, it holds
$\mathcal{S}(\mathbf{x}_{T}^{(j)})=\mathcal{S}_{I}(\mathbf{x}_{T}^{(j)})\cup\mathcal{S}_{II}(\mathbf{x}_{T}^{(j)})$.

We employ the properties of the auxiliary functions in the proof of Theorem 3,
a basic ingredient of which is the fact that in the setting of Theorem 3 the
support of Type II is empty. The proof is similar to the proof of Theorem 1,
given in [13, p. 155]. In fact, we prove a bit more. We show that Theorem 3 is
a consequence of the following result.

######  Theorem 9.

Consider a Fragile multi-CPR Game with $n\geq 1$ players and $m\geq 1$ CPRs
satisfying Assumption 2. Then there exists at most one GNE
$\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$ for which $\mathbf{x}_{i}$
is of Type I, for all $i\in[n]$.

###### Proof.

Let $\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$ be a GNE such that
$\mathbf{x}_{i}$ is of Type I, for all $i\in[n]$ and note that, since
$\mathbf{x}_{i}$ is of Type I, it holds $\sum_{j}x_{ij}<1$, for all $i\in[n]$.
For each $j\in[m]$, let
$\mathcal{S}_{0}(\mathbf{x}_{T}^{(j)}):=\\{i\in[n]:\mathbf{x}_{T}^{(j)}<\omega_{ij}\\}$.
We claim that
$\mathcal{S}_{0}(\mathbf{x}_{T}^{(j)})=\mathcal{S}(\mathbf{x}_{T}^{(j)})$.
Indeed, if there exists
$i\in\mathcal{S}_{0}(\mathbf{x}_{T}^{(j)})\setminus\mathcal{S}(\mathbf{x}_{T}^{(j)})$
then $x_{ij}=0$ and since it holds $\mathbf{x}_{T}^{(j)}<\omega_{ij}$ and
$\sum_{j}x_{ij}<1$, it follows that player $i$ could increase her utility by
investing a suitably small amount, say $\varepsilon>0$, in the $j$-th CPR. But
then this implies that $\mathbf{x}$ cannot be a GNE, a contradiction. Hence
$\mathcal{S}_{0}(\mathbf{x}_{T}^{(j)})=\mathcal{S}(\mathbf{x}_{T}^{(j)})$.

We claim that for any two distinct GNEs, say
$\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$ and
$\mathbf{y}=(\mathbf{y}_{1},\ldots,\mathbf{y}_{n})$, for which
$\mathbf{x}_{i},\mathbf{y}_{i}$ are of Type I for all $i\in[n]$, it holds
$\mathbf{x}_{T}^{(j)}=\mathbf{y}_{T}^{(j)}$, for all $j\in[m]$. Indeed, if the
claim is not true, then there exists $j\in[m]$ such that
$\mathbf{x}_{T}^{(j)}\neq\mathbf{y}_{T}^{(j)}$. Suppose, without loss of
generality, that $\mathbf{x}_{T}^{(j)}<\mathbf{y}_{T}^{(j)}$.

Since $\mathbf{x}_{i}$ is of Type I, for all $i\in[n]$, it follows that
$\mathcal{S}(\mathbf{x}_{T}^{(j)})=\mathcal{S}_{I}(\mathbf{x}_{T}^{(j)})$ and
$\mathcal{S}(\mathbf{y}_{T}^{(j)})=\mathcal{S}_{I}(\mathbf{y}_{T}^{(j)})$.
Moreover, since $\mathbf{x}_{T}^{(j)}<\mathbf{y}_{T}^{(j)}$ it follows that
$\mathcal{S}_{I}(\mathbf{y}_{T}^{(j)})=\mathcal{S}_{0}(\mathbf{y}_{T}^{(j)})\subset\mathcal{S}_{0}(\mathbf{x}_{T}^{(j)})=\mathcal{S}_{I}(\mathbf{x}_{T}^{(j)})$.

