A Game Theoretic Framework for Analysing Coopetition

J van Rooij ANR: 656090

Thesis written in conjunction with an internship at TNO, and submitted in partial fulfillment of the requirements for the degree of Master of Science, and of the Master’s program in Econometrics & Mathematical Economics at the Tilburg School of Economics and Management

Thesis committee: prof. dr. P.E.M. Borm (Supervisor) dr. J.H. Reijnierse (Chairman) dr. M. Groote Schaarsberg (Company supervisor)

TNO The Hague, The Netherlands

Tilburg University Tilburg School of Economics and Management Tilburg, The Netherlands

August 2015 [Page intentionally left blank] Management Summary

In this thesis we develop a game theoretic framework to analyse coopetition. Coopetition stems from a contraction of the words cooperation and competition. When firms are engaged in coopetition this means that the firms are competitors of each other that are working together to achieve a common interest. In real life there are many examples of such situations. One could think of the joint production facility that has been set up by PSA Peugeot Citroen and Toyota Motor Corporation to produce cars, or examples of joint delivery networks. Another example of coopetition is network sharing in the mobile telecommunications industry. In many countries there are several operators that own and operate a mobile network. Upgrading and maintaining such a network is costly, therefore operators are more and more looking for alternatives to reduce costs. One of these alternatives is network sharing: operators jointly invest in network upgrades. Examples of network sharing are already seen in for example Sweden. The theoretical framework that we develop can be used to analyse these scenarios. Our main interest is to find coalitions or coalition structures that are stable. To make predictions about the stability of coalitions we make use of two different stability concepts: individual and core stability. These concepts have to do with the incentives and possibilities for firms to deviate from a certain coalition structure. The framework combines concepts from both cooperative and noncooperative game theory. One of the main advantages of the framework is that it can be used to analyse scenarios with a large (up to 7 or 8) number of players. This is an advantage over many non-cooperative models that can only analyse scenarios with 2 or at most 3 players. Furthermore the framework can be used in a pragmatic way and can be easily adopted to fit different scenarios. Using this framework we analyse theoretical examples of coopetition in a Cournot oligopoly where firms can reduce their marginal costs by working together. We find several results that are of interest. First of all we show that the stability of coalition structures greatly depends on how the firms divide the surplus that arises from cooperation. We also find that coalition structures where all the firms but one work cooperate with each other are often individual stable. The coalition of size n − 1 does not want the single firm to join their coalition. Core stable coalition structures most of the time have a similar structure. Furthermore we will show that getting all the firms to work together is often hard. Due to the competitive nature the gain of one firm often comes at the cost of the loss of another. Finally we show how the framework can be used in a more pragmatic way to analyse network sharing in the mobile telecommunications industry. We discuss an example regarding the Dutch market with three operators. We find that the most likely coalition structure to form is where the two smallest operators invest together and the largest operator invests on its own. Acknowledgements

This thesis marks the end of a period of 9 years as a student. A long ride of which in the end I enjoyed every single bit. During those 9 years, and over the course of my internship at TNO I have got to learn many people that I am grateful for. First of all I am grateful for my supervisor Peter Borm for his excellent guidanc and his efforts in keeping me focussed on the subject instead of diverting and wandering through the beautiful realm of game theory and coalition formation theory. Thank you also for making sure that I actually finished this thesis. I also owe many thanks to my company supervisors Mirjam Groote Schaarsberg and Frank Berkers. Mirjam, thank you for giving me the opportunity to do an internship at TNO and providing me with all the space to pursue my own ideas and to explore all that TNO has to offer. Frank, thank you for always making time to discuss my ideas. I enjoyed every single conversation and discussion. You have always challenged me to look beyond the theory. To all the wonderfull people at the Strategic Business Analysis department of TNO: thank you for making my internship such a pleasant stay with all the outdoor lunches, coffee breaks and improv workshops. Special thanks need to go to my fellow interns Marieke and Tim for supplying me with the occasional motivational coffee break or peptalk. To my friends in Tilburg: I cannot name you all, nonetheless, thanks for everything. Arnoud, Gijs, thank you for being part of my econometrics dream team. You guys kept me going! Marleen, you know all too well how many thanks I owe you. Mom, dad, thank you for always being there. During the past 9 years as a student you have always been there to support me in whatever way possible, I cannot explain in words how grateful I am. I know from time to time you have had your doubt whether or not there would ever come an end to this expedition. Well, to take away the last of them: here it is. Contents

List of Figures

List of Tables

1 Introduction 1

2 Coopetition Framework 3 2.1 Introduction ...... 3 2.2 Overview ...... 4 2.3 Preleminaries and notation ...... 4 2.4 Coopetition Game ...... 5 2.5 Cooperation ...... 6 2.6 Competition ...... 10 2.7 Stability ...... 12 2.8 Computational Complexity ...... 14

3 Cournot Oligopoly: A Theoretical Example 16 3.1 Introduction ...... 16 3.2 Cournot Coopetition Game ...... 16 3.3 Examples ...... 17 3.3.1 Example 1: One Large Firm, Three Small Firms ...... 17 3.3.2 Example 2: Economies of Scale ...... 22 3.4 Beyond a Division Rule ...... 24

4 Case Study: Network Sharing In The Mobile Telecommunications Indus- try 26 4.1 Introduction ...... 26 4.2 The Competition Model ...... 27 4.2.1 Investment in LTE ...... 27 4.2.2 Multinomial Logit Demand Model ...... 29 4.2.3 Linking the LTE Investments and the Logit Demand Model Together 31 4.3 Scenarios ...... 32 4.3.1 Scenario 1: The Dutch Market with 3 Operators without an Outside Good...... 32 4.3.2 Scenario 2: The Dutch Market with 3 Operators and an Outside Good 35

5 Conclusions 38

6 Further Research 39

A Bibliography 40

B Mathematical Proofs 42 B.1 Derivation of the Cournot-Nash equilibrium ...... 42 B.2 Proof of Proposition 1 ...... 43 B.3 Proof of Proposition 2 ...... 45

C Matlab Code 47 C.1 ChiValue.m ...... 47 C.2 StabilityChecker.m ...... 48 C.3 CoreStabilityChecker.m ...... 51 C.4 CournotProﬁt.m ...... 53 C.5 CournotExamples.m ...... 54 C.6 MarketShareSolver.m ...... 55 C.7 BULRIC.m ...... 56 C.8 NewSMARTModel.m ...... 56 List of Figures

1 Schematic overview of the general coopetition model ...... 7 2 Schematic overview of the coopetition model combined with a CS-value . .8 3 Schematic overview of the coopetition model combined with a CS-value and stability concepts ...... 14 4 4 Player Cournot coopetition game: initial conﬁguration ...... 18 5 4 Player Cournot coopetition game: {1234} ...... 19 6 Dutch market with three operators without an outside good ...... 34 7 Dutch market with three operators and an outside good ...... 37

List of Tables

1 Four player Cournot example ...... 19 2 Three player Cournot example: Small and large economies of scale . . . . . 24 3 Coverage and capacity sites ...... 28 4 Dutch market with three operators without an outside good ...... 34 5 Market share of the outside good ...... 35 6 Dutch market with three operators with an outside good ...... 37 [Page intentionally left blank] 1 Introduction

The research for this thesis was conducted as part of an internship at TNO in the Strategic Business Department. TNO is the largest research institution in the Netherlands with over 3,000 employees. Originally this thesis would be connected to the SMART 24 project that was conducted by TNO and commissioned by the European Commission. The SMART 24 project was aimed at analysing the effects of network sharing scenarios in the mobile telecommunications industry. Unfortunately the SMART 24 project was aborted prematurely due to circumstances. This resulted in this thesis more deviating from modelling the telecommunications market to a theoretical and conceptual treatment of coopetition in general. However, we did apply the framework to network sharing scenarios. Our analysis of analysis of these network sharing scenarios uses many ideas and concepts from Offergelt (2011) who also wrote a thesis In a competitive environment firms are often faced with large investments or high costs to keep up with trends in the market. One could think of examples such as investing in a R&D project, building production facilities or investing in a delivery network. A solution to overcome these high costs and save money would be to jointly initiate such projects together with one or more competitors. There indeed exist real life examples where several competitors jointly established a R&D project, build a joint production facility or jointly set up a delivery network. In this thesis we will study such cooperative relations between competitors using game theory. The general idea is that cooperation takes place at a lower level than competition. At this lower level firms can act collectively to create a win-win relationship, but in the market they remain competitors. A situation where competitors are engaged in a cooperative relation is called coopetition. The term coopetition was first introduced by Brandenburger and Nalebuff (1996). In particular we will concentrate on oligopolistic situations where each firm is likely to be aware of the actions of the others. The decisions of one firm therefore influence and are influenced by the decisions of other firms. Given this oligopolistic nature of the market we believe that game theory is the right tool to analyse the coopetitive situation. We will be combining concepts from bot cooperative and non-cooperative game theory since a coopetitive situation entails both cooperation and competition. Eventually we want to be able to predict which coalitions will form. To do this we make use of different stability concepts to assess whether or not certain coalitions or coalition structures are stable. Bogomolnaia and Jackson (2002) and Banerjee et al. (2001) both define stability concepts for hedonic games which we will generalise to fit within our coopetition framework. For TNO it would also be interesting to use this model for large multi-stakeholder innovation projects. In these projects the interests of many stakeholders need to be aligned in order to get them to cooperate. The coopetition framework could provide insight in the considerations of these stakeholders. The buildup of this thesis is as follows. In Chapter 2 we formally will introduce our game theoretical framework for analysing coopetition. To our knowledge this is a completely new model. The definition of a coopetition game will be introduced in this chapter and we will present a method to solve such games using different stability concepts. Chapter 3 takes the framework developed in chapter 2 and applies this approach to the theoretical example of a Cournot oligopoly. In this oligopoly the firms can reduce their marginal costs of production

1 by working together. In chapter 4 we will use our framework in a more pragmatical way to take a look at network sharing in the mobile telecommunications industry. Finally the conclusions are summarised in chapter 5 and chapter 6 points out directions for further research.

2 2 Coopetition Framework

2.1 Introduction A coopetitive situation is different from a cooperative relationship between actors who are not rivals of each other. When a firm saves money because it jointly invested in a production facility these savings can be invested in other activities such as marketing or product development. By doing so the firm strengthens its strategic position. This also means that when a firm works together in a coalition the position of other members of the coalition will be strengthened. When rivals decide to cooperate there are a lot of effects that play a role: Not only direct effects such as the savings itself, but also strategic effects because a firm is able to produce at a lower price. These effects depend on how well firms are able to convert the benefits of a cooperative relationship into strategic advantage. It could happen that a cooperative deal seems to be fair and justifiable at the lower level but has devastating effects for one of the involved firms because all other partners are far better at converting the gains from the collaboration into strategic advantage. The contrary could also be true, a firm might agree on a deal that seems unreasonable at the lower level because otherwise it would be even worse off. Bengtsson and Kock (2000) argue that the most complex, but also the most advantageous relationship between competitors is coopetition. The complexity arises due to the fundamentally different and contradictory logics of interaction that competition and cooperation are built on. We are analysing situations in which firms are engaged in competition with each other. Formally we define competitors as actors that produce and market the same product. Firms try to use the resources available to them in the best way possible to maximise their own interests. The different self-interests are in conflict with each other, which in consequence means that firms compete against each other to best fulfil their own self-interests. A firm’s payoff will be determined by its access to resources, how well it can convert these resources in value and the characteristics of the other firms it is competing with in the market. The concept of cooperation is based on a diametrically opposite assumption; individuals participate in collective actions to achieve common goals. We will think of cooperation as a way of working together - on a lower level than competition - to gain access to more resources or to be able to use already available resources in a more efficient way. We will be more specific regarding what we mean with ‘working together at a lower level than competition’. Let us start by stating what we do not mean by it. We do not consider the formation of cartels where firms maximise their joint profit and redistribute this among the members of the cartel. Regardless of the coalition a firm belongs to, it acts in such a way only to maximise its own interests and not those of the coalition. Firms are not allowed to transfer their individual profits. A good example of such a coopetitive situation is the consortium of PSA Peugeot Citroen and Toyota Motor Corporation. This consortium owns a joint production facility that produces three cars that are, in essence, the same; the Citroen C1, Peugeot 107, and Toyota Aygo. While the cars are produced in the exact same factory and only differ in cosmetics, Toyota and PSA remain competitors in the market. The firms need to negotiate how to split the costs and allocate the capacity of this facility. However, in the market they will compete as fiercely with each other as they would do with other competitors. Since the firms are not allowed to transfer profits, the only compensations they can make are at the lower level by paying more for the joint

3 facility or allocating more capacity to a competitor. Summing up the above discussion we can distinguish two key features of the coopetitive situations we will study: 1. Cooperation takes place at a lower level than competition. Individual profits are nontransferable. Side payments can only be made at the lower level. 2. Cooperation has strategic effects. Market conditions change depending on the coalition structure that is formed and the allocation at the lower level. The next sections will discuss our approach in more detail. We explain the theoretical foundations and introduce definitions and notation.

2.2 Overview In this chapter we will introduce our approach to model coopetition and formally define of a coopetition game. Our model links concepts from both cooperative and non-cooperative game theory. In a coopetition game the firms are basically playing two games at once. At the market level they are competing with each other for the same group of customers and try to maximise their individual profits. However, at a lower level the firms are cooperating with each other in order to exploit synergies and economies of scale. The competition of the firms in the market will be modelled using a non-cooperative, strategic game whereas the cooperation at the lower level will be modelled using a cooperative, coalitional game. The approach assumes that competition, i.e. the strategic game, takes place in the short run, while cooperation, i.e. the coalitional game, occurs more slowly or only occasionally. Therefore we can treat cooperation and the resulting outcome as prior to the determination of each firm’s strategy in the competitive stage. The two different stages are linked in the sense that the outcome of the coalitional (cooperative) game defines the strategic (competitive) game that the firms will play. In a way firms are choosing which strategic game they play in the market by means of cooperation at a lower level. Our main interest is to find out which coalition structures are stable and which are not (with respect to different notions of stability). Unstable coalition structures are less likely to form since there will be incentives and possibilities for firms to deviate.

2.3 Preleminaries and notation Let N = {1, . . . , n} denote the set of firms.. A coalition is a nonempty subset S ⊆ N and the set 2N is the collection of all coalitions. A coalition structure is a collection 1 m k l m k P = {S , ..., S } of coalitions such that S ∩ S = ∅ for all k 6= l and ∪k=1S = N. The set P shall be the collection of all coalition structures of N, and for a coalition S we denote the set of all coalition structures containing S by PS. For i ∈ N and a coalition structure P we denote by P (i) the (unique) coalition S ∈ P with i ∈ S. To condense notation, we shall drop the braces around coalitions in coalition structures, for example, denote {{1}, {2, 3}} by {1, 23}. In case we have variables that depend on a coalition structure P we use the superscript 1 to refer to the coalition structure P = {1, . . . , n} (using the 1 condensed notation), for example πi = πi({1, . . . , n}) refers to the profit of firm i in the absence of any cooperation. Usually we will call this the initial profit of firm i or the profit of firm i in the initial configuration.

