A Model of Oligopoly based on a Network Approach

1 Introduction 4
2 Two-stage oligopoly 8
2.1 The model 8
2.1.1 Strategies 8
2.1.2 Payoff function 9
2.2 Equilibrium at fixed network 10
2.3 Equilibrium in the two-stage game 14
2.4 Preferred equilibria 19
2.5 Sensitivity analysis 21
2.5.1 Regular networks 28
2.5.2 Example 29
2.6 Weighted network 31
3 Cooperative game 33
3.1 Maximin 36
3.1.1 Characteristic function 37
3.1.2 Cooperative solution 40
3.2 Sensitivity analysis 40
3.2.1 Regular networks 41
4 Two-stage oligopoly with offering costs 42
4.1 Equilibrium at fixed network 42
4.2 Equilibrium in the two-stage game 43
4.3 Sensitivity analysis 46
5 Alternative characteristic functions of cooperative game 48
5.1 Maximization of S's payoff with Nash equilibrium strategies
for other individuals 48
5.2 Equilibrium in the game with |N| — |S| + 1 players 50
6 Conclusion 53

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This thesis studies the competition of hrms in one product market with net¬work effect under which costs are dependent upon collaborations between hrms. The idea of research is taken from [14], and concerns the question: what are the incentives of hrms in market competition? The mentioned book covers many cases of market competitions and provides solution techniques. It discovers the topics of monopoly market and optimal behavior in quanti-ties and prices, competition of many hrms in one and many products market, price discrimination, dynamic competitions and so on. We focus on Cournot competition in quantities, the approaches of the research and development adoption of new technologies and cooperative game theory. In the book is proposed different solution concepts and models which concern these issues. The paper [2] provides an example of applying a model with research and development collaborations for non-cooperative and cooperative two-person games. Authors consider a two-stage game where on different stages actions of players represent the value of technological partnership and hnd Nash equi¬libria of the game. Another way of looking at the hrms competition is to look precisely at their collaborations. A collaboration link can be interpreted as a partnership which is costly but lower costs of production of the hrms in¬volved. There can be many incentives for collaboration. Indeed technological partnerships, reduction of transportation and holding costs and others. The collaborations between hrms can be represented by a network with hrms set¬tled in the nodes. In [9] M. Jackson describes social and economic networks, constructs models of behavior and analyses them using game theory and op¬timization methods. He provides allocation rules for cooperative games on networks as well. Mostly he focuses on the topology structure of equilib¬rium and stable networks. He discovers network formation stage, provides the conditions for existence of stable networks. He introduces a one-stage model of the game we discover, the game on a hxed network, but he does not inspect the two-stage game and consequently the issues of the two-stage equilibria and cooperative game. We use dehnitions, concepts and notations from this book in our work. In [8] authors discover a coordination game with the endogenous network structure with and costs of maintaining the collaborations. They examine stochastic stability issue on hxed networks, characterize stochastically stable states and inspect how the endogenous net-works affect stochastic stability. Similar to ours, a non-cooperative model of network formation with link formation costs is investigated in [3]. There is considered one-way and two-way flow of benehts. The strict Nash equilibria are found in both models: for one-way flow model there are empty network and wheel network and for two-way flow model - empty and star networks. Also there is considered dynamic process and is proved that it converges to strict Nash equilibrium. Another close research is done in [6]. There Cournot oligopoly is considered with addition of opportunity for each hrm to form pair-wise collaborative links with other hrms which will lower costs of production of participants. The result is in the characterization of stable networks and comparison them with efficient networks. There is found that the complete network is stable. Authors also show that from a social point of view the complete network is efficient. The comprehensive overview of cooperative games and coalitional formations for applications in economics is provided in [4]. There are discovered general issues of incentives to cooperate, form a coalition, provided analysis of influence groups of coalitions to other coalitions, examined the bargaining issue of total payoff of coalition between players. And there is considered competition of coalitions. In [13] there is an analysis of cooperative game based on network model with costs for estab¬lishing links using an extension of the Myerson value to determine the payoffs in a 3-player symmetric game and the issue of existence coalition proof Nash equilibria in the 3-player symmetric game. In [7] authors develop a model of oligopoly market with the network effect on payoff functions and examine the incentives of firms to form collaborations with other firms. They find the nature of collaboration structures that are stable under different market conditions, and characterized stable structures. Unlike stability issue in [7] here we inspect Nash equilibria. We decide to discover the firms competition from the two points of view at the same time: quantities competition and network formation. As the basis of such analysis we use [11]. The paper pro¬vides analysis of links’ influence on strategy choice of a player for a general payoff function. The issue of dynamic stability of cooperation solutions is examined.
The dissertation is based on these works. As in [11] we consider a two- stage game of n firms where at the first stage players form the network of collaborations and at the second stage the firms chose quantities of production as in [7]. After these two stages payoffs are computed and the game ends. This game illustrates the competition of firms in one-product market. Our first aim is to find equilibria, characterize them by profitability and network topology structure. We establish preferred equilibria and provide sensitivity analysis of the player’s behavior and the market performance. The second goal is to find the cooperative solution of the game and compare it with non-cooperative solution. We examine a two-stage oligopoly model from [7] with offering costs as well. It differs from the previous model in the payoff function in such a way that an incentive to form a collaboration link induces additional costs. In this model we find sufficient condition for equilibria. We should notice that in the papers above the issue of equilibria in two- stages games is not discussed as like as the cooperative solution of hrms competition in two-stages and our work tries to hgure out hrms equilibrium behavior and common laws which helps to better understand how hrms should act in one-product market competition: should they play as singletones or cooperate, how the collaborations inhuence on different players and what concrete actions they should do to beneht.
The paper has the following structure. In section 2 we investigate the non-cooperative two-stage oligopoly. At hrst we dehne the model, strate¬gies and payoff functions. Then we hnd an equilibrium when the network is hxed. After this we construct a hypothesis of equilibrium network topol¬ogy structure and test it. Next we answer the question which equilibria are more prohtable for players and how it concerns other players. In sensitiv¬ity analysis we explore how the adding or removing the link affect player’s equilibrium strategies, payoffs and price function. The special case of regular network is explored in detail and with an example. At the end of this sec¬tion we adopt the cost function for the weighted networks and say how the equilibrium action for hxed network will change. Section 3 is a consideration of cooperative game approach. We investigate both models: with full coop¬eration on two stages and cooperation only on quantities competition stage. We give an overview of methods of construction characteristic function, and introduce solution concepts of the bargaining total payoff which we will use. The characteristic function then is chosen as the value maximin optimization problem. The Shapley value [12] and the center-of-gravity of the imputation set (CIS value) [5] are used as imputations. Finally the sensitivity analysis of cooperative game is provided. Section 4 introduces the model of two-stage oligopoly with offering costs which we examined. The methods of analysis the last model coincide with the previous two-stage oligopoly model.

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