A class of stochastic games and moving free boundary problems
AA CLASS OF STOCHASTIC GAMES AND MOVING FREE BOUNDARYPROBLEMS
XIN GUO, WENPIN TANG, AND RENYUAN XU
Abstract.
In this paper we propose and analyze a class of N -player stochastic games thatinclude finite fuel stochastic games as a special case. We first derive sufficient conditions for theNash equilibrium (NE) in the form of a verification theorem. The associated Quasi-Variational-Inequalities include an essential game component regarding the interactions among players,which may be interpreted as the analytical representation of the conditional optimality forNEs. The derivation of NEs involves solving first a multi-dimensional free boundary problemand then a Skorokhod problem. Finally, we present an intriguing connection between theseNE strategies and controlled rank-dependent stochastic differential equations. Key words : finite fuel problem, free boundary problem, Markovian Nash equilibrium, N -playergames, rank-dependent SDEs, reflected Brownian motion, Skorokhod problem. AMS 2010 Mathematics Subject Classification:
Introduction
Recently there are renewed interests in N -player non-zero-sum stochastic games, inspired bythe rapid growth in the theory of Mean Field Games (MFGs) led by the pioneering work of[29, 39, 40, 41]. In this paper, we formulate and analyze a classical of stochastic N -playergames that originated from the classic finite fuel problem. There are many reasons to considerthis type of games. Firstly, the finite fuel problem (e. g. [6, 7, 33]) is one of the landmarks instochastic control theory, therefore mathematically a game formulation is natural. Secondly, inaddition to the interest of stochastic control theory ( [3, 9, 16, 53, 54]), its simple yet insightfulsolution structures have had a wide range of applications including economics and finance[2, 15, 17, 43], operations research [25, 37], and queuing theory [36]. Thirdly, prior success inanalyzing its stochastic game counterpart has been restricted to the special case of two-playergames ([18, 27, 28, 34, 38, 45]) or without the fuel constraint ([26]).In this paper, we will analyze a class of N -player stochastic game that include the finitefuel stochastic game as a special case. The stochastic game presented in this paper goes asfollows. There are N players whose dynamics XXX t = ( X t , · · · , X Nt ) are governed by the following N -dimensional diffusion process: dX it = b i ( XXX t − ) dt + σ i σ i σ i ( XXX t − ) dBBB ( t ) + dξ i + t − dξ i − t , X i − = x i , ( i = 1 , · · · , N ) , (1.1)where BBB := ( B , · · · , B N ) is a standard N -dimensional Brownian motion in a filtered probabil-ity space (Ω , F , {F t } t ≥ , P ), with drift bbb := ( b , · · · , b N ) and covariance matrix σσσ := ( σ σ σ , · · · , σ N σ N σ N )satisfying appropriate regularity conditions. Player i ’s control ξ i is of finite variation. Eachplayer has access to some or all of M types of resources. Players interact through their objec-tive functions h i ( X t , · · · , X Nt ), as well as their shared resources which are the “fuels” of their Date : February 14, 2020. a r X i v : . [ q -f i n . M F ] F e b XIN GUO, WENPIN TANG, AND RENYUAN XU control. The accessibility of these resources to players and how these resources are consumedby their respective players are governed by a matrix
AAA := ( a ij ) i,j ∈ R N × M . For instance, when M = 1 and AAA = [1 , , · · · , T ∈ R N × , this game ( C p C p C p ) corresponds to the N -player finite fuelgame where the N players share a fixed amount of the same resource. When M = N and AAA = I N I N I N , this is an N -player game ( C d C d C d ) where each player has her individual fixed amount ofresource. In general, this matrix AAA describes the network structure of the N -player game.The goal for player i in the game is to minimize E (cid:90) ∞ e − αt h i ( X t , · · · , X Nt ) dt, over appropriate admissible game strategies, which are specified in Section 2. Note that this N -player game cannot be simply analyzed with an MFG approach as the network structurewould collapse if an aggregation approach was applied.We will analyze the NEs of this stochastic game. We first derive sufficient conditions forthe NE policy in the form of a verification theorem (Theorem 3.3), which reveals an essentialgame element regarding the interactions among players. This is the Hamilton–Jacobi–Bellman(HJB) representation of the conditional optimality for NE in a stochastic game. To understandthe structural properties of the NEs, we proceed further to analyze this stochastic game interms of the game values, the NE strategies, and the controlled dynamics. Mathematically, theanalysis involves solving first a multi-dimensional free boundary problem and then a Skorokhodproblem with a moving boundary. The boundary is “moving” in that it moves in response toboth changes of the system and controls of other players. The analytical solution is derived byfirst exploring the two special games C p C p C p and C d C d C d . Analyzing these two types of games provideskey insights into the solution structure of the general game. Finally, we reformulate the NEstrategies in the form of controlled rank-dependent stochastic differential equations (SDEs),and compare game values between games C p C p C p and C d C d C d . Main contributions. (i) In the verification theorem for N -player games, we obtain the formof the HJB equations for general stochastic games with singular controls. Unlike all previousanalysis that focused on two-player games, we show that in addition to the standard HJBsthat correspond to stochastic control problems, there is an essential term that is unique tostochastic games. This term represents the interactions among players, especially the ones whoare active and those who are waiting. This critical term was hidden in two-player stochasticgames and was previously (mis)understood as a regularity condition (Remark 3.1).(ii) The structural difference between games and control problems is further revealed in theexplicit solution to the NEs for N -player games. In a Markovian control problem, a freeboundary depends on the state of the system; in stochastic games, however, the “face” of theboundary moves based on the action of herself and interaction among players in the game(Figure 4). Note that this free boundary for stochastic games with an infinite time horizon moves in a different sense from the one in [16] for finite time control problems where theboundary is time dependent. Rather it moves due to changes of the system and the competitionin the game.(iii) This difference is further highlighted in the framework of controlled rank-dependent SDEs.To the best of our knowledges, this is the first time a stochastic game is explicitly connectedwith rank-dependent SDEs in a more general form. This new form of rank-dependent SDEspresents a fresh class of yet-to-be studied SDEs (Section 7.2). INITE FUEL GAME 3 (iv) We recast the controlled dynamics of the game solution in the framework of controlled rank-dependent SDEs. Compared with the well-known rank-dependent SDEs, rank-dependentSDEs with an additional control component are new. We establish the existence of the solutionby directly constructing a reflected diffusion process. (See Section 7.2 for further discussions.)(v) Finally, stochastic games considered in this paper are resource allocation games. Resourceallocation problems have a wide range of applications including inventory management, resourceallocation, cloud computing, smart power grid control, and multimedia wireless networks [23,24, 42, 50]. However, the existing literature has been unsuccessful in analyzing the resourceallocation problem in the setting of stochastic games. Besides the technical contributions, ouranalysis provides a useful economic insight: in a stochastic game of resource allocations, sharinghas lower cost than dividing and pooling yields the lowest cost for each player.
Related work.
There are a number of papers on non-zero-sum two-player games with singularcontrols. By treating one player as a controller and the other as a stopper, Karatzas and Li [34]analyze the existence of an NE for the game using a BSDE approach. Hernandez-Hernandez,Simon, and Zervos [28] study the smoothness of the value function and show that the optimalstrategy may not be unique when the controller enjoys a first-move advantage. Kwon andZhang [38] investigate a game of irreversible investment with singular controls and strategicexit. They characterize a class of market perfect equilibria and identify a set of conditions underwhich the outcome of the game may be unique despite the multiplicity of the equilibria. DeAngelis and Ferrari [18] establish the connection between singular controls and optimal stoppingtimes for a non-zero-sum two-player game. Mannucci [45] and Hamadene and Mu [27] considerthe fuel follower problem in a finite-time horizon with a bounded velocity, and establish viadifferent techniques the existence of an NE of the two-player game. Very recently, [26] comparethe N-player game versus the MFG for the fuel follower problem. All these works are withoutthe fuel constraint and are essentially built on one-dimensional stochastic control problems.Furthermore, except for [26], all of these papers are restricted to the case of N = 2. Tothe best of our knowledge, our work is the first to complete the mathematical analysis on an N -player stochastic game based on an original two-dimensional control problem.There has been some works on reflected SDEs in time-dependent or state-dependent domains.[10, 11] consider reflected Brownian motions in smooth time-dependent domains with normalreflection via the heat equation. [12] studies the one-dimensional case through the Skorokhodproblem. Later, [44, 47] give the construction of reflected SDEs in non-smooth time-dependentdomains with oblique reflection. However, none of these works involve controls. [8] considersreflected SDEs in the orthant R d + and focuses on the viscosity solution analysis.In our work the controlled dynamics are recast in the framework of controlled rank-dependentSDEs. Rank-dependent SDEs without controls arise in the “Up the River” problem [1] and instochastic portfolio theory [21], including the well-studied Atlas model for the ergodicity andsample path properties [4, 30, 48, 51, 52] and for the hydrodynamic limit and fluctuations ofthe Atlas model [13, 20, 55].
Notations and organization.
Throughout the paper, we denote vectors/matrices by boldcase letters, e.g., xxx and
XXX . The transpose of a real vector xxx is denoted as xxx T . For a vector xxx , (cid:107) xxx (cid:107) denotes its l norm. For a matrix XXX , (cid:107) XXX (cid:107) denotes its spectral norm.The paper is organized as follows. Section 2 presents the mathematical formulation of the N -player game. Section 3 provides verification theorem for sufficient conditions of the NE ofthe game and the existence of Skorokhod problem for NE strategies. Section 4 studies game XIN GUO, WENPIN TANG, AND RENYUAN XU C p C p C p and Section 5 studies game C d C d C d . With the insight from these two games, Section 6 analyzesthe general N -player game CCC . Section 7 compares games C p C p C p , C d C d C d and CCC , discusses the gamevalues and their economic implications, and unifies their corresponding controlled dynamics inthe framework of the controlled rank-dependent SDEs.2.
Problem Setup
Now we present the mathematical formulation for the stochastic N -player game. Controlled dynamics.
Let ( X it ) t ≥ be the position of player i, ≤ i ≤ N . In the absence ofcontrols, XXX t = ( X t , · · · , X Nt ) is governed by the stochastic differential equation (SDE): dXXX t = bbb ( XXX t ) dt + σσσ ( XXX t ) dBBB ( t ) , XXX − = ( x , · · · , x N ) , (2.1)where BBB := ( B , · · · , B N ) is a standard N -dimensional Brownian motion in a filtered probabil-ity space (Ω , F , {F t } t ≥ , P ), with the drift bbb ( · ) := ( b ( · ) , · · · , b N ( · )) and the covariance matrix σσσ ( · ) := ( σ ij ( · )) ≤ i,j ≤ N . To ensure the existence and uniqueness of the SDE, bbb ( · ) and σσσ ( · ) areassumed to satisfy the usual global Lipschitz condition and linear growth condition : H1.
There exists a constant L > L > (cid:107) bbb ( xxx ) − bbb ( yyy ) (cid:107) + (cid:107) σσσ ( xxx ) − σσσ ( yyy ) (cid:107) ≤ L (cid:107) xxx − yyy (cid:107) , (cid:107) bbb ( xxx ) (cid:107) + (cid:107) σσσ ( xxx ) (cid:107) ≤ L (1 + (cid:107) xxx (cid:107) ) , for all xxx, yyy ∈ R N .Assumption H1 ensures the existence of a strong solution to (2.1) and the solution is square-integrable [35, Theorem 2.9 in Chapter 5]. Here and throughout the rest of the paper, theinfinitesimal generator L is L := (cid:88) i b i ( xxx ) ∂∂x i + 12 (cid:88) i,j ( σσσ ( xxx ) σσσ ( xxx ) T ) i,j ∂ ∂x i ∂x j , (2.2)where σσσ ( xxx ) σσσ ( xxx ) T is assumed to be positive-definite for every xxx ∈ R N . See [35, Chapter 5], or[31, Chapter IV] for background on SDEs.If a control is applied to X it , then X it evolves as dX it = b i ( XXX t − ) dt + σσσ i ( XXX t − ) dBBB ( t ) + dξ i + t − dξ i − t , X i − = x i , (2.3)where σσσ i is the i th row of the covariance matrix σσσ . Here the control ( ξ i + , ξ i − ) is a pair ofnon-decreasing and c`adl`ag processes. In other words, ( ξ i + , ξ i − ) is the minimum decompositionof the finite variation process ξ i such that ξ i := ξ i + − ξ i − . Also note that under assumption H1 , (2.3) has a strong solution [49, Theorem V.7]. Game objective.
The game is for player i to minimize, for all ( ξ i + , ξ i − ) in an appropriateadmissible control set, over an infinite time horizon, the following objective function, E (cid:90) ∞ e − αt h i ( X t , · · · , X Nt ) dt. (2.4)Here α > h i ( xxx ) : R N → R + . H2.
Each h i ( xxx ) is assumed to be twice differentiable, with k ≤ ||∇ h i ( xxx ) || ≤ K for some K > k > INITE FUEL GAME 5
For example, h i ( xxx ) = h ( x i − (cid:80) Nj =1 x j N ) is a distance function between the position of player i and the center of all players.Note that in the objective function (2.4), there is no cost of control. With this formulation,the explicit solution structure of the NE for game (2.4) is clean. It is entirely possible toconsider an N-player game with additional cost of control. For instance, one might studythe game formulation of [33] with a proportional cost of control. We conjecture that thesolution structure would be similar although the analysis will be more involved. This will bean interesting problem for future analysis. Admissible control policies.