4 2.4 Coopetition Game At the start of the game there is no cooperation. Each firm starts with a bundle of resources which might be assets such as intellectual property, production technology, government permits, machinery or simply cash. In a competitive market firms try to use their resources as efficiently as possible in order to maximise their individual profits. We will assume that there is only one type of resources and call this the capital stock of a firm. The vector 1 k = (k1, . . . , kn) denotes the capital stock of all firms i ∈ N, and k denotes the initial level of capital stock. Based on the initial level of capital stock, k1, the firms play a strategic 1 1 1 1 1 game G which is given by G = {(Xi , πi )}i∈N where for each firm i ∈ N, Xi denotes the 1 Q 1 strategy space and πi : j∈N Xj → R the payoff function. We will assume that the game G1 has a unique Nash equilibrium. In mathematical terms this means that there exists a 0 Q 1 xˆ ∈ j∈N Xj such that 1 1 1 1 1 πi (ˆx ) ≥ πi (xi , xˆN\{i}) 1 1 for all xi ∈ Xi , and all i ∈ N. In a Nash equilibrium there are no unilateral deviations that are profitable for the firms. Therefore the result of competition between the firms will be this equilibrium. This is nothing new, we follow a classic approach to modelling competition between firms. One could for example think of a Cournot oligopoly in which the cost functions or the maximum production capacity of the firms depend on the level of capital stock. So both the payoff functions (cost functions) and the strategy spaces (maximum production capacity) of the firms can be functions of the level of capital stock. At some point in time it becomes possible for the firms to cooperate with each other. By cooperating the firms can exploit synergies or economies of scale. In this way they can gain access to more capital stock then they would on their own. The firms need to decide two things: who to cooperate with and how to split the surplus of capital stock arising from their cooperation. We will model this cooperative stage using a coalitional game (N, v) where N, as before, is the set of firms and v : 2N → RN a characteristic function describing the amount of capital stock each coalition can acquire. It is assumed that v ≥ 0 and v(∅) = 0. The outcome of this coalitional game will be an allocation configuration which is defined as follows:

Definition 1. Let (N, v) be a coalitional game. The outcomes of this game can be represented by an allocation configuration. An allocation configuration is a pair α = (P, k) where P ∈ P is a coalition structure of N and k ∈ RN is an allocation vector. Given an allocation configuration α = (P, k) the coalition structure P describes which coalitions have formed and the allocation vector k denotes how the firms split the surplus of capital stock arising from their cooperation. Capital stock is transferable within, and non- transferable between coalitions in a coalition structure. Firms use capital stock efficiently, which means that they use all the capital stock that is available to them. Therefore we re- quire an allocation configuration to be component efficient. An allocation vector k is called P N component efficient with respect to P if i∈S ki = v(S) for all S ∈ P . Let KP (N, v) ⊂ R denote the set of all component efficient allocation vectors with respect to P for (N, v). An allocation configuration α = (P, k) is called component efficient if k is component efficient with respect to P . Let A(N, v) ⊂ RN×P denote the set of all component efficient allocation configurations for (N, v).

5 When firms cooperate at a lower level this changes the competition in the market. By acquiring more capital stock firms might be able to produce at a lower cost, sell new products or improve their marketing strategy. Somehow the outcome of the coalitional game must be linked to a strategic game. In a coopetition game we propose to link the two different stages by a mapping Γ which assigns to each component efficient outcome α = (P, k) of the coalitional game (N, v) a strategic game Gα = Γ(α). As mentioned before both the strategy spaces and the payoff functions can depend on the level of capital stock. Therefore they also both depend on the allocation configuration α, so Gα = Γ(α) = α α {(Xi , πi )}i∈N . Now we have discussed the different ingredients of a coopetition situation we are ready to give a formal definition.

Deﬁnition 2. A coopetition game is a tuple (N, v, Γ) where

(a) N = {1, . . . , n} is a set of ﬁrms;

(b)( N, v) is a coalitional game, v : 2N → RN is a characteristic function, and the outcomes of this game are represented by component eﬃcient allocation conﬁgurations α = (P, k) ∈ A(N, v);

(c) Γ is a mapping from the space A(N, v) of component efficient allocation configurations of (N, v), to the space of strategic games. Each outcome α = (P, k) ∈ A(N, v) of the α α α coalitional game (N, v) maps to a strategic game G = {(Xi , πi )}i∈N = Γ(α); and (d) for each α = (P, k) ∈ A(N, v) the strategic game Gα has a unique Nash equilibrium α Q α α α α α α α α xˆ ∈ j∈N Xj such that πi (ˆx ) ≥ πi (xi , xˆN\i) for all xi ∈ Xi and all i ∈ N. Note that a coopetition game is a generalisation of both the strategic-form non-cooperative and TU cooperative game models. In the absence of competition the model reduces to a coalitional game and in the absence of cooperation the model reduces to a strategic game. Figure 1 shows a schematic overview of our model. In a coopetition game firms both cooperate and compete with each other. Cooperation at a lower level results in different allocation configurations that describe which firms work together and how they divide the proceeds of their cooperation. In general here are many possible allocation configurations. Each allocation configuration α = (P, k) in turn maps to a strategic game Gα = Γ(α). This strategic game describes how the firms compete with each other in the market. Different forms of cooperation lead to different forms of competition: in the coalitional game the firms choose which strategic game they will play in the market. Each strategic game Gα has a unique Nash equilibrium. The result of competition will be this equilibrium and the firms will receive their equilibrium payoffs. In the next two sections that follow we will discuss both the cooperative and competitive stage and their complexities in more detail. We will show how the number of possible allocation configurations can be reduced and how we can use the competitive stage to rank the different outcomes of the cooperative stage.

2.5 Cooperation In this section we will focus on the cooperative part of a coopetition game. The complexities of this stage will be discussed and we will propose a way of dealing with those complexities. By cooperating with each other at a lower level than competition, ﬁrms can gain access

6 Coopetition N, v, Γ Situation

Allocation α = (P, k) with ··· ··· Conﬁguration k ∈ KP (N, v)

Strategic Game ··· Gα ···

Equilibrium ··· πα(ˆxα) ··· Payoﬀ

Figure 1: Schematic overview of the model. The starting point is a coopetition game (N, v, Γ). In the cooperative stage firms play a coalitional game (N, v) to exploit synergies and acquire additional capital stock. The outcome of this coalitional game is an allocation configuration α = (P, k). For each coalition structure P ∈ P the corresponding allocation vectors k are component efficient, l α k ∈ KP l (N, v). Based on the outcome α the firms play a strategic game G = Γ(α) in the competitive stage. Therefore different outcomes of the coalitional game lead to different stategic games. The strategic game Gα has a unique Nash-equilibrium, xˆα, therefore the result of the competitive stage will be a vector of payoffs πα(ˆxα). to more resources than they would be able to individually. The question that needs to be answered in the cooperative stage is the following: which coalitions will form and how will their members split the capital stock? Each coalition can gain acces to an amount of capital stock which depends on the members of the coalition. This relation is given by a characteristic function v : 2N → R. We can also incorporate the initial level of capital stock, k1, in this characteristic function 1 by simply defining v({i}) = ki . Whether or not to do this depends on the context. As explained before, the firms play a coalitional game (N, v) and the outcomes of this game will be allocation configurations denoted by α = (P, k) where P represents the coalition structure that forms and k is a vector describing how capital stock is allocated. Since we assumed that capital stock is transferable within and nontransferable between coalitions and that firms use the available capital stock efficiently, the outcomes of the coalitional game need to be component efficient. The set P = {P 1,...,P m} denotes all possible coalition structures.1 The coalition structure of only singletons will be denoted by P 1 = {{1},... {n}}. When P 1 forms there is no cooperation and firms will only be able to use their initial level of capital stock. Therefore the allocation configuration will be α = (P 1, k1). For all other coalition structures P l, with l ∈ {2, . . . , m}, the firms are free in negotiating the division of capital stock within their coalition. Therefore in general multiple allocation vectors are possible as

1 m = Bn, and Bn is the Bell number which counts the number of partitions of a set that has exactly n elements.

7 l long as they are component efficient with respect to the coalition structure, k ∈ KP l (N, v). The freedom to choose any allocation vector, as long as it is component efficient, makes the model complicated. In principle there is an infinite amount of allocation configurations that can be the result of the cooperative stage. Given a coalition structure we do not know what the corresponding allocation vector will be. Therefore we would like to simplify the model by fixing the allocation of capital stock given a coalition structure. Fixing the allocation also allows us to overcome possible problems concerning bargaining. Take for example a three player coopetition game and suppose that the coalition structure {1, 23} forms. In this case it might be that firm 1 has a very low payoff and would like to propose a new allocation configuration in which it works together with either firm 2 or firm 3. In this case firm 1 only needs to propose a deal in which it is better off than in the current situation even if the deal seems to be unfair. This will result is the formation of the coalition structure {12, 3} and now firm 3 finds itself in the same position as firm 1. Later in section 2.4 we will see that in some examples we can lift this restriction, but for now we will keep the allocation of capital stock fixed. One way of fixing the allocation of capital stock would be to specify a value for games with a given coalition structure (henceforth CS-values and CS-games). Such games and values were first introduced by Aumann and Dreze (1974). A CS-value is an operator ϕ that assigns allocation vectors to all CS-games, ϕ(N, v, P ) ∈ RN . Basically it is the same as a solution concept for TU-games (such as the Shapley value or the Nucleolus) however this time it is not the grand coalition that forms but the coalition structure P . All allocation configurations will be of the form α = (P, ϕ(N, v, P ) for all P ∈ P. Instead of a manifold of allocation vectors there we use a CS-value to fix just one allocation vector for each coalition structure. Figure 2 shows how this changes the model. Instead of an infinite l amount of branches at each coalition structure (one branch for every k ∈ KP l and each l ∈ {1, . . . , m}) there is just one branch per coalition structure.

Coopetition Situation (N, v, Γ),ϕ + CS-value Allocation α1(ϕ) = (P 1, k1) with αm(ϕ) = (P m, km) with ··· Conﬁguration k1(ϕ) = ϕ(N, v, P 1) km(ϕ) = ϕ(N, v, P m)

Strategic Game Gα1(ϕ) ··· Gαm(ϕ)

Equilibrium α1(ϕ) ··· αm(ϕ) Payoﬀ π π

Figure 2: By combining a coopetition situation (N, v, Γ) with a CS-value ϕ ww can simplify the model. Instead of an inﬁte amount of branches there now is a ﬁnite amount of branches, one for each coalition structure P ∈ P.

In this thesis we will use two CS-values in particular. Both values are based on the well known Shapley value. The ﬁrst CS-value is the one introduced by Aumann and Dreze

8 (1974) (henceforth AD-value) which assigns to each player in a coalition structure element the Shapley value of the subgame restricted to that element:

ADi(N, v, P ) = Shi(P (i), v|P (i)), where Sh(N, v) is the Shapley value of (N, v).2 The second CS-value we will investigate is the χ-value of Casajus (2009). This value uses the Shapley value as a yardstick to distribute the allocation within a structural coalition. It compares the sum of the Shapley allocations in a coalition with the worth of that coalition; the diﬀerence, positive or negative, is distributed evenly. The χ-value is deﬁned as: P v(P (i)) − Shj(N, v) χ (N, v, P ) = Sh (N, v) − j∈P (i) . i i |P (i)|

Note that in theory the χ-value could lead to a negative allocation of capital stock. This would mean that a firm pays another firm to cooperate. The intuition behind the χ- value is the following. Suppose that we believe that the Shapley value does a good job at distributing the worth of the grand coalition, v(N). If a group of players decides to split off, these players should all win/lose an equal amount since much. Similar to solution concepts for TU-games, CS-values can be characterised using properties and axioms. We list the following properties from Casajus (2009) and refer the reader for more properties and an in-depth review to his paper:

Axiom 1. (Additivity, A). ϕ(N, v + v0,P ) = ϕ(N, v, P ) + ϕ(N, v0,P ) for all characteristic functions v, v0.

The ﬁrst axiom states that if there are two CS-games with the same coalition structure P that are being played, the sum of the CS-values for both games should equal the CS-value of the game that would consist of putting those two games together. For the second axiom we need to deﬁne what symmetric players are. Two players i and j are called symmetric if v(S ∪ {i}) = v(S ∪ {j}) for all S ∈ 2N \{i, j}. That is, we can change one player for the other in any coalition that contains one of the players and not change that coalitions worth. The next axiom captures the idea that symmetric players should be treated equal when they are part of the same coalition in a coalition structure.

Axiom 2. (Component Restricted Symmetry, CS). If i, j ∈ N are symmetric and j ∈ P (i), we have ϕi(N, v, P ) = ϕj(N, v, P ). Earlier we already defined component efficiency for an allocation configuration. We can do the same for a CS-value. When a CS-value is component efficient this relates to the idea that the coalitions are the productive units; the players within a component work together in order to generate that coalition’s surplus. P Axiom 3. (Component Efficiency, CE). For all S ∈ P , we have i∈S ϕi(N, v, P ) = v(S).

2 P |S|!(|N|−|S|−1)! The Shapley value is deﬁned as: Shi(v) = S∈2N :i/∈S |N|! (v(S ∪ {i}) − v(S)).

9 For the fourth and fifth axiom we again need to define a new type of player, a Null player. Formally a Null player i satisfies v(S ∪ {i}) = v(S) for every S ∈ 2N \{i}. A Null player never contributes anything to a coalition. The following axiom states that therefore this player should also receive nothing.

Axiom 4. (Null Player, N). If i ∈ N is a Null player, then ϕi(N, v, P ) = 0. In some situations the previous axiom might be too restrictive. Therefore this axiom can be replaced with the following axiom which only states that a Null player should recieve nothing when the grand coalition forms.

Axiom 5. (Grand Coalition Null Player, GN) If i ∈ N is a Null player, then ϕi(N, v, {N}) = 0.

A coalition structure P 0 ∈ P is ﬁner than P ∈ P if P 0(i) ⊆ P (i) for all i ∈ N. Intuitively this means that P 0 can be obtained by splitting the elements of P . The ﬁnal axiom now states that if a group of players within a coalition decides to leave that coalition they should all win/lose the same amount.

Axiom 6. (Splitting, SP). If P 0 is ﬁner than P then for all i ∈ N and j ∈ P 0(i), we have

0 0 ϕi(N, v, P ) − ϕ(N, v, P ) = ϕj(N, v, P ) − ϕj(N, v, P ).

Using the axioms deﬁned above it is possible to (uniquely) characterise both the AD- value and the χ-value. The following two theorems do so. For the proofs of these theorems we refer to the original work.

Theorem 1 (Aumann and Dreze (1974)). The AD-value is the unique CS-value that satisﬁes A, CS, CE and N.

Theorem 2 (Casajus (2009)). The χ-value is the unique CS-value that satisﬁes A, CS, CE, GN and SP.

Both the AD-value and the χ-value are based on the Shapley value and whenever the Grand Coalition forms they both distribute the Shapley value amongst the players. Now what is the main difference between both values? Casajus (2009) points out that the AD- value fails to account for outside options. It ignores the fact that a firm could also form a coalition with firms that are not in present in its current coalition structure element since it only uses information from the subgame restricted to that coalition structure element. We are interested in whether or not different CS-values lead to different outcomes and different stable coalition structures. One remark considering the initial level of capital stock k1 and the normalisation of v. Since we are considering two CS-values that both satisfy additivity we can choose whether or not to include k1 in v since ϕ(v + k1) = ϕ(v) + ϕ(k1). So in some cases we will choose to zero-normalise v and explicitly specify k1.