Now notice that (26) implies that
$\mathcal{G}_{ij}(\mathbf{x}_{T}^{(j)})=x_{ij}$, for all
$i\in\mathcal{S}_{I}(\mathbf{x}_{T}^{(j)})$, and
$\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})=y_{ij}$, for all
$i\in\mathcal{S}_{I}(\mathbf{y}_{T}^{(j)})$. Since
$\mathcal{S}_{I}(\mathbf{y}_{T}^{(j)})\subset\mathcal{S}_{I}(\mathbf{x}_{T}^{(j)})$
it holds that

$\sum_{i\in\mathcal{S}_{I}(\mathbf{y}_{T}^{(j)})}\mathcal{G}_{ij}(\mathbf{x}_{T}^{(j)})\leq\mathbf{x}_{T}^{(j)}<\mathbf{y}_{T}^{(j)}=\sum_{i\in\mathcal{S}_{I}(\mathbf{y}_{T}^{(j)})}\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})\,.$
(31)

However, since $\mathcal{G}_{ij}$ is strictly decreasing, it follows that
$\mathcal{G}_{ij}(\mathbf{x}_{T}^{(j)})>\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})$,
for all $i\in\mathcal{S}_{I}(\mathbf{y}_{T}^{(j)})$, which contradicts (31).
We conclude that $\mathbf{x}_{T}^{(j)}=\mathbf{y}_{T}^{(j)}$ and
$\mathcal{S}_{I}(\mathbf{x}_{T}^{(j)})=\mathcal{S}_{I}(\mathbf{y}_{T}^{(j)})$.
Finally, given a total investment $\mathbf{x}$ of the players at a GNE, we
claim that the optimal investment of every player on any CPR is unique.
Indeed, if a player, say $i\in[n]$, has two optimal investments, say $x<z$, on
the $j$-th CPR, then it holds
$\mathcal{G}_{ij}(\mathbf{x})=x<z=\mathcal{G}_{ij}(\mathbf{x})$, a
contradiction. The result follows. ∎

Theorem 3 is a direct consequence of Theorem 9, as we now show.

###### Proof of Theorem 3.

We know from Theorem 2 that the game admits a GNE, and it is therefore enough
to show that it is unique. Since $m=1$, the first condition in Assumption 2
implies that no player invests an amount of $1$ in the CPR which in turn
implies that all coordinates of any GNE are of Type I. The result follows from
Theorem 9. ∎

Observe that a basic ingredient in the proof of Theorem 9 is the fact that
$\mathcal{S}(\mathbf{x}_{T})=\mathcal{S}_{I}(\mathbf{x}_{T}),\mathcal{S}(\mathbf{y}_{T})=\mathcal{S}_{I}(\mathbf{y}_{T})$
and $\mathcal{S}_{I}(\mathbf{y}_{T})\subset\mathcal{S}_{I}(\mathbf{x}_{T})$.
Moreover, observe that the proof of Theorem 9 proceeds in two steps: in the
first step it is shown that any two GNEs admit the same total investment in
the CPR, and in the second step it is shown that, given an optimal total
investment, every player has a unique optimal investment in the CPR. In the
following section we are going to improve upon the aforementioned
observations. A bit more concretely, we are going to prove that the set
consisting of all GNEs of a Fragile multi-CPR Game is “small” via showing that
the set consisting of all total investments at the GNEs is “small”.

##  7 Proof of Theorem 4

Throughout this section, we denote by $G^{(2)}$ a Fragile multi-CPR Game
satisfying Assumption 2. Moreover, given a finite set, $F$, we denote by $|F|$
its cardinality. Now consider the set

$\mathcal{N}(G^{(2)}):=\\{\mathbf{x}\in\mathcal{C}_{n}:\mathbf{x}\text{ is a
GNE of }G^{(2)}\\}$

and, given
$\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathcal{N}(G^{(2)})$,
let

$\mathcal{T}_{I}(\mathbf{x})=\\{i\in[n]:\mathbf{x}_{i}\text{ is of
Type\leavevmode\nobreak\ I}\\}$

and

$\mathcal{T}_{II}(\mathbf{x})=\\{i\in[n]:\mathbf{x}_{i}\text{ is of
Type\leavevmode\nobreak\ II}\\}\,.$

Recall the definition of active CPRs corresponding to $\mathbf{x}_{-i}$, which
is denoted $A(\mathbf{x}_{-i})$ and is defined in (9), as well as the
definition of effective CPRs corresponding to $\mathbf{x}_{-i}$, which is
denoted $J_{\mathbf{x}_{-i}}$ and is defined in (18).