The admissible control set S N ( xxx, yyy ) for this N -player game isgiven by S N ( xxx, yyy ) := (cid:40) ξξξ : ξ i ∈ U iN for 1 ≤ i ≤ N, N (cid:88) i =1 (cid:90) ∞ a ij Y jt − (cid:80) Mk =1 a ik Y kt − d ˇ ξ it ≤ y j , ≤ j ≤ M, P (cid:16) ∆ ξ it ∆ ξ kt (cid:54) = 0 (cid:17) = 0 for all t ≥ i (cid:54) = k (cid:41) , (2.5)where U iN := (cid:110) ( ξ + , ξ − ) : ξ + and ξ − are F BBBt -progressively measurable, c`adl`ag, non-decreasing , with E (cid:90) ∞ e − αt [ ξ ± t ] < ∞ andξ +0 − = ξ − − = 0 (cid:111) , with F BBBt := σ ( BBB s , s ≤ t ) the filtration generated by Brownian motion BBB , and Y jt = y j − N (cid:88) i =1 (cid:90) t a ij Y js − (cid:80) Mk =1 a ik Y ks − d ˇ ξ is ∈ R + and Y j − = y j , (2.6)with a ij = 0 or 1 for 1 ≤ i ≤ N and 1 ≤ j ≤ M , (cid:80) Mj =1 a ij > i = 1 , · · · , N , and (cid:80) Ni =1 a ij > j = 1 , · · · , M . Moreover,ˇ ξ it := ξ i + t + ξ i − t , (2.7)is the accumulative amount of controls/resources consumed by player i up to time t .The non-decreasing and c`adl`ag processes ( ξ i + , ξ i − ) ∈ U iN can be decomposed in the differentialform, dξ i ± t = d ( ξ i ± t ) c + ∆ ξ i ± t , (2.8)where d ( ξ i ± t ) c is the continuous part and ∆ ξ i ± t := ξ i ± t − ξ i ± t − is the jump part of dξ i ± t .Here is the intuition for the admissible control set S N ( xxx, yyy ). In this game, each player i will make a decision based on the current positions of all players and the available resources.In addition to this adaptedness constraint, the admissible control set S N ( xxx, yyy ) specifies theresource allocation policy for each player. For M different types of resources, define AAA :=( a ij ) i,j ∈ R N × M to be the adjacent matrix with a ij = 0 or 1. Then AAA describes the relationshipbetween the players and the types of available resources, with a ij = 1 meaning that resourceof type j is available to player i , and a ij = 0 meaning that resource of type j is inaccessibleto player i . The condition (cid:80) Mj =1 a ij > i = 1 , · · · , N implies that each player i has XIN GUO, WENPIN TANG, AND RENYUAN XU access to at least one resource, and the condition (cid:80) Ni =1 a ij > j = 1 , · · · , M indicatesthat each resource j is available to at least one player. Moreover, when player i would like toexercise control, she will consume resources proportionally to all the resources available to her.She will stop consuming once all the available resources hit level zero. This results in the formof the integrand in the expression of (2.6). Note that the denominator is always no smallerthan the numerator hence the integrand is well-defined with the convention = 0. See Figure1 for illustration. (a) Relationship. (b) Resource allocation policy. Figure 1.
Example of adjacent matrix
AAA , relationship between the players andresources when N = 4 and M = 6.Take an example of N = 4, M = 6, with the matrix AAA defined as
AAA = , , , , , , , , , , , , , , , , , , , , , (Figure 1a). The resource allocation policy is illustrated in Figure 1b, with the amount ofavailable resource y and y of type one and two respectively. When player one wishes to applycontrols of amount ∆, say ∆ ≤ y + y , she will consume resources randomly from type oneand two. So player one will take ∆ y y + y from resource one and ∆ y y + y from resource two.Finally, the condition P (∆ ξ it ∆ ξ kt (cid:54) = 0) = 0 for all t ≥ i (cid:54) = k excludes the possibility ofsimultaneous jumps of any two out of N players, which facilitates designing feasible controlpolicies when controls involve jumps. This condition is not a restriction, and instead shouldbe interpreted as a regularization . See also [5, 26, 38]. Indeed, when there are multiple playerswho would like to jump at the same time, one can simply design a proper order , for instanceby indexing the players and their jump orders, so that they will move sequentially .In this paper, we focus on time-independent open-loop controls with respect to F BBBt . In addition,we focus on Markovian controls in the sense that (
BBB [0 ,t ] , XXX ξξξt ) is jointly Markovian, with XXX ξξξt thecontrolled dynamics under ξξξ as defined in (2.3) and
BBB [0 ,t ] = { BBB s } s ∈ [0 ,t ] . This is consistent withearlier works of [22] which adopts open-loop singular controls for game analysis and [18, 38] forgames with open-loop singular controls of Markovian structures.Within the framework of time-independent Markovian controls, we can define the followingregions to partition the R N × R N + space. Definition 2.1 (Action and waiting regions) . Denote A i ⊆ R N × R M + as the i th player’saction region and W i := ( R N × R M + ) \ A i as the waiting region. Let A − i := ∪ j (cid:54) = i A j , and W − i := ∩ j (cid:54) = i W j . INITE FUEL GAME 7
Then players’ actions are the follows: player i controls if and only if the process enters the set A i . Under condition (2.5), we have A i ∩ A j = ∅ for any i, j = 1 , , · · · , N and i (cid:54) = j . Game formulation and game criterion.
Let ξξξ := ( ξ , · · · , ξ N ) be the controls from theplayers. Let xxx := ( x , · · · , x N ) and yyy := ( y , · · · , y M ). Then the stochastic game is for eachplayer i to minimize J i ( xxx, yyy ; ξξξ ) := E (cid:90) ∞ e − αt h i ( XXX t ) dt, (2.9)subject to the dynamics in (2.3) and (2.6) with the constraint in (2.5). There are two specialgames of particular interest. One is a game where all players pool their resources such that N (cid:88) i =1 ˇ ξ i ∞ < y < ∞ . (2.10)When N = 1, this is a single player game corresponding to the finite fuel control problemwhich is well studied in [7, 33]. We call this game a pooling game C p C p C p . Clearly in terms of theadjacent matrix AAA , this corresponds to M = 1, and AAA = [1 , , · · · , T ∈ R N × . Another is agame where players divide the resource up front such thatˇ ξ i ∞ < y i , (2.11)where y i is the total amount of controls that player i can exercise. This game is called C d C d C d , with M = N , and AAA = I N I N I N . Finally, we refer the game with a general matrix AAA as game
CCC .We will analyze the N -player game under the criterion of NE. See [14] for various conceptsof NE of differential games. Recall the definition of NE of N -player games. Definition 2.2.
A tuple of admissible controls ξξξ ∗ := ( ξ ∗ , · · · ξ N ∗ ) is a Markovian NE of the N -player game (2.9), if for each ξ i ∈ U iN such that ( ξξξ − i ∗ , ξ i ) ∈ S N ( xxx, yyy ) , J i ( xxx, yyy ; ξξξ ∗ ) ≤ J i (cid:0) xxx, yyy ; (cid:0) ξξξ − i ∗ , ξ i (cid:1)(cid:1) , where ξξξ − i ∗ = ( ξ ∗ , · · · , ξ i − ∗ , ξ i +1 ∗ , · · · , ξ N ∗ ) and ( ξξξ − i ∗ , ξ i ) = ( ξ ∗ , · · · , ξ i − ∗ , ξ i , ξ i +1 ∗ , · · · , ξ N ∗ ) .Here the strategies ξ i ∗ and ξ i are functions of XXX t = ( X t , · · · , X Nt ) , and YYY t = ( Y t , · · · , Y Mt ) (with XXX − = xxx and YYY − = yyy ). Controls that give NEs are called the Nash Equilibrium Points(NEPs). The associated value function J i ( xxx, yyy ; ξξξ ∗ ) ( i = 1 , , · · · , N ) is called the game value. Remark 2.3.
There are two well-known criteria for analyzing stochastic N -player games.One is NE, the other is Pareto Optimality (PO). NE is mainly used for analyzing stabilityof competitive games. In contrast, PO is widely adopted to analyze the efficiency and socialoptimality for strategies in collaborative games. In this paper, our focus is NE. NE Game Solution: Verification Theorem and Skorokhod Problem
Quasi-variational Inequalities.
We first derive heuristically the associated quasi-variationalinequalities (QVIs) for game (2.9), with A i ∩ A j = ∅ .First, define the intervene operator Γ asΓ j v i ( xxx, yyy ) = M (cid:88) k =1 a jk y j (cid:80) Ms =1 a js y s v iy k ( xxx, yyy ) , (3.1) XIN GUO, WENPIN TANG, AND RENYUAN XU for ( xxx, yyy ) ∈ R N and i, j = 1 , , · · · , N . Here v iy k := ∂v i ∂y k ( i = 1 , , · · · , N and k = 1 , , · · · , M ).Note that the operator Γ j can be viewed as the impact of player j ’s infinitesimal control onplayer i in terms of resource changes, with the following intuition: suppose player j takes apossibly suboptimal action ∆ ξ j, + >
0, then by the resource allocation policy (2.6) v i ( xxx, yyy ) ≤ v i (cid:32) xxx − j , x j + ∆ ξ j, + , yyy − (cid:32) a j y (cid:80) Mk =1 a jk y k , · · · , a jN y N (cid:80) Mk =1 a jk y k (cid:33) ∆ ξ j, + (cid:33) . (3.2)This is equivalent to0 ≤ v i (cid:32) xxx − j , x j + ∆ ξ j, + , yyy − (cid:32) a j y (cid:80) Mk =1 a jk y k , · · · , a jN y N (cid:80) Mk =1 a jk y k (cid:33) ∆ ξ j, + (cid:33) − v i ( xxx, yyy ) . (3.3)Dividing both sides by ∆ ξ j, + and letting ∆ ξ j, + →
0, we have0 ≤ − Γ j v i ( xxx, yyy ) + v ix j ( xxx, yyy ) . (3.4)Next, we provide the heuristics for deriving the QVIs. Denote ∆ ξ i := ∆ ξ i ( xxx, yyy ) as the controlof player i with joint state position ( xxx, yyy ). When ( xxx, yyy ) ∈ W − i , we have ∆ ξ j = 0 for j (cid:54) = i , bydefinition. Thus the game for player i becomes a classical control problem with three choices:∆ ξ i = 0, ∆ ξ i, + >
0, and ∆ ξ i, − >
0. The first case ∆ ξ i = 0 implies, by simple stochasticcalculus, − αv i + h i ( xxx ) + L v i ≥
0. By similar derivations as in (3.2)-(3.4), the second case∆ ξ i, + > − Γ i v i + v ix i ≥
0. Similarly, the third case ∆ ξ i, − > − Γ i v i − v ix i ≥ Since one of the three choices will be optimal, one of the inequalities willbe an equation. That is, for ( xxx, yyy ) ∈ W − i ,min (cid:8) − αv i + h i ( xxx ) + L v i , − Γ i v i + v ix i , − Γ i v i − v ix i (cid:9) = 0 , (3.5)Since each player i can only control x i and the resources that are available to her, the aboveequation minimizes over ( x i , yyy ).When ( xxx, yyy ) ∈ A j , player j will control with the amount of control being (∆ ξ j, + , ∆ ξ j, − ) (cid:54) = 0.Therefore, v j ( xxx, yyy ) ≤ v j (cid:32) xxx − j , x j + ∆ ξ j, + − , yyy − (cid:32) a j y (cid:80) Mk =1 a jk y k , · · · , a jN y N (cid:80) Mk =1 a jk y k (cid:33) ∆ ξ j, + (cid:33) , (3.6) v j ( xxx, yyy ) ≤ v j (cid:32) xxx − j , x j − ∆ ξ j, − , yyy − (cid:32) a j y (cid:80) Mk =1 a jk y k , · · · , a jN y N (cid:80) Mk =1 a jk y k (cid:33) ∆ ξ j, − (cid:33) , (3.7)and one of the inequalities in (3.6)-(3.7) will be an equality. This leads to the following conditionmin (cid:110) − Γ j v j + v jx j , − Γ j v j − v jx j (cid:111) = 0 . (3.8)For player i (cid:54) = j , we should have, v i ( xxx, yyy ) = v i (cid:32) xxx − j , x j + ∆ ξ j, + − ∆ ξ j, − , yyy − (cid:32) a j y (cid:80) Mk =1 a jk y k , · · · , a jN y N (cid:80) Mk =1 a jk y k (cid:33) (∆ ξ j, + + ∆ ξ j, − ) (cid:33) . We adopt the convention = 0. INITE FUEL GAME 9
This is because A j ∩ A i = ∅ . Thus, − Γ j v i + v ix j = 0 , on { ( xxx, yyy ) ∈ R N × R M + (cid:12)(cid:12) − Γ j v j + v jx j = 0 } , − Γ j v i − v ix j = 0 , on { ( xxx, yyy ) ∈ R N × R M + (cid:12)(cid:12) − Γ j v j − v jx j = 0 } . (3.9)Note that by letting ∆ ξ i, ± →
0, equations (3.5),(3.8) and (3.9) describe the behavior in W i and near boundary ∂ W i ( i = 1 , , · · · , N ). Moreover, we can show that (3.5),(3.8) and (3.9) areconsistent with the jump behaviors in A i ( i = 1 , , · · · , N ). Notice that − (cid:80) Mj =1 a ij y j (cid:80) Mk =1 a ik y k v iy j ± v ix i = 0 has a linear solution v i ( xxx ) = a (cid:16) ± x i + (cid:80) Mj =1 a ij y j (cid:17) + b for some a, b ∈ R . And it iseasy to check that ∀ (cid:80) Mk =1 a ik y k ≥ ∆ > a ij y j − a ij y j (cid:80) Mk =1 a ik y k ∆ (cid:80) Mk =1 a ik y k − ∆ = a ij y j (cid:80) Mk =1 a ik y k , which means that the allocation policy (jump direction) outside the waiting region is linear.Hence the dynamics in (2.6) satisfies the HJB equation (3.5) in A i . The consistency propertyalso holds for (3.9).In summary, we have the following QVIs:min (cid:8) − αv i + h i ( xxx ) + L v i , − Γ i v i + v ix i , − Γ i v i − v ix i (cid:9) = 0 , on ∩ j (cid:54) = i (cid:110)(cid:110) − Γ j v j + v jx j > (cid:111) ∩ (cid:110) − Γ j v j − v jx j > (cid:111)(cid:111) , (3.10a) − Γ j v i + v ix j = 0 , on {− Γ j v j + v jx j = 0 } , (3.10b) − Γ j v i − v ix j = 0 , on {− Γ j v j − v jx j = 0 } , (3.10c)where Γ i and Γ j are defined in (3.1). Remark 3.1.
Note that when N = 2 , the above equation corresponds to the continuity conditionof game values. See Proposition 3.4 in [19] . For general N -player games, it is a mathematicaldescription of interactions between the player in control and those who are not. It guaran-tees that all players control optimally so that they sequentially push the underlying dynamicsuntil reaching the common waiting region. This is consistent with the intuition that NE isconditionally optimal for each player. Remark 3.2.
Under the ‘no simultaneous jump’ assumption in (2.5) , there are only two gra-dient terms in (3.5) corresponding to the actions from player i . If one removes this ‘no si-multaneous jump’ assumption, there will be N − terms for gradient constraints, making theproblem intractable. Similar analysis holds for (3.9) . Verification Theorem.