10 2.6 Competition Once alliances have been made and capital stock has been divided firms will try to maximise their individual profits by competing with each other in the market. Based on the outcome α = (P, k) of the coalitional game (N, v) the firms play a non-cooperative strategic game α α α α G = {(Xi , πi )}i∈N = Γ(α). When playing G firms independently and simultaneously choose their strategies. Furthermore they make their choices irrespective of the coalition they belong to. This means that they compete as fiercely with their fellow coalition members as they do with other firms. We will assume that the game Gα has a unique, strong Nash equilibrium for every allocation configuration α ∈ V (N, v).3 In mathematical terms α Q α this means that for every α there exists ax ˆ ∈ j∈N Xj such that

α α α α α πi (ˆx ) ≥ πi (xi , xˆN\{i})

α α for all xi ∈ Xi , and all i ∈ N. Since for every allocation configuration there exists a unique Nash equilibrium the strategic game reduces to a payoff function. The assumption of a unique Nash equilibrium for every allocation configuration also allows us to use our framework in a more pragmatic way. In many situations competition may be hard to model. Specifying strategy spaces and interactions between firms might be extremely complicated or even impossible. However, there might be analysts or other experts who are able to predict what will happen to the market and the payoffs of the firms in different situations. Most likely their models are not formulated in terms of strategic games and Nash equilibria. We could use the models of these experts to calculate the payoffs for different allocation configurations while keeping all the other steps in our analysis the same. Basically we assume that the experts are able to capture the complex, competitive interactions between the firms in their models without explicitly specifying all the game theoretic specifics. In this way our approach remains useful even if the competitive stage is not formulated as a strategic game with a unique Nash equilibrium. In that case the boxes containing the strategic games in figure 1 and 2 could be viewed as ‘black boxes’ that model the competition in the market. In a later chapter of this thesis we will discuss such an example. Since firms seek to maximise their individual profits the outcome of the second stage can be used to rank the different outcomes of the first stage. The payoff function provides us with a valuation over the different allocation configurations. A firm will prefer one configuration, α = (P, k) over another configuration α0 = (P 0, k0) if and only if it obtains a higher payoff in the competitive stage of the game. This can be formalised in mathematical terms: 0 α α α0 α0 α i α ⇐⇒ πi (ˆx ) ≥ πi (ˆx ), 0 0 where α i α means that player i thinks that the configuration α is at least as good as α . As explained in the previous section we can apply different CS-values, ϕ, to the coalitional game that the firms play in the cooperative stage. For each coalition structure the division of capital stock will then be fixed by ϕ, and outcomes of the cooperative stage will always

3This uniqueness assumption is common in most models in the literature. If the second-stage game has multiple Nash-equilibria the analysis would become problematic. In this case the preferences of the ﬁrms (and stability of conﬁgurations) would depend on their expectations.

11 be of the form α = P, ϕ(N, v, P ). Instead of ranking allocation conﬁgurations we are ϕ 4 now able to derive a proﬁle of preferences (i )i∈N over all coalition structures P ∈ P:

ϕ 0 α α α0 α0 P i P ⇐⇒ πi (ˆx ) ≥ πi (ˆx ),

0 0 0 ϕ where α = P, ϕ(N, v, P ) and α = P , ϕ(N, v, P ) . The profile of preferences (i )i∈N is denoted by ϕ. We will call a pair (N, ϕ) an ordinal game with respect to ϕ. So using a CS-value ϕ we can reduce a coopetition game (N, v, Γ) to an ordinal game (N, ϕ). In an ordinal game all players have their own preferences over the different coalition 0 0 structures. If the preference relations i satisfy P ∼i P whenever P (i) = P (i) the preferences of the firms only depend on the members of the coalition they belong to. Such types of ordinal games are known in the literature as hedonic games and were introduced and studied for the first time by Drèzeand Greenberg (1980). The papers by Banerjee et al. (2001) and Bogomolnaia and Jackson (2002) investigate different stability concepts in the hedonic setting. However, the results from these papers cannot be translated one-on-one since in our setting firms have preferences over coalition structures and not just coalitions. Whether or not a player prefers one coalition over another also depends on the behaviour of the players outside of those coalitions. In a hedonic game the preferences of firms only depend on the coalition they belong to, irrespective of the behaviour of the firms outside of that coalition. This does not have to be the case for an ordinal game. When preferences not only depend on the members of the coalition a firm belongs to, but also on the organisation of the firms outside this coalition there are so-called externalities. An externality is defined as an effect of economic activity affecting those who are not directly involved in it. In this sense cooperation and the changes in capital stock allocation can have different effects on the profits of the firms. When a firm gains more capital stock because it works together with other players in a coalition it is reasonable to assume that this would lead to an increase in profit for that firm. However, the other firms within this coalition also gain more capital stock which in turn has an effect on the firm’s profit. On top of that the formation of other coalitions also affects the firm’s profit. In turn the net effect could be negative, even if the firm gained extra capital stock. Clearly, a coopetition situation exhibits externalities. This makes the ordinal game derived from a coopetition situation more complicated than a hedonic game since firms need to have beliefs regarding the behaviour of their competitors. In the next section we will explain how to deal with different notions of stability for an ordinal game.

2.7 Stability In the previous sections we have discussed our concept of a coopetition game in detail and showed how using a CS-value ϕ a coopetition game (N, v, Γ) can be reduced to an ordinal game (N, ϕ). Up until now we have only discussed what can happen in a coopetition game, but not what will happen. In order to answer this question we will use the concept of stability. Stability has to do with the possibilities and incentives of (groups of) players to deviate from a certain coalition structure. When (a group of) players both have the possibility and the incentive to deviate from a certain coalition structure it is likely that they will do so. Therefore this coalition structure will be unstable. There are various

4 We denote the symmetric part of i by ∼i and the asymmetric part by i.

12 different ways of specifying what possibilities and incentives firms have. In this thesis we will use two different notions of stability: individual and core stability. We will start by looking only at individual deviations. This means that a firm can only deviate on its own and groups of firms cannot deviate. When a firm deviates it leaves the coalition it belongs to and joins one of the other coalitions or simply remains on its own. In case it wants to join another coalition, all firms belonging to that coalition need to agree upon its arrival, i.e. each firm must weakly prefer the new configuration over the old one. Furthermore the deviating firm only does so if it strictly prefers the new configuration. The following definition summarises individual stability.

Deﬁnition 3. Let (N, ϕ) be an ordinal game derived from a coopetition game (N, v, Γ) using a CS-value ϕ. A coalition structure P ∈ P is individually stable with respect to ϕ if there are no i ∈ N and S ∈ P ∪ {∅} with i∈ / S such that

0 ϕ P i P and 0 ϕ P j P for all j ∈ S where P 0 is the coalition structure that is obtained from P by replacing P (i) with P (i)\{i} and S with S ∪ {i}.

Underlying this definition is the assumption that when one firm makes an individual deviation all other, non-deviating firms will stay in their respective coalitions. Contrary to individual deviations we could also look at group deviations. What happens if a group of firms is not satisfied with a certain coalition structure and thinks that it can obtain higher payoffs by working together with each other? Arguably this group would get together to form a coalition. However, whether or not the firms will obtain a higher payoff also depends on the behaviour of the other firms outside of that coalitions. In our definition of core stability we will therefore be rather pessimistic and assume the worst case scenario. Suppose we have a coalition structure P ∈ P and that the coalition S ∈ P is contemplating a deviation. S will only deviate if each firm is at least as well off as in the coalition structure they deviated from and for each firm there is a possibility that it is strictly better off. This leads to the following definition of core stability:

Deﬁnition 4. Let (N, ϕ) be an ordinal game derived from a coopetition game (N, v, Γ) using a CS-value ϕ. A coalition structure P ∈ P is core stable with respect to ϕ if there are no S ∈ 2N such that for all i ∈ S

0 ϕ P i P

0 for all P ∈ PS and 0 ϕ P i P 0 for some P ∈ PS.

N 0 ϕ If there does exists a S ∈ 2 such that for all i ∈ S we have that P i P for 0 0 ϕ 0 all P ∈ PS and P i P for some P ∈ PS we say that S blocks P . So a core stable coalition structure is a coalition structure that is not blocked by any coalition. Note that

13 core stability does not imply individual stability or vice versa. Furthermore, both these definitions are generalisations of the definitions of individual stability and core stability for a hedonic game.5. In case there are no externalities the definitions coincide. The observant reader might note that our definition of core stability reflects a rather pessimistic view of the players on the outcome of deviating. We are aware of the criticism of this point of view by for example Ray and Vohra (1997) who call it bloodthirsty behavior. One of the main differences between individual and core stability is that in our definition of core stability when a group of firms decides to deviate all other firms are allowed to reconfigure themselves whereas in the case case of individual stability all firms remain in their coalitions. We justify this in the following way. When a group of firms blocks a coalition structure and decides to deviate this can be seen as a big event. Firms that previously might have been in different coalitions decide to get together and form a new coalition. Such a move can be seen as disruptive and in response other firms might also contemplate forming new coalitions. In the case of individual deviations there is only one firm that moves from one coalition to another. Such a move can be seen as less disruptive and therefore we assume that all other firms remain in their coalition. Ideally we are looking for coalition structures that are both individual and core stable since in this case their are no incentives for individual firms or groups of firms to deviate. Checking whether or not a coalition structure is stable (with respect to a CS-value) is the final step of our approach. This concludes our game theoretic framework for analysing coopetition. Figure 3 presents a schematic overview of the complete model. After a short note concerning the computational complexity of our model we will continue with the next two chapters where we will use this framework to explore coopetition. First we will do this using the theoretical example of a Cournot oligopoly. After that we will use the model in a more pragmatic way to look at coopetition in the mobile telecommunications market.

(N, v, Γ), ϕ

Allocation α1(ϕ) = (P 1, k1) with αm(ϕ) = (P m, km) with ··· Conﬁguration k1(ϕ) = ϕ(N, v, P 1) km(ϕ) = ϕ(N, v, P m)

Strategic Game Gα1(ϕ) ··· Gαm(ϕ)

Equilibrium α1(ϕ) ··· αm(ϕ) Payoﬀ π π

Stable? Yes/No Yes/No

Figure 3: Schematic overview of the complete model. In this ﬁgure the last step of checking which coalition structure is stable and which not is included. In theory diﬀerent stability concepts could be used. However, in this thesis we will focus on individual and core stability.

5See for example Bogomolnaia and Jackson (2002) for these deﬁnitions.

14 2.8 Computational Complexity Ballester (2004) shows that for a hedonic game the core and the individually stable set have corresponding NP-complete decision problems and, hence, that only slow algorithms are known for solving them. Without going into the details of a formal proof we expect that, since an ordinal game and the associated concepts of individual and core stability are generalisations of a hedonic game and the associated stability concepts, a similar result holds for ordinal games. One of the issues causing these problems to be so hard is the fact that the number of possible coalition structures grows exponentially with the number of players. For n = 3 there are 5 different coalition structures. When the number of players doubles, so n = 6, there are 203 different coalition structures. If we double the number of players again to n = 12 the number of different coalition structures explodes to over 4 million. This is not only bad news from a computational point of view since it limits us in finding stable coalition structures, it also is bad news for the players of the game. We cannot expect that the firms oversee all the consequences of their actions when the number of different coalition structures is large, therefore it may be hard for them to find or reach stable configurations. To check whether or not a coalition structure is stable we have implemented different routines in Matlab. When the number of players remains lower than 8 these routines seem to be able to check whether or not there are stable partitions in a reasonable amount of time (the order of minutes).

15 3 Cournot Oligopoly: A Theoretical Example

3.1 Introduction In this chapter we will demonstrate our approach using the theoretic example of the Cournot competition model. In this model firms compete with each other in the market by setting their output levels. The price is then determined based on the total level of output. By forming coalitions and cooperating at a lower level firms are able to lower their (marginal) costs to increase profitability and become more competitive. The inspiration for this example comes from different sources. The example is closely related to the works of Bloch (1995) and Yi (1997) who both study the formation of associations in a Cournot oligopoly. In their models players can also work together in order to reduce their marginal cost. Another source of inspiration are Farrell and Shapiro (1990) who investigate the effects of accumulating capital stock in a Cournot oligopoly. This chapter will have the same structure as the previous chapter which described our approach and defined the concept of a coopetition game. We will start by discussing competition in a Cournot oligopoly in general and how this can be modelled using strategic game. Then we will discuss the coalitional game (N, v) and how the outcomes of this game change the competition between the firms. The model will then be demonstrated using several examples.

3.2 Cournot Coopetition Game In general a Cournot oligopoly with linear inverse demand can be described by the tuple (N, (Ci(xi, ki))i∈N , (ki)i∈N , b) where N = {1, . . . , n} is the set of firms and Ci(xi, ki) is the cost function describing the costs of producing xi numbers of goods for firm i which owns ki units of capital stock. We will assume that additional capital stock lowers the marginal 2 cost curve: ∂ Ci(xi, ki)/∂ki∂xi < 0. To keep things simple we will assume that the cost functions of the firms have constant marginal costs, so C(xi, ki) = ci(ki)xi and thus that ∂ci(ki)/∂ki < 0. The constant b ≥ 0 is the intercept of the inverse demand function, so that 6 when the total production is xN , the price per unit is given by p = b − xN . Any scaling has already been taken account for in the units of measurement of x. Corresponding to the oligopoly defined above the strategic oligopoly game G = {(Xi, πi)}i∈N is defined by

Xi = [0, ∞) for every i ∈ N, and πi = (b − xN )xi − ci(ki)xi Q for every i ∈ N and every x = (xi)i∈N ∈ j∈N Xj. It can be shown that (if b is large enough) this game has a unique, strong Nash equilibriumx ˆ which is called the Cournot- Nash equilibrium. The derivation of this equilibrium can be found in the appendix. In this equilibrium the output quantities are given by P b − (n + 1)ci(ki) − cj(kj) x = j∈N i (n + 1)

6 P The total level of production is denoted using xN = i xi∈N .

16 for every i ∈ N. The payoﬀs are then given by

P 2 b − (n + 1)ci(ki) + j∈N cj(kj) π = i (n + 1)2 for every i ∈ N. Using this expression we can directly calculate the payoffs of the firms in a Cournot oligopoly with linear inverse demand. Basically the payoffs are (increasing) functions of the level of capital stock of the firms: firms that own more capital stock will realise larger payoffs. Initially the level of capital stock of the firms is fixed by k1. Therefore, in absence of cooperation, the payoffs of the firms are also fixed by k1. To keep things simple for now we will assume that marginal costs are linearly decreasing in the level of capital stock of a firm and that this happens in the same way for all the firms, ci(ki) = c(ki) =c ˜−ki. In the case of constant marginal costs that are decreasing in the level of capital stock the oligopoly can be fully described using the tuple (N, c,˜ (ki)i∈N , b). Without cooperation the 1 1 marginal costs of firm i are given by ci =c ˜ − ki and the corresponding payoffs are given by 2 1 P 1 b − c˜ + (n + 1)ki − j∈N kj π1 = . i (n + 1)2 At some point in time the option to cooperate becomes available to the firms and the firms will play a coalitional game (N, v) at the lower level in order to acquire extra capital stock. The outcome of this game will be a component efficient allocation configuration α = (P, k) ∈ A(N, v). As explained in the previous chapter each outcome α maps to a strategic game Gα that the firms play with each other in the market. This mapping works in the exact same way as the description above of transforming a Cournot oligopoly in a strategic game. If the firms cooperate they will gain access to more capital stock. Naturally each allocation configuration α = (P, k) leads to an oligopoly (N, c,˜ (ki)i∈N , b) which can be transformed into a strategic game. This fully defines the Cournot coopetition game (N, v, Γ). If we refer to a Cournot coopetition game in this thesis we refer to a coopetition game derived from a Cournot oligopoly with linear inverse demand and constant marginal costs that are linearly decreasing in capital stock. Using a CS-value ϕ this Cournot coopetition game can be reduced to an ordinal game (N, ϕ) as we have discussed in the previous chapter. To get some feeling for how the model works we will discuss some examples in the next section.

3.3 Examples 3.3.1 Example 1: One Large Firm, Three Small Firms The first example we will discuss is a four player Cournot coopetition game. In this example there is one large firm, labeled 1, that is the market leader and three, identical, smaller firms labeled 2-4. Initially the market leader is almost twice as large in terms of market share as the smaller firms. However, by cooperating the firms can gain access to more capital stock and use this to increase their market share. We will assume that a combination of small firms can exploit larger synergies than a combination containing the large firm. The goal of showing this example is twofold. One the one hand it demonstrates the overall

17 workings of the model, on the other it shows that different CS-values or allocation rules lead to different results. The marginal costs ci of each firm are linearly decreasing in the level of capital stock of that firm and this happens in the same way for all the firms, so ci(ki) = c(ki) =c ˜ − ki. In this example we will choosec ˜ = 10. The initial level of capital stock is given by k1 = (4, 1, 1, 1), which leads to marginal costs of c1 = (6, 9, 9, 9). In the Cournot-Nash equilibrium payoffs and output levels will then be given by π1 = (43.56, 12.96, 12.96, 12.96) 1 and x = (6.6, 3.6, 3.6, 3.6). We can also calculate the market shares of the firms, si = 1 xi/xN , which leads to s = (37.9%, 20.7%, 20.7, %, 20.7%). Figure 4 graphically shows these results for the initial configuration.

