Implicit cartels : cooperation in a duopoly with endogenous timing of decisions

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Master’s Thesis

Implicit Cartels: Cooperation in a Duopoly

with endogenous timing of decisions

N.C. van Ommeren

Student number: 10017704

Date of final version: December 12, 2015 Master’s programme: Econometrics

Specialisation: Mathematical Econometrics Supervisor: Dhr. dr. R. Ramer

Second reader: Prof. dr. J. Tuinstra

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Statement of Originality

This document is written by Student N.C. van Ommeren who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Introduction

Suppose there is a street with two bakeries, which both try to maximize profit by selling the most bread for the highest price possible. When one of the firms decreases its price, the demand for its bread goes up which could harm the profitability of the other bakery. As a reaction the other firm responds, lowering its price to make sure it is slightly cheaper. As a result both bakers see their profits decrease and unilaterally decide to increase their price at the beginning of the year. In this example both firms are able to choose when they want to set their prices. However, in most competition models firms move either simultaneously or sequentially, which often does not reflect reality. We introduce a model which allows for endogenous timing of decisions to better replicate the real world. Through this model, we investigate whether a collusive solution, where both firms earn more than in a Nash equilibrium, is an outcome of a finitely repeated version of the game.

Whether it is more profitable to move first or second depends on the direction of the reaction curves, that is the best response function for each player. For downward sloping reaction curves as in the Cournot quantity competition model it is more profitable to move first (Gal-Or, 1985). The decision to move sequentially or simultaneously should be result of the player’s decisions and therefore a model should capture this feature (Hamilton & Slutsky, 1990). In one of their proposed models a firm is able to choose to move in the first or second period. Then these decisions are revealed and the firms choose the value of their strategic variable. This is a step forward in modeling a duopoly with endogenous timing, but it could be further extended by releasing the constraint of only two decision periods and introducing a more general model.

The aim of this research is to investigate whether the collusive cartel solution is an outcome of a finitely repeated game with endogenous timing of decisions. In this solution both players do not make formal agreements about the value of their strategic variable and earn more than in the Nash equilibrium of a single period game. The strength of the model is its simplicity and general structure, with only a small adaption from the original model it captures the feature of endogenous timing. Therefore this model describes the behaviour of firms in the market more realistically as it does not restrict firms to moving simultaneously or sequentially. The single strategic variable of the model is not restricted to a specific type since the outcome of

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CHAPTER 1. INTRODUCTION 2 the game is described for each direction of the reaction curves. We take an example with price and quantity as strategic variables, but this could easily be extended to other strategic variables such as location or quality.

The model allows for endogenous timing by giving both players the ability to wait or to set the value of their strategic variables. The initial values are the trading values from the previous game and they have a large influence on the equilibrium outcome. If both firms wait, nothing changes and the initial values hold for another period. When one of the firms waits and the other chooses to set their strategic variables, the game is carried out with sequential actions. This in contrast to the simultaneously timed model where both firms choose to set their value immediately. For different initial values we investigate the outcome of the game using the concept of weak dominance. We also look what happens to the outcome when the game is repeated. The main objective of this paper is to answer whether a collusive cartel is a possible solution to the game when the timing of decisions is not exogenously determined.

Chapter 2 includes an overview of the literature on duopoly competition models with a particular focus on papers that include a collusive outcome. The chapter also explains the importance of these implicit cartel solutions for antitrust policy and provides background in-formation on mathematical methods used in this paper. Chapter 3 introduces the assumptions for the competition model and the notation used in this paper. It specifies the properties of quantity and price competition models in Subsection 3.2.1 and 3.2.2. The model is the basis of the one-stage game and its timing is described in Chapter 4 together with how different games are linked together in the repeated game. Chapter 5 lists the possible equilibria outcomes for the one- and multistage model and describes which steady state occurs for certain specifications of the model, number of stages k and the initial values from history h, all of which are clarified later. In addition, it describes the consequences for quantity and price models mentioned ear-lier and investigates the stability of the steady states under small shocks. Chapter 6 provides a conclusion of the results together with some suggestions for further research.

