Collaborative Network Formation in Spatial Oligopolies

1 Collaborati v e Network F ormation in Spatial Oligopolies Shaun Lichter , T erry Friesz, and Christopher Grif fin Abstract Recently , it has been shown that networks with an arbitrary degree sequence may be a stable solution to a network formation game. Further , in recent years there has been a rise in the number of firms participating in collaborativ e efforts. In this paper , we show conditions under which a graph with an arbitrary degree sequence is admitted as a stable firm collaboration graph. I . I N T RO D U C T I O N Recently there has been a rise in the number of firms participating in collaborativ e efforts. Goyal and Joshi [4] present a model of firm collaboration in aspatial oligopolies and in this paper we e xtend this model to spatial oligopolies. W e in vestig ate the impact of the spatial economy on the collaboration network and the impact of the collaboration network on the spatial economy . Since the 1960s the number of firm’ s participating in collaborativ e agreements has increased significantly [5]–[10]. This collaboration takes v arious forms, one of which is research and de velopment (R & D) that often consists of sharing resources such as equipment, laboratory space, office space, as well as engineers and scientists through separate R & D subcompanies. This collaboration has become very popular within industries that are R & D intensi ve. Hagedoorn sho ws that the number of collaborations has increased since the 1960s and rapidly increased in the 1980s [7]. Firms in R & D intensiv e industries are enabled to flexibly ally themselves to further their business. Nonetheless, the existence of these collaborations is counterintuitiv e because firms should not want to share R & D results or expenditures because it is the foundation of their future products. As a result of this contradiction, this collaboration has spurred a host of literature [5]–[10]. Goyal and Joshi present a model of horizontal firm collaboration in oligopolies, where firms compete in the market after choosing collaborators [4]. The moti vation behind this model November 10, 2021 DRAFT 2 is an examination of the incenti ves for collaboration and the interaction of these incentives with market competition. Firms are able to lo wer production costs by committing some resources to a pair-wise collaboration effort. A particular collaboration network is formed as a result of the collection of pairwise collaborations. For each collaboration network, each firm has a particular production cost which effects the market competition that occurs over this collaboration network. Hence, the oligopoly induces an allocation of value ov er the set of firms for a given collaboration network. I I . C O L L A B O R A T I O N N E T W O R K S A N D C O L L A B O R A T I V E O L I G O P O L I E S In this section we present an introduction to collaboration networks and collaborati v e oligopolies. W e modify the notational con ventions from the common notation in this this body of literature [3], [12], [13] in order to better accomodate the spatial variables needed in later sections of this paper . Let N = { 1 , 2 , . . . n } be the set of nodes in a graph, which will represent players or a group of players. The set of links in the graph is a set of pairs of nodes (subsets of N of size two). A graph g is a set of links (set of subsets of N of size two) and g N is the complete set of all links. The set G is the set of all graphs ov er the nodes N , that is, G = { g : g ⊂ g N } . The value of a graph g is the total v alue produced by agents in the graph; we denote the value of a graph as the function h : G → R and the set of of all such value functions as H . An allocation rule Y : H × G → R N distributes the value h ( g ) among the agents in g . Denote the v alue allocated to agent i as Y i ( h, g ) . Since, the allocation rule must distribute the v alue of the network to all players, it must be balanced ; i.e., P i Y i ( h, g ) = h ( g ) for all ( h, g ) ∈ H × G . The allocation rule gov erns how the value is distributed and thus makes a significant contribution to the model. Jackson and W olinksy use pairwise stability to model stable networks without the use of noncooperativ e games [13]. Definition II.1. A network g with value function h and allocation rule Y is pairwise stable if (and only if): 1) for all ij ∈ g , Y i ( h, g ) ≥ Y i ( h, g − ij ) and 2) for all ij 6∈ g , if Y i ( h, g + ij ) > Y i ( h, g ) , then Y j ( h, g + ij ) < Y j ( h, g ) Pairwise stability implies that in a stable network, for each link that exists, (1) both players must benefit from it and (2) if a link can provide benefit to both players, then it in fact must November 10, 2021 DRAFT 3 exist. Jackson notes that pairwise stability may be too weak because it does not allow groups of players to add or delete links, only pairs of players [12]. Deletion of multiple links simultaneously has been considered in [1]. W e present an application of the network formation game to firm collaboration in spatial oligopolies, which is an extension to the firm collaboration presented by Goyal and Joshi in [4]. A. General Collaborative Oligopoly Model Consider n firms that compete in an oligopoly who may collaborate with any of the other n − 1 firms. Firm i produces a quantity q i . Denote q = ( q 1 , q 2 , . . . q n ) as the vector of quantity production across all firms and q − i = ( q 1 , . . . , q i − 1 , q i +1 , . . . q n ) as the vector containing produc- tion quantities for all firms, but firm i . Collaboration among firms affects the marginal cost of production. Thus a particular (collaboration) graph g induces a mar ginal cost of firm i under collaboration graph g of c i ( q i | g ) . W e consider marginal cost functions of the form (1) where the marginal cost c i ( q i | g ) for firm i is a function of q i , the quantity produced by firm i , and η i ( g ) , the degree of firm i in graph g . c i ( q i | g ) = f i ( q i , η i ( g )) (1) Here, f i ∈ C 1 (Ω i ) where q i ∈ Ω i and Ω i is defined as the feasible region for firm i . Ω i = { q i : 0 ≤ q i } Gi ven a network g , there is an induced set of costs which, along with the demand functions, produces a set of profit functions for each firm, Y i ( g ) (the allocation of payoff for player i ). These profit functions then induce a Nash equilibrium of production, which pro vides the precise allocation rule (i.e., profit) for each firm on the graph. The stability of the collaboration network can then be analyzed using the definition of stability II.1. Denote the market marginal price function as P ( q 1 , q 2 , . . . q n ) . In this paper , we consider a market marginal price function (dependent on quantity produced) giv en by P ( q 1 , q 2 , . . . q n ) = α − X i ∈ N q i (2) This can also be denoted as P ( Q ) = α − Q where Q = P i ∈ N q i The profit for Player i is: Y i ( q i | q − i , g ) = α − X i ∈ N q i ! q i − c ( q i | g ) q i (3) November 10, 2021 DRAFT 4 Gi ven collaboration graph g , firm i will solve the problem max Y i ( q i | q ∗ − i , g ) s.t. q i ∈ Ω i (4) where q ∗ − i is composed of the optimal production quantities for all firms, but i . The gradient of the objecti ve for firm i : ∇ q i Y i ( q i | q ∗ − i , g ) = P ( Q ) − f i ( q i , η i ( g )) − q i − q i ∂ f i ∂ q i Each firm i will solve an equiv alent v ariational inequality by finding q ∗ i ∈ Ω i such that: h∇ q i Y i ( q i | q ∗ − i , g ) , q i − q ∗ i i ≥ 0 (5) where h· , ·i denotes a dot product. In this case: h P ( Q ) − f i ( q i , η i ( g )) − q i − q i ∂ f i ∂ q i , q i − q ∗ i i ≥ 0 (6) The equilibrium for this oligopoly can be found by solving the variational inequality defined as finding q ∗ ∈ Ω such that h∇ q Y ( q | q ∗ , g ) , q − q ∗ i ≥ 0 (7) where [ ∇ q Y ( q | q ∗ , g )] i = P ( Q ) − f i ( q i , η i ( g )) − q i − q i ∂ f i ∂ q i (8) It is difficult to analytically determine which collaboration graphs will be stable because the oligopoly equilibriums are solutions to a variational inequality . One could empirically find stable graphs, but instead we seek to find subcases of the model for which we can find analytical results. B. Pr evious Results on Network Stability in Aspatial Oligopoly In Goyal and Joshi [4], it is assumed that the mar ginal cost of firm i linearly decreases with the number of collaborators for firm i : c i ( g ) = γ 0 − γ η i ( g ) (9) where, as before, η i ( g ) is the number of links for firm i and γ 0 is the marginal cost of production when a firm has no links. Notice that γ 0 is constant for all firms. One e xample that Goyal and November 10, 2021 DRAFT 5 Joshi [4] study is that of a homogenous product oligopoly . W ith the market marginal price function (2) and marginal cost (9), the resulting profit to Player i is: Y i ( g ) = α − X i ∈ N q i ! q i − ( γ 0 − γ η i ( g )) q i = ( α − γ 0 ) q i + X i ∈ N q i ! q i − ( − γ η i ( g )) q i (10) Goyal and Joshi show that with mar ginal cost (9) and market demand (2), the complete network is the unique stable network [4]. C. Results of Nonlinear Cost on Stability In this section we revie w the results from [14], where we show the effect a nonlinear v ariation on the marginal cost function has on the stability of collaboration structures. In particular , we sho w that with cost functions of a particular form, the collaborativ e oligopoly will result in a stable collaboration graph with an arbitrary degree sequence. W e consider a marginal cost function: c i ( g ) = γ 0 + f i ( η i ( g )) (11) where f i is some function f i : R → R . Lemma II.2. Suppose we have an oligopoly consisting of n firms in which collaboration is defined by the graph g and the pr ofit function (allocation rule) for F irm i in that oligopoly is given by: Y i ( g ) = ( α − γ 0 ) q i ( g ) − X j ∈ N q j ! q i ( g ) − f i ( η i ( g )) q i ( g ) (12) then the quantity pr oduced for firm i is: q i ( g ) = α − γ 0 − nf i ( η i ( g )) + P j 6 = i f j ( η j ( g )) n + 1 (13) Pr oof: From [17], for an y oligopoly with profit function of the form: π i ( q ) = aq i − X j ∈ N q j ! q i − b i q i (14) The resulting Cournot equilibrium point on quantities is: q i = a − nb i + P j 6 = i b j n + 1 (15) November 10, 2021 DRAFT 6 In our case, we have: a = α − γ 0 b i = f i ( η i ( g )) ∀ i Substituting these definitions into Expression (15) yields Expression (13). This completes the proof. Remark II.3 . It is worth noting that when for each firm i , b i = − γ η i ( g ) then the cost function (9) and induced equilibrium quantity (15) is equiv alent to that used in Goyal and Joshi. Corollary II.4. Suppose that f is a nonne gative ( f ( η ) ≥ 0 ) con vex function that has a minimum at 0 . Further , suppose f i ( η i ( g )) = f ( η i ( g ) − k i ) wher e k i ∈ { 0 , 1 , . . . , n − 1 } . If the par ameters α and γ 0 and the function f ar e such that: α − γ 0 − n max { f ( n − 1) , f (1 − n ) } − 1 2 ( n − 1) max { f (1) − f (0) , f ( − 1) − f (0) } > 0 (16) and n ≥ 2 , then the Cournot equilibrium quantities (13) ar e nonne gative for all firms and for all collabor ation graphs and the following inequalities hold: 2 q i ( g ) − n − 1 n + 1 [ f (1) − f (0)] > 0 (17) 2 q i ( g ) − n − 1 n + 1 [ f ( − 1) − f (0)] > 0 (18) Pr oof: Since n ≥ 2 and f is con vex and has a minimum at 0 , this implies that n − 1 n +1 [ f (1) − f (0)] and n − 1 n +1 [ f ( − 1) − f (0)] are non-negati ve. If (17) and (18) hold, then q i ( g ) is non-negati ve and hence it suf fices to only sho w that (17) and (18) are implied by (16). For all i , function f i is a con ve x function of the degree of node i in the graph g ; the degree of node i must take an integer value between 0 and n − 1 , which due to the con ve xity of f i and the fact that k i ∈ { 0 , 1 , . . . n − 1 } implies that the maximum of f i is equi valent to max { f i (0) , f i ( n − 1) } and is less than max { f ( n − 1) , f ( − n + 1) } . That is, f i ( η i ( g )) ≤ max { f i ( η i ( g )) } = max { f i (0) , f i ( n − 1) } ≤ max { f ( n − 1) , f ( − n + 1) } This means that (16) implies: α − γ 0 − nf i ( η i ( g )) − 1 2 ( n − 1) max { f (1) − f (0) , f ( − 1) − f (0) } > 0 ∀ i (19) November 10, 2021 DRAFT 7 Since, all f i ( η i ( g )) ≥ 0 , we may add P j 6 = i f j ( η j ( g )) to the left side of (19) without changing the inequality: α − γ 0 − nf i ( η i ( g )) + X j 6 = i f j ( η j ( g )) − 1 2 ( n − 1) max( f (1) − f (0) , f ( − 1) − f (0)) > 0 ∀ i (20) Di viding by n + 1 yields: α − γ 0 − nf i ( η i ( g )) + P j 6 = i f j ( η j ( g )) n + 1 − 1 2 n − 1 n + 1 max( f (1) − f (0) , f ( − 1) − f (0)) > 0 ∀ i (21) From Lemma II.2 this simplifies to: q i ( g ) − 1 2 n − 1 n + 1 max( f (1) − f (0) , f ( − 1) − f (0)) > 0 ∀ i (22) Multiplying through by two yields: 2 q i ( g ) > n − 1 n + 1 max( f (1) − f (0) , f ( − 1) − f (0)) > n − 1 n + 1 [ f (1) − f (0)] ∀ i 2 q i ( g ) > n − 1 n + 1 max( f (1) − f (0) , f ( − 1) − f (0)) > n − 1 n + 1 [ f ( − 1) − f (0)] ∀ i No w (17) and (18) immediately follo w . Remark II.5 . This essentially means that the steeper a function f around zero and on the interv al ( − n + 1 , n − 1) , the greater the quantity α − γ 0 is needed to ensure the theorem prov ed later in this section. It is worth pointing out that this bound may often not be tight (i.e., the inequalities may hold true and production quantities may be positi ve ev en when the condition is not met). Theorem II.6. Suppose that f is a nonne gative ( f ( η ) ≥ 0 ) con vex function that has a minimum at 0 . Further , suppose f i ( η i ( g )) = f ( η i ( g ) − k i ) . Define the change in f as 4 − f i ( k i ) = f i ( k i − 1) − f i ( k i ) = f ( − 1) − f (0) = 4 − f (0) and 4 + f i ( k i ) = f i ( k i + 1) − f i ( k i ) = f (1) − f (0) = 4 + f (0) . Suppose n ≥ 2 firms compete in an oligopoly with market demand P = α − P i ∈ N q i and mar ginal costs c i ( g ) = γ 0 + f i ( η i ( g )) . If the parameters α and γ 0 and the functions f i obe y condition (16), then the equivalence class of graphs [ g ] η such that η i ( g ) = k i is an equivalence class of stable collaboration graphs. Pr oof: Let g be a graph in the equiv alence class of graphs [ g ] η , that is, g has a degree sequence such that η i ( g ) = k i for all firms i . Consider a firm i who may consider dropping its November 10, 2021 DRAFT 8 link with node j . If node i drops its link with node j leading to graph g − ij , then η i ( g − ij ) = k i − 1 and η j ( g − ij ) = k j − 1 , while η r ( g − ij ) = k r for r 6∈ { i, j } . Using Lemma II.2 q i = α − γ 0 − nf i ( η i ( g )) + P j 6 = i ∈ N f j ( η j ( g )) n + 1 (23) Calculate: q i ( g − ij ) = q i ( g ) − 4 − f i ( k i )  n n + 1  + 4 − f j ( k j )  1 n + 1  q j ( g − ij ) = q j ( g ) − 4 − f j ( k j )  n n + 1  + 4 − f i ( k i )  1 n + 1  q r ( g − ij ) = q r ( g ) + 4 − f i ( k i )  1 n + 1  + 4 − f j ( k j )  1 n + 1  It then follows that Q ( g − ij ) = Q ( g ) −  1 n + 1  ( 4 − f i ( k i ) + 4 − f j ( k j )) P ( g − ij ) = P ( g ) +  1 n + 1  ( 4 − f i ( k i ) + 4 − f j ( k j )) c i ( g − ij ) = c i ( g ) + 4 − f i ( k i ) No w , we can calculate Y i ( g − ij ) in terms of Y i ( g ) : Y i ( g − ij ) = q i ( g − ij )[ P ( g − ij ) − c i ( g − ij )] = Y i ( g ) + q i ( g )  2 n + 1  [ 4 − f j ( k j ) − n 4 − f i ( k i )] +  [ 4 − f j ( k j ) − n 4 − f i ( k i )] n + 1  2 Since f i ( η i ( g )) = f ( η i ( g ) − k i ) this implies that 4 − f i ( k i ) = 4 − f j ( k j ) and we obtain (24) and then (25) and (26) through algebraic manipulation. Finally , by the assumptions of the theorem and condition (16) each of the quantities 4 − f i ( k i ) , n − 1 n +1 , and 2 q i ( g ) − n − 1 n +1 4 − f i ( k i ) are nonnegati ve implying (27). Y i ( g − ij ) − Y i ( g ) = 2 q i ( g ) 4 − f i ( k i )  1 − n n + 1  + ( 4 − f i ( k i )) 2  1 − n n + 1  2 (24) = 4 − f i ( k i )  1 − n n + 1   2 q i ( g ) + 1 − n n + 1 4 − f i ( k i )  (25) = −4 − f i ( k i )  n − 1 n + 1   2 q i ( g ) − n − 1 n + 1 4 − f i ( k i )  (26) < 0 (27) November 10, 2021 DRAFT 9 This implies that if firm i attempts to drop link ij , then Y i ( g ) > Y i ( g − ij ) and thus firm i decreases its profit. The same will be true for firm j . Hence, no firm has an incenti ve to drop a link from graph g . No w , we will consider the case where firm i attempts to add a link to the graph g , gi ving g + ij under the assumption that the link ij does not exist in graph g . This analysis will follow closely the analysis for g − ij . First note that η i ( g ) = k i for all firms i and η i ( g + ij ) = k i + 1 and η j ( g + ij ) = k j + 1 , while η r ( g + ij ) = k r for r 6∈ { i, j } . W e define 4 + f i ( k i ) as 4 + f i ( k i ) = f i ( k + 1) − f i ( k ) ; note the subtle dif ference from the definition of 4 − f i ( k i ) . Again using Lemma II.2, we calculate the production quantity for each node in graph g + ij : q i ( g + ij ) = q i ( g ) − 4 + f i ( k i )  n n + 1  + 4 + f j ( k j )  1 n + 1  q j ( g + ij ) = q j ( g ) − 4 + f j ( k j )  n n + 1  + 4 + f i ( k i )  1 n + 1  q r ( g + ij ) = q r ( g ) + 4 + f i ( k i )  1 n + 1  + 4 + f j ( k j )  1 n + 1  W e can then calculate the corresponding total production quantity Q , the market price P and marginal costs for each player for the graph g + ij : Q ( g + ij ) = Q ( g ) −  1 n + 1  ( 4 + f i ( k i ) + 4 + f j ( k j )) P ( g + ij ) = P ( g ) +  1 n + 1  ( 4 + f i ( k i ) + 4 + f j ( k j )) c i ( g + ij ) = c i ( g ) + 4 + f i ( k i ) No w , we can calculate Y i ( g + ij ) in terms of Y i ( g ) : Y i ( g + ij ) = q i ( g + ij )[ P ( g + ij ) − c i ( g + ij )] = Y i ( g ) + q i ( g )  2 n + 1  [ 4 + f j ( k j ) − n 4 + f i ( k i )] +  [ 4 + f j ( k j ) − n 4 + f i ( k i )] n + 1  2 Since f i ( η i ( g )) = f ( η i ( g ) − k i ) this implies that 4 + f i ( k i ) = 4 + f j ( k j ) and we obtain (28) and then (29) and (30) through algebraic manipulation. Finally , by the assumptions of the theorem and condition (16), each of the quantities 4 + f i ( k i ) , n − 1 n +1 , and 2 q i ( g ) − n − 1 n +1 4 + f i ( k i ) are positiv e November 10, 2021 DRAFT 10 implying (31). Y i ( g + ij ) − Y i ( g ) = 2 q i ( g ) 4 + f i ( k i )  1 − n n + 1  + ( 4 + f i ( k i )) 2  1 − n n + 1  2 (28) = 4 + f i ( k i )  1 − n n + 1   2 q i ( g ) + 1 − n n + 1 4 + f i ( k i )  (29) = −4 + f i ( k i )  n − 1 n + 1   2 q i ( g ) − n − 1 n + 1 4 + f i ( k i )  (30) < 0 (31) This implies that if firm i attempts to add a link ij , then Y i ( g ) > Y i ( g + ij ) and the firm decreases its profit. The same will be true for firm j . Hence, no firm has an incentiv e to add a link to graph g . Since no firm has an incentiv e to add or drop a link to graph g , it is stable. This completes the proof. I I I . C O L L A B O R A T I V E S PA T I A L O L I G O P O L I E S Spatial Oligopolies (Oligopolies on spatially separated markets) have been studied extensi vely [2], [11], [15], [16]. In this section we extend the collaborati ve oligopoly model of Goyal and Joshi by applying it to spatially separated markets and we extend the existing literature in spatial oligopolies by allo wing firm collaboration. W e seek to find which graphs g are stable collaboration graphs. As in prior sections, N = { 1 , 2 , . . . n } will denote firms, which are nodes on the collaboration graph. Alternati vely , there is a spatial transport network with nodes denoted as V = { 1 , 2 , . . . v } . Consumer demand at transport node l ∈ V for firm i ∈ N is denoted as d li and the total demand at node l is denoted as D l = P i d li . Denote the vector d i = [ d li ] l ∈ V as the demand vector for firm i across all nodes. The quantity produced by firm i is again denoted as q i . Noting that q i = P l ∈ V d li , we can eliminate q i by formulating all e xpressions in terms of d i = [ d li ] l ∈ V . The induced price at node l is denoted as P l ( D l ) = P l ( d l 1 , d l 2 , . . . , d ln ) . The marginal pro- duction cost is c i ( q i | g ) = f i ( q i , η i ( g )) as before in (9) but is now denoted as c i ( P l ∈ V d li | g ) = f i ( P l ∈ V d li , η i ( g )) . Ho we ver , now there is an additional marginal cost to ship a unit of quantity to node l for firm i denoted as s li 1 . Define Y i ( d i | g ) as the profit for firm i with collaboration 1 Each firm is not explicitly placed on the transport network, b ut its location may be implied through the s li values November 10, 2021 DRAFT 11 graph g : Y i ( d i | g ) = X l ∈ V d li " p l ( D l ) − s li − f i X l ∈ V d li , η i ( g ) !# Hence, the firm i will solve the problem max Y i ( d i | g ) s.t. d i ∈ Θ i (32) where Θ i = { d i : d i ≥ 0 } . W e can calculate the gradient of the objectiv e for firm i : ∇ d i Y i ( d i ) = " P l ( D l ) − s li − f i X l ∈ V d li , η i ( g ) ! − d li − d li ∂ f i ( P l ∈ V d li , η i ( g )) ∂ d li # l ∈ V Each firm i will solve the equiv alent v ariational inequality by finding d ∗ i ≥ 0 such that: h∇ d i Y i ( d ∗ i | g ) , d i − d ∗ i i ≥ 0 (33) W e may no w find an equilibrium to the spatial oligopoly for all firms by solving the single composed v ariational inequality . Find d ∗ ≥ 0 such that: h∇ d Y ( d ∗ | g ) , d − d ∗ i ≥ 0 (34) Where d = [ d i ] i ∈ N and Y ( d ∗ | g ) = [ Y i ( d ∗ i | g )] i ∈ N . W ith such a spatial model, it again becomes difficult to analytically find stable graphs. Stability is difficult to determine analytically because in order to determine if a link should exist, the v alue a node receives from the link must be contrasted from the v alue without the link. This is dif ficult without using sensiti vity analysis for variational inequalities. Instead we seek to show a set of models that do yield analytical results. A. Nonlinear pr oduction costs in Spatial Collabor ative Oligopoly Consider a marginal cost function: c i ( g ) = γ 0 + f i ( η i ( g )) (35) where f i is some function f i : R → R . The marginal cost to ship a unit of quantity to node l for firm i is again denoted as s li . Each firm maximizes its profit by solving its own nonlinear November 10, 2021 DRAFT 12 problem: max X l ∈ V d li [ P l ( d l 1 , d l 2 , . . . , d ln ) − s li − γ 0 − f i ( η i ( g ))] s.t. 0 ≤ d li ∀ i ∈ N , l ∈ V Remark III.1 . This nonlinear program