Coalition Structure: {1,2,3,4} (AD-value and χ-value) Capital Stock Profit Market Shares 70

8 60 Firm 4: 21% 7 = 43.6 50 1 6 Firm 1: 38% π

5 = 4.0 40 Firm 1 1 k k

π Firm 2 4 30 Firm 3 Firm 4 3 Firm 3: 21% = 13.0 = 13.0 = 13.0 20 2 3 4 2 = 1.0 = 1.0 = 1.0 π π π 2 3 4 k k k 10 1 Firm 2: 21%

0 0 1 2 3 4 1 2 3 4 Firm Firm

Figure 4: Graphical overview of the capital stock k, market shares s, and proﬁt π in the initial conﬁguration of the four player Cournot coopetition game. Whether we use the AD-value or χ-value does not matter: since there is no cooperation and thus no capital stock to divide.

Now we are ready to introduce the possibility to cooperate. At the lower level the ﬁrms play a coalitional game (N, v) where v is given by:

S ∅ {1}{2}{3}{4}{1,2}{1,3}{1,4}{2,3}{2,4}{3,4}{1,2,3}{1,2,4}{1,3,4}{2,3,4}{1,2,3,4} v(S) 0 0 0 0 0 1 1 1 4 4 4 5 5 5 9 12

Let us inspect this characteristic function in more detail. First of all it has been zero- normalised. Clearly, coalitions containing firm 1 can gain less capital stock than coalitions not containing firm 1. In this sense firm 1 can be seen as a bad partner. The Shapley 1 7 7 7 value of this game is given by Sh(v) = (1 4 , 3 12 , 3 12 , 3 12 ). So if all firms would decide to work together and divide the surplus of capital stock using the Shapley value the resulting 3 5 5 5 marginal costs will be c = (4 4 , 5 12 , 5 12 , 5 12 ). In this case the payoffs of the firms are given by π = (29.70, 22.88, 22.88, 22.88). Again the results are shown graphically in figure 5. If the grand coalition forms the payoffs of firms 1, 2 and 3 increases while the payoff of firm 1 1 decreases. Although firm 1 increases its capital stock by 1 2 it loses profit. The explanation can be found in the fact that the three smaller firms also gained more capital stock and therefore became more competitive. Here we clearly see that there are different effects of cooperation. Now how do the results look for the other possible coalition structures? In this case the results will be different based on the CS-value of choice. Table 1 compares the results

18 Coalition Structure: {1234} (AD-value and χ-value) Capital Stock Profit Market Shares 70

8 60 Firm 4: 24% 7 Firm 1: 28%

= 5.25 50 6 1 k = 4.58 = 4.58 = 4.58 2 3 4 k k k 40 5 = 29.70 Firm 1 1 k π π Firm 2 4 = 22.88 = 22.88 = 22.88 30 Firm 3 2 3 4 π π π Firm 4 3 20 2 Firm 3: 24% Firm 2: 24% 10 1

0 0 1 2 3 4 1 2 3 4 Firm Firm

Figure 5: Graphical overview of the capital stock k, market shares s, and profit π in the Cournot coopetition game for the coalition structure {1234} where all firms cooperate with each other. Whether we use the AD-value or χ-value does not matter: in both case the Shapley value is being used to distribute the surplus of capital stock. of the AD-value with those of the χ-value. We will discuss the most important differences. Clearly, both values lead to different stable coalition structures.

AD-value χ-value P AD(N, v, P ) πα IS? CS? χ(N, v, P ) πα IS? CS? {1, 2, 3, 4} (0, 0, 0, 0) (43.56, 12.96, 12.96, 12.96) No No (0, 0, 0, 0) (43.56, 12.96, 12.96, 12.96) No No 1 1 2 2 {12, 3, 4} ( 2 , 2 , 0, 0) (47.61, 15.21, 11.56, 11.56) No No (− 3 , 1 3 , 0, 0) (32.87, 25.67, 11.56, 11.56) No No 1 1 2 2 {13, 2, 4} ( 2 , 0, 2 , 0) (47.61, 11.56, 15.21, 11.56) No No (− 3 , 0, 1 3 , 0) (32.87, 11.56, 25.67, 11.56) No No 1 1 2 2 {14, 2, 3} ( 2 , 0, 0, 2 , ) (47.61, 11.56, 11.56, 15.21) No No (− 3 , 0, 0, 1 3 ) (32.87, 11.56, 11.56, 25.67) No No {1, 23, 4} (0, 2, 2, 0) (33.64, 23.04, 23.04, 7.84) No No (0, 2, 2, 0) (33.64, 23.04, 23.04, 7.84) No Yes {1, 24, 3} (0, 2, 0, 2) (33.64, 23.04, 7.84, 23.04) No No (0, 2, 0, 2) (33.64, 23.04, 7.84, 23.04) No Yes {1, 2, 34} (0, 0, 2, 2) (33.64, 7.84, 23.04, 23.04) No No (0, 0, 2, 2) (33.64, 7.84, 23.04, 23.04) No Yes 1 1 2 2 {12, 34} ( 2 , 2 , 2, 2) (37.21, 9.61, 21.16, 21.16) No No (− 3 , 1 3 , 12) (24.34, 18.20, 21.16, 21.16) No No 1 1 2 2 {13, 24} ( 2 , 2, 2 , 2) (37.21, 21.16, 9.61, 21.16) No No (− 3 , 2, 1 3 , 2) (24.34, 21.16, 18.20, 21.16) No No 1 1 2 2 {14, 23} ( 2 , 2, 2, 2 ) (37.21, 21.16, 21.16, 9.61) No No (− 3 , 2, 2, 1 3 ) (24.34, 21.16, 21.16, 18.20) No No 2 1 1 1 4 4 {123, 4} ( 3 , 2 3 , 2 3 , 0) (39.27, 22.72, 22.72, 6.76) Yes No ( 9 , 2 9 , 2 9 , 0) (32.62, 25.45, 25.45, 6.76) No Yes 2 1 1 1 4 4 {124, 3} ( 3 , 2 3 , 0, 2 3 ) (39.27, 22.72, 6.76, 22.72) Yes No ( 9 , 2 9 , 0, 2 9 ) (32.62, 25.45, 6.76, 25.45) No Yes 2 1 1 1 4 4 {134, 2} ( 3 , 0, 2 3 , 2 3 ) (39.27, 6.76, 22.72, 22.72) Yes No ( 9 , 0, 2 9 , 2 9 ) (32.62, 6.76, 25.45, 25.45) No Yes {1, 234} (0, 3, 3, 3) (23.04, 23.04, 23.04, 23.04) Yes Yes (0, 3, 3, 3) (23.04, 23.04, 23.04, 23.04) Yes No 1 7 7 7 1 7 7 7 {1234} (1 4 , 3 12 , 3 12 , 3 12 ) (29.7, 22.88, 22.88, 22.88) Yes No (1 4 , 3 12 , 3 12 , 3 12 ) (29.7, 22.88, 22.88, 22.88) Yes No

Table 1: Results for the four player example. For each coalition structure we can calculate the allocation of capital stock using either the AD-value or the χ-value. This leads to different payoffs which in turn leads to different results regarding the stability of coalition stuctures. IS: individual stable. CS: core stable.

Using the AD-value we find that the following coalition structures are individual stable: {123, 4}, {124, 3}, {134, 2}, {1, 234} and {1234}. All coalition structures where the set of firms is split in a coalition of size three and a coalition of size one are individual stable. The coalitions consisting of three firms profit from the fact that the fourth firm does not accumulate extra capital stock and therefore does not lower its marginal costs. The firm that is left alone would like to join the coalition of three, the other firms however will not accept this, since then they can no longer exploit their relative advantage. For the other individual stable coalition structure, {1234} the story is different. The only deviation that firms can make is leaving the grand coalition to continue the game as a singleton. The

19 resulting coalition structure will consist of two coalitions: a coalition of all the firms except the deviating firm, and the deviating firm as a singleton. A few lines ago we however argued that in such a case the firm that forms a coalition on its own would like to join the other coalition. Therefore no single firm would like to deviate from the grand coalition. When we discuss more examples we will see that the grand coalition is often individually stable in a coopetition game. This can be interpreted as follows. Once you get all the firms in a market to work together, it is very unlikely that there will be a single firm that would like to deviate and leave the grand coalition. However, the story can be very different when we consider deviations for groups of firms as we will see when we discuss core stability! It is interesting to now take a look at the individual stable coalition structures when the χ-value is being used to distribute the capital stock. In this case there are only two individual stable coalition structures, namely {1, 234} and {1234}, which where also individual stable when we used the AD-value. The reasoning behind this is exactly the same as before. A more interesting question is why the coalitions structures {123, 4}, {124, 3} and {134, 2} are no longer individual stable. The answer to this question lies in the division of the capital stock and the fact that the χ-value takes outside options into account. Let us discuss this using the coalition structure {123, 4} as an example (since firms 2, 3 and 4 are symmetric this makes no difference). Firms 2 and 3 do not have to work together with firm 1, the could also form a coalition together with firm 4. When they would do so, the coalition 234 would gain access to an amount of capital stock of 9 whereas the coalition 123 only has a worth of 5. Firms 2 and 3 should be compensated for this when they decide to work together with firm 1. The result is that firm 1 gets a lower allocation of capital stock, indeed it is even negative, meaning that firm 1 should pay firms 2 and 3 some extra capital stock in order to work together. Firm 1 therefore might decide to leave the coalition. The result from this move would be that the coalition structure {1, 23, 4} is formed. In this coalition structure the profit of firm 1 is higher than in the previous coalition structure, namely 33.64 versus 32.62. Therefore the coalition structures {123, 4}, {124, 3} and {134, 2} are not individual stable anymore. One could say that firm 1 does not receive a high enough share of the capital stock in those coalition structures. The observant reader might note the following. Suppose that firm 1 indeed does move from {123, 4} to {1, 23, 4}, what is stopping firm 4 from joining firms 2 and 3? And if this would happen, wouldn’t firm 1 be worse off than in the coalition structure {123, 4}. To answer the latter question, yes firm 1 would be worse off, and unfortunately for firm 1, one could argue that there is nothing stopping firm 4 from joining firms 2 and 3 since their profits do not decrease if this happens. So why is {123, 4} then individually stable? If firm 1 would think two steps ahead instead of one, it would not leave the coalition 123. The fact that in spite of this argument we still call this coalition structure individual stable has to do with that in the definition of individual stability we consider the players to be myopic and not farsighted. By definition the firms only think one step ahead and not two or three. Whether or not this is realistic is indeed questionable. A solution would be to consider farsighted firms. Excellent material concerning farsightedness is written by for example Ray and Vohra (2015). Introducing farsightedness comes at a cost. It introduces more complexity to the model, instead of thinking 1 step ahead the firms think many more steps ahead. One needs to carefully define the rules of the game: what deviations are allowed and given a coalition structure which firm or group of firms has the right to be

20 the first to deviate. One could for example impose that in each consecutive step coalitions can only become smaller and let the first group of players to propose such a deviation be chosen randomly. On top of this we encounter another difficulty: coopetition games have externalities meaning that the payoff of the members of a coalition not only depends on their internal organisation but also on the organisation of the firms outside of this coalition. Considering all this, farsightedness is out of scope for this thesis. Let us, taking the above in consideration, advance with our analysis of the Cournot coopetition game and continue with discussing the core stable coalition structures in this example. In case the AD-value is being used there is only one core stable coalition structure, namely {1, 234}. It is pretty straightforward to see why: the payoffs of firms 1, 2, and 3 are strictly the largest in this coalition structure. Therefore this coalition of firms will always want to work together thereby blocking every other coalition structure. Again we will see that when the χ-value is used the results change. This time there are as many as six core stable coalition structures. Since players 2, 3, and 4 are symmetric, there are two different ‘flavours’ of core stable coalition structures: one where two smaller firms work together with the large firm and the third small firm is on its own, e.g. {123, 4}, and one where two smaller firms work together and both the large firm and the third small firm are on their own, e.g. {1, 23, 4}. We will start with discussing the coalition structure {123, 4}. For firms 2 and 3 the payoff in this coalition structure is the largest they can realise in the entire game. Therefore there will be no coalition that could offer either of the two a possibility to increase its payoff, it can only decrease. Therefore there is no coalition containing firm 2 and/or 3 that blocks this coalition structure. So the only coalitions that could block this coalition structure are coalitions that consist of firms 1 and 4. When a coalition blocks a coalition structure it needs to ensure that, in whatever way the other firms may organise themselves, when this coalition works together the payoffs of the members do not decrease with respect to their payoff in the coalition structure that is being blocked. Even if the other firms outside this coalition organise themselves in such a way that they minimise the payoffs of the coalition. Furthermore there needs to be a chance that their payoffs increase. Looking at table 1 we can verify that coalitions that consist of firm 1 and/or firm 4 cannot guarantee that the payoffs of firms 1 and 4 do not decrease. Therefore there are no coalitions blocking coalition structures {123, 4}, {124, 3}, and {134, 2}. The final core stable coalition structure we need to discuss is {1, 23, 4}. The only way that firms 2 and 3 could obtain a higher payoff is by forming a coalition with firm 1. However, the payoff of firm 1 can only decrease when it joins firms 2 and 3, so there are no blocking coalitions containing firm 2 and 3. The only way firm 1 could improve its profit is if the initial configuration {1, 2, 3, 4} would form. However, firm 1 already is a singleton, so there are also no blocking coalitions containing firm 1. Therefore we can conclude that no firm would like to work together with firm 4, and that there thus is no blocking coalition for this coalition structure. One could ask the question why the coalition 234 does not block this coalition structure. When firms 2, 3 and 4 form a coalition the coalition structure {1, 234} forms as a result of this. In this case the profit of firm 4 would increase, however, the profits of firms 2 and 3 stays the same. In our definition of core stability there needs to be a chance for every deviating firm that its profit increases. Therefore this coalition does not block the coalition structure. So why did we define core stability in such a way? If we

21 would not impose this condition their wouldn’t be a core stable coalition structure in this example because of the symmetry of ﬁrms 2, 3 and 4. This concludes our four player example of a Cournot coopetition game where we looked at the stability of coalition structures using diﬀerent CS-values to divide the capital stock at a lower level. We will now continue with a three player example.