Next we present a verification theorem which gives sufficientconditions of an NEP.To start, we redefine the action and waiting regions ( A i and W i ) in terms of v i ( i = 1 , , · · · , N )as the following: A i = A + i ∪ A − i , (3.11)where A + i := { ( xxx, yyy ) ∈ R N × R M + | − Γ i v i − v ix i = 0 } and A + i := { ( xxx, yyy ) ∈ R N × R M + | − Γ i v i + v ix i = 0 } . Moreover, W i = ( R N × R M + ) \ A i and W − i = ∩ j (cid:54) = i W j . Theorem 3.3 (Verification theorem) . Assume H1 - H2 . Assume A i , W i and W − i are definedaccording to (3.11) . Further assume that A j ∩ A i = ∅ for all i (cid:54) = j . For each i = 1 , · · · , N ,suppose that the i th player’s strategy ξ i ∗ ∈ U iN satisfies the following conditions(i) ξξξ ∗ := ( ξ ∗ , · · · , ξ N ∗ ) ∈ S N ( xxx, yyy ) , (ii) v i ( · ) satisfies the QVIs (3.10) ,(iii) v i ( xxx, yyy ) satisfies the transversality condition lim sup T →∞ e − αT E v i ( XXX T , YYY T ) = 0 , (3.12) for any ( XXX t , YYY t ) under admissible controls.(iv) v i ( xxx, yyy ) ∈ C ( W − i ) and v i is convex for all ( xxx, yyy ) ∈ W − i ,(v) v ix j is bounded in W − i for each j = 1 , , · · · , N ,(vi) for any ξ i ∈ U iN such that ( ξξξ − i ∗ , ξ i ) ∈ S N ( xxx, yyy ) , P (( XXX − i ∗ t , X it , YYY t ) ∈ W − i ) = 1 for all t ≥ , where ( XXX − i ∗ t , X it , YYY t ) is under ( ξξξ − i ∗ , ξ i ) .(vii) For j (cid:54) = i , dξ j ∗ t = 0 when ( XXX − i ∗ t − , X it − , YYY t − ) ∈ W j , (3.13) dξ j ∗ t (cid:54) = 0 when ( XXX − i ∗ t − , X it − , YYY t − ) ∈ A j , (3.14) where ( XXX − i ∗ t , X it , YYY t ) is under ( ξξξ − i ∗ , ξ i ) .Then v i ( xxx, yyy ) ≤ J i ( xxx, yyy ; ( ξξξ − i ∗ , ξ i )) , for all ξ ∈ U iN such that ( ξ i , ξξξ − i ∗ ) ∈ S N , and v i ( xxx, yyy ; ξξξ ∗ ) = J i ( xxx, yyy ; ( ξξξ − i ∗ , ξ i )) . That is, ξξξ ∗ is anNEP with value function v i a solution to (3.10) .Proof of Theorem 3.3. It suffices to prove that for each i = 1 , · · · , N , J i ( xxx, yyy ; ξξξ ∗ ) ≤ J i ( xxx, yyy ; ( ξξξ − i ∗ , ξ i )) , for all ( ξξξ − i ∗ , ξ i ) ∈ S N ( xxx, yyy ). INITE FUEL GAME 11
Recall (2.1) and (2.6). From condition ( vi ), under control ( ξξξ − i ∗ , ξ i ) ∈ S N ( xxx, yyy ), ( XXX − i ∗ t , X it , YYY t ) ∈W − i a.s.. Applying Itˆo-Meyer’s formula [46, Theorem 21] to e − αt v i ( XXX − i ∗ t , X it , YYY t ) yields E [ e − αT v i ( XXX − i ∗ T , X iT , YYY T )] − v i ( xxx, yyy )= E (cid:90) T e − αt (cid:0) L v i − αv i (cid:1) dt + E (cid:90) T e − αt N (cid:88) j =1 v ix j dB jt + N (cid:88) j =1 ,j (cid:54) = i E (cid:90) [0 ,T ) e − αt ( v ix j dξ j ∗ , + t − v ix j dξ j ∗ , − t ) − N (cid:88) j =1 ,j (cid:54) = i E (cid:90) [0 ,T ) e − αt Γ j v i ( XXX − i ∗ t − , X it − , YYY t − ) (cid:16) dξ j ∗ , + t + dξ j ∗ , − t (cid:17) + E (cid:90) [0 ,T ) e − αt ( v ix i dξ i, + t − v ix i dξ i, − t ) − E (cid:90) [0 ,T ) e − αt Γ i v i ( XXX − i ∗ t − , X it − , YYY t − ) (cid:16) dξ i, + t + dξ i, − t (cid:17) + E (cid:88) ≤ t 0. Next we have E (cid:90) [0 ,T ) e − αt ( v ix i dξ i, + t − v ix i dξ i, − t ) − E (cid:90) [0 ,T ) e − αt Γ i v i ( XXX − i ∗ t − , X it − , YYY t − ) (cid:16) dξ i, + t + dξ i, − t (cid:17) = E (cid:90) [0 ,T ) e − αt (cid:2) v ix i ( XXX − i ∗ t − , X it − , YYY t − ) − Γ i v i ( XXX − i ∗ t − , X it − , YYY t − ) (cid:3) dξ i, + t + E (cid:90) [0 ,T ) e − αt (cid:2) − v ix i ( XXX − i ∗ t − , X it − , YYY t − ) − Γ i v i ( XXX − i ∗ t − , X it − , YYY t − ) (cid:3) dξ i, − t ≥ . The last inequality holds due to conditions ( ii ) and ( iv ). More precisely, v i ( xxx ) satisfies theHJB equation (3.10a) in W − i . Along with ( iv ), we have the following with probability one, v ix i ( XXX − i ∗ t − , X it − , YYY t ) − Γ i v i ( XXX − i ∗ t − , X it − , YYY t − ) ≥ ,v ix i ( XXX − i ∗ t − , X it − , YYY t − ) − Γ i v i ( XXX − i ∗ t − , X it − , YYY t − ) ≥ . For each j (cid:54) = i , almost surely, we have dξ j ∗ t (cid:54) = 0 only when ( XXX t , YYY t ) ∈ ∂ W − i ∩ ∂ A j . Along withthe condition ( ii ) and (3.10b)-(3.10c), E (cid:90) [0 ,T ) e − αt ( v ix j ( XXX − i ∗ t − , X it − , YYY t ) dξ j, + t − v ix j ( XXX − i ∗ t − , X it − , YYY t ) dξ j, − t ) − E (cid:90) [0 ,T ) e − αt Γ j v i ( XXX − i ∗ t − , X it − , YYY t ) (cid:16) dξ j, + t + dξ j, − t (cid:17) = E (cid:90) [0 ,T ) e − αt (cid:2) v ix j − Γ j v i (cid:3) ( XXX − i ∗ t − , X it − , YYY t ) dξ j ∗ , + t + (cid:2) − v ix j − Γ j v i (cid:3) ( XXX − i ∗ t − , X it − , YYY t ) dξ j ∗ , − t = 0 . Condition ( ii ) also implies L v i − αv i ≥ h . Combining all of the above, e − αT E v i ( XXX − i ∗ T , X iT , YYY T ) + E (cid:90) T e − αt h (cid:0) XXX − i ∗ t , X it (cid:1) dt ≥ v i ( xxx, yyy ) . (3.15)By letting T → ∞ , the inequality (3.15) and condition ( iii ) lead to the desirable inequality.The equality holds in (3.15) when ξ i = ξ i ∗ such that dξ i ∗ t = 0 when ( XXX i ∗ t − , YYY ∗ t − ) ∈ W i , (3.16) dξ i ∗ t (cid:54) = 0 when ( XXX − i ∗ t − , YYY ∗ t − ) ∈ A i , (3.17)and P (cid:16) ( XXX i ∗ t , YYY ∗ t ) ∈ ∩ Ni =1 W i (cid:17) = 1 for all t ≥ , where ( XXX ∗ t , YYY ∗ t ) is the dynamics under ξξξ ∗ . (cid:3) Remark 3.4. Note that, unlike the usual stochastic control problem which requires C regularityin the whole space R N , in the N -player game (2.9), the minimum regularity needed is C in W − i . This is due to the game nature and interactions among players. Suppose the game value v i ( i = 1 , , · · · , N ) that satisfies the verification theorem (Theorem3.3) are given, the next step is to construct the corresponding NE strategies. This is by solvinga Skorokhod problem, introduced in the next subsection.3.3. Skorokhod Problem. Let G = ∩ i ∈I G i be a nonempty domain in R n + m , where I is anonempty finite index set and for each i ∈ I , G i is a nonempty domain in R n + m . For simplicity,we assume that I = { , , · · · , I } , with |I| = I . For each i ∈ I , let nnn i : R n + m → R n + m bethe unit normal vector field on ∂G i that points into G i . And denote rrr i ( · ) : R n + m → R n + m asthe reflection direction on ∂G i . Fix bbb ∈ R n and σσσ ∈ R n × n as the drift and covariance of thediffusion process without reflection. Let ν denote a probability measure on ( G, B ( G )), where B ( G ) is the Borel σ -algebra on G .A Skorokhod problem is to find a reflected diffusion process in G such that the initial distri-bution follows ν , the diffusion parameters are ( bbb, σσσ ), and the reflection direction is rrr i on face ∂G i . For each reflection direction rrr i ( i ∈ I ), denote rrr + i := ( r i, , · · · , r i,n ) as the vector of thefirst n components of rrr i and denote rrr − i := ( r i,n +1 , · · · , r i,n + m ) as the vector of the next m components of rrr i . Note that r − i,k = r i,k + n by the usual index rule ( k = 1 , · · · , m ). Definition 3.5 (Constrained semimartingale reflecting Brownian motion) . A constrained semi-martingale reflecting Brownian motion (SRBM) associated with the data ( G, bbb, σσσ, { rrr i } Ii =1 , ν ) is INITE FUEL GAME 13 an {F t } -adapted, n -dimensional process XXX defined on some filtered probability space (Ω , F , {F t } , P ) such that: (i) P -a.s., XXX t = WWW t + (cid:80) i ∈I (cid:82) [0 ,t ) rrr + i ( XXX s , YYY s ) dη is for all t ≥ , (ii) under P , WWW t is an n -dimensional F t -Brownian motion with drift vector bbb , covari-ance matrix σσσ and initial distribution ν , (iii) dY jt = (cid:80) i ∈I (cid:82) [0 ,t ) rrr − i,j ( XXX t , YYY t ) dη it and Y jt ≥ for j = 1 , , · · · , m , (iv) for each i ∈ I , η i is a one-dimensional process such that P -a.s.,(a) η i = 0 ,(b) η i is continuous and nondecreasing,(c) η it = (cid:82) (0 ,t ] { W s ∈ ∂G i ∩ ∂G } dη is for all t ≥ , (v) P -a.s., ( XXX t , YYY t ) has continuous paths and ( XXX t , YYY t ) ∈ G for all t ≥ , Remark 3.6. We would like to point out that the solution approach presented in this sectionis largely inspired by Kang and Williams [32] , with appropriate modifications to our problemframework. Specific to the stochastic game in this paper, XXX t is the controlled diffusion pro-cess and YYY t is the resource levels. The domain G restricts the dynamics of both XXX t and YYY t .Note that the constrained SRBM is slightly different from the standard SRBM (see Kang andWilliams [32] ) in the sense that the reflection domain depends on both the diffusion process XXX t and the resource process YYY t . For each ( xxx, yyy ) ∈ R n + m , let I ( xxx, yyy ) = { i ∈ I : ( xxx, yyy ) ∈ ∂G i } . Let U (cid:15) ( S ) denote the closedset { ( xxx, yyy ) ∈ R n + m : dist (( xxx, yyy ) , S ) ≤ (cid:15) } for any (cid:15) > S ⊂ R n + m . If S = ∅ , set U (cid:15) ( S ) = ∅ for any (cid:15) > 0. We propose the following assumptions on domain G and reflection directions { rrr i , i ∈ I} : A1. G is the nonempty domain in R n + m such that G = ∩ i ∈I G i , (3.18)where for each i ∈ I , G i is a nonempty domain in R n + m , G i (cid:54) = R m + n and theboundary ∂G i is C . A2. For each (cid:15) ∈ (0 , 1) there exists R ( (cid:15) ) > i ∈ I , ( xxx, yyy ) ∈ ∂G i ∩ ∂G and ( xxx (cid:48) , yyy (cid:48) ) ∈ G satisfying (cid:107) ( xxx, yyy ) − ( xxx (cid:48) , yyy (cid:48) ) (cid:107) < R ( (cid:15) ), we have (cid:10) nnn i ( xxx, yyy ) , ( xxx (cid:48) , yyy (cid:48) ) − ( xxx, yyy ) (cid:11) ≥ − (cid:15) (cid:107) ( xxx, yyy ) − ( xxx (cid:48) , yyy (cid:48) ) (cid:107) . A3. The function D : [0 , ∞ ) → [0 , ∞ ] is such that D (0) = 0 and D ( (cid:15) ) = sup I ∈I , I (cid:54) = ∅ sup { dist (( xxx, yyy ) , ∩ i ∈I ( ∂G i ∩ ∂G )) : ( xxx, yyy ) ∈ ∩ i ∈I U (cid:15) ( ∂G i ∩ ∂G ) } , for (cid:15) > D ( (cid:15) ) → (cid:15) → A4. There is a constant L > i ∈ I , rrr i ( · ) is a uniformly Lips-chitz continuous function from R n + m into R n + m with Lipschitz constant L and (cid:107) rrr i ( xxx, yyy ) (cid:107) = 1 for each ( xxx, yyy ) ∈ R n + m . A5. There is a constant a ∈ (0 , ccc ( · ) = ( c ( · ) , · · · , c I ( · ))and ddd ( · ) = ( d ( · ) , · · · , d I ( · )) from ∂G into R I + such that for each ( xxx, yyy ) ∈ ∂G , (i) (cid:80) i ∈I ( xxx,yyy ) c i ( xxx, yyy ) = 1,min k ∈I ( xxx,yyy ) (cid:42) (cid:88) i ∈I ( xxx,yyy ) c i ( xxx, yyy ) nnn i ( xxx, yyy ) , rrr k ( xxx, yyy ) (cid:43) ≥ a, (ii) (cid:80) i ∈I ( xxx,yyy ) d i ( xxx, yyy ) = 1,min k ∈I ( xxx,yyy ) (cid:42) (cid:88) i ∈I ( xxx,yyy ) d i ( xxx, yyy ) rrr i ( xxx, yyy ) , nnn k ( xxx, yyy ) (cid:43) ≥ a. Theorem 3.7. Given assumptions A1 - A5 . Then there exists a constrained SRBM associatedwith the data ( G, bbb, σσσ, { rrr i , i ∈ I} , ν ) . The proof of Theorem 3.7 is adapted from Kang and Williams [32, Theorem 5.1] and combinedwith [32, Theorem 4.3]. More precisely, we construct a sequence of approximation (randomwalks) to the constrained SRBM