aa r X i v : . [ c s . G T ] F e b A Fragile multi-CPR Game *Christos Pelekis † Panagiotis Promponas ‡ Juan Alvarado § Eirini Eleni Tsiropoulou ¶ Symeon Papavassiliou || February 26, 2021
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 play-ers. Each player has a fixed initial endowment and is faced with the task of investingin 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 investigatean extended version of a Fragile CPR Game, in which players are allowed to sharemultiple common-pool resources that are also prone to failure due to overuse. Werefer to this game as a Fragile multi-CPR Game. Our main result states that, undersome mild assumptions, a Fragile multi-CPR Game admits a Generalized Nash equi-librium. Moreover, we show that, when there are more players than common-poolresources, the set consisting of all Generalized Nash equilibria of a Fragile multi-CPRGame is of Lebesgue measure zero.
Keywords and phrases : CPR games; prospect theory; Generalized Nash equilibrium
MSC(2010) : 91A06; 90C25 * 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 procurementof high-cost research equipment grant? (Project Number: HFRI-FM17-2436). † School of Electrical and Computer Engineering, National Technical University of Athens, Zografou,Greece, 15780, e-mail: [email protected] ‡ School of Electrical and Computer Engineering, National Technical University of Athens, Zografou,Greece, 15780, e-mail: [email protected] § KU Leuven, Department of Computer Sciences, Celestijnenlaan 200A, 3001, Belgium, e-mail:[email protected] ¶ Department of Electrical and Computer Engineering, University of New Mexico, New Mexico, USA,87131, e-mail: [email protected] || School of Electrical and Computer Engineering, National Technical University of Athens, Zografou,Greece, 15780, e-mail: [email protected] Prologue, related work and main results
In this article we shall be concerned with a resource sharing game . Such games model in-stances in which a common-pool resource (henceforth CPR), which is prone to failure dueto overuse, is shared among several users who are addressing the problem of choosinghow much to exploit from / invest in the CPR without forcing it to fail. Resource shar-ing games arise in a variety of problems ranging from economics to computer science.Examples of CPRs include arable lands, forests, fisheries, groundwater basins, spectrumand computing resources, the atmosphere, among many others. Such CPRs are, on theone hand, usually regenerative but, on the other hand, subject to failure when severalagents exploit the resource in an unsustainable manner. Each agent exploits / invests inthe CPR in order to obtain an individual benefit. However, it has been observed that ac-tions which are individually rational (e.g. Nash equilibria) may result in outcomes thatare collectively irrational, thus giving rise to a particular social dilemma known as “thetragedy of the commons” (see [12]). It is thus of interest to investigate equilibrium pointsof resource sharing games, in order to better understand situations where such a socialdilemma arises. This is a topic that has drawn considerable attention, both from a theo-retical 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 usremark that most results in the literature appear to focus on games in which players investon a single CPR. In this article we investigate a resource sharing game in which playersare allowed to invest in several CPRs, whose performances are mutually independent. Tothe best of our knowledge our work appears to be among the first to consider resourcesharing games on more than one CPR.We shall be interested in a multi-version of a particular resource sharing game, whichis referred to as a
Fragile CPR Game . It is initially introduced in [13] and is played by sev-eral players, each of whom has a fixed initial endowment and must decide how much toinvest in the CPR without forcing it to fail. The return from the CPR is subject to uncer-tainty, 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 anextended version of a Fragile CPR Game in which players are allowed to share multipleCPRs. We refer to the corresponding game as a
Fragile multi-CPR Game and investigateits
Generalized Nash equilibria . Our main result states that the set consisting of all Gener-alized Nash equilibria of a Fragile multi-CPR Game is non-empty and, when there aremore players than CPRs, “small” in a measure-theoretic sense. In the next subsection weintroduce the Fragile CPR Game and state the main result from [13]. We then proceed, inSubsection 1.2, with defining the Fragile multi-CPR Game, which is the main target of thiswork, and stating our main results. 2 .1 Fragile CPR game
Throughout the text, given a positive integer n , we denote by [ n ] the set { , . . . , n } . Inthis article we extend a particular resource sharing game to the case where the players areallowed to share multiple resources, by determining how to distribute/invest their initialfixed endowment in the available CPRs. The resource sharing game under considerationis 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 areassumed to be indexed by the set [ n ] . It is also assumed that there is a single CPR, andeach player has to decide how much to invest in the CPR. Each player has an availableendowment, which, without loss of generality, is assumed to be equal to . Every player,say i ∈ [ n ] , invests an amount x i ∈ [0 , in the CPR. The total investment of all playersin the CPR is denoted x T = P i ∈ [ n ] x i . The return from the CPR is subject to uncertainty,that is there is a probability p ( x T ) that the CPR will fail, and this probability depends onthe total investment of the players in the CPR. In case the CPR fails, the players lose theirinvestment in the CPR. In case the CPR does not fail, then there is a rate of return from theCPR which depends on the total investment of all players, and is denoted by R ( x T ) . Therate of return is assumed to satisfy R ( x T ) > , for all x T .In other words, player i ∈ [ n ] gains x i · R ( x T ) − x i with probability − p ( x T ) , andgains − x i with probability p ( x T ) . The situation is modelled through a prospect-theoreticperspective, in the spirit of [16]. More precisely, let x ( i ) = P j ∈ [ n ] \{ i } x j ; hence it holds x i + x ( i ) = x T . Then the utility of player i ∈ [ n ] is given by the following utility function: V i ( x i , x ( i ) ) = ( ( x i · ( R ( x T ) − a i , with probability − p ( x T ) , − k i x a i i , with probability p ( x T ) . (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 > then a player weighs losses more than gains, a behaviour which is referred to as“loss averse”. On the other hand, when k i ∈ [0 , then a player weighs gains more thanlosses, a behaviour which is referred to as “gain seeking”. Capturing behaviours of thistype 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 ∈ [ n ] is risk neutral .Each player of the Fragile CPR game is an expected utility maximizer, and thereforechooses x i ∈ [0 , that maximizes the expectation of V ( x i , x ( i ) ) , i.e, that maximizes the utility of player i ∈ [ n ] which is given by E (cid:16) V i ( x i , x ( i ) ) (cid:17) = x a i i · F i ( x T ) , where F i ( x T ) = ( R ( x T ) − a i · (1 − p ( x T )) − k i · p ( x T ) (2)3s the effective rate of return to payer i ∈ [ n ] .The main result in [13] establishes, among other things, the existence of a unique Nashequilibrium for the Fragile CPR game, provided the following hold true. Assumption 1.
Consider a Fragile CPR game that satisfies the following properties.1. It holds p (0) = 0 and p ( x T ) = 1 , whenever x T ≥ .2. a i ∈ (0 , and k i > , for all i ∈ [ n ] .3. For all i ∈ [ n ] and all x T ∈ (0 , it holds ∂∂ x T F i ( x T ) , ∂ ∂ x T F i ( x T ) < , where F i is givenby (2) . In other words, the third condition in Assumption 1 states that the effective rate ofreturn of all players is a strictly decreasing and concave function. An example of an effec-tive rate of return F i satisfying the conditions of Assumption 1 is obtained by choosing a i < / , p ( x T ) = x T , and R ( x T ) = 2 − e x T − , as can be easily verified.Before proceeding with the main result from [13], let us recall here the notion of Nashequilibrium, adjusted to the setting of the Fragile CPR Game. Definition 1. (Nash Equilibrium) A
Nash equilibrium for a Fragile CPR Game is a strategyprofile ( x , . . . , x n ) ∈ [0 , n such that for all i ∈ [ n ] it holds: E (cid:16) V ( x i , x ( i ) ) (cid:17) ≥ E (cid:16) V ( z i , x ( i ) ) (cid:17) , for all z i ∈ [0 , . In other words, ( x , . . . , x n ) ∈ [0 , n is a Nash equilibrium for a Fragile CPR Gameif no player can increase her utility by unilaterally changing strategy. The main result inHota et al. [13] reads as follows. Theorem 1 ([13]) . Consider a Fragile CPR Game that satisfies Assumption 1. Then the gameadmits a unique
Nash equilibrium.
We now proceed with defining the
Fragile multi–CPR Game , whose equilibria are themain target of the present article.