3.3.2 Example 2: Economies of Scale In the previous example we have seen that the way capital stock is being divided amongst the firms can have significant effects on the stability of coalition structures. In this section we will take a look at another aspect of the Cournot coopetition game. This time we look at the characteristic function v in more detail, or in other words, the synergies that firms can exploit by working together. Again we will look at the effect on the stability of coalition structures. We will do this using the example of a three player Cournot coopetition game. In this example there are three, identical firms. Since the firms are symmetric there will be no difference between the AD-value and the χ-value. Furthermore we will set the initial level of capital stock to be zero for all firms. Let the characteristic function v be given by:

S ∅ {1}{2}{3}{1,2}{1,3}{2,3}{1,2,3} v(S) 0 0 0 0 1 1 1 a

Using this characteristic function we can investigate different economies of scale. We will explain what we mean with this. When the size of a coalition grows, the relative benefits of working together, i.e. the amount of capital stock per coalition member, can either in- or decrease. Using the parameter a we can determine the economies of scale. When a is large, it is profitable to work together with all the firms, when it is small, it may be that smaller coalitions of firms prefer to work together. A reader that is familiar with cooperative game theory and the concept of balancedness 1 might notice that if and only if a ≥ 1 2 the TU-game defined by (N, v) is balanced and has a nonempty core. However, as we will see in case of a coopetition game the results are different. Because the firms not only cooperate but also compete with each other it might be the case that there is no core stable coalition structure even though the allocation vector lies in the core of the TU-game (N, v). The competitive pressure destabilises cooperation. Before we will analyse the two different scenarios we first show that we can use a much simpler expression to compare different allocation configurations. Remember that a firm prefers one allocation configuration over another if it obtains a higher payoff. Therefore we could in principle take any monotonic transformation of the payoffs and compare those transformations since they preserve the ordering. In the case of a Cournot oligopoly with linear inverse demand and constant marginal costs that are linearly decreasing in capital

22 stock the following simpliﬁcation can be used:

0 3α i α ⇐⇒ α α α0 α0 πi (ˆx ) ≥ πi (ˆx ) ⇐⇒ 2 2 P 0 P 0 b − (n + 1)ci(ki) + j∈N cj(kj) b − (n + 1)ci(ki) + j∈N cj(kj) ≥ ⇐⇒ (n + 1)2 (n + 1)2 2 2 P 0 P 0 b − c˜ + (n + 1)ki − j∈N kj b − c˜ + (n + 1)ki − j∈N kj (1) ≥ ⇐⇒ a(n + 1)2 a(n + 1)2 X 0 X 0 (n + 1)ki − kj ≥ (n + 1)ki − kj ⇐⇒ j∈N j∈N X 0 X (n + 1)ki − v(S) ≥ (n + 1)ki − v(S). S∈P S∈P 0

where α = (P, k) and α0 = (P 0, k0). Let us take the expression on the last line and P deﬁne νi(α) = (n + 1)ki − S∈P v(S) with α = (P, k). No we have that

0 0 α i α ⇐⇒ νi(α) ≥ νi(α ).

So instead of the cluttered expression for the payoff in the Cournot-Nash equilibrium we can use a more simple and elegant expression to derive a profile of preferences over the coalition structures for the firms. Let us see if it is possible to give some interpretation to this expression. One the one hand we see that νi is increasing in ki, it is multiplied by (n + 1). So the more firms there are in the market, the larger the extra profit a firm gains when it increases its capital stock. However, on the other hand we see that νi is decreasing P in S∈P v(S), all the extra capital stock that is gained by the firms from cooperation. We should make the remark that νi can only be used to derive a profile of preferences for a player α α α α and not to compare the payoffs of two players, i.e., νi(α) ≥ νj(α) 6⇐⇒ πi (ˆx ) ≥ πj (ˆx ). We will now analyse the influence of the parameter a using (1) in the following two 1 scenarios, a = 1 2 and a = 3. The results are presented in table 2. We observe quite some 1 differences between the two different cases. When a = 1 2 there is no core stable allocation configuration. When the coalition structure containing the grand coalition forms each coalition of size 2 has an incentive to deviate from this coalition structure. Furthermore all coalition structures except {1, 2, 3} are individually stable. In case a = 3 only the coalition structure containing the grand coalition is individually stable. However, this time almost all coalition structures are core stable. Suppose now that we would take a > 3. In this case only the coalition structure {123} a a a a would be core stable. To see this observe that in this case νi({123}, ( 3 , 3 , 3 )) = 3 > 1 and therefore all firms will prefer {123} over any other coalition structure. Contrary, for any a < 3 there are no core stable coalition structures. In case a = 3 all coalition structures except {1, 2, 3} are core stable. So where for a normal TU-game without competition we 1 would need to have a ≥ 1 2 in order to have a core stable allocation configuration the competitive pressure in a Cournot oligopoly raises this to a ≥ 3.

23 1 Small Economies of Scale, a = 1 2 Large Economies of Scale, a = 3 Coalition Structure ϕ(N, v, P ) να IS? CS? ϕ(N, v, P ) να IS? CS? {1, 2, 3} (0, 0, 0) (0, 0, 0) No No (0, 0, 0) (0, 0, 0) No No 1 1 1 1 {12, 3} ( 2 , 2 , 0) (1, 1, −1) Yes No ( 2 , 2 , 0) (1, 1, −1) No Yes 1 1 1 1 {13, 2} ( 2 , 0, 2 ) (1, −1, 1) Yes No ( 2 , 0, 2 ) (1, −1, 1) No Yes 1 1 1 1 {1, 23} (0, 2 , 2 )(−1, 1, 1) Yes No ( 2 , 0, 2 )(−1, 1, 1) No Yes 1 1 1 1 1 1 {123} ( 2 , 2 , 2 )( 2 , 2 , 2 ) Yes No (1, 1, 1) (1, 1, 1) Yes Yes

Table 2

3.4 Beyond a Division Rule In the previous sections we restricted ourselves to allocation configurations where the allocation vector was determined using a CS-value. Previously we argued that without a CS-value the model is very complicated. However, in the simple case of a Cournot oligopoly with linear inverse demand and constant marginal costs that are linearly decreasing in capital stock we can analyse such situations. This is possible because the expression for νi(α) is fairly simple. In this section we lift the restriction that allocation vectors should be the result of a CS-value and let the firms propose any allocation vector as long as it is component efficient with respect to the coalition structure that forms. Note that these results are quite theoretical and might not be of interest to all readers. Our previous definition of individual stability was formulated with respect to a CS-value ϕ. This time there can be more than one allocation vector per coalition structure so we don’t have preferences over coalitions structures anymore. Instead preferences are given over allocation configurations now. Therefore we need to revise the definition of individual stability in the following way: Definition 5. Let (N, v, Γ) be a coopetition game. An allocation configuration α = (P, k) is called individually stable if the are no i ∈ N and S ∈ P ∪ {∅} with i∈ / S such that

0 α i α and 0 α j α for all j ∈ S where α0 = (P 0, k0) and P 0 is the coalition structure that is obtained from P by replacing P (i) with P (i) \{i} and S with S ∪ {i}, and k0 can be any component efficient allocation 0 configuration k ∈ KP 0 (N, v). No we have altered the definition of individual stability in such a way that firms could propose any allocation vector we can derive the following result: Proposition 1. Let (N, v, Γ) be a Cournot coopetition game. There exist an individual stable allocation configuration if and only if there exists a coalition structure P ∈ P such that for all S ∈ P X nn − |T | |T | + 1 o v(S) ≥ max v(T ∪{i})−v(T )+ v(S)−v(S\{i}) : T ∈ P ∪{∅},T 6= S . n + 1 n + 1 i∈S (2)

24 A proof of this theorem can be found in the appendix. Note that this result is similar to the top-partition property of Lazarova et al. (2011) for coalitional games without competition or externalities. We can also derive a similar result for core stability. However, ﬁrst we also need to adjust the deﬁnition of core stability. This can be done in the following way:

Deﬁnition 6. Let (N, v, Γ) be a coopetition game. An allocation conﬁguration α = (P, k) is called core stable is there are no S ∈ 2N such that for all i ∈ S

0 α i α for all α0 ∈ A(N, v) and 0 α i α 0 0 0 0 0 0 0 for some α ∈ A(N, v) where α = (P , k ) with P any coalition structure P ∈ PS and k 0 any component eﬃcient allocation vector k ∈ KP 0 . Now we are able to derive the following proposition:

Proposition 2. Let (N, v, Γ) be a Cournot coopetition game with v superadditive and P v(N) > S∈P v(S) for all P ∈ P. If there exists a core stable allocation conﬁguration it is of the form α = ({N}, k). Such a core stable allocation conﬁguration exists if and only if N there exists a k ∈ KN such that for all S ∈ 2

X n + 1 − |S| |S| k ≥ v(S) + v(N) − v(N \ S). (3) i n + 1 n + 1 i∈S

A proof can be found in the appendix. If we look closer at (3) we observe similarities with the definition of the core for a cooperative game. The right side of the equation now is a convex combination of v(S) and v(N) − v(N \ S) whereas in the definition of the core for a TU-game this would be v(S). This result may look complicated at first sight, however, the interpretation might be simpler than one thinks. Consider a three player Cournot coopetition game with symmet- rical firms. Suppose that the firms 1 and 2 are working together. This means that firm 3 finds itself in the worst position it could be. Therefore firm 3 decides to offer firm 1 a deal that is slightly better than the deal firm 1 currently has with firm 2. Since the deal is better, firm 1 will accept is. However, now firm 2 finds itself in the position where firm 3 was previously. Therefore this loop would keep on endlessly going. This is why if we don’t fix the division of capital stock using a CS-value that when there exist a core stable allocation configuration, it must be that in that configuration the grand coalition forms. By fixing the division of capital stock we prevent firm 3 of offering firm 1 a better deal.

25 4 Case Study: Network Sharing In The Mobile Telecom- munications Industry

Starting Remarks The research done for this thesis was partly connected to the SMART 24 project carried out by TNO. An extensive part of this project was developing a model to simulate competition in the mobile telecommunications market. Unfortunately, due to circumstances, the SMART 24 project was terminated prematurely. Therefore the model that is used in this chapter to simulate competition in the mobile telecommunications market is far from complete. However, to demonstrate the applicability of the coopetition framework we make use of models developed in the SAPHYRE project (see Oﬀergelt (2011)) and early stages of the SMART 24 project.

4.1 Introduction In chapter 3 we have shown how we can use the coopetition framework in a Cournot oligopoly where players have the possibility to lower their production costs by means of cooperation. This chapter will again demonstrate the coopetition framework using a different example. We will take a look at network sharing in the mobile telecommunications industry. In this example we will use the coopetition framework in a more pragmatic way. After explaining the model we will take a look at different scenarios. To get the reader familiar with network sharing in the mobile telecommunications industry we will provide a very brief introduction. In this introduction we will explain the main concepts and introduce some terminology. For a more rigorous treatment of the technical aspects of network sharing we refer the reader to Offergelt (2011). In most countries there are many companies that offer mobile services, however few own and operate their own network. Companies that have frequency allocations, and all required infrastructure to run a an independent mobile telecommunications network are called mobile network operator (MNO). In the Netherlands there are three operators that own and operate their own network, namely: KPN, Vodafone and T-Mobile. Next to these MNOs there are numerous other companies that offer mobile services. These providers make use of the networks of the MNOs. Such providers are called Mobile Virtual Network Operators (MVNOs). Examples of MVNOs are companies such as: Telfort, Ben, Simpel or Simyo. In order to keep up with the increasing mobile data demand operators need to constantly invest in new technologies. Since investments are costly operators explore new alternatives to save costs. One of these alternatives is network sharing: operators share their mobile networks instead of building and maintaining their own private network. In this chapter we will take a look at investments in LTE (long term evolution). When we speak of network sharing we mean sharing of the RAN network combined with radio spectrum sharing. The RAN (Radio Access Network) connects the mobile phone of a user with the core network of an operator and consist of different parts: the mast with antennas, the base transceiver station and the base station controller. Radio spectrum sharing means that operators make use of each others licenses for different spectrum bands. As we will see, RAN sharing leads to significant cost reductions.

26 4.2 The Competition Model This section will explain how the mobile telecommunications market is being modelled. The model consist of two parts. The first part is used to estimate the potential savings of a coalition of operators. The second part makes predictions about the market shares and profits of the operators based on certain characteristics of these operators. One of the key elements in this model is that the savings that are realised will be used to lower the subscription fees of the operators. Unlike Offergelt (2011) we do not address the question of whether or not to invest. Instead we assume that each operator investd in LTE technology, and that they all have sufficient financial capital to do so. Therefore the question is not whether or not an operator will invest, but together with who will an operator invest?

4.2.1 Investment in LTE In order to offer LTE to their customers the operators need to upgrade their existing or build new sites. We distinguish two categories of sites: Coverage sites and capacity sites. Coverage sites are needed to ensure that everywhere in a country the network can be accessed. On the other hand capacity sites are needed to ensure that all customers that want to access the network are able to do so. Some areas are more densely populated than others, this impacts the number of capacity sites that are needed. Compare the city centre of Amsterdam with a meadow in Friesland. To ensure geographical coverage in Amsterdam only a small number of sites is needed, however, these site will not be able to process all traffic. Therefore additional capacity sites need to be installed. On the other hand, in the meadow in Friesland the coverage sites will easily be able to process all traffic. To take the difference between different areas into account we will make distinction between three different area-types: urban, suburban and rural. To capture all this we will, similar as Offergelt (2011), use (a simplified version of) the BULRIC model to calculate the number of sites that need to be upgraded for each operator. In this model the number of sites that need to be upgraded for operator i is given by

3 X #Sitesi = max {si(Covj + Capj), Covj} j=1 (4) 3 X Covj = max s , (Cov + Cap ) , i Cov + Cap j j j=1 j j where si is the market share of ﬁrm i. Covj and Capj correspond to the total number of coverage and capacity sites that are needed to serve 100% of the market in area-type j where j = 1, 2, 3 corresponds to urban, suburban, and rural respectively. For the Netherlands these numbers are given in table 3 below. Now let us try to get some feeling for how this model works by examining an example. Consider two dutch operators that are thinking of jointly investing in LTE technology: operator one and two have a marketshare of s1 = 0.5 and s2 = 0.3 respectively. Using (4) and the numbers in table 3 we can calculate the number of sites that the operators need to upgrade: #Sites1 = 1520 and #Sites2 = 1467. If the operators would decide to jointly invest in a LTE network that serves their combined marketshare of 0.8 they

27 Cov Cap Urban 212 318 Suburban 808 105 Rural 447 0

Table 3: Number of coverage and capacity sites needed to serve 100% of the dutch mobile market. would only need to build 1679 sites. By cooperating the operators can decrease the total number of sites by almost 44% which is a signiﬁcant saving. Now why is this such a large decrease? This is mainly caused by the suburban and rural areas. Since the operators need to build a relatively high number of coverage sites in those areas they basically install a large overcapacity. By cooperating the operators can more eﬃciently make use of the capacity of the coverage sites. Suppose now that both operators have an equal marketshare of s1 = s2 = 0.4 which again sums up to 0.8. In this case the operators both would need to build 1467 sites if they decide to invest in LTE on their own. Again, if the operators would jointly invest they would only need to build 1679 sites. This time the total number of sites is decreased by almost 43% which is slightly lower than in the previous example. From this example we learn that the savings not only depend on the total market share of a coalition of operators, but also on the distribution of the individual market shares within a coalition. More precisely, the savings w(S) of a coalition S are given by X w(S) = #Sitesi − #SitesS i∈S 3 ( )! X X Covj X Covj = max si, − max si, (Covj + Capj) , Covj + Capj Covj + Capj j=1 i∈S i∈S where #SitesS denotes the number of sites needed to serve the combined marketshares of P the members of S, i∈S si. We could view w(S) as a cost savings game in terms of number of sites. In case we have three (dutch) operators with market shares of s1 = 0.5, s2 = 0.3, and s3 = 0.2, w(S) is given by:

S ∅ {1}{2}{3}{1,2}{1,3}{2,3}{1,2,3} ν(S) 0 0 0 0 1308 1361 1414 2564

Looking at these numbers it becomes clear that the grand coalition can save the most. Comparing the savings for the coalitions of size two we notice that it are the two smallest operators, 2 and 3, that can save the most. This is because these two operators need to install relatively more coverage sites in comparison with operator 1. For example with more than 3 operators the results are similar: the more members a coalition has, and the smaller the market shares of operators working together, the higher their savings. So far we have only focused on savings in terms of number of sites but not in monetary terms. We will now shift focus from the number of sites to the actual costs. We distinguish two types of costs; capital expenditures (RAN-CAPEX), and operational expenditures

28 (RAN-OPEX)7. All costs will be speciﬁed as costs per year. These costs include: • RAN-CAPEX;

– Depreciation costs of e12,000 per site per year; – Integration costs of e2,000 per shared site per year;

• RAN-OPEX;

– Operating costs of e10,000 per individual site per year; – Operating costs of e11,000 per shared site per year (10% markup for shared sites).