that each firm will solve has been decoupled, such that no w at each transport node l , the firms participate in oligopolistic competition that is independent from the competition at each other node. Ho wev er , at each node, each firm has a different cost due to the v ariability of the shipment cost to that node for each firm. Lemma III.2. Suppose we have an oligopoly consisting of n firms in which collaboration is defined by the graph g , the demand function at node l is P l ( d l 1 , d l 2 , . . . , d ln ) = α l − P i ∈ N d li , and the pr ofit function (allocation rule) for F irm i in that oligopoly is given by: Y i ( g , d 1 i , d 2 i , . . . , d li ) = X l ∈ V d li [ α l − X j ∈ N d lj − s li − γ 0 − f i ( η i ( g ))] (36) then the demand met at node l by firm i is: d li = α l − γ 0 − n ( s li + f i ( η i ( g ))) + P j 6 = i [ s lj + f j ( η j ( g ))] n + 1 (37) Pr oof: The profit for firm i can be rearranged: Y i ( g , d 1 i , d 2 i , . . . , d li ) = X l ∈ V d li [ α l − X j ∈ N d lj − s li − γ 0 − f i ( η i ( g ))] = X l ∈ V ( α l − γ 0 ) d li − X j ∈ N d lj ! d li − ( s li + f i ( η i ( g ))) d li From [17], for any oligopoly with profit function of the form: Y i ( q ) = aq i − X j ∈ N q j ! q i − b i q i (38) The resulting Cournot equilibrium point on quantities is: q i = a − nb i + P j 6 = i b j n + 1 (39) In our case, we have an oligopoly at each location l and quantities d li with parameters : a = α l − γ 0 b i = s li + f i ( η i ( g )) ∀ i November 10, 2021 DRAFT 13 Substituting these definitions into Expression (39) yields Expression (37). This completes the proof. Corollary III.3. Suppose that f is a nonne gative ( f ( η ) ≥ 0 ) con vex function that has a minimum at 0 . Further , suppose f i ( η i ( g )) = f ( η i ( g ) − k i ) wher e k i ∈ { 0 , 1 , . . . , n − 1 } . If the function f and par ameters α , γ 0 , and s ar e such that: α − γ 0 − n  max l ∈ V ,i ∈ N s li + max { f ( n − 1) , f (1 − n ) }  − 1 2 ( n − 1) max { f (1) − f (0) , f ( − 1) − f (0) } > 0 (40) and n ≥ 2 , then the Cournot equilibrium quantities (37) ar e nonne gative for all firms at all locations and for all collaboration gr aphs and the following inequalities hold: 2 d li ( g ) − n − 1 n + 1 [ f (1) − f (0)] > 0 ∀ i ∈ N , l ∈ V (41) 2 d li ( g ) − n − 1 n + 1 [ f ( − 1) − f (0)] > 0 ∀ i ∈ N , l ∈ V (42) Pr oof: Since n ≥ 2 and f is con vex and has a minimum at 0 , this implies that n − 1 n +1 [ f (1) − f (0)] and n − 1 n +1 [ f ( − 1) − f (0)] are non-negati ve. If (41) and (42) hold, then d li ( g ) is non-negati v e and hence it suf fices to only sho w that (41) and (42) are implied by (40). For all i , function f i is a con ve x function of the degree of node i in the graph g ; the degree of node i must take an integer value between 0 and n − 1 , which due to the con ve xity of f i and the fact that k i ∈ { 0 , 1 , . . . n − 1 } implies that the maximum of f i is equi valent to max { f i (0) , f i ( n − 1) } and is less than max { f ( n − 1) , f ( − n + 1) } . That is, f i ( η i ( g )) ≤ max { f i ( η i ( g )) } = max { f i (0) , f i ( n − 1) } ≤ max { f ( n − 1) , f ( − n + 1) } ∀ i ∈ N Further , s li ≤ max l ∈ V ,i ∈ N s li . This means that (40) implies: α − γ 0 − n [ s li + f i ( η i ( g ))] − 1 2 ( n − 1) max { f (1) − f (0) , f ( − 1) − f (0) } > 0 ∀ i ∈ N , l ∈ V (43) Since, all f i ( η i ( g )) ≥ 0 and all s li ≥ 0 , we may add P j 6 = i [ s lj + f j ( η j ( g ))] to the left side of (43) without changing the inequality: α − γ 0 − n [ s li + f i ( η i ( g ))] + X j 6 = i [ s lj + f j ( η j ( g ))] − 1 2 ( n − 1) max { f (1) − f (0) , f ( − 1) − f (0) } > 0 ∀ i ∈ N , l ∈ V (44) November 10, 2021 DRAFT 14 Di viding by n + 1 yields: α − γ 0 − n [ s li + f i ( η i ( g ))] + P j 6 = i [ s lj + f j ( η j ( g ))] n + 1 − 1 2 n − 1 n + 1 max { f (1) − f (0) , f ( − 1) − f (0) } > 0 ∀ i ∈ N , l ∈ V (45) From Lemma III.2 this simplifies to: d li ( g ) − 1 2 n − 1 n + 1 max { f (1) − f (0) , f ( − 1) − f (0) } > 0 ∀ i ∈ N , l ∈ V (46) Multiplying through by two yields: 2 d li ( g ) > n − 1 n + 1 max { f (1) − f (0) , f ( − 1) − f (0) } > n − 1 n + 1 [ f (1) − f (0)] ∀ i ∈ N , l ∈ V 2 d li ( g ) > n − 1 n + 1 max { f (1) − f (0) , f ( − 1) − f (0) } > n − 1 n + 1 [ f ( − 1) − f (0)] ∀ i ∈ N , l ∈ V No w (41) and (42) immediately follo w . Remark III.4 . It should be noted that this bound will often not be tight and hence demand quantities may be positi v e even when