######  Lemma 5.

Let $\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathcal{N}(G^{(2)})$
and suppose that $i\in\mathcal{T}_{I}(\mathbf{x})$, for some $i\in[n]$. Then
it holds $J_{\mathbf{x}_{-i}}=A(\mathbf{x}_{-i})$.

###### Proof.

Recall from Theorem 8 that $J_{\mathbf{x}_{-i}}$ is such that $x_{ij}>0$ if
and only if $j\in J_{\mathbf{x}_{-i}}$. Suppose, towards arriving at a
contradiction, that there exists $j\in A(\mathbf{x}_{-i})\setminus
J_{\mathbf{x}_{-i}}$. Since $i\in\mathcal{T}_{I}(\mathbf{x})$, it follows that
$\sum_{j\in J_{\mathbf{x}_{-i}}}x_{ij}<1$ and thus player $i$ can increase her
utility by investing a suitably small amount $\varepsilon>0$ in the $j$-th
CPR. This contradicts the fact that $\mathbf{x}$ is a GNE, and the lemma
follows. ∎

######  Lemma 6.

Let $\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$ and
$\mathbf{y}=(\mathbf{y}_{1},\ldots,\mathbf{y}_{n})$ be two elements from
$\mathcal{N}(G^{(2)})$ such that
$\mathbf{x}_{T}^{(j)}\leq\mathbf{y}_{T}^{(j)}$, for all $j\in[m]$. Then the
following hold true:

  1. 1.

If $i\in\mathcal{T}_{I}(\mathbf{x})$, then $J_{\mathbf{y}_{-i}}\subset
J_{\mathbf{x}_{-i}}$.

  2. 2.

It holds $\mathcal{T}_{I}(\mathbf{x})\subset\mathcal{T}_{I}(\mathbf{y})$.

###### Proof.

Fix $i\in[n]$ such that $i\in\mathcal{T}_{I}(\mathbf{x})$ and notice that
Lemma 5 implies that $\mathbf{x}_{T}^{(j)}\geq\omega_{ij}$, for all
$j\in[m]\setminus J_{\mathbf{x}_{-i}}$. Since
$\mathbf{x}_{T}^{(j)}\leq\mathbf{y}_{T}^{(j)}$, for all $j\in[m]$, it holds
$\mathbf{y}_{T}^{(j)}\geq\omega_{ij}$, for all $j\in[m]\setminus
J_{\mathbf{y}_{-i}}$, and we conclude that $J_{\mathbf{y}_{-i}}\subset
J_{\mathbf{x}_{-i}}$. The first statement follows.

We proceed with the second statement. Let $i\in[n]$ be such that
$\mathbf{x}_{i}$ is of Type I. We have to show that $\mathbf{y}_{i}$ is also
of Type I. Suppose that this is not true; hence $\mathbf{y}_{i}$ is of Type
II, and thus it holds $\sum_{j\in J_{\mathbf{y}_{-i}}}y_{ij}=1$. Since
$\mathbf{y}_{i}$ is of Type II, it follows from (27) that
$\mathcal{H}_{ij}(\mathbf{y}_{T}^{(j)};\kappa_{0})=y_{ij}$, for all $j\in
J_{\mathbf{y}_{-i}}$ and some $\kappa_{0}\geq 0$. Since $\mathbf{x}_{i}$ is of
Type I and $\mathcal{H}_{ij}$ is decreasing, we may apply (26) and conclude

$x_{ij}=\mathcal{G}_{ij}(\mathbf{x}_{T}^{(j)})\geq\mathcal{H}_{ij}(\mathbf{x}_{T}^{(j)};\kappa_{0})\geq\mathcal{H}_{ij}(\mathbf{y}_{T}^{(j)};\kappa_{0})=y_{ij},\,\text{
for all }\,j\in J_{\mathbf{y}_{-i}}\,.$

Hence $1>\sum_{j\in J_{\mathbf{y}_{-i}}}x_{ij}\geq\sum_{j\in
J_{\mathbf{y}_{-i}}}y_{ij}=1$, a contradiction. The result follows. ∎

######  Lemma 7.

Assume that $m\leq n$. Then it holds
$\mathcal{T}_{I}(\mathbf{x})\neq\emptyset$, for every
$\mathbf{x}\in\mathcal{N}(G^{(2)})$.