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Chapter 2

Literature Review

The aim of this research is to extend the basic duopoly competition model to allow for endoge-nous timing of decisions. We investigate the stability of the implicit cartel solution, which is of importance in the detection of explicit cartels. Section 2.1 provides an overview of the literature on oligopoly models. The main point of interest is whether there is a collusive solution to the game. The influence of the collusive outcome on the detection of illegal cartels is described in Section 2.2. The last section explains and motivates the game theoretic methods used later in this paper.

2.1 Oligopoly Competition Models

This section provides an overview of the literature on oligopoly models. We will keep the order chronological within each subsection to provide a context of the evolution of these models. It starts with models with simultaneous timing of decisions, continues with sequential models and ends with models that allow for endogenous timing of production or decisions. We will refer to these models as hybrid models.

2.1.1 Simultaneous Models

One of the earliest duopoly models is the quantity competition model of Cournot. In his original example there are two firms who both sell the homogeneous good spring water. Firms are not able to set their price, but the price depends on the total quantity of spring water available in the market. Each firm attempts to maximize its profit by independently and simultaneously choosing the amount of water to sell (Cournot, 1838). In his theory of oligopoly, Cournot describes an equilibrium which occurs when each firm chooses the output that maximizes its profits given the output of the other firms. In the analysis of the stability of the equilibrium Cournot introduces the concept of best response dynamics. The equilibrium is the simultaneous solution of the best response functions of each firm, which occurs when the two functions intersect. We will refer to this as the Nash equilibrium. The intention of Cournot was to describe competition among firms. He shows that when the number of firms increases, the

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CHAPTER 2. LITERATURE REVIEW 4 aggregate output is larger which leads to a lower price. When the number of firms is unlimited, the Cournot model describes the situation of perfect competition where price equals marginal costs.

One year after Cournot published his theory, an American engineer Ellet wrote a recom-mendation to the US government about the toll rates of canals and railways. He argues that business decision making should be derived on mathematical principles. The toll rate should not be a uniform tariff, but should depend on factors such as the type and amount of goods transported and the length and part of the route. It was written in a time when transportation of large heavy articles was commonplace.In order to maximize profits, the transportation of commodities with a small value, for example stone, should not incur too high a cost. Charging too much makes it unprofitable to ship, which leads to a decline in demand and possibly a fall in tolls and profits. He advises regulating the market in such a way that results in the imposition of a tariff that leads to maximum profits. In addition he proposes a restriction that prohibits proprietors of the road to ask more than the tariff that maximizes their profit Calsoyas (1950). Bertrand criticizes the model of Cournot. In a review to his book he proposes that firms should compete on price instead of quantity (Bertrand, 1883). In his price competition model firms have constant marginal costs and act individually. The goods are homogeneous and all consumers buy from the firm that offers products for the lowest price. In the Bertrand model firms should satisfy all demand for the price they ask, which is a reasonable assumption when the cost of losing customers is high or when it is enforced by consumer protection laws (Vives, 1999).

If we assume firms have equal marginal costs and get an equal share of the total demand when they charge the same price, the outcome of the game is a Nash equilibrium in which price equals marginal costs. In this equilibrium none of the firms makes profits, therefore it is also called the Bertrand paradox. When one firm asks a higher price than the other, he loses all demand. Therefore he has an incentive to slightly undercut the price of the competitor, thereby attracting all demand. Firms have an incentive to undercut each other until price equals marginal costs, because after this point firms make negative profits with each product they sell. Under these conditions the equilibrium is unique, but it is a weak equilibrium as firms earn no profits which is the same as if they ask a higher price (Tirole, 1988). When all goods are homogeneous the price in the Nash equilibrium of the price competition model is lower than the price in the quantity competition model.

The Bertrand paradox does not hold when we relax the assumption that all demand has to be satisfied (Edgeworth, 1925), when products are differentiated or when there are fixed costs, which results in negative profit. In reality products are not always homogeneous as sellers differentiate their products by introducing different brands or qualities.