In this article we introduce and investigate a multi-version of the Fragile CPR game. Inorder 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 = ( x , . . . , x m ) ∈ [0 , m : X i ∈ [ m ] x i ≤ . (3)4oreover, let C n denote the Cartesian product Q i ∈ [ n ] C m and let C − i = Q [ n ] \{ i } C m denotethe Cartesian product obtained from C n by deleting its i -th component. Elements in C − i are denoted by x − i , as is customary, and an element x = ( x , . . . , x n ) ∈ C n is occasionallywritten x = ( x i , x − i ) , for i ∈ [ n ] , x i ∈ C m and x − i ∈ C − i . We now proceed with definingthe Fragile multi-CPR Game .Suppose that there are n players, indexed by the set [ n ] , each having an initial endow-ment equal to . Assume further that there are m available CPRs, where m ≥ is aninteger. Every player has to decide how much to invest in each CPR. More precisely, ev-ery player, say i ∈ [ n ] , chooses an element x i = ( x i , . . . , x im ) ∈ C m and invests x ij in the j -th CPR. Given strategies x i = ( x i , . . . , x im ) ∈ C m , i ∈ [ n ] , of the players and an integer j ∈ [ m ] , set x ( j ) T = X i ∈ [ n ] x ij and x j | iT = X ℓ ∈ [ n ] \{ i } x ℓj . (4)Hence it holds x ( j ) T = x ij + x j | iT , for all i ∈ [ n ] . In other words, x ( j ) T equals the totalinvestment of the players in the j -th CPR and x j | iT equals the total investment of all playersexcept player i in the j -th CPR. As in the case of the Fragile CPR Game, we assume that theperformance of each CPR is subject to uncertainty, and that each CPR has a correspondingrate of return, both depending on the total investment of the players in each CPR. Moreprecisely, for j ∈ [ m ] , let R j ( x ( j ) T ) denote the return rate of the j -th CPR and let p j ( x ( j ) T ) denote the probability that the j -th CPR fails . We assume that R j ( x ( j ) T ) > holds true, forall x ( j ) T .The utility of player i ∈ [ n ] from the j -th CPR is given, as in the case of the Fragile CPRgame, via the following prospect-theoretic utility function: V ij ( x ij , x j | iT ) = ( ( x ij · ( R j ( x ( j ) T ) − a i , with probability − p j ( x ( j ) T ) , − k i x a i ij , with probability p j ( x ( j ) T ) . (5)We assume that the performance of each CPR is independent of the performances of allremaining CPRs. Players in the Fragile multi-CPR Game are expected utility maximizers.If player i ∈ [ n ] plays the vector x i = ( x i , . . . , x im ) ∈ C m , and the rest of the players play x − i ∈ C − i then her expected utility from the j -th CPR is equal to E ij ( x ij ; x j | iT ) := E (cid:16) V ij ( x ij , x j | iT ) (cid:17) = x a i ij · F ij ( x ( j ) T ) , (6)where F ij ( x ( j ) T ) := ( R j ( x ( j ) T ) − a i (1 − p j ( x ( j ) T )) − k i p j ( x ( j ) T ) (7)is the effective rate of return to the i -th player from the j -th CPR. Notice that, since weassume that the performance of each CPR is independent of the performances of the re-maining CPRs, E ij depends only on the values of x ij , x j | iT and does not depend on the5alues of x ik , x k | iT , for k = j . In other words, the (total) prospect-theoretic utility of player i ∈ [ n ] in the Fragile multi-CPR Game is given by: V i ( x i ; x − i ) = X j ∈ [ m ] E ij ( x ij ; x j | iT ) . (8)In this article we establish the existence of a Generalized Nash equilibrium for theFragile multi-CPR game, provided the following holds true. Assumption 2.
Consider a Fragile multi-CPR Game that satisfies the following properties:1. For every j ∈ [ m ] it holds p j (0) = 0 and p j ( x ( j ) T ) = 1 , whenever x ( j ) T ≥ .2. It holds a i ∈ (0 , and k i > , for all i ∈ [ n ] .3. For all i ∈ [ n ] and all j ∈ [ m ] it holds ∂∂ x ( j ) T F ij ( x ( j ) T ) , ∂ ∂ ( x ( j ) T ) F ij ( x ( j ) T ) < , where F ij isgiven by (7) . Notice that, similarly to the Fragile CPR Game, the third condition in Assumption 2states that the effective rate of return of every player from any CPR is a strictly decreasingand concave function. An example of an effective rate of return satisfying Assumption 2 isobtained by choosing, for j ∈ [ m ] , the return rate of the j -th CPR to be equal to R j ( x ( j ) T ) = c j + 1 , where c j > is a constant, and the probability that the j -th CPR fails to be a strictlyincreasing and convex, on the interval [0 , , function such that p j ( x ( j ) T ) = 1 , when x ( j ) T ≥ .Before stating our main result, let us proceed with recalling the notion of GeneralizedNash equilibrium (see [11]).Consider the, above-mentioned, Fragile multi-CPR Game, denoted G . Assume furtherthat, for each player i ∈ [ n ] , there exists a correspondence ϑ i : C − i → C m mapping everyelement x − i ∈ C − i to a set ϑ i ( x − i ) ⊂ C m . The set-valued correspondence ϑ i is referredto as a constraint policy and may be thought of as determining the set of strategies thatare feasible for player i ∈ [ n ] , given x − i ∈ C − i . We refer to the tuple ( G, { ϑ i } i ∈ [ n ] ) as the Constrained Fragile multi-CPR Game with constraint policies { ϑ i } i ∈ [ n ] . Corresponding to aconstrained game is the following notion of Constrained Nash equilibrium (or
GeneralizedNash equilibrium ): Definition 2 (GNE) . A Generalized Nash equilibrium for a Constrained Fragile multi-CPRGame ( G, { ϑ i } i ∈ [ n ] ) is a strategy profile x ∗ = ( x ∗ , . . . , x ∗ n ) ∈ C n such that1. For all i ∈ [ n ] , it holds x ∗ i ∈ ϑ i ( x ∗− i ) , for all i ∈ [ n ] , and2. For all i ∈ [ n ] , it holds V i ( x ∗ i ; x ∗− i ) ≥ V i ( x i ; x ∗− i ) , for all x i ∈ ϑ i ( x ∗− i ) , where V i ( · ; · ) isthe utility function of the i -th player in a Fragile multi-CPR Game, given in (8) .
6n other words, x ∗ = ( x ∗ , . . . , x ∗ n ) ∈ C n is a GNE if no player can increase her utility byunilaterally changing her strategy to any other element of the set ϑ i ( x ∗− i ) . We may nowproceed with stating our main results. Theorem 2.
Consider a Fragile multi-CPR game, G , with n ≥ players and m ≥ CPRs, whichsatisfies Assumption 2. Then there exist constraint policies { ϑ i } i ∈ [ n ] such that the ConstraintFragile multi-CPR Game ( G, { ϑ i } i ∈ [ n ] ) admits a Generalized Nash equilibrium. Given Theorem 2, it is natural to ask about the “size” of the set consisting of all GNEsof a Fragile multi-CPR Game. Let us note that it is a well known fact that GeneralizedNash equilibrium problems tend to possess infinitely many GNEs (see [11, p. 192]). In thecase of a single CPR, i.e., when m = 1 , the corresponding Constrained Fragile CPR Gameadmits a unique GNE. Theorem 3.
Consider a Fragile multi-CPR Game with n ≥ players and m = 1 CPR satisfyingAssumption 2. Then the game admits a unique GNE.
The proof of Theorem 3 is based upon a “first order condition” which is satisfied bythe best response correspondence in a Fragile multi-CPR Game. It turns out that the afore-mentioned “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 gen-eral statement (i.e., Theorem 9 below) which provides an upper bound on the numbers ofGNEs in a Fragile multi-CPR Game, subject to the assumption that best response of everyplayer is of the first type.For general m we are unable to determine the exact “size” of the set of GNEs. Weconjecture its size is always finite. Our main result, which is valid when there are moreplayers 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 ≥ players and m ≥ CPRs,which satisfies Assumption 2. Assume further that m ≤ n , and let N ( G (2) ) be the set consistingof all Generalized Nash equilibria of G (2) . Then the ( n · m ) -dimensional Lebesgue measure of N ( G (2) ) is equal to zero. As mentioned already, and despite the fact that GNE problems tend to possess in-finitely many solutions, we speculate that the “size” of the set N ( G (2) ) in Theorem 4 canbe reduced significantly. Conjecture 1.