###### Proof.

Suppose that the conclusion is not true; hence there exists
$\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathcal{N}(G^{(2)})$
such that $\mathcal{T}_{II}(\mathbf{x})=[n]$, which in turn implies that
$\sum_{j\in[m]}\mathbf{x}_{T}^{(j)}=n\geq m$. Hence there exists $k\in[m]$
such that $\mathbf{x}_{T}^{(k)}\geq 1$. We now claim that
$\mathbf{x}_{T}^{k|i}\geq\omega_{ik}$, for all
$i\in\mathcal{S}_{II}(\mathbf{x}_{T}^{(k)})$, where $\mathcal{S}_{II}(\cdot)$
is defined in (30) and $\omega_{ik}$ is given by Lemma 1. To prove the claim,
notice that if there exists $i\in\mathcal{S}_{II}(\mathbf{x}_{T}^{(k)})$ such
that $\mathbf{x}_{T}^{k|i}<\omega_{ik}$ then, since $x_{ik}$ is a best
response of player $i$ in the $k$-th CPR, by Remark 1, it would assume a value
for which $x_{ik}+\mathbf{x}_{T}^{k|i}<\omega_{ij}$, which contradicts the
fact that $\mathbf{x}_{T}^{(k)}\geq 1$. The claim follows.

However, since $x_{ik}$ is a best response and
$\mathbf{x}_{T}^{k|i}\geq\omega_{ik}$, for all
$i\in\mathcal{S}_{II}(\mathbf{x}_{T}^{(k)})$, it follows that $x_{ik}=0$, for
all $i\in\mathcal{S}_{II}(\mathbf{x}_{T}^{(k)})$. This contradicts the fact
that $\mathbf{x}_{T}^{(k)}\geq 1$, and the result follows. ∎

######  Lemma 8.

Assume that $m\leq n$. Then there do not exist distinct elements
$\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$ and
$\mathbf{y}=(\mathbf{y}_{1},\ldots,\mathbf{y}_{n})$ in $\mathcal{N}(G^{(2)})$
for which it holds $\mathbf{x}_{T}^{(j)}\leq\mathbf{y}_{T}^{(j)}$, for all
$j\in[m]$, and
$\sum_{j\in[m]}\mathbf{x}_{T}^{(j)}<\sum_{j\in[m]}\mathbf{y}_{T}^{(j)}$.

###### Proof.

Suppose that such GNEs do exist. Since $m\leq n$, it follows from Lemma 7 that
$\mathcal{T}_{I}(\mathbf{x})\neq\emptyset$, for every
$\mathbf{x}\in\mathcal{N}(G^{(2)})$.

Notice that Lemma 6 implies that
$\mathcal{T}_{I}(\mathbf{x})\subset\mathcal{T}_{I}(\mathbf{y})$ and
$J_{\mathbf{y}_{-i}}\subset J_{\mathbf{x}_{-i}}$, and (26) implies that
$\mathcal{G}_{ij}(\mathbf{x}_{T}^{(j)})=x_{ij}$, for all
$i\in\mathcal{T}_{I}(\mathbf{x})$ and all $j\in J_{\mathbf{x}_{-i}}$.
Similarly, it holds $\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})=y_{ij}$, for all
$i\in\mathcal{T}_{I}(\mathbf{y})$ and all $j\in J_{\mathbf{y}_{-i}}$. Hence we
may write

$\sum_{j\in[m]}\mathbf{x}_{T}^{(j)}=\sum_{i\in\mathcal{T}_{I}(\mathbf{x})}\sum_{j\in
J_{\mathbf{x}_{-i}}}\mathcal{G}_{ij}(\mathbf{x}_{T}^{(j)})+|\mathcal{T}_{II}(\mathbf{x})|$

as well as

$\sum_{j\in[m]}\mathbf{y}_{T}^{(j)}=\sum_{i\in\mathcal{T}_{I}(\mathbf{y})}\sum_{j\in
J_{\mathbf{y}_{-i}}}\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})+|\mathcal{T}_{II}(\mathbf{y})|\,.$

Since $\sum_{j}\mathbf{x}_{T}^{(j)}<\sum_{j}\mathbf{y}_{T}^{(j)}$,
$|\mathcal{T}_{I}(\mathbf{x})|\leq|\mathcal{T}_{I}(\mathbf{y})|$ and
$|\mathcal{T}_{II}(\mathbf{x})|\geq|\mathcal{T}_{II}(\mathbf{y})|$ hold true,
it follows that