In economic theory the collusive solution describes the situation in which the market price is close to the monopoly price. This occurs in the quantity competition model for example when two players earn higher profits than in the Nash equilibrium by producing a lower amount, which

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CHAPTER 2. LITERATURE REVIEW 5 yields a higher price and higher profits. Previous game-theoretic research states that such a collusive outcome without formal agreements is never an outcome of the one-stage game as both players have a unilateral motive to deviate. We use a general prisoner’s dilemma to illustrate this theory. Firms can set their production levels relatively high or low with α > β > γ > δ:

Table 2.1: Basic Cournot prisoner’s dilemma

Player i

Player j

low high low β, β δ, α high α, δ γ, γ

The solution (γ, γ) where both players produce a low amount of goods is not an outcome of this game, because each player has an incentive to deviate as α > β. This results in both players producing a high amount of goods with strictly lower profits than when they would produce a low amount of goods as β > γ. In this one-shot model the only way to earn the higher payoff β is when the players make formal agreements by forming a explicit cartel.

When this game is repeated an infinite amount of times the Folk theorem applies, which states that if players are sufficiently patient the cooperative solution is a possible equilibrium outcome (Friedman, 1971). For the game in Table 2.1 this corresponds to the case where both players produce the low amount of goods and receive payoff β. The profit from deviating once and receiving payoff α once and payoff γ in the future is lower than the payoff from cooperating in which case they would receive β forever. The collusive solution is a possible outcome of this game, because firms can threaten each other with the production of a high amount of goods when the other player deviates from the collusive equilibrium. However, for this theorem to hold the game should be played an infinite amount of times; an assumption lacking grounding in reality. In addition, in punishing another player, an agent must stick to the strategy ad infinitum. If players were able to communicate and reconsider their options, they are likely to fall back to mutual cooperation, as in these games the Nash equilibrium not only harms the opponent, but also themselves (Farrell & Maskin, 1989).

There are other game theoretical methods in which a credible threat can provide stability of the collusive outcome in a game with a finite number of periods. However, these equilibria do not satisfy the properties of subgame perfectness and are therefore eliminated by backwards induction (Osborne & Rubinstein, 1994). The fact that the collusive outcome is never a solution to the game, independent of the number of firms on the market, is already criticized by Radner (1980). He proposes a model where each firm does not have to play its best-response, but needs to stay within epsilon utility of the value that maximizes its profit. This so-called epsilon-equilibrium leads to stability of the cartel solutions for a sufficient number of periods depending on the number of firms on the market. Kreps, Milgrom, Roberts, & Wilson (1982) also notices that in experiments the Nash equilibrium is not always played in the finitely repeated prisoner’s game and they show how imperfect information can lead to observed cooperation. We show

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CHAPTER 2. LITERATURE REVIEW 6 that even when players are allowed to play their best response and have perfect information, the endogenous timing ensures that for a sufficient number of repetitions the cartel solution is an outcome of the game.

2.1.2 Sequential Models

The models in the previous subsection assume that firms make decisions about the value of their strategic variable simultaneously. The extent to which this assumption is realistic is questionable since sequential play could be more profitable for a firm. The slope of the reaction curves decides whether it is more profitable to be the follower or the leader. Gal-Or (1985) found that for downward sloping reaction curves as in the quantity competition model, it is more profitable to be the leader. In the price competition model with upward sloping reaction functions the follower is better off as he is able to respond to the price posted by the other player.

One of the best known leader-follower models is the Stackelberg model in which two firms compete on quantity. It starts with one firm choosing the amount that optimizes its profit, then the other firm chooses its optimal quantity given the choice of the leader. Unfortunately, the question which of the two firms will eventually gain the leader position cannot be answered theoretically when the two firms are symmetric. Stackelberg (1934) believed that the equilibrium where one firm is the leader and the other the follower can never be stable as both firms will continue to fight for the leader position. It can only be stable when one firm has an advantage over the other firm or when the market is regulated.