The set N ( G (2) ) is finite. 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 certainadditional technicalities arise that are substantially different from those addressed in the7roof of Theorem 1 in [13]. First and foremost, in a Fragile multi-CPR Game the strategyspace of each player consists of m -dimensional vectors, a setting which requires conceptsand ideas from multi-variable calculus.In [13] the existence of a Nash equilibrium in a Fragile CPR Game is established intwo ways: the first approach employs Brouwer’s fixed point theorem, and the secondapproach employs ideas from a particular class of games known as Weak Strategic Substi-tute Games (see [7]). The first approach requires, among other things, the best responsecorrespondence to be single-valued. The second approach requires the best-response cor-respondence 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 forthe Fragile multi-CPR Game by showing that it belongs to a particular class of “convexconstrained games” which are known to possess Generalized Nash equilibria.In [13] the uniqueness of the Nash equilibrium for a Fragile CPR Game is establishedby showing that a particular auxiliary function, corresponding to the fact that the best re-sponse 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 areemployed in the proofs of Theorems 3 and 4. However, the corresponding “first orderconditions” are more delicate to characterise, and we do so by employing the KKT con-ditions 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 equa-tions, having unique solution, and results in two types of “first order conditions” (seeTheorem 8 below). Having established the first order conditions in a Fragile multi-CPRGame, we complete the proofs of our main results by employing monotonicity propertiesof certain auxiliary functions, in a way which may be seen as an extension of the approachtaken in the proof of Theorem 1 in [13].
The remaining part of our article is organised as follows. In Section 2 we show that theutility function of each player in a Fragile multi-CPR Game is concave on a particularsubset of the strategy space. In Section 3 we prove Theorem 2, namely, we show that aFragile multi-CPR Game admits a GNE. In Section 4 we show that the best response ofeach player in a Fragile multi-CPR Game satisfies certain “first order conditions”, whichare then used, in Section 5, in order to define suitable auxiliary functions whose mono-tonicity properties play a key role in the proofs of Theorems 3 and 4. Theorem 3 is provenin 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 thenemployed in order to formulate a conjecture which is equivalent to Conjecture 1. Ourpaper ends with Section 9 which includes some concluding remarks and conjectures.8
Concavity of utility function
In this section we show that the utility function, given by (8), of each player in a Fragilemulti-CPR Games is concave in some particular subset of C m . Before proceeding withthe details let us mention that this particular subset will be used to define the constraintpolicies in the corresponding Constrained Fragile multi-CPR Game.We begin with the following result, which readily follows from [13, Lemma 1]. Recallthe definition of x ( j ) T and x j | iT , given in (4), and the definition of the effective rate of return, F ij , given in (7). Lemma 1 (see [13], Lemma 1) . Let i ∈ [ n ] and x − i ∈ C − i be fixed. Then, for every j ∈ [ m ] ,there exists a real number ω ij ∈ (0 , such that F ij ( x ( j ) T ) > , whenever x ( j ) T ∈ (0 , ω ij ) , and F ij ( ω ij ) = 0 . Furthermore, provided that x j | iT < ω ij , the function E ij ( · ; x j | iT ) is concave in theinterval (0 , ω ij − x j | iT ) .Proof. We repeat the proof for the sake of completeness. Notice that F ij (0) > . Moreover,Assumption 2 implies that F ij (1) < . Since F ij is continuous, the intermediate valuetheorem implies that there exists ω ij ∈ (0 , such that F ij ( ω ij ) = 0 . Since F ij is assumedto be decreasing, the first statement follows, and we proceed with the proof of the secondstatement. To this end, notice that (6) yields ∂ ∂x ij E ij ( x ij ; x j | iT ) = a i ( a i − x a i − ij F ij ( x ( j ) T ) + 2 a i x a i − ij ∂∂x ij F ij ( x ( j ) T ) + x a i ij ∂ ∂x ij F ij ( x ( j ) T ) . Notice also that ∂∂x ij F ij ( x ( j ) T ) = ∂∂ x ( j ) T F ij ( x ( j ) T ) as well as ∂ ∂x ij F ij ( x ( j ) T ) = ∂ ∂ ( x ( j ) T ) F ij ( x ( j ) T ) .Moreover, Assumption 2 guarantees that ∂ ∂x ij F ij ( x ( j ) T ) , ∂∂x ij F ij ( x ( j ) T ) < as well as that a i − ≤ . Since F ij ( x ( j ) T ) > when x ( j ) T ∈ (0 , ω ij ) , we conclude that ∂ ∂x ij E ij ( x ij ; x j | iT ) < and therefore E ij ( · ; x j | iT ) is concave in the interval (0 , ω ij − x j | iT ) , as desired.In other words, given the choices of all players except player i , the utility of the i -thplayer 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 eachplayer in the Fragile multi-CPR Game, namely, V i ( x i ; x − i ) , given by (8).Given i ∈ [ n ] and x − i ∈ C − i , let A ( x − i ) := { j ∈ [ m ] : x j | iT < ω ij } , (9)where ω ij , j ∈ [ m ] , is provided by Lemma 1. We refer to A ( x − i ) as the set of active CPRs corresponding to i and x − i . 9 heorem 5. Fix i ∈ [ n ] and x − i ∈ C − i . Let A ( x − i ) be the set of active CPRs corresponding to i and x − i , and consider the set R A ( x − i ) = Q j ∈ A ( x − i ) (0 , ω ij − x j | iT ) . Then the function V A ( x − i ) := P j ∈ A ( x − i ) E ij ( x ij ; x j \ iT ) is concave in R A ( x − i ) .Proof. If | A ( x − i ) | = 1 , then the result follows from Lemma 1 so we may assume that | A ( x − i ) | ≥ . The set R A ( x − i ) is clearly convex. Let j, k ∈ A ( x − i ) be such that j = k andnotice that ∂ V A ( x − i ) ∂x ij ∂x ik = 0 . (10)Moreover, by Lemma 1, we also have ∂ V A ( x − i ) ∂x ij = ∂ E ij ( x ij ; x j | iT ) ∂x ij < , for all x ij ∈ (0 , ω ij − x j | iT ) . (11)Given x ∈ R A ( x − i ) , denote by H ( x ) = (cid:18) ∂ V A ( x − i ) ( x ) ∂x ij ∂x ik (cid:19) j,k ∈ A ( x − i ) the Hessian matrix of V A ( x − i ) evaluated at x , and let ∆ k ( x ) , for k ∈ A ( x − i ) , be the principal minors of H ( x ) (see [4, p. 111]). Notice that (10) implies that H ( x ) is a diagonal matrix. Therefore, us-ing (11), it follows that ( − k · ∆ k ( x ) > , when x ∈ R A ( x − i ) . In other words, H ( · ) isnegative definite on the convex set R A ( x − i ) and we conclude (see [4, Theorem 3.3, p. 110])that V A ( x − i ) is concave in R A ( x − i ) , as desired. In this section we show that the Fragile multi-CPR Game possesses a Generalized Nashequilibrium. Recall that the notion of Generalized Nash equilibrium depends upon thechoice of constraint policies. Thus, before presenting the details of the proof, we firstdefine the constrained policies under consideration.Let i ∈ [ n ] and x − i = ( x , . . . , x i − , x i +1 , . . . , x n ) ∈ C − i , where x j = ( x j , . . . , x jm ) ∈ C m , for j ∈ [ n ] \ { i } , be fixed. Recall that x j | iT = P ℓ ∈ [ n ] \{ i } x ℓj and consider the set of activeindices corresponding to i and x − i , i.e., consider the set A ( x − i ) , defined in (9).Define the constraint policy ϑ i ( · ) that maps each element x − i ∈ C − i to the set ϑ i ( x − i ) = C m \ Y j ∈ A ( x − i ) [0 , ω ij − x j | iT ] × Y j ∈ [ m ] \ A ( x − i ) { } , (12)where { ω ij } j ∈ A ( x − i ) is given by Lemma 1. Notice that, for every x − i ∈ C − i , the set ϑ i ( x − i ) is non-empty, compact and convex . Fig. 1 provides a visualization of the aforementionedconstraint policy, in the case of m = 2 . 10 ω ˆ ω C ϑ i ( x − i ) CP R C P R Figure 1: Visualization of an instance of the constraint policy ϑ i ( · ) (blue shaded region) ofplayer i in the case of m = 2 , where we denote ˆ ω = ω i − x | iT and ˆ ω = ω i − x | iT .We aim to show that the Constrained Fragile multi-CPR Game, with constraint policiesgiven by (12), admits a Generalized Nash equilibrium. In order to do so, we employ thefollowing theorem. Recall (see [15, p. 32–33]) that a set-valued correspondence φ : X → Y is upper semicontinuous if for every open set G ⊂ Y , it holds that { x ∈ X : φ ( x ) ⊂ G } isan open set in X . A set-valued correspondence φ : X → Y is lower semicontinuous if everyopen set G ⊂ Y , it holds that { x ∈ X : φ ( x ) ∩ G = ∅} is an open set in X . Recall also that,given S ⊂ R s , a function f : S → R is quasi-concave if f ( λ x +(1 − λ ) y ) ≥ min { f ( x ) , f ( y ) } ,for all x = y in S and λ ∈ (0 , . Clearly, a concave function is also quasi-concave. Theorem 6.