$\displaystyle\sum_{i\in\mathcal{T}_{I}(\mathbf{x})}\,\sum_{j\in
J_{\mathbf{x}_{-i}}}\mathcal{G}_{ij}(\mathbf{x}_{T}^{(j)})+|\mathcal{T}_{II}(\mathbf{x})\setminus\mathcal{T}_{II}(\mathbf{y})|$
$\displaystyle<$
$\displaystyle\sum_{i\in\mathcal{T}_{I}(\mathbf{x})}\,\sum_{j\in
J_{\mathbf{y}_{-i}}}\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})$ (32)
$\displaystyle+$
$\displaystyle\sum_{i\in\mathcal{T}_{I}(\mathbf{y})\setminus\mathcal{T}_{I}(\mathbf{x})}\,\,\sum_{j\in
J_{\mathbf{y}_{-i}}}\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})\,.$

However, the fact that $\mathcal{G}_{ij}$ is decreasing implies that
$\mathcal{G}_{ij}(\mathbf{x}_{T}^{(j)})\geq\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})$,
for all $i\in\mathcal{T}_{I}(\mathbf{x})$ and all $j\in J_{\mathbf{y}_{-i}}$;
hence it holds

$\sum_{i\in\mathcal{T}_{I}(\mathbf{x})}\,\sum_{j\in
J_{\mathbf{x}_{-i}}}\mathcal{G}_{ij}(\mathbf{x}_{T}^{(j)})\geq\sum_{i\in\mathcal{T}_{I}(\mathbf{x})}\,\sum_{j\in
J_{\mathbf{y}_{-i}}}\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})\,.$ (33)

Moreover, since $\sum_{j\in
J_{\mathbf{y}_{-i}}}\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})<1$, for all
$i\in\mathcal{T}_{I}(\mathbf{y})\setminus\mathcal{T}_{I}(\mathbf{x})$, it
holds

$\sum_{i\in\mathcal{T}_{I}(\mathbf{y})\setminus\mathcal{T}_{I}(\mathbf{x})}\,\,\sum_{j\in
J_{\mathbf{y}_{-i}}}\mathcal{G}_{ij}(\mathbf{y}_{T}^{(j)})<|\mathcal{T}_{I}(\mathbf{y})\setminus\mathcal{T}_{I}(\mathbf{x})|=|\mathcal{T}_{II}(\mathbf{x})\setminus\mathcal{T}_{II}(\mathbf{y})|\,.$
(34)

Now notice that (33) and (34) contradict (32). The result follows. ∎

Finally, the proof of Theorem 4 requires the following measure-theoretic
results. Here and later, given a positive integer $k\geq 1$, $\mathcal{L}^{k}$
denotes $k$-dimensional Lebesgue measure. Moreover, given a function
$f:\mathbb{R}^{k}\to\mathbb{R}^{m}$ and a set $B\subset\mathbb{R}^{m}$, we
denote $f^{-1}(B):=\\{\mathbf{x}\in\mathbb{R}^{k}:f(\mathbf{x})\in B\\}$ the
preimage of $B$ under $f$.

######  Lemma 9.

Let $f:\mathbb{R}^{d}\to\mathbb{R}^{m}$ be a continuously differentiable
function for which $\mathcal{L}^{d}(\\{\mathbf{x}\in\mathbb{R}^{d}:\nabla
f(\mathbf{x})=0\\})=0$. Then it holds $\mathcal{L}^{d}(f^{-1}(A))=0$, for
every $A\subset\mathbb{R}^{m}$ for which $\mathcal{L}^{m}(A)=0$.

###### Proof.

See [23, Theorem 1]. ∎

Let $m\geq 1$ be an integer. A set $A\subset[0,1]^{m}$ is called an
_antichain_ if it does not contain two distinct elements
$\mathbf{x}=(x_{1},\ldots,x_{m})$ and $\mathbf{y}=(y_{1},\ldots,y_{m})$ such
that $x_{j}\leq y_{j}$, for all $j\in[m]$.

######  Lemma 10.

Let $A\subset[0,1]^{m}$ be an antichain. Then $\mathcal{L}^{m}(A)=0$.