and use the invariance principle to establish the weak conver-gence. The main difference is that the constrained SRMB problem in this paper depends notonly on the diffusion process XXX t but also on a degenerate process YYY t indicating the remainingresource levels. The detailed proof of Theorem 3.7 is provided in Appendix A.Next, we solve explicitly the game CCC , based on sufficient conditions in the above verificationtheorem. We will first analyze games C p C p C p and C d C d C d to gain insight into the solution structure. Forgeneral bbb and σσσ , explicit solution is almost impossible. Therefore we consider the following bbb , σσσ , with a general hhh for the rest of this paper. That is, we assume H1 (cid:48) . b i = 0 , i = 1 , , · · · , N, and σσσ = III N . Moreover, we assume that h i ( xxx ) := h (cid:18) x i − (cid:80) Nj =1 x j N (cid:19) , such that H2 (cid:48) . h is symmetric, h (0) ≥ h (cid:48)(cid:48) is non-increasing and k ≤ h (cid:48)(cid:48) ≤ K for some 0 < k < K .4. Nash Equilibrium for Game C p C p C p This section analyzes the Markovian NE of game C p C p C p . Section 4.1 derives the solution to theHJB equations. Section 4.2 constructs the controlled process from the HJB solution. Section4.3 derives the NE for the game C p C p C p and specifies the NE for the two-player game with h ( x ) = x .Recall that in game C p C p C p , AAA = [1 , , · · · , T ∈ R N × , and Y t = y − N (cid:88) i =1 ˇ ξ it and Y − = y. (4.1)4.1. Solving HJB equations. Define (cid:101) x i := x i − (cid:80) j (cid:54) = i x j N − ≤ i ≤ N, (4.2)to be the relative position from x i to the center of ( x j ) j (cid:54) = i . For game C p C p C p , if A i ∩ A j = ∅ , theHJB system simplifies to (HJB- C p ) min − αv i + h (cid:18) N − N (cid:101) x i (cid:19) + 12 N (cid:88) j =1 v ix j x j , − v iy + v ix i , − v iy − v ix i = 0 , for ( xxx, y ) ∈ W − i , min (cid:8) − v iy + v ix j , − v iy − v ix j (cid:9) = 0 , for ( xxx, y ) ∈ A j , j (cid:54) = i. INITE FUEL GAME 15 Now we look for a threshold function f N : R → R with f N ( − x ) = f N ( x ) such that the actionregion A i and the waiting region W i of the i th player are defined by A i := ( E + i ∪ E − i ) ∩ Q i and W i := ( R N × R + ) \ A i , (4.3)where E + i := (cid:8) ( xxx, y ) ∈ R N × R + : (cid:101) x i ≥ f − N ( y ) (cid:9) and E − i := (cid:8) ( xxx, y ) ∈ R N × R + : (cid:101) x i ≤ − f − N ( y ) (cid:9) , (4.4)and Q i := { ( xxx, y ) ∈ R N × R + : | (cid:101) x i | ≥ | (cid:101) x k | for k < i, | (cid:101) x i | > | (cid:101) x k | for k > i } . Note here the partition { Q i } ≤ i ≤ N is introduced to avoid simultaneous jumps by multipleplayers so that A i ∩ A j = ∅ . The key idea of designing the partition is that if several playersare in E + i ∪ E − i , the player who is the farthest away from the center controls. If ties occur,the player with the largest index controls. It is easy to see that W i (cid:54) = ∅ for 1 ≤ i ≤ N , and A i ∩ A j = ∅ for i (cid:54) = j .We seek a solution v i ( xxx, y ) ∈ C ( W − i ) such that if | (cid:101) x i | < f − N ( y ), it is of the form, v i ( xxx, y ) = p N ( (cid:101) x i ) + A N ( y ) cosh (cid:32)(cid:101) x i (cid:114) N − αN (cid:33) , (4.5)where p N ( x ) := E (cid:90) ∞ e − αt h (cid:32) N − N x + (cid:114) N − N B t (cid:33) dt, (4.6)with B t being a one-dimensional Brownian motion. Note that p N ( (cid:101) x i ) is a solution to − αv i + h ( N − N (cid:101) x i ) + (cid:80) Nj =1 v ix j x j = 0, which corresponds to the waiting region, and cosh( (cid:113) N − αN (cid:101) x i )is a solution to − αv i + (cid:80) Nj =1 v ix j x j = 0. If there is no resource, then v i ( xxx, y ) = p N ( (cid:101) x i ), so A N (0) = 0. The smooth-fit principle states that, along the boundary y = f N (˜ x i ) betweenthe continuation set W and the action set A i , v i has certain regularity properties across thehyperplane. Now applying the smooth-fit principle, we get v ix i x i = v iyy = − v ix i y at the boundary y = f N ( (cid:101) x i ) with (cid:101) x i > 0. This follows from v x i + v y = 0 and we expect v i ∈ C ( W − i ). A (cid:48) N ( f N ) = − p (cid:48) N cosh (cid:32) x (cid:114) N − αN (cid:33) + p (cid:48)(cid:48) N (cid:115) N N − α sinh (cid:32) x (cid:114) N − αN (cid:33) ,A N ( f N ) = p (cid:48) N (cid:115) N N − α sinh (cid:32) x (cid:114) N − αN (cid:33) − p (cid:48)(cid:48) N N N − α cosh (cid:32) x (cid:114) N − αN (cid:33) . (4.7)As a consequence, f (cid:48) N ( x ) = p (cid:48) N − N N − α p (cid:48)(cid:48)(cid:48) N p (cid:48)(cid:48) N (cid:113) N N − α tanh (cid:18) x (cid:113) N − αN (cid:19) − p (cid:48) N , (4.8) and A N ( y ) = p (cid:48) N (cid:115) N N − α sinh (cid:32) x (cid:114) N − αN (cid:33) − p (cid:48)(cid:48) N N N − α cosh (cid:32) x (cid:114) N − αN (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = f − N ( y ) . (4.9)Moreover, the curve y = f N ( x ) intersects { x > } at x such that A N ( f N ( x )) = 0. That is,under Assumptions H1 (cid:48) - H2 (cid:48) , x is the unique positive root of (cid:114) N − αN tanh (cid:32) z (cid:114) N − αN (cid:33) = p (cid:48)(cid:48) N ( z ) p (cid:48) N ( z ) . (4.10)The proof of the unique positive root of (4.10) is provided in Appendix C.Specializing to the case h ( x ) = x , we get p sqN ( x ) = (cid:18) N − N (cid:19) x α + N − N α , (4.11) f sqN ( x ) = (cid:90) | x |∧ c (cid:113) N N − α c (cid:113) N N − α (cid:32) z (cid:115) N N − α tanh (cid:32) z (cid:114) N − αN (cid:33) − (cid:33) − dz, (4.12)where c is the unique positive root of z tanh z = 1, and A sqN ( y ) = − NN − α (cosh z − z sinh( z )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = f − N ( y ) (cid:113) N − αN . (4.13)4.2. Controlled dynamics. Given the candidate solution to (HJB- C p ), we derive the corre-sponding NEP by showing the existence of a weak solution ( XXX t , Y t ) to a Skorokhod problemwith an unbounded domain, where the boundary of the domain depends on both the diffusionterm XXX t and the degenerate term YYY t .To start, let W NE : = { ( xxx, y ) ∈ R N +1 : | (cid:101) x i | < f − N ( y ) for 1 ≤ i ≤ N } = (cid:40) ( xxx, y ) ∈ R N +1 : nnn i · xxx > − (cid:114) N − N f − N ( y ) for 1 ≤ i ≤ N (cid:41) (4.14)= ∩ Ni =1 (cid:0) E − i ∪ E + i (cid:1) c . The normal direction of each face is given by ( i = 1 , , · · · , N ) nnn i = c i (cid:18) − N − , · · · , − N − , , − N − , · · · , − N − , ( f − N ) (cid:48) ( y ) (cid:19) ,nnn i + N = c i + N (cid:18) N − , · · · , N − , − , N − , · · · , N − , ( f − N ) (cid:48) ( y ) (cid:19) , with the i th component to be ± c i and c N + i are normalizing constants such that (cid:107) nnn i (cid:107) = (cid:107) nnn N + i (cid:107) = 1. INITE FUEL GAME 17 Note that W NE is an unbounded domain in R N +1 with 2 N boundaries. For i = 1 , , · · · , N ,define the 2 N faces of W NE F i = { ( xxx, y ) ∈ ∂ W NE | ( xxx, y ) ∈ ∂E + i } ,F i + N = { ( xxx, y ) ∈ ∂ W NE | ( xxx, y ) ∈ ∂E − i } . Denote the reflection direction on each face as rrr i = c (cid:48) i (0 · · · , − , · · · , − ,rrr N + i = c (cid:48) N + i (0 · · · , , · · · , − , with the i th component to be ± c (cid:48) i and c (cid:48) N + i are normalizing constants such that (cid:107) rrr i (cid:107) = (cid:107) rrr N + i (cid:107) = 1. NE strategy is defined as follows. Case 1: ( XXX − , Y − ) = ( xxx, y ) ∈ W NE . One can check that W NE defined in (4.14) and { rrr i } Ni =1 defined above satisfies assumptions A1 - A5 . According to Theorem 3.7, there exists a weaksolution to the Skorokhod problem with data (cid:0) W NE , { rrr i } Ni =1 , bbb, σσσ, xxx ∈ W NE (cid:1) . (See AppendixB for the satisfiability of A1 - A5 .) Case 2: ( XXX − , Y − ) = ( xxx, y ) / ∈ W NE , that is, there exists i ∈ { , · · · , N } such that ( XXX − , Y − ) ∈A i . We show that the controlled process ( XXX, Y ) jumps sequentially to a point ( (cid:98) xxx, (cid:98) y ) ∈ W NE for some 0 ≤ (cid:98) y < y , and then follows the solution to the Skorokhod problem starting at( (cid:98) xxx, (cid:98) y ) ∈ W NE . In this case, the jumps will either stop in finite steps, or converge to a limitpoint ( (cid:98) xxx, (cid:98) y ) ∈ W NE for 0 ≤ (cid:98) y < y .For each k ≥ 1, let xxx k = ( x k , · · · , x Nk ) be the positions, and y k be the remaining resource afterthe k th jump. If ( xxx k , y k ) ∈ A i , then the i th player will jump until XXX hits ∂E + i ∪ ∂E − i . Supposethat the jumps do not stop in finite steps. At the k th step, let x (1) k ≤ · · · ≤ x ( N ) k be the orderstatistics of xxx k . Note that only the player with position x (1) k or x ( N ) k intervenes. Then ( x (1) k ) k ≥ is non-decreasing and bounded from above by x ( N )0 , therefore ( x (1) k ) k ≥ converges, and so does( x ( N ) k ) k ≥ . Hence ( xxx k ) k ≥ converges. Since ( y k ) k ≥ is decreasing and bounded below by 0, itconverges to some point ˆ y . Now suppose that ( xxx k , y k ) → ( (cid:98) xxx, (cid:98) y ) / ∈ ∂ W NE . Let i ∗ ∈ { , · · · , N } such that (cid:98) xxx ∈ A i ∗ . For k sufficiently large, we have | xxx k − (cid:98) xxx | < ε and by the triangle inequality, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x i ∗ k − (cid:80) j (cid:54) = i ∗ x jk N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ max ≤ i ≤ N (cid:40)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:98) x ik − (cid:80) j (cid:54) = i (cid:98) x jk N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − f − N ( (cid:98) y ) (cid:41) − ε. Thus the i th ∗ player should jump at least (cid:18) max ≤ i ≤ N (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)(cid:98) x ik − (cid:80) j (cid:54) = i (cid:98) x jk N − (cid:12)(cid:12)(cid:12)(cid:12) − f − N ( (cid:98) y ) (cid:27) − ε (cid:19) ∧ (cid:98) y inthe ( k + 1) th step. It suffices to take ε sufficiently small to get a contradiction.In summary, the controlled process inherits a rich structure from the candidate solution. • If starting at a point in the common waiting region of all N players, then the controlledprocess is a reflected Brownian motion with an evolving free boundary. • If staring at a point outside the common waiting region, then the controlled processfollows rank-dependent dynamics with a moving origin. NE for the N -player game. Combining the results in Sections 4.1 and 4.2, and basedon the verification theorem developed in Section 3, we have the following theorem of the NEfor the N -player game (2.9) with constraint (4.1). Theorem 4.1 (NE for the N -player game C p C p C p ) . Assume H1 (cid:48) - H2 (cid:48) . Let v i : R N × R + → R bedefined by v i ( xxx, y ) = p N ( (cid:101) x i ) + A N ( y ) cosh (cid:18)(cid:101) x i (cid:113) N − αN (cid:19) if ( xxx, y ) ∈ W − i ∩ W i ,v i (cid:18) xxx − i , x i + + (cid:80) k (cid:54) = i x k N − , f N ( x i + ) (cid:19) if ( xxx, y ) ∈ W − i ∩ E + i ,v i (cid:18) xxx − i , (cid:80) k (cid:54) = i x k N − − x i − , f N ( x i − ) (cid:19) if ( xxx, y ) ∈ W − i ∩ E − i ,v i (cid:18) xxx − j , x j + + (cid:80) k (cid:54) = j x k N − , f N ( x j + ) (cid:19) if ( xxx, y ) ∈ A j ∩ E + j for j (cid:54) = i,v i (cid:18) xxx − j , (cid:80) k (cid:54) = j x k N − − x j − , f N ( x j − ) (cid:19) if ( xxx, y ) ∈ A j ∩ E − j for j (cid:54) = i, (4.15) where • A i and W i are given in (4.3) , and E ± i is given in (4.4) with f N ( · ) defined by (4.8) - (4.10) , • (cid:101) x i is defined by (4.2) , and A N ( · ) is defined by (4.9) , • x i + is the unique positive root of z − f N ( z ) = (cid:101) x i − y , and x i − is the unique negative rootof z + f N ( z ) = (cid:101) x i + y .Then v i is the game value associated with an NEP ξξξ ∗ = ( ξ ∗ , · · · , ξ N ∗ ) . That is, v i ( xxx, y ) = J iC p ( xxx, y ; ξξξ ∗ ) . Moreover, the controlled process ( XXX ∗ , Y ∗ ) under ξξξ ∗ is given in Section 4.2.Proof. Now we check that conditions (i)-(vii) in Theorem 3.3 are satisfied.