Let n players be characterized by strategy spaces X i , i ∈ [ n ] , constraint policies φ i , i ∈ [ n ] , and utility functions V i : Q i X i → R , i ∈ [ n ] . Suppose further that the following holdtrue for every i ∈ [ n ] :1. X i is non-empty, compact, convex subset of a Euclidean space.2. φ i ( · ) is both upper semicontinuous and lower semicontinuous in X − i .3. For all x − i ∈ X − i , φ i ( x − i ) is nonempty, closed and convex.4. V i is continuous in Q i X i .5. For every x − i ∈ X − i , the map x i
7→ V i ( x i , x − i ) is quasi-concave on φ i ( x − i ) . hen 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, k · k d denotes d -dimensional Euclidean distance, and B d ( ε ) := { x ∈ R d : k x k d ≤ ε } is the closed ball of radius ε centered at the origin. Moreover,given A ⊂ R d and ε > , we denote by { A } ε the set A + B d ( ε ) := { a + b : a ∈ A and b ∈ B d ( ε ) } and by (1 − ε ) · A the set { (1 − ε ) · a : a ∈ 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 6holds true. The third condition also holds true, by (12). Moreover, the fourth condition ofTheorem 6 is immediate from the definition of utility, given in (8), while the fifth conditionfollows from Theorem 5.It remains to show that the second condition of Theorem 6 holds true, i.e., that for each i ∈ [ n ] the constrained policy ϑ i ( · ) , given by (12), is both upper and lower semicontinuous.Towards this end, fix i ∈ [ n ] and let G ⊂ C m be an open set. Consider the sets G + := { x − i ∈ C − i : ϑ i ( x − i ) ⊂ G } and G − := { x − i ∈ C − i : ϑ i ( x − i ) ∩ G = ∅} . We have to show that both G + and G − are open subsets of C − i . We first show that G + isopen.If G + is empty then the result is clearly true, so we may assume that G + = ∅ . Let y = ( y , . . . , y i − , y i +1 , . . . , y n ) ∈ G + ; hence ϑ i ( y ) ⊂ G . We have to show that there exists ε > such that for every x ∈ C − i with k x − y k ( n − m < ε , we have ϑ i ( x ) ⊂ G . Since ϑ i ( y ) is a compact subset of the open set G , it follows that there exists ε > such that { ϑ i ( y ) } ε ⊂ G . Since summation is continuous, there exists ε > such that for every x ∈ C − i with k x − y k ( n − m < ε it holds x ∈ { ϑ i ( y ) } ε . The desired ε is given by ε .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 ∈ [ n ] , let g i : C − i → R m ≥ be thecontinuous function whose j -th coordinate, for j ∈ [ m ] , is given by g ij ( x − i ) = ( ω ij − x j | iT , if ω ij − x j | iT > , if ω ij − x j | iT ≤ , where ω ij is given by Lemma 1. Let h : R m ≥ → C m be the set-valued function definedby h ( z , . . . , z m ) = Q j ∈ [ m ] [0 , z j ] , with the convention [0 ,
0] := { } . Clearly, it holds that ϑ i = h ◦ g i , for all i ∈ [ n ] .We claim that h is lower semicontinuous. If the claim holds true then it follows thatthe set H := { z ∈ R m ≥ : h ( z ) ∩ G = ∅} is open. Notice that G − = ∅ implies that H = ∅ .12ince g i is continuous, it follows that the preimage of H under g i , i.e., g − i ( H ) , is open. Inother words, the set { x ∈ C − i : h ◦ g i ( x ) ∩ G = ∅} = { x ∈ C − i : ϑ i ( x ) ∩ G = ∅} is open andthe proof of the theorem is complete.It remains to prove the claim, i.e., that h is lower semicontinuous. To this end, let G ⊂ C m be an open set, and let G ∗ := { z ∈ R m ≥ : h ( z ) ∩ G = ∅} . We have to show that G ∗ is open; that is, we have to show that for every z ∈ G ∗ there exists ε > such that w ∈ G ∗ ,for all w with k z − w k m < ε . Fix z ∈ G ∗ . Since h ( z ) is compact and G is open, it followsthat there exists ε > such that (1 − ε ) · h ( z ) ∩ G = ∅ . Now choose ε > such that forevery w ∈ C m for which k z − w k m < ε it holds (1 − ε ) · h ( z ) ⊂ h ( w ) . In other words, forthis particular choice of ε > it holds h ( w ) ∩ G = ∅ , for every w with k z − w k m < ε . Theclaim follows. Having established the existence of a GNE for a Fragile multi-CPR Game, we now pro-ceed with the proofs of Theorems 3 and 4. The proofs will be obtained in two steps. Inthe first step we deduce certain “first order conditions” which are satisfied by the bestresponse correspondence of each player in the game. In the second step we employ thefirst order conditions in order to define certain auxiliary functions, whose monotonicitywill be employed in the proofs of the aforementioned theorems. In this section we collectsome results pertaining to the first step. We begin with recalling the notion of the bestresponse correspondence (see [19]).Given i ∈ [ n ] and x − i ∈ C − i , let ϑ i ( · ) denote the constraint policy given by (12), andconsider the best response of the i -th player in the Fragile multi-CPR Game defined asfollows: B i ( x − i ) = arg max x i ∈ ϑ i ( x − i ) V i ( x i ; x − i ) , (13)where V i is the utility of the i -th player, given by (8). Notice that B i ( · ) is a correspondence B i : C − i → C m , where C m denotes the class consisting of all subsets of C m . For j ∈ [ m ] ,we denote by B ij ( x − i ) the j -th component of B i ( x − i ) ; hence we have B i ( x − i ) = ( B i ( x − i ) , . . . , B im ( x − i )) . Remark 1.
Notice that Definition 2 implies that if x = ( x , . . . , x n ) ∈ C n is a GNE of a Con-strained Fragile multi-CPR Game, with constraint policies given by (12) , then for each i ∈ [ n ] itholds x i ∈ B i ( x − i ) . Recall that A ( x − i ) denotes the set of active CPRs corresponding to x − i , defined in (9),and notice that B ij ( x − i ) = 0 , for all j ∈ [ m ] \ A ( x − i ) .For x ij ∈ [0 , , let ψ ij ( x ij ; x j | iT ) be the function defined via ψ ij ( x ij ; x j | iT ) = x ij · ∂∂x ij F ij ( x ij + x j | iT ) + a i F ij ( x ij + x j | iT ) . (14)13 emma 2. Fix i ∈ [ n ] and x − i ∈ C − i and let R A ( x − i ) = Q j ∈ A ( x − i ) (0 , ω ij − x j | iT ) , where ω ij isprovided by Lemma 1. Then a global maximum of the function V x − i := P j ∈ A ( x − i ) E ij ( x ij ; x j | iT ) defined on the set R A ( x − i ) is given by the unique solution of the following system of equations: ψ ij ( x ij ; x j | iT ) = 0 , for j ∈ A ( x − i ) , (15) where ψ ij ( x ij ; x j | iT ) is defined in (14) .Proof. To simplify notation, we write ψ ij ( · ) instead of ψ ij ( · ; x j | iT ) . Using (6) and (8), it isstraightforward to verify that for every j ∈ A ( x − i ) it holds ∂ V x − i ∂x ij = ∂ E ij ( x ij ; x j | iT ) ∂x ij = x a i − ij · ψ ij ( x ij ) . (16)Now notice that ψ ij (0) > as well as ψ ij ( ω ij − x j | iT ) < . Moreover, Assumption 2 readilyimplies that ψ ij ( · ) is strictly decreasing on the interval (0 , ω ij − x j | iT ) . The intermediatevalue theorem implies that there exists unique λ ij ∈ (0 , ω ij − x j | iT ) such that ψ ij ( λ ij ) =0 . Hence, it follows from (16) that the points λ ij , for j ∈ A ( x − i ) , are critical points ofthe function V x − i , which is concave on the open and convex set R A ( x − i ) , by Theorem 5.It follows (see [4, Theorem 2.4, p. 132]) that { λ ij } j ∈ A ( x − i ) is a global maximum of V x − i on R A ( x − i ) . We conclude that V x − i is maximized when x ij = λ ij , for j ∈ A ( x − i ) , asdesired. Remark 2.