it is not met. Suppose that f is a con v ex function that has a minimum at 0 . Further , suppose f i ( η i ( g )) = f ( η i ( g ) − k i ) . Define the change in f as 4 − f i ( k i ) = f i ( k i − 1) − f i ( k i ) = f ( − 1) − f (0) = 4 − f (0) and 4 + f i ( k i ) = f i ( k i + 1) − f i ( k i ) = f (1) − f (0) = 4 + f (0) . Suppose n ≥ 2 firms compete in an oligopoly with market demand P = α − P i ∈ N q i and mar ginal costs c i ( g ) = γ 0 + f i ( η i ( g )) . If the parameters α and γ 0 and the functions f i obey condition (16), then the equiv alence class of graphs [ g ] η such that η i ( g ) = k i is an equiv alence class of stable collaboration graphs. The induced price at node l is denoted as P l ( D l ) = P l ( d l 1 , d l 2 , . . . , d ln ) . The marginal pro- duction cost is c i ( q i | g ) = f i ( q i , η i ( g )) as before in (9) but is now denoted as c i ( P l ∈ V d li | g ) = f i ( P l ∈ V d li , η i ( g )) . Ho we ver , now there is an additional marginal cost to ship a unit of quantity to node l for firm i denoted as s li Suppose we ha ve an oligopoly consisting of n firms in which collaboration is defined by the graph g , the demand function at node l is P l ( d l 1 , d l 2 , . . . , d ln ) = α l − P i ∈ N d li , and the profit function (allocation rule) for Firm i in that oligopoly is giv en by: Theorem III.5. Suppose that f is a nonne gative ( f ( η ) ≥ 0 ) con vex function that has a minimum at 0 . Further , suppose f i ( η i ( g )) = f ( η i ( g ) − k i ) . Define the chang e in f as 4 − f i ( k i ) = f i ( k i − 1) − November 10, 2021 DRAFT 15 f i ( k i ) = 4 − f (0) = f ( − 1) − f (0) and 4 + f i ( k i ) = f i ( k i + 1) − f i ( k i ) = 4 + f (0) = f (1) − f (0) . Suppose n ≥ 2 firms compete in an oligopoly with market demand P l ( d l 1 , d l 2 , . . . , d ln ) = α l − P i ∈ N d li , mar ginal pr oduction cost of c i ( g ) = γ 0 + f i ( η i ( g )) , and mar ginal shipping cost of s li . If the parameters α , γ 0 , and s as well as the function f obey condition (40), then the equivalence class of graphs [ g ] η such that η i ( g ) = k i is an equivalence class of stable collaboration graphs. Pr oof: Let g be a graph in the equiv alence class of graphs [ g ] η , that is, g has a degree sequence such that η i ( g ) = k i for all firms i . Consider a firm i who may consider dropping its link with node j . If node i drops its link with node j leading to graph g − ij , then η i ( g − ij ) = k i − 1 and η j ( g − ij ) = k j − 1 , while η r ( g − ij ) = k r for r 6∈ { i, j } . Using Lemma III.2 d li = α l − γ 0 − n ( s li + f i ( η i ( g ))) + P j 6 = i [ s lj + f j ( η j ( g ))] n + 1 (47) Calculate: d li ( g − ij ) = d li ( g ) − 4 − f i ( k i )  n n + 1  + 4 − f j ( k j )  1 n + 1  d lj ( g − ij ) = d lj ( g ) − 4 − f j ( k j )  n n + 1  + 4 − f i ( k i )  1 n + 1  d lr ( g − ij ) = d lr ( g ) + 4 − f i ( k i )  1 n + 1  + 4 − f j ( k j )  1 n + 1  It then follows that D l ( g − ij ) = D l ( g ) −  1 n + 1  ( 4 − f i ( k i ) + 4 − f j ( k j )) P l ( g − ij ) = P l ( g ) +  1 n + 1  ( 4 − f i ( k i ) + 4 − f j ( k j )) c i ( g − ij ) = c i ( g ) + 4 − f i ( k i ) Define Y i ( g ) = P l ∈ V y li ( g ) where y li ( g ) = d li ( g )[ P l ( g ) − c i ( g ) − s li ] . No w , we can calculate y li ( g − ij ) in terms of y li ( g ) : y li ( g − ij ) = d li ( g − ij )[ P l ( g − ij ) − c i ( g − ij ) − s li ] = y li ( g ) + d li ( g )  2 n + 1  [ 4 − f j ( k j ) − n 4 − f i ( k i )] +  [ 4 − f j ( k j ) − n 4 − f i ( k i )] n + 1  2 Since f i ( η i ( g )) = f ( η i ( g ) − k i ) this implies that 4 − f i ( k i ) = 4 − f j ( k j ) yielding (48) and then (49) and (50) through algebraic manipulation. Finally , 4 − f i ( k i ) and n − 1 n +1 are non-negati ve and November 10, 2021 DRAFT 16 by Corollary III.3, the term  2 d li ( g ) − n − 1 n +1 4 − f i ( k i )  is non-negati v e as well. Hence, this implies (51). y li ( g − ij ) − y li ( g ) = 2 d li ( g ) 4 − f i ( k i )  1 − n n + 1  + ( 4 − f i ( k i )) 2  1 − n n + 1  2 (48) = 4 − f i ( k i )  1 − n n + 1   2 d li ( g ) + 1 − n n + 1 4 − f i ( k i )  (49) = −4 − f i ( k i )  n − 1 n + 1   2 d li ( g ) − n − 1 n + 1 4 − f i ( k i )  (50) < 0 (51) Since y li ( g − ij ) − y li ( g ) < 0 for all i , we can sum over all transport nodes l , to see that this implies that node i does not ha ve an incenti ve to drop a link. y li ( g − ij ) − y li ( g ) < 0 X l y li ( g − ij ) − X l y li ( g ) < 0 Y i ( g − ij ) − Y i ( g ) < 0 This implies that if firm i attempts to drop link ij , then Y i ( g ) > Y i ( g − ij ) and thus firm i decreases its profit. The same will be true for firm j . Hence, no firm has an incenti ve to drop a link from graph g . No w , we will consider the case where firm i attempts to add a link to the graph g , gi ving g + ij under the assumption that the link ij does not exist in graph g . This analysis will follow closely the analysis for g − ij . First note that η i ( g ) = k i for all firms i and η i ( g + ij ) = k i + 1 and η j ( g + ij ) = k j + 1 , while η r ( g + ij ) = k r for r 6∈ { i, j } . W e define 4 + f i ( k i ) as 4 + f i ( k i ) = f i ( k + 1) − f i ( k ) ; note the subtle dif ference from the definition of 4 − f i ( k i ) . Again using Lemma II.2, we calculate the production quantity for each node in graph g + ij : d li ( g + ij ) = d li ( g ) − 4 + f i ( k i )  n n + 1  + 4 + f j ( k j )  1 n + 1  d lj ( g + ij ) = d lj ( g ) − 4 + f j ( k j )  n n + 1  + 4 + f i ( k i )  1 n + 1  d lr ( g + ij ) = d lr ( g ) + 4 + f i ( k i )  1 n + 1  + 4 + f j ( k j )  1 n + 1  November 10, 2021 DRAFT 17 W e can then calculate the corresponding total production quantity Q , the market price P and marginal costs for each player for the graph g + ij : D l ( g + ij ) = D l ( g ) −  1 n + 1  ( 4 + f i ( k i ) + 4 + f j ( k j )) P l ( g + ij ) = P l ( g ) +  1 n + 1  ( 4 + f i ( k i ) + 4 + f j ( k j )) c i ( g + ij ) = c i ( g ) + 4 + f i ( k i ) No w , we can calculate y li ( g + ij ) in terms of y li ( g ) : y li ( g + ij ) = d li ( g + ij )[ P l ( g + ij ) − c i ( g + ij ) − s li ] = y li ( g ) + d li ( g )  2 n + 1  [ 4 + f j ( k j ) − n 4 + f i ( k i )] +  [ 4 + f j ( k j ) − n 4 + f i ( k i )] n + 1  2 Since f i ( η i ( g )) = f ( η i ( g ) − k i ) this implies that 4 + f i ( k i ) = 4 + f j ( k j ) yielding (52) and then (53) and (54) through algebraic manipulation. Finally , 4 + f i ( k i ) and n − 1 n +1 are non-negati ve and by Corollary III.3, the term  2 d li ( g ) − n − 1 n +1 4 + f i ( k i )  is non-negati v e as well. Hence, this implies (55). y li ( g + ij ) − y li ( g ) = 2 d li ( g ) 4 + f i ( k i )  1 − n n + 1  + ( 4 + f i ( k i )) 2  1 − n n + 1  2 (52) = 4 + f i ( k i )  1 − n n + 1   2 d li ( g ) + 1 − n n + 1 4 + f i ( k i )  (53) = −4 + f i ( k i )  n − 1 n + 1   2 d li ( g ) − n − 1 n + 1 4 + f i ( k i )  (54) < 0 (55) Since y li ( g + ij ) − y li ( g ) < 0 for all i , we can sum ov er all transport nodes l , to see that this implies that node i does not ha ve an incenti ve to drop a link. y li ( g + ij ) − y li ( g ) < 0 X l y li ( g + ij ) − X l y li ( g ) < 0 Y i ( g + ij ) − Y i ( g ) < 0 This implies that if firm i attempts to add a link ij , then Y i ( g ) > Y i ( g + ij ) and the firm decreases its profit. The same will be true for firm j . Hence, no firm has an incentiv e to add a link to graph g . Since no firm has an incentiv e to add or drop a link to graph g , it is stable. This completes the proof. November 10, 2021 DRAFT 18 I V . N U M E R I C A L E X A M P L E W e present a numerical example of Theorem III.5. Let N = 5 firms compete in an oligopoly with in verse demand function P l = 103 − P i d li , fix ed cost γ 0 = 5 , shipping costs s li = 1 ∀ l i , and f i ( η i ( g )) = ( η i ( g ) − k i ) 2 + ψ where k = [2 , 3 , 4 , 3 , 2] T and ψ = 2 . W e want to test the stability of a graph g with η i ( g ) = k i and f i ( η i ( g )) = ( η i ( g ) − k i ) 2 + ψ for each node i . Note that f ( η i ( g )) = ( η i ( g )) 2 + ψ . The follo wing calculations will be need: α 103 γ 0 5 n 5 max l ∈ V ,i ∈ N s li 1 f ( n − 1) = f (4) = 4 2 + 2 18 f (1 − n ) = f ( − 4) = ( − 4) 2 + 2 18 f (1) = 1 2 + 2 3 f ( − 1) = ( − 1) 2 + 2 3 f (0) = 0 2 + 2 2 In order to in vok e Corollary III.3, we must ensure condition (40) holds: α − γ 0 − n  max l ∈ V ,i ∈ N s li + max { f ( n − 1) , f (1 − n ) }  − 1 2 ( n − 1) max { f (1) − f (0) , f ( − 1) − f (0) } > 0 Plugging in the appropriate values: 103 − 5 − 5 · [1 + max(18 , 18)] − 1 2 (4) max { 3 − 2 , 3 − 2) } > 0 98 − 5 · 19 − 2 · 1 > 0 98 − 95 − 2 > 0 1 > 0 Condition (40) is met for this set of parameters and function f . T wo stable graphs, shown in Figure 1 and Figure 2, have a degree sequence equi v alent to k . November 10, 2021 DRAFT 19 1 2 3 4 5 Fig. 1. Collaboration Network 1 2 3 4 5 Fig. 2. Collaboration Network 2 V . 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