Using these numbers we can transform the savings in number of sites w(S) to monetary savings. Let v : 2N → R be the characteristic function that assigns to each coalition of operators the savings in monetary units. In our coopetition framework (N, v) is the coalitional game that the operators play at the lower level. For the example we discussed previously v(S) (in millions e) is given by:

S ∅ {1}{2}{3}{1,2}{1,3}{2,3}{1,2,3} v(S) 0 0 0 0 24.8 26.0 27.2 54.7

Recall that we assumed that every operator has suﬃcient ﬁnancial capital to invest in, and operate LTE on its own. When operators decide to cooperate within a coalition they save money which they can use for other purposes. In our approach we use a CS-value ϕ to divide the savings among the operators. Given a coalition structure P and a coalitional game (N, v) the savings ∆Ii(P ) of operator i are

∆Ii(P ) = ϕi(N, v, P ) and the investment Ii(P ) made by operator i will be

1 Ii(P ) = Ii − ∆Ii(P ),

1 where Ii denotes the investment made by operator i without cooperation.

4.2.2 Multinomial Logit Demand Model To model competition between the operators we will make use of the multinomial logit demand model. Using this model we can derive the demand for each operator based on certain characteristics of that operator. In our case we will make a distinction between two separate characteristics, namely all the non-financial benefits of subscribing to an operator such as the perceived brand value or quality of service which are aggregated in one single variable and the price charged by that operator. Note that this model is not formulated as a strategic game. The model describes what we observe in the market which is the effect of an underlying mechanism of competition. It does not describe the competition

7In this thesis we use the term investment to refer to both the RAN-CAPEX and RAN-OPEX. This is not completely correct since the RAN-OPEX are not actual investments.

29 itself. Therefore it is a more pragmatic way of using our coopetition framework since we do not explicitly define a strategic game. The logit demand model allows us to pragmatically attach a profit function to each of the different allocation configurations which can be used to derive an ordinal game. We will consider two different forms of the logit demand model: one with, and one without an outside good. When there is no outside good, the market is completely satu- rated: the utility that consumers derive from having a mobile subscription is high enough so that all consumers that could subscribe to a network, do so. Each consumer subscribes to exactly one operator. In this case the total number of subscriptions cannot grow. The contrary holds when there exists an outside good. In this case consumers might prefer the outside good over a mobile subscription. So some consumers may decide not to take a subscription and opt for the outside good instead. If the operators start offering subscriptions at a lower price level this would increase the utility derived from a subscription. The consumers that previously chose the outside good might change their mind and subscribe after all. The result is that the total number of subscriptions grows. This is similar to the Cournot example where the total number of customers also increases when the price decreases. All market shares will be expressed as the fraction of the consumers that is served by an operator. In case there is an outside good, s0 will denote the market share of this outside good. The market consist of m consumers that are each modelled to subscribe to at most one operator i ∈ N. A consumer j ∈ {1, . . . , m} subscribed to operator i obtains the following utility uj,i = αi + βpi + εj,i, where αi is a fixed utility term derived from subscribing to operator i which for example reflects the brand-value of service quality, pi denotes the subscription fee charged by operator i, and β < 0 is a price sensitivity coefficient. These terms are non-stochastic and reflect the populations taste. The εj,i are random taste shocks and are assumed to be i.i.d. across operators and consumers and have the“type 2 extreme value” distribution. The utility derived by consumer l choosing the outside good will be

uj,0 = α0 + εj,0. It can be shown (see for example McFadden (1973)) that under these assumptions and in case there is no outside good the market share si of operator i is given by

eαi+βpi si = , PN αj +βpj j=1 e and when there is an outside good by

eαi+βpi si = N . α0 P αj +βpj e + j=1 e When there is an outside good the percentage of consumers that does not subscribe to an operator is given by eα0 s0 = N . α0 P αj +βpj e + j=1 e

30 From this expression we readily observe that if the prices charged by the operators decrease, the market share of the outside good will also decrease. When we use this approach to model a mobile telecommunications market we need to choose values for the parameters αi, β and the price levels pi. It is most convenient to fix the initial distribution of market shares and price levels. These are the markets shares and prices when there is no cooperation. Given a value for β we can then calculate the αi’s. The parameter β determines the level of price sensitivity in the market. It can be easily verified that ∂ ln si β = /(1 − si) ∂pi In other words, β may be interpreted as the increase in the market share of firm i, due to a unit price decrease, expressed as a function of the percentage of the market, not yet captured by the firm. Therefore the parameter β is often referred to as the price penetration rate. In our examples we will use different values for β to show the impact of changing this parameter. When β is small, the price level has less impact and the αi’s will be relatively more important. On the other hand, when β is large, price will have a bigger impact and a price cut could lead to significant shiftings in market shares. Let us now define a measure for the initial profit of the operators. We have assumed that all operators will invest in LTE. Initially the profit of the operators is given by 1 1 1 1 πi = si Mpi − Ii

4.2.3 Linking the LTE Investments and the Logit Demand Model Together Now we have discussed both the LTE investments and the logit demand model we need link these two together. How do outcomes of the coalitional game change the competition between the firms, or in terms of the coopetition framework: what does the mapping Γ look like? As stated before, in this model we do not define a strategic game. We use the coopetition framework more pragmatically. What we will do is explain how the outcome of the coalitional game changes the profit of the operators. To do this we need to make assumptions on what the operators do with the savings they realise when cooperating with each other. We will assume that the operators use these savings to cut their prices. More specific, each operator divides their savings evenly over all its subscribers, so given a coalition structure P the new price p(P ) will be

1 ∆Ii(P ) pi(P ) = pi − . si(P )M Of course one could argue whether or not this is the best strategy to follow for each operator. Maybe it is smarter for an operator to cut the prices by less or even not at all. However, Using this new price we can calculate the proﬁt πi(P ) of each operator for every coalition structure P ,

πi(P ) = si(P )Mpi(P ) − Ii(P ) 1 ∆Ii(P ) = si(P )M pi − − Ii(P ) si(P )M (5) 1 = si(P )Mpi − ∆Ii(P ) − Ii(P ) 1 1 = si(P )Mpi − Ii .

31 Notice that the savings are divided over the total number of customers after the price cut. Since market shares depends on the new prices, the total number of consumers subscribed to an operator will also be aﬀected by this, which in turn changes the price cut per subscriber. In order to determine the market shares of the operators we therefore need to solve the following systems of equations:

0 ∆Ii(P ) αi+β p − e i si(P )M si(P ) = , i ∈ N, (6) 0 ∆Ij (P ) N αj +β pi − s (P )M u0 P j e + j=1 e in case there is an outside good, and

1 ∆Ii(P ) αi+β p − e i si(P )M si(P ) = , i ∈ N, (7) 1 ∆Ij (P ) αj +β p − PN i sj (P )M j=1 e in case there is no outside good. These systems of equations cannot be solved analytically, however using a numerical solver a solution can be easily found. We can plug this solution back into (5) to calculate the proﬁts of the operators for each coalition structure. This expression gives more insight in what happens when operators decide to cooperate. In case there is no outside good the market shares of all operators must add up to one. This means that if the market share of one operator goes up the markets share of another operator must go down. Therefore when the grand coalition forms there is at least one operator who would prefer that everybody invests on their own over working together. When there is an outside good it could happen that the marketshare of (multiple) operators goes up while no operator’s market share decreases. This can happen when the outside option loses market share and the total number of subscriptions rises.

4.3 Scenarios 4.3.1 Scenario 1: The Dutch Market with 3 Operators without an Outside Good In this example is very similar to the example discussed by Offergelt (2011). We will take a look at the Dutch market with three operators, namely KPN (K), Vodafone (V ) and T-Mobile (T ). The Dutch market consists of in total 20 million consumers, how many consumers have a mobile subscription depends on whether or not there is an outside good and its attractiveness. Regarding the coverage and capacity sites we will use the same data as before (see table 3). We will set β = −0.3. We have tested different values for β and, unless the value is unrealistically high or low, the results are similar. Both the AD-value and the χ-value have been used to divide the savings. For this particular example the choice of the CS-value seems to play less of a role: results regarding stable coalition structures are similar. The numbers that are presented in this section are calculated using the AD-value as a division rule. The initial market shares of the operators are set to sK = 0.5, sV = 0.3, and sT = 0.2 and the initial prices charged by the operators are pK = e29, pV = e31, and pT = e26. In the first example there will be no outside good.

32 The results for these parameters are presented in table 4 and figure 6. Using these parameters there are two individually stable coalition structures, namely: {K,VT }, and {KVT }. Let us analyse these two coalition structures a bit further starting with {KVT }. In this coalition structure all operators work together and there is a single LTE network. As a result of the savings the average price for a mobile subscription decreases by almost 10%. The operators profiting the most from this are Vodafone and T-Mobile who both prefer this coalition structure above the initial configuration. KPN would rather see everybody investing on its own, however when KPN would decide to leave the grand coalition it would need to compete with the two other operators that are both sharing their networks. Similar arguments hold for Vodafone and T-Mobile. Therefore it is obvious that {KVT } is individually stable. The coalition structure {K,VT } is also individually stable. Although KPN would like to join the coalition of Vodafone and T-Mobile, its arrival will not be accepted. Vodafone and T-Mobile on the other hand do not want to leave their coalition to either join KPN or invest alone. Since for both Vodafone and T-Mobile the profit they realise in this coalition is strictly the largest, the coalition VT blocks any coalition structure but {K,VT }. Since there is no coalition that blocks {K,VT } we have that {K,VT } is also core stable.

33 I (emillions) π(P ) − π1 (emillions) p s P KVTKVTKVTKVT {1, 2, 3} 33.4 32.3 32.3 0 0 0 e29.0 e31.0 e26.0 50.0% 30.0% 20.0% {12, 3} 21.6 20.4 32.2 1.6 23.8 -21.4 e27.8 e29.2 e26.0 50.6% 23.7% 25.7% {13, 2} 20.9 32.2 19.7 3.4 -39.1 29.7 e27.8 e31.0 e23.6 50.3% 33.8% 15.9% {1, 23} 33.4 19.0 19.0 -62.1 28.5 31.8 e29.0 e29.1 e23.5 39.3% 34.6% 26.1% {123} 17.2 15.3 14.7 -35.2 8.2 24.7 e27.2 e28.3 e22.4 43.9% 31.3% 24.8%

Table 4: Results for the Dutch market with three operators, without an outside good.

Coalition Structure: {1,2,3}

7 Investment Prices ×10 Market Shares 45 5 T: 20% 40 = 31.0 35 = 29.0 V 4 = 3.3e+07 p K = 3.2e+07 = 3.2e+07 = 26.0 I K p I V I T 30 T p

3 25 I

K: 50% p

20 2 15 V: 30% 10 1

0 0 KVT KVT Operator Operator

Coalition Structure: {1,23}

7 Investment Prices ×10 Market Shares 45 5 40 T: 28% 35 = 29.0 = 29.2 4 = 3.3e+07 K V K: 35% p I K p 30 = 23.7 T p 3 25 KPN I p = 1.9e+07 = 1.9e+07 Vodafone 20 I V I T T-Mobile 2 15

10 1 V: 36% 5

0 0 KVT KVT Operator Operator

Coalition Structure: {123}

7 Investment Prices ×10 Market Shares 45 5 40 T: 26% 35 = 28.3 4 = 27.1 V K p 30 p K: 42% = 22.6 T 3 25 p I p

= 1.7e+07 20 = 1.5e+07 2 I K = 1.5e+07 I V I T 15

10 1 V: 32% 5

0 0 KVT KVT Operator Operator

Figure 6: Graphical overview of the results for the Dutch market with three operators and without an outside good. The following coalition structures are shown: {K,V,T }, the initial conﬁguration; {KVT } which is individual stable; and {K,VT } which is both individual and core stable.

34 4.3.2 Scenario 2: The Dutch Market with 3 Operators and an Outside Good In the previous example the market share of the outside good is set to zero. So operators can only ‘steal’ consumers from each other and not enlarge the total amount of mobile subscriptions. It is interesting to investigate what happens if there is an outside good. In this case the operators can ‘steal’ consumers from the outside good. In the next scenario the 1 market share of the outside good is set to be s0 = 0.2, so only 80% of the consumers have a mobile subscription. By cooperating the operators can lower their prices and attract customers that previously prefered the outside good. To keep the relative size of the operators the same we will set their market shares to s1 = (0.4, 0.24, 0.16). All other parameters stay the same. The results for this scenario are shown in table 6 and figure 7. Again we have that {KVT } and {K,VT } both are individually stable and the latter coalition structure is also core stable. What is different in this example is that the total amount of consumers that choses a mobile subscription grows when operators work together. The market share of the outside good is clearly being reduced. However, this does not lead to different stable coalition structures. If the market share of the outside good is high, there are more consumers that can be potentially ‘stolen’ by the operators if they would cooperate and lower their prices. 1 In this sense the parameter s0 determines how profitable it is to work together. We are 1 interested whether or not the results do change when s0 would become larger. Testing 1 1 different values of s0 we find that to have any influence s0 needs to be relatively high. The results for different ranges are shown in table 5.

Market Share Outside Good Core Stable Individual Stable 1 s0 ≥ 0.589 {KVT }{KVT } 1 0.543 ≤ s0 < 0.589 {KVT }, {K,VT }{KVT }, {K,VT } 1 s0 < 0.543 {K,VT }{KVT }, {K,VT }

Table 5

We observe something similar as in the Cournot example. For the grand coalition to be core stable the benefits of cooperation need to be high. Cooperating at a lower level has two different effects. On the one hand it helps operators to get a larger piece of the pie, but in this case the gain of one operator needs to be compensated by the loss of the other. The second effect of cooperation is that it can enlarge the size of the pie that needs to be divided. The grand coalition will therefore only be core stable if the benefits of working together are high enough. In both three player examples we find the same stable coalition structures. Using the coopetition framework one might conclude that it is likely that Vodafone and T-Mobile would work together. However, in the real world we do not yet see network sharing in such extent. The explanation for this can be found in different other factors. When Vodafone and T-Mobile would cooperate the two operators cannot differentiate on network quality since it is the same network customers are using. There may also be less rational explanations. It could for example be that the two operators simply do not get along with each other which would complicate the formation of a coalition. Furthermore it could be that both

35 operators have different objectives than simply profit maximisation. To also take effects such as the above into account one could propose to capture all these elements in the valuation of a coalition structure and use this number to derive a profile of preferences over the coalition structures for each operator. However, given the impact and importance of decisions regarding network sharing it is hard to imagine that all considerations and valuations of effects can be captured in a single number. This problem might be a returning problem of the coopetition framework, or game theory in general, when it is used to analyse real-world decisions. In this way the coopetition framework must be seen as a supportive tool, it can give insight in some part of the considerations that operators have when they are faced with network sharing decisions.

36 I (emillions) π(P ) − π1 (emillions) p s P K V T K V T K V T K V T Outside Good {1, 2, 3} 32.3 32.3 32.3 0 0 0 e29.0 e31.0 e26.0 40.0% 24.0% 16.0% 20.0% {12, 3} 19.9 19.9 32.3 20.1 43.2 –24.1 e27.6 e29.0 e26.0 43.4% 31.0% 11.4% 14.2% {13, 2} 19.4 32.3 19.4 32.3 -43.2 45.6 e27.5 e31.0 e23.4 44.0% 17.0% 24.8% 14.2% {1, 23} 32.3 18.3 18.3 -68.8 51.3 49.5 e29.0 e28.8 e23.3 28.1% 32.3% 25.5% 14.1% {123} 14.9 13.8 13.2 -14.8 31.2 42.6 e26.7 e27.8 e22.4 37.5% 29.0% 24.2% 9.3%

Table 6: Results for the Dutch market with three operators, with an outside good (initial market share of 20%).