Let us remark that the solution of the system of equations given by (15) may not belongto the set C m . More precisely, it could happen that the solution of the system of equations (15) , say { λ ij } j ∈ A ( x − i ) , satisfies P j ∈ A ( x − i ) λ ij > . This is a crucial difference between the Fragile CPRGame and the Fragile multi-CPR Game. Now notice that, given x − i ∈ C − i , the best response of player i is a local maximum ofthe following program:maximize { x ij } j ∈ A ( x − i ) V x − i := X j ∈ A ( x − i ) E ij ( x ij ; x j | iT ) subject to X j ∈ A ( x − i ) x ij ≤ ≤ x ij ≤ ω ij − x j | iT , for all j ∈ A ( x − i ) . Equivalently, the best response of player i is a local minimum of the following program:14inimize { x ij } j ∈ A ( x − i ) − X j ∈ A ( x − i ) E ij ( x ij ; x j | iT ) subject to X j ∈ A ( x − i ) x ij ≤ ≤ x ij ≤ ω ij − x j | iT , for all j ∈ A ( x − i ) . (17)Notice that since E ij ( · ; x j | iT ) is concave on (0 , ω ij − x j | iT ) , by Lemma 1, it follows thatProblem (17) is a separable convex knapsack program (see [20, 24]). We are going to de-scribe the optima of Problem (17) using the KKT conditions. The KKT conditions pertainto the Lagrangian corresponding to Problem (17), which is defined as the following quan-tity: L := −V x − i + κ · X j ∈ A ( x − i ) x ij − + X j ∈ A ( x − i ) µ j · ( x ij + x j | iT − ω ij ) + X j ∈ A ( x − i ) ν j · ( − x ij ) , where κ , { µ j } j , { ν j } j are real numbers. The KKT conditions corresponding to prob-lem (17) read as follows (see [18, Theorem 3.8]). Theorem 7 (KKT conditions for Problem (17)) . If { x ij } j ∈ A ( x − i ) is a local minimum of Prob-lem (17) , then there exist non-negative real numbers κ , { µ j } j ∈ A ( x − i ) , and { ν j } j ∈ A ( x − i ) suchthat:1. For all j ∈ A ( x − i ) it holds − x a i − ij · ψ ij ( x ij ; x j | iT ) + κ + µ j − ν j = 0 , where ψ ij is givenby (14) .2. κ · (cid:16)P j ∈ A ( x − i ) x ij − (cid:17) = 0 .3. µ j · ( x ij + x j | iT − ω ij ) = 0 , for all j ∈ A ( x − i ) .4. ν j · x ij = 0 , for all j ∈ A ( x − i ) .5. ≤ x ij ≤ ω ij − x j | iT , for all j ∈ A ( 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 ispresumably reported somewhere in the literature but, lacking a reference, we include aproof for the sake of completeness.
Lemma 3.
Fix a positive integer s and, for each j ∈ [ s ] , let f j : R → R be a strictly decreasingfunction. Then there exists at most one vector ( c, x , . . . , x s ) ∈ R s +1 such that f j ( x j ) = c, for all j ∈ [ s ] , and X j ∈ [ s ] x j = 1 . roof. Suppose that there exist two distinct vectors, say ( c, x , . . . , x s ) and ( d, y , . . . , y s ) .If c = d , then there exists j ∈ [ s ] such that x j = y j and f j ( x j ) = c = d = f j ( y j ) , contrariwise to the assumption that the function f i ( · ) is strictly decreasing. Hence c = d .Since f j ( · ) , j ∈ [ s ] , is strictly decreasing, it is injective and therefore it follows that it isinvertible. Let us denote its inverse by f − j ( · ) . We then have x j = f − j ( c ) and y j = f − j ( d ) , for all j ∈ [ s ] , which in turn implies that x j = y j , for all j ∈ [ 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 ∈ [ s ] ,and therefore P j ∈ [ s ] x j > P j ∈ [ s ] y j = 1 , a contradiction. The result follows.We may now proceed with describing the best responses of each player in the Fragilemulti-CPR Game via a system of “first order conditions”. Theorem 8.
Let i ∈ [ n ] and x − i ∈ C − i be fixed. Suppose that { x ij } j ∈ A ( x − i ) is a best response ofplayer i in the Fragile multi-CPR Game. Then { x ij } j ∈ A ( x − i ) is either of the following two types: • Type I:
There exists J x − i ⊂ A ( x − i ) such that x ij = 0 , when j ∈ A ( x − i ) \ J x − i , and { x ij } j ∈ J x − i satisfy the following inequality, and are given by the unique solution of thefollowing system of equations: X j ∈ J x − i x ij < and ψ ij ( x ij ; x j | iT ) = 0 , for j ∈ J x − i , where ψ ij ( · ; x j | iT ) is defined in (14) . • Type II:
There exists J x − i ⊂ A ( x − i ) and a real number κ ≥ such that x ij = 0 , when j ∈ A ( x − i ) \ J x − i , and { x ij } j ∈ J are given by the unique solution of the following systemof equations: X j ∈ J x − i x ij = 1 and x a i − ij · ψ ij ( x ij ; x j | iT ) = κ , for j ∈ J x − i , where ψ ij ( · ; x j | iT ) is defined in (14) .Proof. Let { x ij } j ∈ A ( x − i ) be a best response of player i ∈ [ n ] . Then { x ij } j ∈ A ( x − i ) is a localminimum of Problem (17); hence it satisfies the KKT Conditions of Theorem 7.If x ij = ω ij − x j | iT , for some j ∈ A ( x − i ) , then Lemma 1 and (6) imply that E ij ( x ij ; x j | iT ) =0 . Hence player i could achieve the same utility from the j -th CPR by choosing x ij = 0 .16hus we may assume that x ij < ω ij − x j | iT , for all j ∈ A ( x − i ) and therefore Theorem 7.(3)implies that µ j = 0 , for all j ∈ A ( x − i ) . Now let J x − i = { j ∈ A ( x − i ) : x ij = 0 } , (18)and notice that Theorem 7.(4) implies that ν j = 0 for j ∈ J x − i . We distinguish two cases.Suppose first that P j ∈ J x − i x ij < . Then Theorem 7.(2) yields κ = 0 , and thereforeTheorem 7.(1) implies that { x ij } i ∈ J x − i is given by the unique solution of the followingsystem of equations: ψ ij ( x ij ; x j | iT ) = 0 , for j ∈ J x − i . In other words, if P j ∈ J x − i x ij < then { x ij } j ∈ A ( x − i ) is of Type I.Now assume that P j ∈ J x − i x ij = 1 . Then Theorems 7.(1) and 7 .(2) imply that thereexists κ ≥ such that − x a i − ij · ψ ij ( x ij ; x j | iT ) = − κ , for all j ∈ J x − i . In other words, { x ij } j ∈ J x − i and κ are given by the solution of the following system of equations: X j ∈ J x − i x ij = 1 and x a i − ij · ψ ij ( x ij ; x j | iT ) = κ , for all j ∈ J x − i . (19)Since the functions f ij ( x ij ) := x a i − ij · ψ ij ( x ij ; x j | iT ) , for j ∈ J x − i , are strictly decreasing,Lemma 3 implies that the system of equations in (19) has a unique solution. Hence { x ij } j ∈ A ( x − i ) is of Type II and the result follows.We refer to the set J x − i provided by Theorem 8, defined in (18), as the set of effective CPRs corresponding to i ∈ [ n ] and x − i ∈ C − i . In the next section we employ Theorem 8in order to define auxiliary functions (i.e., (24) and (25) below) whose monotonicity willplay a key role in the proof of Theorem 4. In this section we define and state basic properties of certain auxiliary functions, whosemonotonicity will be used in the proofs of Theorems 3 and 4, and whose definition de-pends upon the “first order conditions” provided by Theorem 8.Let us begin with some notation and remarks. Fix i ∈ [ n ] and x − i ∈ C − i , and recallfrom (13) that B i ( x − i ) denotes a best response of player i and that B ij ( x − i ) is its j -thcomponent. To simplify notation, let us denote b ij := B ij ( x − i ) . From Theorem 8 we knowthat there exists J x − i ⊂ A ( x − i ) such that b ij = 0 , for j ∈ A ( x − i ) \ J x − i , and either X j ∈ J x − i b ij < and ψ ij ( b ij ; x − i ) = 0 , for all j ∈ J x − i , (20)17r X j ∈ J x − i b ij = 1 and b a i − ij · ψ ij ( b ij ; x − i ) = κ , for all j ∈ J x − i and some κ ≥ . (21)In particular, it holds b ij > , for all j ∈ J x − i . Using (14), it follows that the secondstatement of (20) is equivalent to b ij · ∂∂x ij F ij ( b ij + x j | iT ) + a i F ij ( b ij + x j | iT ) = 0 , for all j ∈ J x − i , (22)and that the second statement of (21) is equivalent to b a i − ij · (cid:18) b ij · ∂∂x ij F ij ( b ij + x j | iT ) + a i F ij ( b ij + x j | iT ) (cid:19) = κ , for all j ∈ J x − i . (23)Now, given x − i ∈ C − i , j ∈ J x − i and κ ≥ , define for each i ∈ [ n ] the functions G ij ( x ij + x j | iT ) := − a i F ij ( x ij + x j | iT ) ∂∂x ij F ij ( x ij + x j | iT ) , for x ij ∈ (0 , ω ij − x j | iT ) (24)and H ij ( x ij + x j | iT ; κ ) := − a i F ij ( x ij + x j | iT ) − κ x aiij + ∂∂x ij F ij ( x ij + x j | iT ) , for x ij ∈ (0 , ω ij − x j | iT ) . (25)Notice that (22) implies that when b ij is of Type I it holds G ij ( b ij + x j | iT ) = b ij , (26)while (23) implies that when b ij is of Type II it holds H ij ( b ij + x j | iT ; κ ) = b ij , . (27)Observe also that it holds G ij ( x ij + x j | iT ) ≥ H ij ( x ij + x j | iT ; κ ) , for all x ij ∈ [0 , ω ij − x j | iT ] . Letus, for future reference, collect a couple of observations about the functions G ij , H ij . Lemma 4.