(i) Based on the analysis in Section 4.2, when ( xxx, y ) ∈ W NE , the NE strategy is a solutionto the Skorokhod problem specified in Case 2, which is a continuous process. When( xxx, y ) / ∈ W NE , the sequential push specified in Case 1 satisfies the “no simultaneousjump” condition.(ii) Solution (4.15) satisfies the derivation in Section 4.1 and hence satisfies the HJB for W − i . More specifically, when ( xxx, yyy ) ∈ W − i , solution (4.15) takes the form of (4.5), with A N defined in (4.9). By the smooth-fit principle leading to (4.7), we have v i ∈ C ( W )where v i is defined in (4.3).(iii) Since (cid:107)∇ v i (cid:107) ≤ K , and the control ξξξ ∈ S N ( xxx, y ) has finite variations, the transver-sality condition ( iii ) is satisfied. More specifically, there exists constant C such that | v ( xxx, yyy ) | ≤ C (1 + (cid:107) xxx (cid:107) + (cid:107) yyy (cid:107) ) given the explicit form of 4.15. Thus, E [ v ( XXX ∗ T , YYY ∗ T )] ≤ C E (1 + (cid:107) XXX ∗ T (cid:107) + (cid:107) yyy (cid:107) ) ≤ C (1 + (cid:107) y (cid:107) ) + C E ( (cid:107) ξξξ ∗ T (cid:107) + (cid:107) BBB T (cid:107) + (cid:107) xxx (cid:107) ) . For each i ( i = 1 , , · · · , N ), suppose E [ ˇ ξ i ∗ t ] (cid:54) = o ( e αt ) as t → ∞ , then standard argumentas in [56, P39] shows that E [ (cid:82) ∞ e − αt d ˇ ξ i ∗ t ] = ∞ , which contradicts the definition ofadmissible set U iN . Finally, we have lim sup T →∞ e − αt E [ v i ( XXX ∗ T , YYY ∗ T )] = 0. INITE FUEL GAME 19 (iv) Solution (4.15) satisfies the smooth-fit principle in Section 4.1, therefore, v i ∈ C ( W − i ). v i is convex in W − i since p N ( (cid:101) x i ) + A N ( y ) cosh (cid:18)(cid:101) x i (cid:113) N − αN (cid:19) is convex.(v) Since f − N is non-increasing, in W − i , ˜ x i ≤ f − N ( y ) ≤ f − N (0) < ∞ . This implies that˜ x i is bounded in W − i . By the definition of A N ( y ) in (4.5), A N ( y ) is bounded in W − i .Hence v ix j is bounded in W − i by definition (4.9).(vi) By the construction of Case 1 and Case 2, when ( xxx, y ) / ∈ W − i , there is a sequential pushat time 0 to move the joint position to some point (ˆ xxx, ˆ y ) ∈ ∂ W − i . when ( xxx, y ) ∈ W − i ,( ξξξ − i ∗ , , ξ i ) forms a solution to the Skorokhod problem in ∩ j (cid:54) = i ( E − j ∪ E + j ) c . It is easy toverify that ∩ j (cid:54) = i ( E − j ∪ E + j ) c ⊂ W − i and the Skorokhod problem with ∩ j (cid:54) = i ( E − j ∪ E + j ) c has a weak solution. Therefore condition ( vi ) is satisfied.(vii) Since v i has the same value before and after player j ’s control, equation (3.9) is triviallysatisfied. (cid:3) To illustrate, we specialize Theorem 4.1 to the case N = 2 and h ( x ) = x . In this case, wecan also construct the strong solution of NE strategies. Corollary 4.2 (NE for the two-player game C p C p C p ) . Assume H1 (cid:48) - H2 (cid:48) . The following controls ξ ∗ , + t = 0 ,ξ ∗ , − t = 0 ,ξ ∗ , + t = max (cid:110) , max ≤ s ≤ t { , x − x + B s − B s − ξ ∗ , + s + ξ ∗ , − s − ( f sq ) − ( y − ξ ∗ , + s − ξ ∗ , − s ) } (cid:111) ,ξ ∗ , − t = max (cid:110) , max ≤ u ≤ t { , x − x + B s − B s + ξ ∗ , + s − ξ ∗ , − s − ( f sq ) − ( y − ξ ∗ , + s − ξ ∗ , − s ) } (cid:111) , give an NEP for the two-player game (2.9) with (4.1) and h ( x ) = x , where ( f sq ) − is definedin (4.12). Moreover, let v and v be the associated values of the above NEP ( ξ ∗ , ξ ∗ ) , then v ( x , x , y ) = ( x − x ) α + α + A ( y ) cosh (cid:0) ( x − x ) √ α (cid:1) if | x − x | ≤ ( f sq ) − ( y ) ,v ( x , x + x , f ( x )) if x ≤ x − ( f sq ) − ( y ) ,v ( x , x − x − , f ( x − )) if x ≥ x + ( f sq ) − ( y ) , (4.16) and v ( x , x , y ) = ( x − x ) α + α + A ( y ) cosh (cid:0) ( x − x ) √ α (cid:1) if | x − x | ≤ ( f sq ) − ( y ) ,v ( x , x − x − , f ( x − )) if x ≤ x − ( f sq ) − ( y ) ,v ( x , x + x , f ( x ))) if x ≥ x + ( f sq ) − ( y ) , (4.17) where A ( y ) = − α (cosh( z ) − z sinh( z )) | z = √ α ( f sq ) − ( y ) , (4.18) and x is the unique root of z − f sq ( z ) = x − y , and x − is the unique root of z + f sq ( z ) = x + y . Note that under partition { Q i } i =1 , , we have A = ∅ , hence ( ξ ∗ , + , ξ ∗ , − ) = (0 , (a) No control from player one. (b) Control from player two. Figure 2. Case C p C p C p : NEP when N = 2.5. Nash Equilibrium For Game C d C d C d In this section, we study the NEP of the N -player game C d C d C d . That is A = I N I N I N ∈ R N × N , and Y it = y i − ˇ ξ it with Y i − = y i . (5.1)Recall that the major difference between game C p C p C p and game C d C d C d is that, in the former all N players share a fixed amount of the same resource, while in the latter each player has her ownindividual fixed resource constraint. This difference is reflected in ( HJ B − C p ) and ( HJ B − C d )in terms of their dimensionality, and in each player’s control based on the remaining resources.In particular, ( HJ B − C p ) and the state space ( xxx, y ) of C p C p C p are of dimension N + 1, whereas( HJ B − C d ) and the state space ( xxx, yyy ) of C d C d C d are of dimension 2 N . Moreover, in game C p C p C p , thegradient constraint is − v iy ± v ix i for player i . In contrast, in game C d C d C d , each player controls herown resource level, the gradient constraint becomes − v iy i ± v ix i for player i . So if A i ∩ A j = ∅ ,the HJB equation for v i ( xxx, yyy ) in game CCC d is as follows. (HJB- C d ) min − αv i + h (cid:18) N − N (cid:101) x i (cid:19) + 12 N (cid:88) j =1 v ix j x j , − v iy i + v ix i , − v iy i − v ix i = 0 , for ( xxx, yyy ) ∈ W − i , min (cid:110) − v iy j + v ix j , − v iy j − v ix j (cid:111) = 0 , for ( xxx, yyy ) ∈ A j , j (cid:54) = i. Note that the control policy of the i th player only depends on ( xxx, y i ) in W − i . As seen inSection 4, for the controlled process of type C p C p C p , upon hitting the boundary of the polyhedron,the polyhedron will expand in all directions. While for the controlled process of type C d C d C d , onlyone direction of the the polyhedron will move once hit.To proceed, similar to Section 4, define the action region A i ∈ R N × R N + and the waitingregion W i of the i th player by A i := ( E + i ∪ E − i ) ∩ Q i and W i := R N × R N + \ A i , (5.2)where Q i := (cid:110) ( xxx, yyy ) ∈ R N × R N + : | (cid:101) x i | − f − N ( y i ) ≥ | (cid:101) x k | − f − N ( y k ) for k < i, | (cid:101) x i | − f − N ( y i ) > | (cid:101) x k | − f − N ( y k ) for k > i (cid:111) , INITE FUEL GAME 21 and E + i := (cid:8) ( xxx, yyy ) ∈ R N × R N + : (cid:101) x i ≥ f − N ( y i ) (cid:9) and E − i := (cid:8) ( xxx, yyy ) ∈ R N × R N + : (cid:101) x i ≤ − f − N ( y i ) (cid:9) . (5.3)Recall the definition of the threshold function f N ( · ) from (4.8)-(4.10), we now investigate controlof player i which only depends on ( xxx, y i ) in W − i . That is, for | (cid:101) x i | < f − N ( y i ), v i ( xxx, yyy ) = p N ( (cid:101) x i ) + A N ( y i ) cosh (cid:32)(cid:101) x i (cid:114) N − αN (cid:33) , (5.4)is a solution to (HJB- C d ), where p N ( · ) is defined by (4.6), and A N ( · ) defined by (4.9).The next step is to construct the controlled process ( XXX, YYY ) corresponding to the HJB solution(5.4). Let W NE : = { ( xxx, yyy ) ∈ R N × R N + : | (cid:101) x i | < f − N ( y i ) for 1 ≤ i ≤ N } = ∩ Ni =1 (cid:0) E − i ∪ E + i (cid:1) c . (5.5)The normal direction on each face is given by nnn i = c i (cid:18) N − , · · · , N − − , N − · · · , N − , · · · , , ( f − N ) (cid:48) (cid:0) y i (cid:1) , , · · · , (cid:19) ,nnn N + i = c N + i (cid:18) − N − , · · · , − N − , , − N − , · · · , − N − , · · · , , ( f − N ) (cid:48) (cid:0) y i (cid:1) , , · · · , (cid:19) , with the i th component to be ± N + i ) th component to be ( f − N ) (cid:48) ( y i ). c i and c N + i are normalizing constants such that (cid:107) nnn i (cid:107) = (cid:107) nnn N + i (cid:107) = 1.Note that W NE is an unbounded domain in R N with 2 N boundaries. For i = 1 , , · · · , N ,define the 2 N faces of W NE F i = { ( xxx, yyy ) ∈ ∂ W NE | ( xxx, yyy ) ∈ ∂E + i } ,F i + N = { ( xxx, yyy ) ∈ ∂ W NE | ( xxx, yyy ) ∈ ∂E − i } . Denote the reflection direction on each face as rrr i = c (cid:48) i (0 · · · , , − , , · · · 0; 0 , · · · , , − , , · · · , ,rrr N + i = c (cid:48) N + i (0 · · · , , , , · · · 0; 0 , · · · , , − , , · · · , , with the i th component to be ± N + i ) th component to be 1. c (cid:48) i and c (cid:48) N + i arenormalizing constants such that (cid:107) rrr i (cid:107) = (cid:107) rrr N + i (cid:107) = 1.The NE strategy is defined as follows. Case 1: ( XXX − , YYY − ) = ( xxx, yyy ) ∈ W NE . One can check that W NE defined in (5.5) and { rrr i } Ni =1 defined above satisfies assumptions A1 - A5 . Therefore, there exists a weak solution to theSkorokhod problem with data (cid:0) W NE , { rrr i } Ni =1 , bbb, σσσ, xxx ∈ W NE (cid:1) . (See Appendix B for the satisfi-ability of A1 - A5 .) Case 2: ( XXX − , YYY − ) = ( xxx, yyy ) / ∈ W NE . There exists i ∈ { , · · · , N } such that ( XXX − , YYY − ) ∈ A i .For each k ≥ 1, let xxx k = ( x k , · · · , x Nk ) be the positions, and yyy k = ( y k , · · · , y Nk ) be the resourceremaining after the k th control. If ( xxx k , yyy k ) ∈ A i , then the i th player will control until XXX hits ∂E + i ∪ ∂E − i . The argument in Section 4.2 shows that the controlled process XXX controlssequentially to a point ( (cid:98) xxx, (cid:98) yyy ) ∈ W NE for 000 ≤ (cid:98) y (cid:98) y (cid:98) y ≤ yyy . Then ( XXX, YYY ) follows the solution to theSkorokhod problem starting at ( (cid:98) xxx, (cid:98) yyy ).In summary, the NE for the N -player game (2.9) with constraint C d C d C