Coalition Structure: {1,2,3}

7 Investment Prices ×10 Market Shares 45 5

4.5 Outside Good: 20% 40 = 31.0 4 35 = 29.0 V = 3.2e+07 = 3.2e+07 = 3.2e+07 p K p = 26.0 3.5 I K I V I T K: 40% 30 T p 3 25 I p 2.5 T: 16% 20 2 15 1.5

10 1

0.5 V: 24% 5

0 0 KVT KVT Operator Operator

Coalition Structure: {1,23}

7 Investment Prices ×10 Market Shares 45 5 Outside Good: 14% 4.5 40 K: 28% 4 35 = 29.0 = 28.8 = 3.2e+07 K V I K p p 3.5 30 = 23.3 T 3 p KPN 25 I p Vodafone 2.5 = 1.8e+07 = 1.8e+07 T: 26% 20 T-Mobile I V I T 2 Outside Good 15 1.5

10 1

0.5 V: 32% 5

0 0 KVT KVT Operator Operator

Coalition Structure: {123}

7 Investment Prices ×10 Market Shares 45 5 Outside Good: 9%

4.5 40

4 35 = 27.8 = 26.7 K: 37% V 3.5 K p 30 p T: 24% = 22.1 3 T 25 p I p 2.5 20 = 1.5e+07 2 = 1.4e+07 = 1.3e+07 I K I V I T 15 1.5

10 1

0.5 V: 29% 5

0 0 KVT KVT Operator Operator

Figure 7: Graphical overview of the results for the Dutch market with three operators with an outside good (initial market share of 20%). The following coalition structures are shown: {K,V,T }, the initial conﬁguration; {KVT } which is individual stable; and {K,VT } which is both individual and core stable.

37 5 Conclusions

In this thesis we proposed a new game theoretical framework for analysing coopetition, e.g. cooperation amongst competitors. We made a clear distinction between cooperation at a lower level and competition at the market level. Furthermore we introduced a new type of games: a coopetition game (N, v, Γ). A coopetition game is a generalisation of both a strategic game and a coalitional game. (N, v) represents the coalitional game that the firms play at a lower level and Γ maps the outcomes of the coalitional game to strategic games at the market level. A coopetition game can be reduced to an ordinal game by using a CS-value ϕ to fix the division of capital stock at the lower level for each coalition structure. To analyse these ordinal games we introduced two different stability concepts: individual stability and core stability. This framework makes it possible to analyse various forms of coopetition in both a theoretical and pragmatic way. We analysed Cournot coopetition games in which players can lower their marginal costs by cooperating. Whilst these examples are theoretical they give a lot of insight in the different considerations of firms facing the decision to cooperate with a competitor. We will briefly highlight the most important findings: • We often see that there are coalition structures where all firms but one work together and that firm forms a coalition on its own, are individually stable.

• The grand coalition most of the times is individually since no ﬁrm wants to leave this coalition to face competition with all the other ﬁrms when they are working together.

• The CS-value that is used to divide the capital stock at a lower level can have significant impact on the results. Different CS-values lead to different stable coalition structures.

• It is hard to get all the firms to work together: the grand coalition is often blocked by other coalitions and therefore not core stable. Furthermore we showed that in a three player Cournot coopetition game with symmetric players the value of the grand coalition needs to be twice as large as in a TU-game for it to be core stable due to the competitive pressure. For the Cournot coopetition examples we also lifted the restriction that the division of capital stock should be fixed using a CS-value. In this case firms can propose any component efficient allocation configuration. We showed that if there exists a core stable allocation configuration that in these configurations the grand coalition forms. We also used the framework in a less rigorous and more pragmatic way to analyse network sharing in the Dutch mobile telecommunications market. Using some simple concepts we constructed a model that is able to make (crude) predictions of what happens to the Dutch mobile market when operators invest together in new types of technology (LTE). In this case we found that most of the time the coalition structure where the n − 1 smallest firms cooperate are both core and individual stable. The grand coalition is individual stable in all the scenarios we have discussed. The choice of the CS-value seemed to have less of an impact in these scenarios. The coopetition framework does a good job in analysing these scenarios, however one can question whether or not the model to predict what happens to the market is accurate enough.

38 Another advantage of the proposed framework is that it is possible to analyse situation with a relatively high number of firms. In the case of the Cournot coopetition games and the network sharing scenarios it is possible to analyse situations with 6 or 7 players. We can identify two types of bottlenecks. The first one would be the strategic games that are a result of the mapping Γ. Strategic games are often very hard to solve for a high number of players. If this is not a bottleneck, for example when a more pragmatic approach is being used, the total number of players is limited by the fact that determining whether or an ordinal game has individual or core stable coalition structures is at least NP-hard. Overall we therefore conclude that the coopetition framework is a useful tool for analysing coopetion. It would be interesting to use the coopetition framework in different settings. An interesting topic would for example be large multistakeholder innovation projects.

6 Further Research

We will point out two different kind of directions for further research. The first one will consider game theory and coalition formation and the second set of directions will focus more on the modelling of mobile telecommunications markets. As mentioned in section 3.3 our definitions of core and individual stability state that the players are myopic. They only think one step ahead. It would be interesting to investigate whether or not farsightedness could be include in the coopetition framework. A good starting point would be the survey by Ray and Vohra (2015). As mentioned before we expect that this will be a difficult an complex undertaking. The approach we took to coalition formation in this thesis was focused on stability concepts. We did not explicitly specify any processes of coalition formation. Like for example Bloch (1995) it is also possible to specify a process of coalition formation. One could specify this process in an extensive form. However, we expect that when doing so the number of players will be limited and/or it would be difficult to incorporate asymmetric players. Another interesting approach is the one recently proposed by Karos (2015) in his working paper. This paper uses concepts closely related to bargaining to analyse stable partitions or games with non-transferable utilities and externalities. In a way a coopetition game also is a game with non-transferable utilities and externalities since players can only compensate each other at the lower level and not transfer profits. With regard to the model used to make predictions about the Dutch telecommunications market we believe that also here there are improvements possible. Currently the estimates are very crude. There are two areas where improvement could be made. Firstly the cost savings could be modelled more precisely. This could for example be done by taking into account more detailed information regarding the operators, their networks, spectrum licenses and the geography. On the other hand improvements can be made in better modelling the market itself and how cost savings can be used to gain more profit. Now we assumed that operators simply use the savings for a price cut. A possible direction worth exploring would be to explicitly formulate a strategic price competition game using logit demand functions as been done by Aksoy-Pierson et al. (2013).

39 A Bibliography

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41 B Mathematical Proofs

B.1 Derivation of the Cournot-Nash equilibrium This derivation is similar to the one in Pham Do and Norde (2007). Consider a Cournot oligopoly with n players defined as in section 3.2. Given the output levels x1, x2, . . . , xn of all firms, the profit of firm i is given by ! X Πi(x1, x2, . . . , xn) = P (Q)xi − Ci(xi) = b − a xj xi − cixi. j∈N

firm i’s optimal response to the output levels set by all other firms is defined by the first order condition: ∂πi(x1, x2, . . . , xn) X = b − a xj − axi − ci = 0. ∂xi j∈N The optimal response maximises firm i’s profit given the choices of all other firms.8 A Cournot-Nash equilibrium is a vector (x1, . . . , xn) such that each firm’s action xi is a best response. This equilibrium is the solution of the (linear) system of equations defined above. To find the solution we first sum the FOC over all firms to obtain P X X X nb − ci nb − a(n + 1) x − c = 0 ⇐⇒ x = i∈N . i i i a(n + 1) i∈N i∈N i∈N When we plug this expression back into the FOC we find that in the Cournot-Nash equilibrium the quantity produced by firm i is P b − (n + 1)ci − cj x = j∈N . i a(n + 1)

To ﬁnd the equilibrium proﬁt we should multiply this with the price in equilibrium: ! X πi = xiP (Q) = xi b − a xj j∈N P 2 b − (n + 1)ci + j∈N cj = . a(n + 1)2

To make sure that all ﬁrms choose to produce a positive quantity in equilibrium we need to ensure that the equilibrium price exceeds the largest marginal cost. Therefore we assume P b+ j∈N cj that n+1 > maxi∈N ci.

2 8 ∂ πi Since 2 = −2a < 0, the solution of the FOC is always a maximum. ∂xi

42 B.2 Proof of Proposition 1 Proof. Let us ﬁrst repeat that from (1) we have that:

0 0 α i α ⇐⇒ νi(α ) ≥ νi(α) Let (N, v) be the coalitional game the is being played at the lower level in a Cournot coopetition game. Now consider an allocation configuration α = (P, k) and suppose that player i ∈ N is thinking of moving from his own coalition structure element P (i) to another coalition structure element S ∈ P ∪{∅}. Let P 0 denote the coalition structure created from P by replacing P (i) with P (i)\{i}, and S with S∪{i}, let k0 be any allocation vector that is component efficient with respect to P 0, and let α0 = (P 0, k0) be the allocation configuration which arises as a result of i’s move. This move will only be profitable for player i if

0 α i α, ⇐⇒

νi(α) > νi(α), ⇐⇒ 0 X X (n + 1)ki − v(S) > (n + 1)ki − v(S), ⇐⇒ S∈P 0 S∈P 0 X X (n + 1)(ki − ki) > v(S) − v(S), ⇐⇒ S∈P 0 S∈P 0 (n + 1)(ki − ki) > v(S ∪ {i}) − v(S) − v(P (i)) − v(P (i) \{i}) . All firms j ∈ S need to accept the arrival of firm i. They do so if and only if for every j ∈ S 0 α j α, so, 0 (n + 1)(kj − kj) ≥ v(S ∪ {i}) − v(S) − v(P (i)) − v(P (i) \{i}) . If we now sum up all these inequalities for all j ∈ S ∪ {i} we find that

X 0 (n + 1) (kj − kj) > (|S| + 1) v(S ∪ {i}) − v(S) − (|S| + 1) v(P (i)) − v(P (i) \{i}) . j∈S∪{i} Since k and k0 are component efficient this reduces to (n+1) v(S∪{i})−v(S)−ki > (|S|+1) v(S∪{i})−v(S) −(|S|+1) v(P (i))−v(P (i)\{i}) . For i’s move to S to be profitable and to be accepted by S it therefore needs to hold that n − |S| |S| + 1 k < v(S ∪ {i}) − v(S) + v(P (i)) − v(P (i) \{i}). i n + 1 n + 1 On the contrary, there is no individual deviation possible for player i if for every S ∈ P we have that n − |S| |S| + 1 k ≥ v(S ∪ {i}) − v(S) + v(P (i)) − v(P (i) \{i}). i n + 1 n + 1 For a allocation configuration (P, k) to be individually stable this needs to hold for every firm i ∈ N. Since k is component efficient, we can derive the following result. For a

43 coalition structure P ∈ P there exists an allocation vector k such that (P, k) is individually stable if for every S ∈ P we have that

X nn − |T | |T | + 1 o v(S) ≥ max v(T ∪ {i}) − v(T ) + v(S) − v(S \{i}) : T ∈ P,T 6= S . n + 1 n + 1 i∈S

44 B.3 Proof of Proposition 2 Proof. Let (N, v, Γ) be a Cournot coopetition game with v superadditive and v(N) > P S∈P v(S) for all P ∈ P. Let α = (P, k) be an allocation conﬁguration. Suppose that a coalition T/∈ P decides to deviate form the current coalition structure P . The allocation 0 0 0 0 conﬁguration that will result from this deviation is α = (P , k ) with P ∈ PT . For T to block P we need to have that for every i ∈ T

0 0 νi(P , k ) ≥ νi(P, k),

0 for all P ∈ PT and 0 0 νi(P , k ) > νi(P, k), 0 for some P ∈ PT . We can derive the following result:

0 0 νi(P , k ) > νi(P, k) ⇐⇒ 0 X X (n + 1)ki − v(S) > (n + 1)ki − v(S) ⇐⇒ S∈P 0 S∈P 0 X X (n + 1)(ki − ki) > v(S) − v(S) ⇐⇒ S∈P 0 S∈P " # 1 X X k0 > k + v(S) − v(S) . i i n + 1 S∈P 0 S∈P We can now sum these inequalities over all i ∈ T . So for T to block P we need to have that " # X |T | X X v(T ) ≥ k + v(S) − v(S) , i n + 1 i∈T S∈P 0 S∈P 0 for all P ∈ PT and " # X |T | X X v(T ) > k + v(S) − v(S) , i n + 1 i∈T S∈P 0 S∈P

0 for some P ∈ PT . This implies that in particular the grand coalition N is a positive deviation from an allocation configuration structure (P, k) if " # X n X v(N) > k + v(N) − v(S) ⇐⇒ i n + 1 i∈N S∈P 1 1 X v(N) > v(S) ⇐⇒ n + 1 n + 1 S∈P X v(N) > v(S). S∈P P Since we have assumed that v(N) > S∈P v(S) the grand coalition blocks any allocation configuration but the ones containing the grand coalition. So either the core is empty, or it consists of allocation configurations of the form α = ({N}, k). In order for such a

45 conﬁguration to be core stable we need to have that no coalition T blocks ({N}, k). This is the case if " # X |T | X v(T ) ≤ k + v(S) − v(N) . i n + 1 i∈T S∈P 0

0 for every P ∈ PT . Since v is super additive the payoff of all the members of T is minimised when the other firms form the coalition N \ T .9 So the Cournot game has a core stable allocation configuration α = ({N}, k) if for all T ∈ 2N

X n + 1 − |T | |T | k ≥ v(T ) + v(N) − v(N \ T ). i n + 1 n + 1 i∈T

9 0 0 P When v is superadditive it can be easily shown that arg minP ∈PT (n + 1)ki − S∈P 0 v(S) = N \ T .

46 C Matlab Code

Here we present the Matlab scripts that have been used to ﬁnd the numerical results in this thesis. Next to these scripts we have also used online available Matlab Toolbox MatTuGames which implements diﬀerent routines to analyse cooperative games.

C.1 ChiValue.m

1 f u n c t i o n chi v l=ChiValue(v,cs)

2 % AD VALUE computes the Aumann−Dreze valuew.r.t.

3 %a coalition structure.

4 %

5 % Usage: ad vl=ADvalue(v,cs)

6 %

7 % Define variables:

8 % output:

9 % ad vl −− The Aumann−Dreze valuew.r.t toa coalition s t r u c t u r e.

10 %

11 % input:

12 %v −− A Tu−Gamev of length 2ˆn −1.

13 % cs −− A coalition structure like [34 8]

14 % for { [1,2],[3],[4] } . Must bea partition.