The aggregate output in the Stackelberg equilibrium is higher and the price lower than in the Nash equilibrium. When firms compete on quantity the leader profit is always higher than the profit in the competitive Nash equilibrium; a result which does not necessarily hold when there are more than two firms on the market (Anderson & Engers, 1992).

Bargaining games, where all players bargain for a share of a trade, provide another example of sequential models. In this type of game there is a maximum amount M to divide between all players. When the sum of all requests is lower than the maximum amount, all players get their request. On the other hand when the total exceeds M , none of the players gets anything. Bargaining games are usually found in small markets without standard prices; likely when the goods are unique (Harrington, 2009). The analysis of these kind of models falls outside the scope of this paper.

2.1.3 Hybrid Models

This subsection expands the basic simultaneous and sequential models with an extra endogenous factor. In the first two examples this is related to the type of strategic variables, which is directly related to the timing of the production. In the last model we consider an example with endogenous timing of decisions.

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CHAPTER 2. LITERATURE REVIEW 7 derived from the market structure. Quantity competition is more likely in the case of perishable products. In this case firms want to sell all their products since afterwards the products lose their value. On the other hand price competition is more suitable when products are made per order, as prices are set first and then products are created.

Instead of making a choice for either quantity or price competition, Kreps and Scheinkman propose a model that combines this decision. In the first stage of their two-stage game a firm chooses the quantity it will produce, the so called capacity. In the second stage it competes on prices where all firms are limited by the capacities they have chosen in the first stage. If both firms choose the quantities from the Cournot-Nash equilibrium as their capacity levels, the prices that firms choose in the second stage are also equal to the prices of the Nash equilibrium in the Cournot quantity competition model. Hence, it results in the same equilibrium as the Cournot quantity model, but it gives a more satisfactory description of the game. However, it was never their intention to provide an accurate model for a duopoly. Their aim is to emphasize the importance of the selection of the strategic variable and timing of production for the outcome of the game (Kreps & Scheinkman, 1983).

The timing of the production has a large influence upon the relevance of each strategic variable and could therefore be modeled as an endogenous factor. Klemperer & Meyer (1986) show that in a symmetric model with simultaneous reactions, firms are indifferent between price and quantity as strategic variable when there is no uncertainty about demand. As an illustration they take a consultancy firm which could either charge a certain price per hour or a price per consultation. In the latter case, the firm can increase its profits by reducing the time spent on one consultation when the office is busy. This naturally increases the price per minute of a consult, which could be seen as a form of quantity competition. When we consider differentiated products firms prefer quantity competition when products are substitutes and price when products are complements (Singh & Vives, 1984). The choice of strategic variables is not limited to quantity or price but could also be, for example, quality or location. In this paper we take quantity and price as an example, but we can easily apply the model to each set of strategic variables as long as the direction of the reaction curves is known and the assumptions are satisfied.

The timing of production is not the only important element in determining the outcome of the game, the timing of decisions is also vital. Hamilton & Slutsky (1990) propose a model which consists of two periods, where each player sets the value of their strategic variable either in the first or in the second period. Choices are simultaneous when both players choose the same period for their decision and sequential when they choose different periods. In the first model players start with an announcement in which period they reveal their decision before making the decision itself. When the strategic variable is quantity, both prefer to be the leader and choose to move in the first period. When both firms move in the same period, we are back in the game of simultaneous play where both firms play the quantity belonging to the Nash equilibrium. In the case of price competition it is more profitable to be the follower and both sequential Stackelberg

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CHAPTER 2. LITERATURE REVIEW 8 outcomes are pure equilibria of this game. In their second model players are not aware in which period the opponent moves before choosing a value for their strategic variable. In this model the only two undominated pure equilibria are the two leader-follower equilibria. The Cournot-Nash equilibrium is never an outcome in undominated strategies. This paper releases the constraint that there are only two decision periods and introduces the possibility that both players keep waiting in which case nothing happens. In our model the collusive solution is always a possible outcome of the game. This has interesting implications for the detection of cartels which we will discuss in the following section.