Let i ∈ [ n ] and j ∈ [ m ] be fixed. Then the functions G ij ( · ) and H ij ( · ; κ ) , definedin (24) and (25) respectively, are strictly decreasing in the interval [0 , ω ij ] . roof. To simplify notation, let F := F ij ( x ij + x j | iT ) , F ′ := ∂∂x ij F and F ′′ := ∂ ∂x ij F . For x ij ∈ (0 , ω ij − x ( j ) T ) , we compute ∂∂x ij H ij ( x ij + x j | iT ; κ ) = − a i F ′ · ( − κ x aiij + F ′ ) + a i F · ( − a i − κ x ai +1 ij + F ′′ )( − κ x aiij + F ′ ) = a i · κ x a i − ij ( x ij F ′ + a i F ) − x a i ij ( F ′ ) + x a i ij F · F ′′ ( − κ + x a i ij F ′ ) < a i · κ x a i − ij ( x ij F ′ + a i F ) − x a i ij ( F ′ ) ( − κ + x a i ij F ′ ) , where the last estimate follows from the fact that, by Assumption 2, it holds F ′′ < . If κ = 0 , then it readily follows that ∂∂x ij H ij ( x ij + x j | iT ; κ ) < and therefore H ij is strictlydecreasing; thus G ij is strictly decreasing as well. So we may assume that κ > . If x ij F ′ + a i F < , then it also follows that H ij is strictly decreasing; thus we may alsoassume that A := x ij F ′ + a i F ≥ . Now notice that ∂A∂x ij = F ′ + x ij F ′′ + a i F ′ < , anddefine the function H ( x ij ) := κ x a i − ij · A − x a i ij ( F ′ ) ; hence it holds ∂∂x ij H ij ( x ij + x j | iT ; κ ) < a i · H ( x ij )( − κ + x aiij F ′ ) . Moreover, it holds ∂∂x ij H ( x ij ) = ( a i − κ x a i − ij · A + κ x a i − ij · ∂A∂x ij − a i x a i − ij ( F ′ ) − x a i ij F ′ F ′′ . Since a i ≤ , A ≥ and F ′ , F ′′ , ∂A∂x ij < , it readily follows that all addends in the previ-ous equation are negative, and therefore ∂∂x ij H ( x ij ) < . In other words, H ( · ) is strictlydecreasing in [0 , ω ij − x j | iT ] and, since H (0) = 0 , H ( ω ij − x j | iT ) < , we conclude that H ( x ij ) ≤ , for all x ij ∈ [0 , ω ij − x j | iT ] . This implies that ∂∂x ij H ij ( x ij + x j | iT ; κ ) < for x ij ∈ [0 , ω ij − x j | iT ] . Since ∂∂ x ( j ) T H ij ( x ( j ) T ; κ ) = ∂∂x ij H ij ( x ij + x j | iT ; κ ) , and similarly for G ij , we conclude that both G ij and H ij are strictly decreasing in the interval [0 , ω ij ] , asdesired. In this section we prove Theorem 3. We first introduce some notation. Consider a GNE,say x = ( x , . . . , x n ) ∈ C n , where x i = ( x i , . . . , x im ) ∈ C m , of a Fragile multi-CPR Gamesatisfying Assumption 2. Given j ∈ [ m ] , let S ( x ( j ) T ) = { i ∈ [ n ] : x ( j ) T < ω ij and x ij > } (28)19e the support of the j -th CPR and let S I ( x ( j ) T ) = { i ∈ S ( x ( j ) T ) : x i is of Type I } (29)be the support of Type I , consisting of those players in the support of the j -th CPR whosebest response is of Type I, and S II ( x ( j ) T ) = { i ∈ S ( x ( j ) T ) : x i is of Type II } (30)be the support of Type II , consisting of those players in the support of the j -th CPR whosebest response is of Type II. Clearly, in view of Theorem 8, it holds S ( x ( j ) T ) = S I ( x ( j ) T ) ∪S II ( x ( j ) T ) .We employ the properties of the auxiliary functions in the proof of Theorem 3, a basicingredient of which is the fact that in the setting of Theorem 3 the support of Type II isempty. The proof is similar to the proof of Theorem 1, given in [13, p. 155]. In fact, weprove 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 ≥ players and m ≥ CPRs satisfyingAssumption 2. Then there exists at most one GNE x = ( x , . . . , x n ) for which x i is of Type I, forall i ∈ [ n ] .Proof. Let x = ( x , . . . , x n ) be a GNE such that x i is of Type I, for all i ∈ [ n ] and notethat, since x i is of Type I, it holds P j x ij < , for all i ∈ [ n ] . For each j ∈ [ m ] , let S ( x ( j ) T ) := { i ∈ [ n ] : x ( j ) T < ω ij } . We claim that S ( x ( j ) T ) = S ( x ( j ) T ) . Indeed, if there exists i ∈ S ( x ( j ) T ) \ S ( x ( j ) T ) then x ij = 0 and since it holds x ( j ) T < ω ij and P j x ij < , it followsthat player i could increase her utility by investing a suitably small amount, say ε > ,in the j -th CPR. But then this implies that x cannot be a GNE, a contradiction. Hence S ( x ( j ) T ) = S ( x ( j ) T ) .We claim that for any two distinct GNEs, say x = ( x , . . . , x n ) and y = ( y , . . . , y n ) ,for which x i , y i are of Type I for all i ∈ [ n ] , it holds x ( j ) T = y ( j ) T , for all j ∈ [ m ] . Indeed, ifthe claim is not true, then there exists j ∈ [ m ] such that x ( j ) T = y ( j ) T . Suppose, without lossof generality, that x ( j ) T < y ( j ) T .Since x i is of Type I, for all i ∈ [ n ] , it follows that S ( x ( j ) T ) = S I ( x ( j ) T ) and S ( y ( j ) T ) = S I ( y ( j ) T ) . Moreover, since x ( j ) T < y ( j ) T it follows that S I ( y ( j ) T ) = S ( y ( j ) T ) ⊂ S ( x ( j ) T ) = S I ( x ( j ) T ) .Now notice that (26) implies that G ij ( x ( j ) T ) = x ij , for all i ∈ S I ( x ( j ) T ) , and G ij ( y ( j ) T ) = y ij ,for all i ∈ S I ( y ( j ) T ) . Since S I ( y ( j ) T ) ⊂ S I ( x ( j ) T ) it holds that X i ∈S I ( y ( j ) T ) G ij ( x ( j ) T ) ≤ x ( j ) T < y ( j ) T = X i ∈S I ( y ( j ) T ) G ij ( y ( j ) T ) . (31)20owever, since G ij is strictly decreasing, it follows that G ij ( x ( j ) T ) > G ij ( y ( j ) T ) , for all i ∈S I ( y ( j ) T ) , which contradicts (31). We conclude that x ( j ) T = y ( j ) T and S I ( x ( j ) T ) = S I ( y ( j ) T ) .Finally, given a total investment x of the players at a GNE, we claim that the optimalinvestment of every player on any CPR is unique. Indeed, if a player, say i ∈ [ n ] , has twooptimal investments, say x < z , on the j -th CPR, then it holds G ij ( x ) = x < z = G ij ( x ) , acontradiction. 