d is stated as follows. Theorem 5.1 (NE for the N -player game C d C d C d ) . Assume H1 (cid:48) - H2 (cid:48) . Let v i : R N × R N + → R bedefined by v i ( xxx, yyy ) = p N ( (cid:101) x i ) + A N ( y i ) cosh (cid:18)(cid:101) x i (cid:113) N − αN (cid:19) if ( xxx, yyy ) ∈ W − i ∩ W i ,v i (cid:18) xxx − i , x i + + (cid:80) k (cid:54) = i x k N − , f N ( x i + ) (cid:19) if ( xxx, yyy ) ∈ W − i ∩ E + i ,v i (cid:18) xxx − i , (cid:80) k (cid:54) = i x k N − − x i − , f N ( x i − ) (cid:19) if ( xxx, yyy ) ∈ W − i ∩ E − i ,v i (cid:18) xxx − j , x j + + (cid:80) k (cid:54) = j x k N − , y i (cid:19) if ( xxx, yyy ) ∈ A j ∩ E + j for j (cid:54) = i,v i (cid:18) xxx − j , (cid:80) k (cid:54) = j x k N − − x j − , y i (cid:19) if ( xxx, yyy ) ∈ A j ∩ E − j for j (cid:54) = i, (5.6) where • A i and W i are given in (5.2) , and E ± i is given in (5.3) with f N ( · ) defined by (4.8) - (4.10) , • (cid:101) x i is defined by (4.2) , and A N ( · ) is defined by (4.9) , • x i + is the unique positive root of z − f N ( z ) = (cid:101) x i − y , and x i − is the unique negative rootof z + f N ( z ) = (cid:101) x i + y .Then v i is the game value associated with an NEP ξξξ ∗ = ( ξ ∗ , · · · , ξ N ∗ ) . That is, v i ( xxx, yyy ) = J iC d ( xxx, yyy ; ξξξ ∗ ) . Moreover, the controlled process ( XXX ∗ , YYY ∗ ) under ξξξ ∗ is given in this section: Case 1 if ( xxx, yyy ) ∈W NE , and Case 2 if ( xxx, yyy ) / ∈ W NE . Theorem 5.1 can be verified in a similar way as Theorem 4.1. Specializing to the two-playergame with h ( x ) = x , we have the following result. Corollary 5.2 (NE for N = 2 for game C d C d C d ) . Assume H1 (cid:48) - H2 (cid:48) . The following controls ξ ∗ , + t := ∆ ξ ∗ , +0 + (cid:90) t ∧ τ { XXX ∗ s ∈ F ( Y ∗ s ) } { Y ∗ s >Y ∗ s } dη s ,ξ ∗ , − t := ∆ ξ ∗ , − + (cid:90) t ∧ τ { XXX ∗ s ∈ F ( Y ∗ s ) } { Y ∗ s >Y ∗ s } dη s ,Y ∗ t := y − ˆ ξ ∗ t , τ := inf { t ≥ Y ∗ t = 0 } ,ξ ∗ , + t := ∆ ξ ∗ , +0 + (cid:90) t ∧ τ { XXX ∗ s ∈ F ( Y ∗ s ) } { Y ∗ s ≥ Y ∗ s } dη s ,ξ ∗ , − t := ∆ ξ ∗ , − + (cid:90) t ∧ τ { XXX ∗ t ∈ F ( Y ∗ s ) } { Y ∗ s ≥ Y ∗ s } dη s ,Y ∗ t := y − ˆ ξ ∗ t , τ := inf { t ≥ Y ∗ t = 0 } , (5.7) INITE FUEL GAME 23 give an NEP for the two-player game C d C d C d with h ( x ) = x , where • F ( y ) = F ( y ) = (cid:110) ( x , x ) : x − x = − ( f sq ) − ( y ) (cid:111) , • F ( y ) = F ( y ) = (cid:110) ( x , x ) : x − x = ( f sq ) − ( y ) (cid:111) , • η i ∗ t are non-decreasing processes with η i ∗ − = 0 ( i = 1 , , , , • ∆ ξ ∗ , +0 = (cid:40) x − , if y ≥ y and x ≤ x − ( f sq ) − ( y ) ,x − , if y < y and x ≤ x − ( f sq ) − ( y ) , ∆ ξ ∗ , − = (cid:40) x , if y ≥ y and x ≥ x − ( f sq ) − ( y ) ,x , if y < y and x ≥ x − − ( f sq ) − ( y ) , ∆ ξ ∗ , +0 = (cid:40) x − , if y > y and x ≤ x − ( f sq ) − ( y ) ,x − , if y < y and x ≤ x − ( f sq ) − ( y ) , ∆ ξ ∗ , − = (cid:40) x , if y > y and x ≥ x − ( f sq ) − ( y ) ,x , if y < y and x ≥ x − − ( f sq ) − ( y ) , • x i + is the unique root of z − f sq ( z ) = x j − y , x i − is the unique root of z + f sq ( z ) = x j + y ,with f sq ( · ) is given by (4.12) . ( i, j = 1 , and i (cid:54) = j ).Moreover, let v and v be the corresponding values of the above NEP ( ξ ∗ , ξ ∗ ) . Then if y > y , v ( x , x , y ) = ( x − x ) α + α + A ( y ) cosh (cid:0) ( x − x ) √ α (cid:1) if | x − x | ≤ ( f sq ) − ( y ) ,v ( x − , x − x − , f sq ( x − )) if x ≤ x − ( f sq ) − ( y ) ,v ( x , x + x , f sq ( x )) if x ≥ x + ( f sq ) − ( y ) ,v ( x , x , y ) = ( x − x ) α + α + A ( y ) cosh (cid:0) ( x − x ) √ α (cid:1) if | x − x | ≤ ( f sq ) − ( y ) ,v ( x , x , y ) if x ≤ x − ( f sq ) − ( y ) ,v ( x − , x , y ) if x ≥ x + ( f sq ) − ( y );(5.8) and if y ≤ y , v ( x , x , y ) = ( x − x ) α + α + A ( y ) cosh (cid:0) ( x − x ) √ α (cid:1) if | x − x | ≤ ( f sq ) − ( y ) ,v ( x , x , y ) if x ≤ x − ( f sq ) − ( y ) ,v ( x , x − , y ) if x ≥ x + ( f sq ) − ( y ) ,v ( x , x , y ) = ( x − x ) α + α + A ( y ) cosh (cid:0) ( x − x ) √ α (cid:1) if | x − x | ≤ ( f sq ) − ( y ) ,v ( x , x + x , f sq ( x )) if x ≤ x − ( f sq ) − ( y ) ,v ( x , x − x − , f sq ( x − )) if x ≥ x + ( f sq ) − ( y ) , (5.9) where A ( · ) is given by (4.18) . Comparison of Corollary 4.2 and Corollary 5.2. Consider N = 2 and h ( x ) = x . Ingame C p C p C p , only player two controls the two separating hyperplanes whereas player one doesnothing, see Figure 2. In game C p C p C p , player one controls the two separating hyperplanes when y > y and she does nothing when y ≥ y . See Figure 3. (a) y ≤ y : no control from player one. (b) y > y : player one controls.(c) y ≤ y : player two controls. (d) y > y : no control from player two. Figure 3. Case C d C d C d : NEP when N = 2.6. Nash Equilibrium for game CCC In the previous two sections, we have dealt with two special games C p C p C p and C d C d C d . Analysisof these two games provides important insight into the solution structure of the general game CCC . Namely, the NE strategy depends on the positions of players and their remaining resourcelevels. With these two special cases in mind, now recall that in game CCC , dY jt = − N (cid:88) i =1 a ij Y jt − (cid:80) Mk =1 a ik Y kt − d ˇ ξ it and Y j − = y j ≥ . (6.1)For the HJB equation ( HJ B − C ), the gradient constraint is more complicated than the twospecial cases C p C p C p and C d C d C d . When A i ∩ A j = ∅ , (HJB- C ) min (cid:40) − αv i + h + 12 N (cid:88) j =1 v ix j x j , Γ i v i + v ix i , − Γ i v i − v ix i (cid:41) = 0 , for ( xxx, yyy ) ∈ W − i , min (cid:8) − Γ j v i + v ix j , − Γ j v i − v ix j (cid:9) = 0 , for ( xxx, yyy ) ∈ A j , j (cid:54) = i. INITE FUEL GAME 25 In particular, if AAA = [1 , , · · · , T ∈ R N × , then ( HJB − C ) becomes ( HJB − C p ); and if AAA = I N I N I N , thenit is ( HJB − C d ).Similar to Section 4, define the action region A i ∈ R N × R M + and the waiting region W i of the i th player by A i := ( E + i ∪ E − i ) ∩ Q i and W i := R N × R N + \ A i , (6.2)where Q i := ( xxx, yyy ) ∈ R N × R M + : | (cid:101) x i | − f − N M (cid:88) j =1 a ij y j ≥ | (cid:101) x k | − f − N M (cid:88) j =1 a kj y j for k < i, | (cid:101) x i | − f − N M (cid:88) j =1 a ij y j > | (cid:101) x k | − f − N M (cid:88) j =1 a kj y j for k > i , and E + i := ( xxx, yyy ) ∈ R N × R M + : (cid:101) x i ≥ f − N M (cid:88) j =1 a ij y j and E − i := ( xxx, yyy ) ∈ R N × R M + : (cid:101) x i ≤ − f − N M (cid:88) j =1 a ij y j . (6.3)From the analysis in Sections 4 and 5, and the “guess” that the control policy of player i only dependson ( xxx, (cid:80) Mj =1 a ij y j ) when in W − i , we get for | (cid:101) x i | < f − N ( (cid:80) Mj =1 a ij y j ), v i ( xxx, yyy ) = p N ( (cid:101) x i ) + A N M (cid:88) j =1 a ij y j cosh (cid:32)(cid:101) x i (cid:114) N − αN (cid:33) , (6.4)is a solution to (HJB- C ), where p N ( · ) is defined by (4.6), and A N ( · ) defined by (4.9).The next step is to construct the controlled process ( XXX, YYY ) corresponding to the HJB solution (6.4). W NE : = ( xxx, yyy ) ∈ R N × R M + : | (cid:101) x i | < f − N M (cid:88) j =1 a ij y j for 1 ≤ i ≤ N = ∩ Ni =1 (cid:0) E − i ∪ E + i (cid:1) c . (6.5)The normal direction on each face is given by nnn i = c i N − , · · · , − , · · · , N − f − N ) (cid:48) M (cid:88) j =1 a ij y j a i , · · · , ( f − N ) (cid:48) M (cid:88) j =1 a ij y j a iM ,nnn N + i = c N + i − N − , · · · , , · · · , − N − f − N ) (cid:48) M (cid:88) j =1 a ij y j a i , · · · , ( f − N ) (cid:48) M (cid:88) j =1 a ij y j a iM , with the i th component being ± 1, and c i and c N + i the normalizing constants such that (cid:107) nnn i (cid:107) = (cid:107) nnn N + i (cid:107) =1. Note that W NE is an unbounded domain in R N with 2 N boundaries. For i = 1 , , · · · , N , define the2 N faces of W NE F i = { ( xxx, yyy ) ∈ ∂ W NE | ( xxx, yyy ) ∈ ∂E + i } ,F i + N = { ( xxx, yyy ) ∈ ∂ W NE | ( xxx, yyy ) ∈ ∂E − i } . Denote the reflection direction on each face as rrr i = c (cid:48) i (cid:32) · · · , − , · · · − a i y (cid:80) Mj =1 a ij y j , · · · , − a iM y M (cid:80) Mj =1 a ij y j (cid:33) ,rrr N + i = c (cid:48) N + i (cid:32) · · · , , · · · − a i y (cid:80) Mj =1 a ij y j , · · · , − a iM y M (cid:80) Mj =1 a ij y j (cid:33) , with the i th component to be ± c (cid:48) i and c (cid:48) N + i are normalizing constants such that (cid:107) rrr i (cid:107) = (cid:107) rrr N + i (cid:107) = 1.NE strategy is defined as follows. Case 1: ( XXX − , YYY − ) = ( xxx, yyy ) ∈ W NE . One can check that W NE defined in (6.5) and { rrr i } Ni =1 definedabove satisfies assumptions A1 - A5 . Therefore, there exists a weak solution to the Skorokhod problemwith data (cid:0) W NE , { rrr i } Ni =1 , bbb, σσσ, xxx ∈ W NE (cid:1) . (See Appendix B for the satisfiability of A1 - A5 .) Case 2: ( XXX − , YYY − ) = ( xxx, yyy ) ∈ W NE . There exists i ∈ { , · · · , N } such that ( XXX − , YYY − ) ∈ A i . Foreach k ≥ 1, let xxx k = ( x k , · · · , x Nk ) be the positions, and yyy k = ( y k , · · · , y Mk ) be the remaining resourcelevel after the k th jump. If ( xxx k , yyy k ) ∈ A i , then the i th player will jump until XXX hits ∂E + i ∪ ∂E − i .The argument in Section 4.2 shows that the controlled process ( XXX, YYY ) jumps sequentially to a point( (cid:98) xxx, (cid:98) yyy ) ∈ W NE for 000 ≤ (cid:98) yyy ≤ yyy . Then ( XXX, YYY ) follows the solution to the Skorokhod problem starting at( (cid:98) xxx, (cid:98) yyy ).The NE for the N -player game (2.9) with constraint CCC is stated as follows. Theorem 6.1 (NE for the N -player game CCC ) . Assume H1 (cid:48) - H2 (cid:48) . Let v i : R N × R M + → R be defined by v i ( xxx, yyy ) = p N ( (cid:101) x i ) + A N ( (cid:80) Mj =1 a ij y j ) cosh (cid:18)(cid:101) x i (cid:113) N − αN (cid:19) if ( xxx, yyy ) ∈ W − i ∩ W i ,v i (cid:18) xxx − i , x i + + (cid:80) k (cid:54) = i x k N − , f N ( x i + ) (cid:19) if ( xxx, yyy ) ∈ W − i ∩ E + i ,v i (cid:18) xxx − i , (cid:80) k (cid:54) = i x k N − − x i − , f N ( x i − ) (cid:19) if ( xxx, yyy ) ∈ W − i ∩ E − i ,v i (cid:18) xxx − j , x j + + (cid:80) k (cid:54) = j x k N − , y i (cid:19) if ( xxx, yyy ) ∈ A j ∩ E + j for j (cid:54) = i,v i (cid:18) xxx − j , (cid:80) k (cid:54) = j x k N − − x j − , y i (cid:19) if ( xxx, yyy ) ∈ A j ∩ E − j for j (cid:54) = i, (6.6) where • A i and W i are given in (6.2) , and E ± i is given in (6.3) with f N ( · ) defined by (4.8) - (4.10) , • (cid:101) x i is defined by (4.2) , and A N ( · ) defined by (4.9) , • x i + is the unique positive root of z − f N ( z ) = (cid:101) x i − (cid:80) Mj =1 a ij y j , and x i − is the unique negativeroot of z + f N ( z ) = (cid:101) x i + (cid:80) Mj =1 a ij y j .Then v i is the value associated with a NEP ξξξ ∗ = ( ξ ∗ , · · · , ξ N ∗ ) . That is, v i ( xxx, yyy ) = J iC ( xxx, yyy ; ξξξ ∗ ) . Moreover, the controlled process ( XXX ∗ , YYY ∗ ) under ξξξ ∗ is a solution to a Skorokhod problem as described in Case 1 if ( xxx, yyy ) ∈ W NE , and described as Case 2 if ( xxx, yyy ) / ∈ W NE . Remark 6.2. Since each player makes decisions based on the total available resource and is indifferentto the resource identity, we assume the boundary in the smooth-fit principle satisfies A N ( y , · · · , y M ) = A N ( (cid:80) Mj =1 a ij y j ) for player i . Note that the value function depends on yyy only through (cid:80) j a ij y j . Therefore INITE FUEL GAME 27 if we denote ˜ v i ( xxx, z ) := v i ( xxx, yyy ) and z = (cid:80) Nj =1 a ij y j , it is easy to verify that (cid:80) Mj =1 a ij y j (cid:80) Mk =1 a ik y k v iy j = (cid:80) Mj =1 a ij y j (cid:80) Mk =1 a ik y k a ij ˜ v iz = ˜ v iz . Hence the calculation is reduced to that for Theorem 4.1. Comparing Games C p , C d and C In this section, we compare the games C p C p C p , C d C d C d and CCC . We will first compare their game values anddiscuss their economic implications. We will then discuss their difference in terms of the NEP. Finally,we discuss their perspective NEs in the framework of controlled rank-dependent SDEs.To make the games comparable, let us assume y = (cid:80) Nj =1 y j . Let us also consider a special sharinggame C s C s C s which can be connected with both C d C d C d and C p C p C p : C s C s C s : M = N and a ii = 1 for i = 1 , , · · · , N .7.1. Pooling, Dividing, and Sharing. Denote the game value and waiting region for each player i as v iC p and W C p i respectively for game C p C p C p . Similar notations are defined for C d C d C d and C s C s C s . Comparing game values.Proposition 7.1 (Game values comparison) . Assume H1 (cid:48) - H2 (cid:48) . For each ( xxx, yyy ) ∈ R N × R N + , if ( xxx, y ) ∈W C p i , and ( xxx, yyy ) ∈ W C d i ∩ W C s i , then, v iC p ( xxx, y ) ≤ v iC s ( xxx, y ) ≤ v iC d ( xxx, yyy ) , i = 1 , , · · · , N. Proof. The comparison is by direct computation. Indeed, recall that in case C p C p C p , when ( xxx, y ) ∈ W C p i , v iC p ( xxx, y ) = p N ( (cid:101) x i ) + A N ( y ) cosh (cid:32)(cid:101) x i (cid:114) N − αN (cid:33) , for i = 1 , , · · · , N , where (cid:101) x i is defined in (4.2) and A N is defined in (4.9).Similarly, in case C d C d C d , when ( xxx, yyy ) ∈ W C d i , v iC d ( xxx, yyy ) = p N ( (cid:101) x i ) + A N ( y i ) cosh (cid:32)(cid:101) x i (cid:114) N − αN (cid:33) , for each i = 1 , , · · · , N . And, in case C s C s C s , when ( xxx, yyy ) ∈ W C s i , v iC s ( xxx, yyy ) = p N ( (cid:101) x i ) + A N N (cid:88) j =1 a ij y j cosh (cid:32)(cid:101) x i (cid:114) N − αN (cid:33) , for each i = 1 , , · · · , N . By elementary calculations, A (cid:48) N ( y ) < . Therefore, when y = (cid:80) Nj =1 y j , ( xxx, y ) ∈ W C p i , and ( xxx, yyy ) ∈ W C d i ∩ W C s i , v iC p ( xxx, y ) ≤ v iC s ( xxx, y ) ≤ v iC d ( xxx, yyy ) . The first inequality holds because y = (cid:80) Ni =1 y i ≥ (cid:80) Ni =1 a ij y j and the equality holds if and only if a ij = 1for each j = 1 , , · · · , N . The second inequality holds because a ii = 1 and the equality holds if and onlyif a ij = 0 for each j (cid:54) = i . (cid:3) This result has a clear economic interpretation. In a stochastic game where players have the optionsto share resources, versus the possibility to divide resources in advance, sharing will have lower costthan dividing. Pooling yields the lowest cost for each player.(a) C p C p C p (b) C d C d C d (c) CCC Figure 4. Comparison of projected evolving boundaries for C p C p C p , C d C d C d , CCC when N = 3. Define the projected common waiting region W NE ( yyy ) := xxx ∈ R N : | (cid:101) x i | < f − N M (cid:88) j =1 a ij y j for 1 ≤ i ≤ N , for any fixed resource level yyy . Then W NE ( yyy ) is a polyhedron with 2 N boundary faces. Figure 4a showsa pooling game C p C p C p . After one player exercises controls, all the faces of the boundary move. Figure 4bcorresponds to a dividing game C d C d C d . After player i exercises controls, her faces of F i and F i + N move.Here i = 1 , N = 3. For a sharing game CCC , shown in Figure 4c, after one player exercises her controls,the faces of the players who are connected with her will move, while the faces for other players remainunchanged. Here i = 2 and player 2 and 3 are connected.7.2. NEs for the games and controlled rank-dependent SDEs. In the previous sections, thecontrolled dynamics is constructed directly via the reflected Brownian motion. This class of SDEs canalso be cast in the framework of rank-dependent SDEs. Indeed, the controlled dynamics of NE in theaction regions of the N -player can be written as a controlled rank-dependent SDEs : dX it = N (cid:88) j =1 F i ( XXX t ,YYY t )= F ( j ) ( XXX t ,YYY t ) (cid:16) δ j dt + σ j dB jt + dξ j, + t − dξ j, − t (cid:17) ,dY jt = − N (cid:88) i =1 a ij Y js − (cid:80) Mj =1 a ij Y js − d ˇ ξ is , with ( ξ i, + , ξ i, − ) the controls, F i : R N × R M + → R a rank function depending on both XXX and YYY , F (1) ≤ · · · ≤ F ( N ) the order statistics of ( F i ) ≤ i ≤ N , and δ i ∈ R , σ i ≥ 0. In game C p C p C p , the controlled INITE FUEL GAME 29 dynamics in the action regions satisfies the SDEs with F iC p ( xxx, yyy ) = | x i − (cid:80) j (cid:54) = i x j N − | , δ i = 0 and σ i = 0 foreach i = 1 , · · · N , and ξ i, ± = 0 for each i = 1 , · · · , N − ξ N, ± (cid:54) = 0 . In game C d C d C d , F iC d ( xxx, yyy ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x i − (cid:80) j (cid:54) = i x j N − − f − N ( y i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For the general game CCC , the controlled process in the action regions is governed by the rank-dependentdynamics with F iC ( xxx, yyy ) = | x i − (cid:80) j (cid:54) = i x j N − − f − N ( (cid:80) Mj =1 a ij y j ) | where f N is a threshold function defined in(4.8)-(4.10), and δ i , σ i and ξ i, ± satisfy the same condition as before.Note that the special case without controls, i.e., F i ( xxx, yyy ) = x i and ξ i, ± = 0, corresponds to the rank-dependent SDEs . In particular, the rank-dependent SDEs with δ = 1, δ = · · · δ N = 0 is knownas the Atlas model . To the best of our knowledge, rank-dependent SDEs with additional controls or ageneral rank function F i has not been studied before. 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(7.1)By (7.1), [32, Lemma 2.1] and the fact that nnn i is continuous on ∂G i for each i ∈ I , we have that foreach ( xxx, yyy ) ∈ ∂G , there is r xxx,yyy ∈ (0 , δ ) such that for each ( xxx (cid:48) , yyy (cid:48) ) ∈ B r xxx,yyy ( xxx, yyy ) ∩ ∂G , l ( xxx (cid:48) , yyy (cid:48) ) ⊂ l ( xxx, yyy ) , (7.2)and min j ∈I ( xxx,yyy ) (cid:42) (cid:88) i ∈I ( xxx,yyy ) d i ( xxx, yyy ) rrr i ( xxx, yyy ) , nnn j ( xxx (cid:48) , yyy (cid:48) ) (cid:43) ≥ a . (7.3)Since ∂G i is C , for each ( xxx, yyy ) ∈ ∂G , there is m ( xxx, yyy ) > r xxx,yyy ∈ (0 , δ ) ( r xxx,yyy can be chosen evensmaller if necessary) such that for each ( xxx (cid:48) , yyy (cid:48) ) ∈ B r xxx,yyy ( xxx, yyy ) ∩ ∂G , (7.2)-(7.3) hold and( xxx (cid:48) , yyy (cid:48) ) + λ (cid:88) i ∈I ( xxx,yyy ) d i ( xxx, yyy ) rrr i ( xxx, yyy ) ∈ G for all λ ∈ (0 , m ( xxx, yyy )) . (7.4)Let B or xxx,yyy ( xxx, yyy ) denote the interior of the closed ball B r xxx,yyy ( xxx, yyy ). There exists a countable set { ( xxx k , yyy k ) } such that ∂G ∈ ∪ k B r xxxk,yyyk and { ( xxx k , yyy k ) } ∩ B N (0) is a finite set for each integer N ≥ 1. We canfurther choose the set { ( xxx k , yyy k ) } to be minimal in the sense that for each strict subset C of { xxx k , yyy k } , { B r xxx,yyy : ( xxx, yyy ) ∈ C } does not cover ∂G . Let D k = (cid:16) B r xxxk,yyyk (cid:17) \ (cid:0) ∪ k − i =1 B r xxxi,yyyi (cid:1) ∩ ∂G for each k . Then D k (cid:54) = ∅ for each k , { D k } is a partition of ∂G , and for each ( xxx, yyy ) ∈ ∂G there is a unique index i ( xxx, yyy )such that ( xxx, yyy ) ∈ D i ( xxx,yyy ) . For each i ( xxx, yyy ) ∈ R n + m , let(¯ xxx, ¯ yyy ) = (cid:26) ( xxx, yyy ) , if ( xxx, yyy ) / ∈ ∂G, ( xxx i ( xxx,yyy ) , yyy i ( xxx,yyy ) ) , if ( xxx, yyy ) ∈ ∂G. Note that for all ( xxx, yyy ) ∈ R n + m , (cid:107) ( xxx, yyy ) − (¯ xxx, ¯ yyy ) (cid:107) < δ. (7.5)For each i ∈ l and ( xxx, yyy ) ∈ R n + m , let rrr δi ( xxx, yyy ) = rrr i (¯ xxx, ¯ yyy ) . (7.6)We construct ( XXX δ , YYY δ ) as follows. Let WWW be defined on some filtered probability space (Ω , F , {F t } , P ) bea d -dimensional {F t } -Brownian motion with drift bbb and covariance matrix σσσ such that WWW is continuousalmost surely and W has distribution ν . Let τ := inf { t ≥ WWW t ∈ ∂G } and XXX δt = WWW t , ηηη δt = 0 , and YYY δt = 0 , for 0 ≤ t < τ . Note that XXX δτ − exists on { τ < ∞} since WWW has continuous paths and in the case that τ = 0, XXX δ − = WWW . On { τ < ∞} , define η i,δτ = , i / ∈ l (cid:16) XXX δτ − , YYY δτ − (cid:17) ,d i (cid:16) XXX δτ − , YYY δτ − (cid:17) (cid:18) m (cid:16) XXX δτ − ,YYY δτ − (cid:17) ∧ δ (cid:19) , i ∈ l (cid:16) XXX δτ − , YYY δτ − (cid:17) , INITE FUEL GAME 33 XXX δτ = XXX τ + m (cid:16) XXX δτ − , YYY δτ − (cid:17) ∧ δ (cid:88) i ∈ l (cid:16) XXX δτ − ,YYY δτ − (cid:17) d i (cid:16) XXX δτ − , YYY δτ − (cid:17) rrr + ,δi ( XXX δτ − , YYY δτ − ) , and Y j,δτ = Y j,δτ − + (cid:88) i ∈ l r − ,δij (cid:0) XXX δτ − , YYY δτ − (cid:1) (cid:16) η i,δτ − η i,δτ − (cid:17) for j = 1 , , · · · m. So XXX δ , ηηη δ and YYY δ have been defined on [0 , τ ) and at τ on { τ < ∞} , such that (i) XXX δt = WWW t + (cid:80) i ∈ l rrr + ,δi ( XXX δ − ) η i,δ + (cid:80) i ∈ l (cid:82) (0 ,t ] rrr + ,δi ( XXX δs − , YYY δs − ) dη i,δs and YYY δt = (cid:80) i ∈ l rrr − ,δi ( YYY δ − ) η i,δ + (cid:80) i ∈ l (cid:82) (0 ,t ] rrr − ,δi ( XXX δs − , YYY δs − ) dη i,δs for all t ∈ [0 , τ ] ∩ [0 , ∞ ),where XXX δ − = WWW and YYY δ − = 0. (ii) ( XXX δt , YYY δt ) ∈ G (iii) for i ∈ l , (a) η i,δ ≥ (b) η i,δ is nondecreasing on [0 , τ ] ∩ [0 , ∞ ), (c) η i,δ = η i,δ + (cid:82) (0 ,t ] { ( XXX δs ,YYY δs ) ∈ U δ ( ∂G ∩ ∂G i ) } dη i,δs for t ∈ [0 , τ ] ∩ [0 , ∞ ), (iv) (cid:107) ∆ ηηη δt (cid:107) = (cid:107) ηηη δt − ηηη δt − (cid:107) ≤ δ for t ∈ [0 , τ ] ∩ [0 , ∞ ), where where ηηη δ − = 000.Proceeding by induction, we assume that for some n ≥ τ ≤ · · · ≤ τ n − have been defined, and( XXX δ , YYY δ , ηηη δ ) has been defined on [0 , τ t − ) and at τ t − on { τ t − < ∞} , such that ( i ) − ( iv ) above holdwith τ n − in place of τ . Then we define τ n = ∞ on { τ n − = ∞} and on { τ n − < ∞} we define τ n = inf { t ≥ τ n − : (cid:16) XXX δτ n − + WWW t − WWW τ n − , YYY δτ n − (cid:17) ∈ ∂G } . Note that between τ n − and τ n , the resource level YYY δt