2.2 Antitrust Policy

In this paper we investigate the possible formation of implicit cartels. In a implicit cartel firms collaborate without formal agreements and earn higher profits than one would expect in a competitive market. Collaboration with formal agreements is forbidden in the United States by the Sherman Antitrust Act since 1890 (Bishop & Walker, 2010) and in Europe by article 101 in the Treaty on the Functioning of the European Union. A cartel is successful if it can maintain a Pareto efficient outcome, where all firms are better off than in the Nash equilibrium. This solution decreases the prosperity of consumers, which is why competition laws forbid private cartels. The identification and elimination is an important part of the competition policy in many countries.

The main goal of the European antitrust policy is to prevent and discourage the forming of collusive agreements. Unannounced inspections at the headquarters of the firm play an important role in the detection of cartels. The European Commission can execute such an inspection when they have sufficient evidence for the existence of a illegal cartel. Evidence is often provided by a leniency application from an involved company, whistle-blowers within the firm or more indirect signs of collusive behavior reported by customers such as the maintenance of high prices. The leniency program provides total immunity or reduction in fines for the involved firm that self-reports and hands in the evidence. This has proven to be successful as it makes the cartel less stable and gives firms an incentive to defect. From 2002 to the end of 2005 approximately two-thirds of all statements of objections in the EU were based on cases involving this program. Additionally, inspection decisions are made on more indirect evidence such as customer responses for example in May and December 2006 in the energy sector. The Commission also uses indirect evidence in the decision to impose a fine on cartel members or to complete the standard of proof (Friederiszick & Maier-Rigaud, 2008).

A key question is whether patterns of high prices be used as part of the evidence. One of the problems arises from the fact that it is hard to derive economic criteria to detect whether collusion is implicit or explicit, since they rely on the same economic principles. Especially in the case when cartel members are not able to write enforceable contracts, both parties have an incentive to defect. This incentive could be even larger in an explicit cartel as members have a risk to be exposed, resulting in further costs, in the case that they do not defect themselves

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CHAPTER 2. LITERATURE REVIEW 9 with a leniency application (Hinloopen & Soetevent, 2008). The result of this paper is therefore of importance in the pursuit of cartels as it contradicts the underlying assumption that a cartel solution never arises without previous reconciliation.

2.3 Game Theoretic Concepts

In this paper we use various game theoretic concepts for the selection of the equilibrium out-comes. This section motivates and explains which concepts we use to derive our results.

In this paper we prefer pure equilibria over mixed equilibria in which the players assign probabilities to all pure strategies and select one of them randomly. This is because the economic interpretation of mixed equilibria is less clear as decisions are conditional on a random lottery executed by a player before the game starts Rubinstein (1991).

To select the pure equilibrium outcomes we use the elimination of weakly dominated strate-gies. Strategy A weakly dominates strategy B if there is at least one action for which strategy A is superior and all other actions yield the same payoff as B for strong dominance all actions of strategy A needs to be superior to B. In this paper dominance refers to weak dominance, in the case of strong dominance it is explicitly mentioned.

In addition, equilibria outcomes also satisfy the subgame perfect refinement, which rules out non-credible threats as these equilibria are also equilibria in each of the subgames. The equilibrium outcomes in the multistage version never contain dominated strategies. The game in this paper assumes perfect information and considers a finitely repeated game, therefore the subgame perfect equilibrium always exists and we can find it by backward induction (Maschler, Solan, & Zamir, 2013).

Backward induction requires that each player at every stage of the game only looks at the opponent’s behavior in the future and ignores all information from before this stage. So each player takes the past behavior of the other players as given and does not draw any conclusions from it. On the other hand with forward induction players use observed past choices of the other players and try to clarify their behavior. Subsequently they base their predictions about their opponents’ future decisions on their past behavior (Perea, 2010). In this paper we assume perfect information, where each player knows what type the other player is without observing his past actions. Therefore we use the concept of backward induction to select the subgame perfect equilibria of the game.