15 %

18 % Author: HolgerI. Meinhardt(hme)

19 %E −Mail: Holger. [email protected] −k a r l s r u h e.de

20 % Institution: University of Karlsruhe(KIT)

21 %

22 % Record of revisions:

23 % Date Version Programmer

24 % ======

25 % 07/17/2013 0.4 hme

26 %

27 i f nargin < 2

28 e r r o r(’A game and coalition structure must be given!’);

29 e l s e i f nargin==2

30 N=length(v);

31 [ ˜ ,n]=log2(N);

32 i f (2ˆn −1)˜=N

33 e r r o r(’Game has not the correct size!’);

34 end

35 end

37 J=1:n;

47 38 i n t=0: −1:1−n;

39 csm=rem(floor(cs(:) ∗pow2(int)) ,2)==1;

40 l c s=length(cs);

41 c h i v l=zeros(1,n);

42 sh=ShapleyValue(v);

43 f o rk=1:lcs

44 Tk=csm(k,:);

45 idx=J(Tk);

46 c h i v l(idx)=sh(idx)+(v(cs(k)) −sum(sh(idx)))/sum(csm(k,:));

47 end C.2 StabilityChecker.m

1 f u n c t i o n partStable= StabilityChecker(P,V,print)

2 % STABILITYCHECKERs

3 %P isa list containing partitions

4 %V isa matrix containing the valuation of each partition

6 n=size(V,1);

7 J=1:n;

8 i n t=0: −1:1−n;

10 i f print

11 c l c;

12 end

14 f o rp=1:length(P)

15 partStable(p) = 1;

16 cs=clToMatlab(P {p});

17 csm=rem(floor(cs(:) ∗pow2(int)) ,2)==1;

18 i f print

19 s t r 1=’’;

20 f o ri=1:length(P {p})

21 s t r 2=’’;

22 f o rj=1:length(P {p}{ i })−1

23 s t r 2=[str2, num2str(P {p}{ i }(j)),’,’];

24 end

25 s t r 2=[str2, num2str(P {p}{ i }(end))];

26 s t r 1=[str1,’ { ’,str2,’ } ’];

27 end

28 end

30 i f print

31 f p r i n t f(’ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n’)

48 32 f p r i n t f([’Checking partition:’, str1,’: \ n’])

33 end

35 lp=size(csm,1);

36 ind=findPartition(cs,P,1);

38 f o ri=1:n

39 j=find(csm(:,i )==1) ;

40 csmFrom=(rem(floor(cs(j)’ ∗pow2(int)) ,2)==1). ∗ [ 1 :n];

41 csmFrom(find(csmFrom==0)) = [ ] ;

42 count=0;

43 f o rl=1:lp

44 csNew=cs;

45 i fl ˜=j

46 csNew(l)=csNew(l )+2ˆ(i −1) ;

47 csNew(j)=csNew(j) −2ˆ(i −1) ;

48 csmTo=(rem(floor(cs(l)’ ∗pow2(int)) ,2)==1). ∗ [ 1 :n];

49 csmTo(find(csmTo==0)) = [ ] ;

50 e l s e

51 csNew(j)=csNew(j) −2ˆ(i −1) ;

52 csNew=[csNew 2ˆ(i −1) ] ;

53 csmTo=i;

54 end

55 i f csNew(j )==0

56 csNew(j)=[];

57 end

59 csNew=sort(csNew);% New partition that is the result of playeri in coalitionj joining coalitionl

60 indNew= findPartition(csNew,P,1);% Index of the new p a r t i t i o n

62 i f˜isequal(csmFrom,csmTo)

63 f l a g=0;

64 i f print

65 strFrom=’’;

66 f o rk=1:length(csmFrom) −1

67 strFrom=[strFrom, num2str(csmFrom(k)), ’,’];

68 end

69 strFrom=[strFrom, num2str(csmFrom(end))];

71 strTo=’’;

72 f o rk=1:length(csmTo) −1

73 strTo=[strTo, num2str(csmTo(k)),’,’];

49 74 end

75 strTo=[strTo, num2str(csmTo(end))];

76 end

79 i fV(i,indNew) >V(i,ind)

80 f l a g=1;

81 i f print

82 f p r i n t f([’ \n Player’, num2str(i),’ wants to deviate from { ’,strFrom,’ } to { ’, strTo,’ } : \n’]);

83 end

85 f o rk=1:length(csmTo)

86 i fV(csmTo(k),indNew) >=V(csmTo(k),ind)

87 i f print

88 f p r i n t f([’ Player’, num2str( csmTo(k)),’ accepts’, num2str(i),’s arrival!!! \n’ ]);

89 end

90 e l s e

91 f l a g=0;

92 i f print

93 f p r i n t f([’ Player’, num2str( csmTo(k)),’ does not accept’ , num2str(i),’s arrival... \n ’]);

94 end

95 end

96 end

97 e l s e i f print

98 f p r i n t f([’ \n Player’, num2str(i),’ does not want to deviate from { ’,strFrom,’ } to { ’, num2str(csmTo),’ } ... \n’])

99 end

100

101 i f flag

102 count=count+1;

103 partStable(p) = 0;

104 end

105 end

106 end

107 i f˜partStable(p) && print

50 108 f p r i n t f(’ \n There are%d possible deviations for player%d. \n’, count,i);

109 end

110 end

111 i f partStable(p) && print

112 f p r i n t f(’ \n Partition%sIS individual stable \n’, str1);

113 e l s e i f print

114 f p r i n t f(’ \n Partition%sIS NOT individual stable \n’, s t r 1);

115 end

116 end

117

118 m=sum(partStable==1) ;

119 i f print

120 f p r i n t f(’ \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n’);

121 f p r i n t f(’The%d individual stable partition(s) are: \ n’,m) ;

122 options.elesep=’,’;

123 DispPartObj(P(partStable==1)) ;

124 end

125

126 end C.3 CoreStabilityChecker.m

1 f u n c t i o n Stab= CoreStabilityChecker(P,V);

3 n=size(V,1)

4 p r i n t =1;

6 lp=length(P)

8 J=1:n;

9 i n t=0: −1:1−n;

11 pws=PowerSet(J);

12 lpws=length(pws);

13 f o ri=1:lpws

14 lpws;

15 c l=clToMatlab(pws { i });

16 clm=(rem(floor(cl’ ∗ pow2(int)) ,2)==1). ∗ [ 1 :n];

17 clm(find(clm==0)) = [ ] ;

18 ind= findCoalition(cl,P,1);

19 tempPi=V(:,ind);

20 maxPi(:,i)=max(tempPi,[],2);

51 21 minPi(:,i)=min(tempPi,[],2);

23 end

25 f o rp=1:lp

26 Stab(p)=1;

27 c l=clToMatlab(P {p});

28 i f print

29 s t r 1=’’;

30 f o ri=1:length(P {p})

31 s t r 2=’’;

32 f o rj=1:length(P {p}{ i })−1

33 s t r 2=[str2, num2str(P {p}{ i }(j)),’,’];

34 end

35 s t r 2=[str2, num2str(P {p}{ i }(end))];

36 s t r 1=[str1,’ { ’,str2,’ } ’];

37 end

38 end

40 i f print

41 f p r i n t f(’ \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n’)

42 f p r i n t f([’Checking partition:’, str1,’: \ n’])

43 end

44 f o rj=1:lpws

45 c l=clToMatlab(pws { j });

46 clm=(rem(floor(cl’ ∗ pow2(int)) ,2)==1). ∗ [ 1 :n];

47 clm(find(clm==0)) = [ ] ;

48 i f sum(V(clm,p) <=minPi(clm,j ) )==length(pws { j }) && sum(V( clm,p)

49 f p r i n t f(’ \ nCoalition%s blocks this partition’, num2str(pws { j }));

50 Stab(p)=0;

51 end

52 end

53 end

55 m=sum(Stab==1) ;

56 i f print

57 f p r i n t f(’ \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n’);

58 f p r i n t f(’The%d core stable partition(s) are: \ n’,m) ;

52 59 DispPartObj(P(Stab==1)) ;

60 end

61 end C.4 CournotProﬁt.m

1 f u n c t i o n[Pi,q,P,W, CS] = CournotProfit(b,a,c)

2 % COURNOTPROFIT calculates the profit, output quantities, price, w e l f a r e

3 % and consumer surplus in the(unique) Nash equilibrium ofa Cournot

4 % oligopoly with linear inverse demand and constant marginal c o s t s.

6 % Define variables:

7 % output:

8 % Pi −− P r o f i t

9 %q −− Output quantities

10 %P −− Equilibrium price

11 %W −− Welfare

12 % CS −− Consumer surplus

13 %

14 % input:

15 %b −− Intersection of the inverse demand curve

16 %a −− Slope of the inverse demand curve

17 %c −− Marginal costs of the playerss

18 %

19 % Author: Job van Rooij

20 %E −mail: [email protected]

22 n= length(c);

24 % Calculate output quantities

25 q=(b −(n+1) ∗c+sum(c))/(a ∗(n+1)) ;

26 i fq <0

27 e r r o r(’Negative quantities!’);

28 end

30 % Calculate price

31 P=b −a∗sum(q);

32 i fP <0

33 e r r o r(’Negative price!’);

34 end

36 % Calculate Profit

37 Pi=q. ∗ (P−c);

53 38

39 % Calculate CS

40 CS = (1/2) ∗sum(q)ˆ2;

42 % Calculate Welfare

43 W= CS+ sum(Pi);

45 end C.5 CournotExamples.m

1 c l e a r all

3 k= [4, 1, 1, 1];

4 %k= [0, 0, 0, 0];

5 n= length(k);

6 a= 1;

7 b= 30;

9 switchn

11 case3

12 clm= { [1],[2],[3],[1 2],[1 3],[2 3],[1 2 3] } ;

13 vlm=[0 00111 2];

15 case4

16 clm= { [1],[2],[3],[4],[1 2],[1 3],[1 4],[2 3],[2 4],[3 4],[1 2 3],[1 2 4],[1 3 4],[2 3 4],[1 2 3 4] } ;

17 vlm=[0 00011144455510 12];

18 %vlm= [4111666555 11 11 11 12 19];

19 %vlm= [11114444449999 16];

21 case5

22 clm= { [1],[2],[3],[4],[5],[1 2],[1 3],[1 4],[1 5],[2 3],[2 4],[2 5],[3 4],[3 5],[4 5],[1 2 3],[1 2 4],[1 2 5],...

23 [1 3 4],[1 3 5],[1 4 5],[2 3 4],[2 3 5],[2 4 5],[3 4 5],[1 2 3 4],[1 2 3 5],[1 2 4 5],[1 3 4 5],[2 3 4 5],[1 2 3 4 5] } ;

24 vlm=[0 000055555555551010101010101010 10 10 15 15 15 15 15 20];

26 case6

27 clm= PowerSet(1:n);

28 f o ri=1:length(clm)

29 vlm(i)= length(clm { i }) ˆ2;

30 end

54 32 case7

33 clm= PowerSet(1:n);

34 f o ri=1:length(clm)

35 vlm(i)= length(clm { i }) ˆ2;

36 end

38 end

40 [v sS] = vclToMatlab(clm,vlm);

42 P= SetPartition(n);

43 f o ri=1:length(P)

44 pm= clToMatlab(P { i });

45 AD(i,:) = ADvalue(v,pm);

46 Chi(i,:) = ChiValue(v,pm);

47 K(i,:) =k+AD(i,:);

48 C(i,:) =10 −K(i,:);

49 [Pi(i,:),q(i,:), Pr(i,:),W(i,:),CS(i,:)] = CournotProfit(b, a,C(i,:));

50 u=(n+1) ∗AD−repmat(sum(AD,2) ,1,n);

51 K1(i,:) =k+Chi(i,:);

52 C1(i,:) =10 −K1(i,:);

53 [Pi1(i,:),q1(i,:), Pr1(i,:), W1(i,:),CS1(i,:)] = CournotProfit(b,a, C1(i,:));

54 u1=(n+1) ∗Chi−repmat(sum(Chi,2) ,1,n);

55 end

57 %Stab= StabilityChecker(P,u’,1)

58 %Stab1= StabilityChecker(P,u1’,1)

61 s=q./repmat(sum(q,2) ,1,n);

62 s1= q1./repmat(sum(q1,2) ,1,n);

64 H= sum(s.ˆ2,2);

65 H1= sum(s1.ˆ2,2); C.6 MarketShareSolver.m

1 f u n c t i o ns= MarketShareSolver(alpha, beta, p0, deltaI, MarketSize, uOutsideGood)

2 % Initialize

3 n= length(p0);

4 s= sym(’s’,[n, 1]);

6 % Construct system of equations

55 7 u= alpha+beta ∗(p0−d e l t a I./(s ∗ MarketSize));

8 i f nargin <6

9 eqn=[s==exp(u)/(sum(exp(u)))];

10 e l s e

11 eqn=[s==exp(u)/(sum(exp(u))+exp(uOutsideGood))];

12 end

14 % Solve system of equations

15 s o l= vpasolve(eqn);

17 % Convert solution to numerics

18 s o l= struct2cell(sol);

19 s= zeros(n,1);

20 f o ri=1:n

21 s(i)=eval(sol { i });

22 end

24 end C.7 BULRIC.m

1 f u n c t i o n NumberOfSites= BULRIC(s,CovCap)

2 i f isrow(s)

3 s=s’;

4 e l s e i f˜isrow(s’)

5 e r r o r(’s is nota vector!’);

6 end

8 n=length(s);

9 NumberOfSites= max(repmat(CovCap(:,1) ,1,n),sum(CovCap,2) ∗s’);

11 end C.8 NewSMARTModel.m

1 c l e a r all

3 % Country Parameters

4 MarketSize=20e6;

5 OutsideGood = 0.1;

6 s0 = [0.5; 0.3; 0.2];% Market Shares

7 n= length(s0);

8 s0 = [(1 − OutsideGood). ∗ s0]

9 u0= log([s0; OutsideGood]./min([s0; OutsideGood]));

10 uOutsideGood= u0(n+1);

11 u0(n+1) = [];

12 p0 = [31; 26; 25];% ARPU

56 13 DprPrd= 1;% Depriciation Period

14 CovCap = [212 318; 808 105; 447 0];

15 CostPerSite = 12000;

16 NetworkIntegrationCostsParameter=1e6 ∗[0 100 125 140 150 155];

17 NetworkIntegrationCostsParameter=0 ∗[0 100 125 140 150 155];

18 RANOPEXFixedParameter= [100 75 60 50 45 30];

19 RANOPEXVariableParameter = 50;

20 n= length(s0);

22 J=1:n;

23 i n t=0: −1:1−n;

24 P=SetPartition(n);

25 lp=length(P);

27 f o ri=1:lp

28 cs=clToMatlab(P { i });

29 csm=rem(floor(cs(:) ∗pow2(int)) ,2)==1;

30 CoalitionSize(:,i)=sum(csm. ∗ repmat(sum(csm,2) ,1,n),1) ’;

31 end

33 % Calculate the RAN−OPEX costs in each coalition structure

34 RANOPEX= RANOPEXFixedParameter(CoalitionSize)+ RANOPEXVariableParameter∗repmat(s0,1,lp);

35 RANOPEX=1e5 ∗RANOPEX;

37 % Calculate the RAN−CAPEX savings for each coalition

38 pws=PowerSet(J);

39 lpws=length(pws);

40 f o ri=1:lpws

41 c l=clToMatlab(pws { i });

42 clm=rem(floor(cl ∗pow2(int)) ,2)==1;

43 NumberOfSitesNoCoop= sum(clm. ∗ sum(BULRIC(s0,CovCap),1));

44 NumberOfSitesCoop= sum(BULRIC(clm ∗s0,CovCap),1);

45 NumberOfSitesSaving(i)= NumberOfSitesNoCoop − NumberOfSitesCoop;

46 Savings(i)= CostPerSite ∗ NumberOfSitesSaving(i) − NetworkIntegrationCostsParameter(sum(clm));

47 end

49 [Savings sS] = vclToMatlab(pws,Savings);

51 % Divide the savings using the AD−value and the Chi−value

52 f o ri=1:lp

53 cs=clToMatlab(P { i });

54 AD(:,i)= ADvalue(Savings,cs);

57 55 Chi(:,i)= ChiValue(Savings,cs);

56 end

58 RANCAPEXAD= CostPerSite ∗repmat(sum(BULRIC(s0,CovCap),1) ’,1,lp) − AD

59 RANCAPEXChi= CostPerSite ∗repmat(sum(BULRIC(s0,CovCap),1) ’,1,lp) − Chi

61 b=−1

62 f o rk=1:length(b)

63 beta=b(k);

64 alpha= u0 −beta ∗p0;

65 d e l t a I= AD/(DprPrd);

66 priceCut= deltaI;

68 f o ri=1:lp

69 s(:,i)= MarketShareSolver(alpha,beta,p0,priceCut(:,i), MarketSize, uOutsideGood);

70 end

72 p=repmat(p0,1,lp) −priceCut./(s ∗ MarketSize);

73 pi=s. ∗ p∗ MarketSize − RANCAPEXAD/(DprPrd);

74 pi−repmat(pi(:,end),1,lp)

75 % IndvStab(k,:)= StabilityChecker(P,pi,0)

76 % CoreStab(k,:)= CoreStabilityChecker(P,pi)

77 end