###### Proof.

The result is an immediate consequence of Lebesgue’s density theorem.
Alternatively, it follows from the main result in [9], and from [10, Theorem
1.3]. ∎

Now given
$\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathcal{C}_{n}$, let
$\mathbf{v}_{\mathbf{x}}$ denote the vector

$\mathbf{v}_{\mathbf{x}}:=(\mathbf{x}_{T}^{(1)},\ldots,\mathbf{x}_{T}^{(m)})\in[0,1]^{m},$
(35)

where $\mathbf{x}_{T}^{(j)},j\in[m]$, is defined in (4). Finally, given
$N\subset\mathcal{C}_{n}$, define the set

$W_{N}:=\bigcup_{\mathbf{x}\in N}\mathbf{v}_{\mathbf{x}}\,.$ (36)

The proof of Theorem 4 is almost complete.

###### Proof of Theorem 4.

To simplify notation, let us set $N:=\mathcal{N}(G^{(2)})$. We have to show
that $\mathcal{L}^{nm}(N)=0$.

Now let $f$ denote the map $f:\mathcal{C}_{n}\to[0,1]^{m}$ given by
$f(\mathbf{x})=\mathbf{v}_{\mathbf{x}}$, where $\mathbf{v}_{\mathbf{x}}$ is
defined in (35). It is straightforward to verify that
$\\{\mathbf{x}\in\mathcal{C}_{n}:\nabla f(\mathbf{x})=0\\}=\emptyset$.

Now consider the set $W_{N}$, defined in (36), and notice that Lemma 8 implies
that $W_{N}$ is an antichain; hence it follows from Lemma 10 that
$\mathcal{L}^{m}(W_{N})=0$. Therefore, Lemma 9 yields

$\mathcal{L}^{nm}(N)=\mathcal{L}^{nm}(f^{-1}(W_{N}))=0\,,$

as desired. ∎

##  8 A restricted version of the game

Let $G^{(2)}$ denote a Fragile multi-CPR Game satisfying Assumption 2. In this
section we show that $G^{(2)}$ admits finitely many GNEs, subject to the
constraint that the total investment in each CPR is fixed. We then use this
result, in the next section, in order to formulate a conjecture which is
equivalent to Conjecture 1. Before being more precise, we need some extra
piece of notation.

Given a set $F\subset[m]$ and real numbers $\\{r_{j}\\}_{j\in F}\subset[0,1]$,
indexed by $F$, we denote by $W(\\{r_{j}\\}_{j\in F})$ the set

$W(\\{r_{j}\\}_{j\in
F}):=\\{\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})\in\mathcal{C}_{n}:\mathbf{x}_{T}^{(j)}=r_{j},\text{
for }j\in F\\}\,,$

where $\mathbf{x}_{T}^{(j)}$ is defined in (4). In other words,
$W(\\{r_{j}\\}_{j\in F})$ consists of those strategy profiles for which the
total investment in the CPRs corresponding to elements in $F$ is fixed, and
equal to the given numbers $\\{r_{j}\\}_{j\in F}$.

In this section we prove the following.

######  Theorem 10.

Fix real numbers $r_{1},\ldots,r_{m}\in[0,1]$. Then the set
$W:=W(r_{1},\ldots,r_{m})$ contains at most $2^{n\cdot(m+1)}$ GNEs of
$G^{(2)}$.

The proof requires a couple of observations which we collect in the following
lemmata.

######  Lemma 11.

Suppose that $\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$ and
$\mathbf{y}=(\mathbf{y}_{1},\ldots,\mathbf{y}_{n})$ are two GNEs of $G^{(2)}$
such that $\mathbf{x},\mathbf{y}\in W:=W(r_{j})$ and $0<x_{ij}<y_{ij}$, for
some $i\in[n]$, $j\in[m]$ and $r_{j}\in[0,1]$. Then either $\mathbf{x}_{i}$ is
of Type II or $\mathbf{y}_{i}$ is of Type II.