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 there-fore enough to show that it is unique. Since m = 1 , the first condition in Assumption 2implies that no player invests an amount of in the CPR which in turn implies that allcoordinates 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 S ( x T ) = S I ( x T ) , S ( y T ) = S I ( y T ) and S I ( y T ) ⊂ S I ( x T ) . Moreover, observe that the proof of The-orem 9 proceeds in two steps: in the first step it is shown that any two GNEs admit thesame total investment in the CPR, and in the second step it is shown that, given an op-timal total investment, every player has a unique optimal investment in the CPR. In thefollowing section we are going to improve upon the aforementioned observations. A bitmore concretely, we are going to prove that the set consisting of all GNEs of a Fragilemulti-CPR Game is “small” via showing that the set consisting of all total investments atthe GNEs is “small”. Throughout this section, we denote by G (2) a Fragile multi-CPR Game satisfying Assump-tion 2. Moreover, given a finite set, F , we denote by | F | its cardinality. Now consider theset N ( G (2) ) := { x ∈ C n : x is a GNE of G (2) } and, given x = ( x , . . . , x n ) ∈ N ( G (2) ) , let T I ( x ) = { i ∈ [ n ] : x i is of Type I } and T II ( x ) = { i ∈ [ n ] : x i is of Type II } . Recall the definition of active CPRs corresponding to x − i , which is denoted A ( x − i ) and isdefined in (9), as well as the definition of effective CPRs corresponding to x − i , which isdenoted J x − i and is defined in (18). 21 emma 5. Let x = ( x , . . . , x n ) ∈ N ( G (2) ) and suppose that i ∈ T I ( x ) , for some i ∈ [ n ] . Thenit holds J x − i = A ( x − i ) .Proof. Recall from Theorem 8 that J x − i is such that x ij > if and only if j ∈ J x − i . Suppose,towards arriving at a contradiction, that there exists j ∈ A ( x − i ) \ J x − i . Since i ∈ T I ( x ) ,it follows that P j ∈ J x − i x ij < and thus player i can increase her utility by investing asuitably small amount ε > in the j -th CPR. This contradicts the fact that x is a GNE, andthe lemma follows. Lemma 6.
Let x = ( x , . . . , x n ) and y = ( y , . . . , y n ) be two elements from N ( G (2) ) such that x ( j ) T ≤ y ( j ) T , for all j ∈ [ m ] . Then the following hold true:1. If i ∈ T I ( x ) , then J y − i ⊂ J x − i .2. It holds T I ( x ) ⊂ T I ( y ) .Proof. Fix i ∈ [ n ] such that i ∈ T I ( x ) and notice that Lemma 5 implies that x ( j ) T ≥ ω ij , forall j ∈ [ m ] \ J x − i . Since x ( j ) T ≤ y ( j ) T , for all j ∈ [ m ] , it holds y ( j ) T ≥ ω ij , for all j ∈ [ m ] \ J y − i ,and we conclude that J y − i ⊂ J x − i . The first statement follows.We proceed with the second statement. Let i ∈ [ n ] be such that x i is of Type I. We haveto show that y i is also of Type I. Suppose that this is not true; hence y i is of Type II, andthus it holds P j ∈ J y − i y ij = 1 . Since y i is of Type II, it follows from (27) that H ij ( y ( j ) T ; κ ) = y ij , for all j ∈ J y − i and some κ ≥ . Since x i is of Type I and H ij is decreasing, we mayapply (26) and conclude x ij = G ij ( x ( j ) T ) ≥ H ij ( x ( j ) T ; κ ) ≥ H ij ( y ( j ) T ; κ ) = y ij , for all j ∈ J y − i . Hence > P j ∈ J y − i x ij ≥ P j ∈ J y − i y ij = 1 , a contradiction. The result follows. Lemma 7.
Assume that m ≤ n . Then it holds T I ( x ) = ∅ , for every x ∈ N ( G (2) ) .Proof. Suppose that the conclusion is not true; hence there exists x = ( x , . . . , x n ) ∈N ( G (2) ) such that T II ( x ) = [ n ] , which in turn implies that P j ∈ [ m ] x ( j ) T = n ≥ m . Hencethere exists k ∈ [ m ] such that x ( k ) T ≥ . We now claim that x k | iT ≥ ω ik , for all i ∈ S II ( x ( k ) T ) ,where S II ( · ) is defined in (30) and ω ik is given by Lemma 1. To prove the claim, noticethat if there exists i ∈ S II ( x ( k ) T ) such that x k | iT < ω ik then, since x ik is a best response ofplayer i in the k -th CPR, by Remark 1, it would assume a value for which x ik + x k | iT < ω ij ,which contradicts the fact that x ( k ) T ≥ . The claim follows.However, since x ik is a best response and x k | iT ≥ ω ik , for all i ∈ S II ( x ( k ) T ) , it followsthat x ik = 0 , for all i ∈ S II ( x ( k ) T ) . This contradicts the fact that x ( k ) T ≥ , and the resultfollows. 22 emma 8. Assume that m ≤ n . Then there do not exist distinct elements x = ( x , . . . , x n ) and y = ( y , . . . , y n ) in N ( G (2) ) for which it holds x ( j ) T ≤ y ( j ) T , for all j ∈ [ m ] , and P j ∈ [ m ] x ( j ) T < P j ∈ [ m ] y ( j ) T .Proof. Suppose that such GNEs do exist. Since m ≤ n , it follows from Lemma 7 that T I ( x ) = ∅ , for every x ∈ N ( G (2) ) .Notice that Lemma 6 implies that T I ( x ) ⊂ T I ( y ) and J y − i ⊂ J x − i , and (26) implies that G ij ( x ( j ) T ) = x ij , for all i ∈ T I ( x ) and all j ∈ J x − i . Similarly, it holds G ij ( y ( j ) T ) = y ij , for all i ∈ T I ( y ) and all j ∈ J y − i . Hence we may write X j ∈ [ m ] x ( j ) T = X i ∈T I ( x ) X j ∈ J x − i G ij ( x ( j ) T ) + |T II ( x ) | as well as X j ∈ [ m ] y ( j ) T = X i ∈T I ( y ) X j ∈ J y − i G ij ( y ( j ) T ) + |T II ( y ) | . Since P j x ( j ) T < P j y ( j ) T , |T I ( x ) | ≤ |T I ( y ) | and |T II ( x ) | ≥ |T II ( y ) | hold true, it follows that X i ∈T I ( x ) X j ∈ J x − i G ij ( x ( j ) T ) + |T II ( x ) \ T II ( y ) | < X i ∈T I ( x ) X j ∈ J y − i G ij ( y ( j ) T )+ X i ∈T I ( y ) \T I ( x ) X j ∈ J y − i G ij ( y ( j ) T ) . (32)However, the fact that G ij is decreasing implies that G ij ( x ( j ) T ) ≥ G ij ( y ( j ) T ) , for all i ∈ T I ( x ) and all j ∈ J y − i ; hence it holds X i ∈T I ( x ) X j ∈ J x − i G ij ( x ( j ) T ) ≥ X i ∈T I ( x ) X j ∈ J y − i G ij ( y ( j ) T ) . (33)Moreover, since P j ∈ J y − i G ij ( y ( j ) T ) < , for all i ∈ T I ( y ) \ T I ( x ) , it holds X i ∈T I ( y ) \T I ( x ) X j ∈ J y − i G ij ( y ( j ) T ) < |T I ( y ) \ T I ( x ) | = |T II ( x ) \ T II ( 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. Hereand later, given a positive integer k ≥ , L k denotes k -dimensional Lebesgue measure.Moreover, given a function f : R k → R m and a set B ⊂ R m , we denote f − ( B ) := { x ∈ R k : f ( x ) ∈ B } the preimage of B under f . 23 emma 9. Let f : R d → R m be a continuously differentiable function for which L d ( { x ∈ R d : ∇ f ( x ) = 0 } ) = 0 . Then it holds L d ( f − ( A )) = 0 , for every A ⊂ R m for which L m ( A ) = 0 .Proof. See [23, Theorem 1].Let m ≥ be an integer. A set A ⊂ [0 , m is called an antichain if it does not containtwo distinct elements x = ( x , . . . , x m ) and y = ( y , . . . , y m ) such that x j ≤ y j , for all j ∈ [ m ] . Lemma 10.