remains constant while X δt behaves like a Brownianmotion.For τ n − ≤ t < τ n , let ηηη δt = ηηη δτ n − ,YYY δt = YYY δτ n − ,XXX δt = XXX δτ n − + WWW t − WWW τ n − . On { τ n < ∞} , let η i,δτ n = η i,δτ n − , i / ∈ l (cid:16) XXX δτ n − , YYY δτ n − (cid:17) ,d i (cid:16) XXX δτ n − − , YYY δτ n − − (cid:17) (cid:32) m (cid:16) XXX δτn − − ,YYY δτn − − (cid:17) ∧ δ (cid:33) , i ∈ l (cid:16) XXX δτ n − , YYY δτ n − (cid:17) ,XXX δτ n = XXX τ n − + m (cid:16) XXX δτ n − , YYY δτ n − (cid:17) ∧ δ (cid:88) i ∈ l (cid:16) XXX δτn − ,YYY δτn − (cid:17) d i (cid:16) XXX δτ n − , YYY δτ n − (cid:17) rrr + ,δi ( XXX δτ n − , YYY δτ n − ) , and Y j,δτ n = Y j,δτ n − + (cid:88) i ∈ l r − ,δij (cid:0) XXX δτ n − , YYY δτ n − (cid:1) (cid:16) η i,δτ n − η i,δτ n − (cid:17) for j = 1 , , · · · m. In this way, XXX δ , ηηη δ and YYY δ have been defined on [0 , τ n ) and at τ n on { τ n < ∞} such that (i)-(iv) holdwith τ n in place of τ . By construction { τ n } ∞ n =1 is a nondecreasing sequence of stopping times. Let τ = lim n →∞ τ n . On { τ = ∞} , the construction of ( XXX δ , ηηη δ , YYY δ ) is complete. A similar argument in [32,Theorem 5.1] shows that { τ < ∞} = ∅ .Consider a sequence of sufficiently small δ ’s, denoted by { δ n } , such that δ n ↓ n → ∞ . For each δ n ,let ( XXX δ n , YYY δ n , ηηη δ n ) be the tuple constructed as above for the same diffusion process WWW with drift bbb andcovariance matrix σσσ . Assumption 4.1 in [32] is satisfied with α n = 0, β n = 0 and and 2 δ n in place of δ n .Denote WWW n = WWW + (cid:80) i ∈ l rrr + ,δ n i ( XXX δ n − , YYY δ n − ) η i,δ n . Consequently, {ZZZ δ n } ∞ n =1 := { ( WWW δ n , XXX δ n , YYY δ n , ηηη δ n ) } ∞ n =1 is C -tight and any weak limit point ZZZ of this sequence satisfies conditions (i), (iii), (iv) and (v) and inDefinition 3.5 with F t = σ ( ZZZ s : 0 ≤ s ≤ t ), t ≥ WWW n converges to Brownian motion with drift bbb in D . In addition, MMM δ n := { WWW δ n t − WWW δ n − bbbt, t ≥ } = { WWW t − WWW − bbbt, t ≥ } is a martingale with respect to WWW which ( XXX δ n , YYY δ n , ηηη δ n )is adapted to. Therefore MMM δ n is a martingale with respect to ( WWW δ n , XXX δ n , YYY δ n , ηηη δ n )) and it is alsouniformly integrable.Hence by Proposition 4.1 in [32], any weak limit point of { ZZZ δ n } ∞ n =1 is an extended constrained SRBMwith data ( G, bbb, σσσ, { rrr i , i ∈ l } , ν ). (cid:3) Appendix B Take n = N , m = M and I = 2 N in Definition 3.5. We then check the satisfiability for Assumptions A1 - A5 for game CCC . C p C p C p and C d C d C d are two special cases. A1. Assumption A1 is trivially satisfied by definition. We write G = ∩ Nj =1 G j , where G i = ( xxx, yyy ) ∈ R N + M (cid:12)(cid:12) ˜ x i ≤ f − N M (cid:88) j =1 a ij y j ,G N + i = ( xxx, yyy ) ∈ R N + M (cid:12)(cid:12) ˜ x i ≥ − f − N M (cid:88) j =1 a ij y j , for i = 1 , , · · · , N . The boundary of G i is smooth since f − N is smooth. A2. Assumption A2 is satisfied since f − N is smooth and decreasing. It satisfies the uniform exteriorcone condition. At any boundary point ( xxx , yyy ) ∈ ∂G j , we can put a truncated closed right circularcone V ( xxx ,yyy ) satisfying V ( xxx ,yyy ) ∩ ¯ G = { ( xxx , yyy ) } . A3. Assumption A3 can be shown by contradiction. The proof is inspired from that of [32, Lemma(A.2)] which is for bounded region with tightness argument. We modify the proof via a shifting argument.Suppose that Assumption A3 does not hold. Then, since there are only finite many l ∈ l , l (cid:54) = ∅ ,there is an (cid:15) > 0, a nonempty set l ⊂ l , a sequence { (cid:15) n } ⊂ (0 , ∞ ) with (cid:15) n → n → ∞ , a sequence { ( xxx n , yyy n ) } ⊂ R N + M such that for each n , ( xxx n , yyy n ) ∈ ∩ j ∈ l U (cid:15) n ( ∂G j ∩ ∂G ) and dist(( xxx n , yyy n ) , ∩ j ∈ l ( ∂G j ∩ ∂G )) ≥ (cid:15) .By exploiting the special structure of region G , dist(( xxx, yyy ) , ∩ j ∈ l ( ∂G j ∩ ∂G )) = dist(( xxx − a , yyy ) , ∩ j ∈ l ( ∂G j ∩ ∂G )) for any a ∈ R and ( xxx, yyy ) ∈ R N + M . Here ∈ R N is a vector with all ones. Intuitively, this isbecause for any fixed yyy , the projection of G onto xxx -space is a polyhedron unbounded along the directionsof ± ∈ R N . This is consistent with the model where we only look at the relative distance betweenpositions. INITE FUEL GAME 35 Therefore, for each ( xxx n , yyy n ), there exists a n ∈ R such that (cid:107) xxx n − a n (cid:107) ≤ 1. Denote ˜ xxx n = xxx n − a n .Hence (˜ xxx n , yyy n ) is a bounded sequence in R N + M and dist((˜ xxx n , yyy n ) , ∩ j ∈ l ( ∂G j ∩ ∂G )) ≥ (cid:15) . WLOG,we may assume that (˜ xxx n , yyy n ) → ( xxx, yyy ) as n → ∞ for some ( xxx, yyy ) ∈ R N + M . It follows that ( xxx, yyy ) ∈∩ j ∈ l ( ∂G j ∩ ∂G ), since for each j ∈ l ,dist(( xxx, yyy ) , ∂G j ∩ ∂G ) ≤ (cid:107) (˜ xxx n , yyy n ) − ( xxx, yyy ) (cid:107) + dist((˜ xxx n , yyy n ) , ∂G j ∩ ∂G ) ≤ (cid:107) (˜ xxx n , yyy n ) − ( xxx, yyy ) (cid:107) + (cid:15) n → , as n → ∞ . This contradicts with the fact that (˜ xxx n , yyy n ) → ( xxx, yyy ) and dist((˜ xxx n , yyy n ) , ∩ j ∈ l ( ∂G j ∩ ∂G )) ≥ (cid:15) . A4. Recall that for i = 1 , , · · · , N , rrr i = c (cid:48) i (cid:32) · · · , − , · · · − a i y (cid:80) Mj =1 a ij y j , · · · , − a iM y M (cid:80) Mj =1 a ij y j (cid:33) ,rrr N + i = c (cid:48) N + i (cid:32) · · · , , · · · − a i y (cid:80) Mj =1 a ij y j , · · · , − a iM y M (cid:80) Mj =1 a ij y j (cid:33) , where c (cid:48) j is a normalizing constant such that (cid:107) rrr j (cid:107) = 1 ( j = 1 , , · · · , N ).On each face j = 1 , , · · · , N , rrr j is a function of yyy , which is bounded. Moreover, rrr j is smooth and D yyy rrr j is bounded. Therefore, rrr j ( · ) is uniformly Lipschitz continuous function. Note that when the adjacentmatrix A = { a kj } ≤ k,j ≤ N is an identity matrix or matrix with all ones, rrr i is constant on ∂G i for all i ∈ l . A5. Denote g := f − N . First we show that g is a non-negative decreasing function on [0 , y ] where y := (cid:80) Mj =1 y j is the total resource.Recall that f (cid:48) N ( x ) = p (cid:48) N − N N − α p (cid:48)(cid:48)(cid:48) N p (cid:48)(cid:48) N (cid:113) N N − α tanh (cid:18) x (cid:113) N − αN (cid:19) − p (cid:48) N . We claim that f (cid:48) N ( x ) < x ≥ x ↓ f (cid:48) N ( x ) = −∞ . Since h (cid:48) ( x ) ≥ h (cid:48)(cid:48)(cid:48) ( x ) ≤ x ≥ p (cid:48) N − N N − α p (cid:48)(cid:48)(cid:48) N ≥ x ≥ 0. Denote q ( x ) = p (cid:48)(cid:48) N (cid:113) N N − α tanh (cid:18) x (cid:113) N − αN (cid:19) − p (cid:48) N . It is easyto check that q (0) = 0. Moreover, q (cid:48) ( x ) = p (cid:48)(cid:48)(cid:48) N (cid:113) N N − α tanh (cid:18) x (cid:113) N − αN (cid:19) + p (cid:48)(cid:48) N (cid:18) x (cid:113) N − αN (cid:19) − p (cid:48)(cid:48) N < x > q (cid:48) ( x ) = 0 for x = 0. This is because h (cid:48)(cid:48)(cid:48) ≤ x ≥ x ) ≥ x ≥ x ) = 1 if and only if x = 0. Moreover, given the fact that lim x ↓ f N ( x ) = ∞ , f (cid:48) N ( x ) is not boundedas x ↓ 0, we have lim x ↓ f (cid:48) N ( x ) = −∞ .Combining all above, f (cid:48) N ( x ) < x ≥ 0. Therefore, there exists 0 < ˜ k ( y ) < ˜ K ( y ) < ∞ suchthat −∞ < − ˜ K ( y ) < f (cid:48) N ( z ) < − ˜ k ( y ) < z ∈ [ x, x ]. Here x = g ( y ) > x = g (0). Notethat g (cid:48) ( · ) = f (cid:48) ( f − ( · )) , therefore − k ( y ) ≤ g (cid:48) ( w ) ≤ − K ( y ) when w ∈ [0 , y ]. Now let k ( y ) := K ( y ) and K ( y ) := k ( y ) . Next, Recall that nnn i = c i N − , · · · , − , · · · , N − g (cid:48) M (cid:88) j =1 a ij y j a i , · · · , g (cid:48) M (cid:88) j =1 a ij y j a iM ,rrr i = c (cid:48) i (cid:32) · · · , − , · · · − a i y (cid:80) Mj =1 a ij y j , · · · , − a iM y M (cid:80) Mj =1 a ij y j (cid:33) ,nnn N + i = c N + i − N − , · · · , , · · · , − N − g (cid:48) M (cid:88) j =1 a ij y j a i , · · · , g (cid:48) M (cid:88) j =1 a ij y j a iM ,rrr N + i = c (cid:48) N + i (cid:32) · · · , , · · · − a i y (cid:80) Mj =1 a ij y j , · · · , − a iM y M (cid:80) Mj =1 a ij y j (cid:33) , where ± i -th position. Obviously all the latter M components in nnn j and rrr j are non-positive(1 ≤ j ≤ N ).By simple calculation, we have (cid:113) NN − + K ( y ) N ≤ c j ≤ (cid:113) NN − + (cid:15) and (cid:113) NN +1 ≤ c (cid:48) j ≤ √ for all 1 ≤ j ≤ N .Similar to the definition of rrr + j and rrr − j , denote nnn + j as the first N components in nnn j and nnn − j as the latter M components in nnn j .Since face i and N + i are parallel to each other ( i = 1 , , · · · , N ), there are at most N faces intersectingwith each other. It suffices to consider ( xxx, yyy ) such that |I (( xxx, yyy )) | = N . For these points, consider c i = N and d i = N ( i = 1 , , · · · , N ). Therefore, for i ∗ ∈ { i, N + i } with i = 1 , , · · · , N , (cid:42) (cid:80) Ni =1 nnn i ∗ N , rrr i ∗ (cid:43) ≥ N (cid:104) nnn − i ∗ , rrr − i ∗ (cid:105) = 1 N c (cid:48) i ∗ c i ∗ (cid:104) nnn − i ∗ , rrr − i ∗ (cid:105) = − c (cid:48) i ∗ c i ∗ g (cid:48) M (cid:88) j =1 a ij y j ≥ (cid:113) N +1 N − + ( N + 1) K ( y ) k ( y ) . Similarly, for i ∗ ∈ { i, N + i } with i = 1 , , · · · , N , (cid:42) (cid:80) Ni =1 rrr i ∗ N , n i ∗ (cid:43) ≥ N (cid:104) nnn − i ∗ , rrr − i ∗ (cid:105) = 1 N (cid:104) nnn − i ∗ , rrr − i ∗ (cid:105) = − c (cid:48) i ∗ c i ∗ g (cid:48) M (cid:88) j =1 a ij y j ≥ (cid:113) N +1 N − + ( N + 1) K ( y ) k ( y ) . Appendix C The unique positive root to (4.10) . Define q ( z ) = p (cid:48)(cid:48) N ( z ) p (cid:48) N ( z ) where p N ( x ) is defined in (4.6). Note that q (0) = p (cid:48)(cid:48) N (0) p (cid:48) N (0) = E (cid:82) ∞ e − αt h (cid:48)(cid:48) (cid:16)(cid:113) N − N B t (cid:17) dt E (cid:82) ∞ e − αt h (cid:48) (cid:16)(cid:113) N − N B t (cid:17) dt . Under Assumption H2 (cid:48) , p (cid:48) N (0) = 0, kα < p (cid:48)(cid:48) N (0) < Kα , and q (cid:48) ( z ) = p (cid:48)(cid:48)(cid:48) N ( z ) p (cid:48) N ( z ) − ( p (cid:48)(cid:48) N ( z )) ( p (cid:48) N ( z )) . Moreover, Assumption H2 (cid:48) implies that h (cid:48)(cid:48)(cid:48) ( z ) ≤ h (cid:48) ( z ) ≥ z ≥ 0. Therefore, q (0) = ∞ and q (cid:48) ( z ) ≤ 0. Furthermore, since k ≤ h (cid:48)(cid:48) ≤ K and h (cid:48) ≥ kx + c for some constant c , we have lim x →∞ q ( x ) = 0. INITE FUEL GAME 37 On the other hand, define f ( x ) = (cid:113) N − αN tanh (cid:18) z (cid:113) N − αN (cid:19) . It is easy to check that f (0) = 0, f (cid:48) ( x ) > x ≥ 0, and lim x →∞ f ( x ) = (cid:113) N − αN . Therefore, f ( x ) = q ( x ) has a unique positivesolution. Department of Industrial Engineer and Operations Research, UC Berkeley. E-mail address : [email protected] Department of Mathematics, UCLA. E-mail address : [email protected] Department of Industrial Engineer and Operations Research, UC Berkeley. E-mail address ::