###### Proof.

Suppose, towards arriving at a contradiction, that the conclusion is not true.
Then both $\mathbf{x}_{i}$ and $\mathbf{y}_{i}$ are of Type I, and thus (26)
implies that $\mathcal{G}_{ij}(r)=x_{ij}$ and $\mathcal{G}_{ij}(r)=y_{ij}$.
Hence it holds
$\mathcal{G}_{ij}(r_{j})=x_{ij}<y_{ij}=\mathcal{G}_{ij}(r_{j})$, a
contradiction. The result follows. ∎

######  Lemma 12.

Suppose that $\mathbf{x}=(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$ and
$\mathbf{y}=(\mathbf{y}_{1},\ldots,\mathbf{y}_{n})$ are two GNEs of $G^{(2)}$
for which it holds $\mathbf{x},\mathbf{y}\in W:=W(r_{l},r_{\ell})$ and
$0<x_{ij}<y_{ij}$ and $x_{i\ell}>y_{i\ell}>0$, for some $i\in[n]$ and
$\\{j,\ell\\}\subset[m]$. Then either $\mathbf{x}_{i}$ is of Type I or
$\mathbf{y}_{i}$ is of Type I.

###### Proof.

Suppose, towards arriving at a contradiction, that both $\mathbf{x}_{-i}$ and
$\mathbf{y}_{-i}$ are of Type II. Recall the definition of
$\psi_{ij}(\,\cdot\,;\,\cdot\,)$, given in (14), and notice that, since both
$\mathbf{x}_{i},\mathbf{y}_{i}$ are of Type II, Theorem 8 implies the
existence of $\kappa_{x},\kappa_{y}\geq 0$ such that

$\kappa_{x}=x_{ij}^{a_{i}-1}\cdot\psi_{ij}(x_{ij};r_{j}-x_{ij})=x_{i\ell}^{a_{i}-1}\cdot\psi_{i\ell}(x_{i\ell};r_{\ell}-x_{i\ell})$

and

$\kappa_{y}=y_{ij}^{a_{i}-1}\cdot\psi_{ij}(y_{ij};r_{j}-y_{ij})=y_{i\ell}^{a_{i}-1}\cdot\psi_{i\ell}(y_{i\ell};r_{\ell}-y_{i\ell})\,.$

Now notice that, for all $k\in[m]$, the function
$\Psi_{k}(x):=x^{a-1}\cdot\psi_{ik}(x;r-x)$ is decreasing in $x$, for fixed
$r>0$ and $a\in(0,1]$. Hence, $x_{ij}<y_{ij}$ implies that
$\kappa_{x}>\kappa_{y}$, and $x_{i\ell}>y_{i\ell}$ implies that
$\kappa_{x}<\kappa_{y}$, a contradiction. The result follows. ∎

We may now proceed with the proof of the main result of this section.

###### Proof of Theorem 10.

For every $i\in[n]$, define the set

$N_{i}:=\\{\mathbf{x}_{i}\in C_{m}:(\mathbf{x}_{i},\mathbf{x}_{-i})\in
W,\text{ for some }\mathbf{x}_{-i}\in\mathcal{C}_{-i}\\}\,.$

We first show that the cardinality of $N_{i}$, denoted $|N_{i}|$, is at most
$2^{m+1}$.

Let $\mathbf{x}_{i}\in N_{i}$, and recall from Theorem 8, and (18), that there
exists $J\subset[m]$ such that $x_{ij}>0$, when $j\in J$, and $x_{ij}=0$ when
$j\in[m]\setminus J$. In other words, to every $\mathbf{x}_{i}\in N_{i}$ there
corresponds a set $J\subset[m]$ such that $x_{ij}>0$ if and only if $j\in J$.
Now, given $J\subset[m]$, let

$N_{J}:=\\{\mathbf{x}_{i}\in N_{i}:x_{ij}>0\text{ if and only if }j\in
J\\}\,.$

Assume first that $|J|\geq 2$. In this case we claim that $|N_{J}|\leq 2$.
Indeed, if $|N_{J}|\geq 3$, then there are two elements, say
$\mathbf{x}^{(1)},\mathbf{x}^{(2)}\in N_{J}$, which are either both of Type I,
or both of Type II. If both $\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$ are of
Type I, then there exists $j\in J$ such that, without loss of generality, it
holds $x_{ij}^{(1)}<x_{ij}^{(2)}$; which contradicts Lemma 11. If both
$\mathbf{x}^{(1)}$ and $\mathbf{x}^{(2)}$ are of Type II, then there exist
$j,\ell\in J$ such that $x_{ij}<y_{ij}$ and $x_{i\ell}>y_{i\ell}$; which
contradicts Lemma 12. The claim follows.