Let A ⊂ [0 , m be an antichain. Then L m ( A ) = 0 .Proof. The result is an immediate consequence of Lebesgue’s density theorem. Alterna-tively, it follows from the main result in [9], and from [10, Theorem 1.3].Now given x = ( x , . . . , x n ) ∈ C n , let v x denote the vector v x := ( x (1) T , . . . , x ( m ) T ) ∈ [0 , m , (35)where x ( j ) T , j ∈ [ m ] , is defined in (4). Finally, given N ⊂ C n , define the set W N := [ x ∈ N v x . (36)The proof of Theorem 4 is almost complete. Proof of Theorem 4.
To simplify notation, let us set N := N ( G (2) ) . We have to show that L nm ( N ) = 0 .Now let f denote the map f : C n → [0 , m given by f ( x ) = v x , where v x is definedin (35). It is straightforward to verify that { x ∈ C n : ∇ f ( x ) = 0 } = ∅ .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 L m ( W N ) = 0 . Therefore, Lemma 9yields L nm ( N ) = L nm ( f − ( W N )) = 0 , as desired. Let G (2) denote a Fragile multi-CPR Game satisfying Assumption 2. In this section weshow that G (2) admits finitely many GNEs, subject to the constraint that the total invest-ment in each CPR is fixed. We then use this result, in the next section, in order to formulatea conjecture which is equivalent to Conjecture 1. Before being more precise, we need someextra piece of notation. 24iven a set F ⊂ [ m ] and real numbers { r j } j ∈ F ⊂ [0 , , indexed by F , we denote by W ( { r j } j ∈ F ) the set W ( { r j } j ∈ F ) := { x = ( x , . . . , x n ) ∈ C n : x ( j ) T = r j , for j ∈ F } , where x ( j ) T is defined in (4). In other words, W ( { r j } j ∈ F ) consists of those strategy profilesfor which the total investment in the CPRs corresponding to elements in F is fixed, andequal to the given numbers { r j } j ∈ F .In this section we prove the following. Theorem 10.
Fix real numbers r , . . . , r m ∈ [0 , . Then the set W := W ( r , . . . , r m ) containsat most n · ( m +1) GNEs of G (2) . The proof requires a couple of observations which we collect in the following lemmata.
Lemma 11.
Suppose that x = ( x , . . . , x n ) and y = ( y , . . . , y n ) are two GNEs of G (2) suchthat x , y ∈ W := W ( r j ) and < x ij < y ij , for some i ∈ [ n ] , j ∈ [ m ] and r j ∈ [0 , . Then either x i is of Type II or y i is of Type II.Proof. Suppose, towards arriving at a contradiction, that the conclusion is not true. Thenboth x i and y i are of Type I, and thus (26) implies that G ij ( r ) = x ij and G ij ( r ) = y ij . Henceit holds G ij ( r j ) = x ij < y ij = G ij ( r j ) , a contradiction. The result follows. Lemma 12.
Suppose that x = ( x , . . . , x n ) and y = ( y , . . . , y n ) are two GNEs of G (2) forwhich it holds x , y ∈ W := W ( r l , r ℓ ) and < x ij < y ij and x iℓ > y iℓ > , for some i ∈ [ n ] and { j, ℓ } ⊂ [ m ] . Then either x i is of Type I or y i is of Type I.Proof. Suppose, towards arriving at a contradiction, that both x − i and y − i are of Type II.Recall the definition of ψ ij ( · ; · ) , given in (14), and notice that, since both x i , y i are ofType II, Theorem 8 implies the existence of κ x , κ y ≥ such that κ x = x a i − ij · ψ ij ( x ij ; r j − x ij ) = x a i − iℓ · ψ iℓ ( x iℓ ; r ℓ − x iℓ ) and κ y = y a i − ij · ψ ij ( y ij ; r j − y ij ) = y a i − iℓ · ψ iℓ ( y iℓ ; r ℓ − y iℓ ) . Now notice that, for all k ∈ [ m ] , the function Ψ k ( x ) := x a − · ψ ik ( x ; r − x ) is decreasingin x , for fixed r > and a ∈ (0 , . Hence, x ij < y ij implies that κ x > κ y , and x iℓ > y iℓ implies that κ x < κ 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 ∈ [ n ] , define the set N i := { x i ∈ C m : ( x i , x − i ) ∈ W, for some x − i ∈ C − i } .
25e first show that the cardinality of N i , denoted | N i | , is at most m +1 .Let x i ∈ N i , and recall from Theorem 8, and (18), that there exists J ⊂ [ m ] such that x ij > , when j ∈ J , and x ij = 0 when j ∈ [ m ] \ J . In other words, to every x i ∈ N i therecorresponds a set J ⊂ [ m ] such that x ij > if and only if j ∈ J . Now, given J ⊂ [ m ] , let N J := { x i ∈ N i : x ij > if and only if j ∈ J } . Assume first that | J | ≥ . In this case we claim that | N J | ≤ . Indeed, if | N J | ≥ , thenthere are two elements, say x (1) , x (2) ∈ N J , which are either both of Type I, or both ofType II. If both x (1) and x (2) are of Type I, then there exists j ∈ J such that, without lossof generality, it holds x (1) ij < x (2) ij ; which contradicts Lemma 11. If both x (1) and x (2) areof Type II, then there exist j, ℓ ∈ J such that x ij < y ij and x iℓ > y iℓ ; which contradictsLemma 12. The claim follows.If | J | = 1 , say J = { j } , we claim that | N J | ≤ . Indeed, suppose that | N J | ≥ holds true and notice that every element of N J is of Type I. However, the assumptionthat | N J | ≥ implies that there exist x (1) , x (2) ∈ N J such that < x (1) ij < y (1) ij ; whichcontradicts Lemma 11. The second claim follows.Since there are m subsets J ⊂ [ m ] , and for each J it holds | N J | ≤ , it follows thatthere are at most m +1 elements in N i . Since there are n players in the game, the resultfollows. Let G (2) denote a Fragile multi-CPR Game satisfying Assumption 2, and let N ( G (2) ) bethe set consisting of all GNEs of G (2) . So far we have proven that the ( n · m ) -dimensionalLebesgue measure of N ( G (2) ) equals zero, but there are several problems and questionsthat remain open. First and foremost, we believe that the following holds true. Conjecture 2.
Let N := 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 holdstrue as well. Since the converse is clearly true, it follows that Conjecture 1 and Conjec-ture 2 are equivalent. The exact number of GNEs in a Fragile multi-CPR Game appearsto depend on the relation between the number of players, n , and the number of CPRs, m .When n ≥ m we conjecture that that for every GNE the players choose best responses ofType I and therefore, provided this is indeed the case, Theorem 9 would imply that thegame admits a unique GNE. Conjecture 3. If n ≥ m , then |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 initial26nvestment in the CPRs and proceed with updating their investment, over rounds, in sucha way that in the t -th round player i ∈ [ n ] invests b ( t ) i := B i ( x ( t ) − i ) , where B i ( · ) is definedin (13) and x ( t ) − i ∈ 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., whetherthere exists a round t such that b ( t ) i = b ( t ) i , for all t ≥ t and all i ∈ [ 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 CPRGame converge to its Nash Equilibrium. This is obtained as a consequence of the fact thatthe best response correspondence is single-valued and decreasing in the total investmentin the CPR (see the remarks following [13, Proposition 7]). Moreover, it is not difficult toverify that the Nash equilibrium of the Fragile CPR Game is also the Generalized Nashequilibrium. Hence, the best response dynamics of a Fragile CPR Game converge to theGeneralized Nash equilibrium. When m ≥ , the best response correspondence need nolonger be decreasing in each CPR. It is decreasing for those players whose best response isof Type I, as can be easily seen using the fact that the auxiliary function G ij is decreasing.This monotonicity may no longer be true when a player moves from a best response ofType II to a best response of Type I, or from a best response of Type II to a best response ofthe 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 holisticvalidity of Conjecture 4. However, our numerical experiments suggest that Conjecture 4holds true, and we expect that we will be able to report on that matter in the future. References [1] S. Aflaki,
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