If $|J|=1$, say $J=\\{j\\}$, we claim that $|N_{J}|\leq 1$. Indeed, suppose
that $|N_{J}|\geq 2$ holds true and notice that every element of $N_{J}$ is of
Type I. However, the assumption that $|N_{J}|\geq 2$ implies that there exist
$\mathbf{x}^{(1)},\mathbf{x}^{(2)}\in N_{J}$ such that
$0<x_{ij}^{(1)}<y_{ij}^{(1)}$; which contradicts Lemma 11. The second claim
follows.

Since there are $2^{m}$ subsets $J\subset[m]$, and for each $J$ it holds
$|N_{J}|\leq 2$, it follows that there are at most $2^{m+1}$ elements in
$N_{i}$. Since there are $n$ players in the game, the result follows. ∎

##  9 Concluding remarks and conjectures

Let $G^{(2)}$ denote a Fragile multi-CPR Game satisfying Assumption 2, and let
$\mathcal{N}(G^{(2)})$ be the set consisting of all GNEs of $G^{(2)}$. So far
we have proven that the $(n\cdot m)$-dimensional Lebesgue measure of
$\mathcal{N}(G^{(2)})$ equals zero, but there are several problems and
questions that remain open. First and foremost, we believe that the following
holds true.

######  Conjecture 2.

Let $N:=\mathcal{N}(G^{(2)})$. Then the antichain $W_{N}$, defined in (36), is
finite.

Notice that if Conjecture 2 holds true then, in view of Theorem 10, Conjecture
1 holds true as well. Since the converse is clearly true, it follows that
Conjecture 1 and Conjecture 2 are equivalent. The exact number of GNEs in a
Fragile multi-CPR Game appears to depend on the relation between the number of
players, $n$, and the number of CPRs, $m$. When $n\geq m$ we conjecture that
that for every GNE the players choose best responses of Type I and therefore,
provided this is indeed the case, Theorem 9 would imply that the game admits a
unique GNE.

######  Conjecture 3.

If $n\geq m$, then $|\mathcal{N}(G^{(2)})|=1$.

Another line of research is to investigate the _best response dynamics_ of a
Fragile multi-CPR Game, which may be seen as a behavioral rule along which
players fix an initial investment in the CPRs and proceed with updating their
investment, over rounds, in such a way that in the $t$-th round player
$i\in[n]$ invests ${}_{i}^{(t)}:=B_{i}(\mathbf{x}_{-i}^{(t)})$, where
$B_{i}(\cdot)$ is defined in (13) and
$\mathbf{x}_{-i}^{(t)}\in\mathcal{C}_{-i}$ is the strategy profile of all
players except player $i$ in the $t$-th round. A natural question to ask is
whether the best response dynamics converge, i.e., whether there exists a
round $t_{0}$ such that ${}_{i}^{(t)}=_{i}^{(t_{0})}$, for all $t\geq t_{0}$
and all $i\in[n]$.

######  Conjecture 4.

The best response dynamics of $G^{(2)}$ converge.

When $m=1$, it is shown in [13] that the best response dynamics of the Fragile
CPR Game converge to its Nash Equilibrium. This is obtained as a consequence
of the fact that the best response correspondence is single-valued and
decreasing in the total investment in the CPR (see the remarks following [13,
Proposition 7]). Moreover, it is not difficult to verify that the Nash
equilibrium of the Fragile CPR Game is also the Generalized Nash equilibrium.
Hence, the best response dynamics of a Fragile CPR Game converge to the
Generalized Nash equilibrium. When $m\geq 2$, the best response correspondence
need no longer be decreasing in each CPR. It is decreasing for those players
whose best response is of Type I, as can be easily seen using the fact that
the auxiliary function $\mathcal{G}_{ij}$ is decreasing. This monotonicity may
no longer be true when a player moves from a best response of Type II to a
best response of Type I, or from a best response of Type II to a best response
of the same type. Furthermore, Theorem 8 does not guarantee that the set of
effective CPR, defined in (18), is unique. Hence, the best response
correspondence may not be single-valued. So far our theoretical analysis does
not provide sufficient evidence for the holistic validity of Conjecture 4.
However, our numerical experiments suggest that Conjecture 4 holds true, and
we expect that we will be able to report on that matter in the future.

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