We also analyze whether the equilibrium outcomes are stable under small shocks, which could for example occur by trembling hands of the players. In this concept firms only play totally mixed strategies in which all pure strategies are played with a non-zero profitability. This can be viewed as the possibility that players make a slight mistake and play a different strategy than what they were intended to play (Maschler, Solan, & Zamir, 2013).

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Chapter 3

The Basic Model

The origin of the competition model used in this paper is simple and quite general with one single strategic variable for each player which is not restricted to one type. Section 3.1 lists the assumptions of the general model together with some remarks on the notation we use in this paper. The strategy space is divided into subsets which differ in terms of profitability and these are described Section 3.2 with a description of its implications for the quantity and price model in Subsections 3.2.1 and 3.2.2. In this chapter we only consider basic models, the following chapter introduces our model with endogenous timing of decisions.

3.1 General Assumptions

The strategic variables si and sj for player i and j respectively are continuous. For simplicity

we assume they have values on the closed unit interval I = [0, 1]. As a consequence the values of all parameters and solutions of the game are equal to or lie between 0 and 1. Since often profit functions or strategies are identical for player 1 and 2, we use the more general notation of i and j for the different players with i = 1, 2, j = 1, 2 and i 6= j as in Kreps (1990). We assume each player tries to maximize his profit πi and profit functions that are continuous,

strictly quasi-concave and increasing in player i’s own strategic variable si. A player is always

able to set his strategic variable equal to zero which gives πi(0) = 0 irrespective of sj, therefore

profits are non-negative.

All players of the game have complete information, which means they are aware of the strategies and payoffs of the other players. When they are making any decision they are fully informed of all events that have occurred in the past. Reaction functions ri(sj), which we also

call best response functions, are continuous and have a unique intersection point at the Nash equilibrium n ∈ I2. Following the notation of Ramer (n.d.) the graphs of the reaction functions are denoted by Ri _{and R}j_.

n = R1∩ R2 _{= (n}

1, n2) (3.1)

Because we do not assume that players move simultaneously, another equilibrium that could

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CHAPTER 3. THE BASIC MODEL 11 arise is the Stackelberg equilibrium. One player, the leader, sets his strategic variable first and the other player, the follower, reacts with his best-response function. Therefore leader i chooses the value on the reaction graph Rj of player j that maximizes his profits. The two unique Stackelberg equilibria of sequential play with player 1 and 2 as leader are denoted by:

s1 = (l1, f2), s2 = (f1, l2) (3.2)

We assume that the Nash and Stackelberg equilibria differ from each other as in Hamilton & Slutsky (1990). Their profits are expressed by:

π(n) = (πn₁, π₂n), π(s1) = (π₁l, π₂f), π(s2) = (πf₁, π₂l)

3.2 Disjoint Subsets

The strategy space I2 is divided into five disjoint subsets depending on the profitability for each player. For certain specifications of the model subsets can be empty, this depends on the direction of the reaction functions and the specification of the profit functions. In the quantity duopoly model both reaction functions are downward sloping and the consequences for the subsets are described in Subsection 3.2.1. The reaction functions are upward sloping when price is taken as the strategic variable, which is illustrated Subsection 3.2.2.

Before we describe the various subsets it is important to note that this paper uses the following notation to compare vectors:

X ≥ Y means xi≥ yi ∀i

X > Y means xi≥ yi with ∃i xi > yi

X >> Y means xi > yi ∀i

The main interest of this paper lies in the collusive outcome as a solution to the game. In this solution both players earn a higher profit than in the competitive Nash equilibrium. When both players earn at least their leader payoff this situation arises, but can be caused by other stimuli. Call Ui the set where player i earns at least their leader profit:

Ui= { s ∈ I2 | πi(s) ≥ πil }

We denote the better set B as the set where both players earn at least their leader payoff:

B = { s ∈ I2 | π(s) ≥ (πl₁, π₂l) } = U1∩ U2 (3.3) All points in B are possible candidates for the cartel solutions, they are given the following notation:

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CHAPTER 3. THE BASIC MODEL 12 The subset where both players have strictly lower profits than their leader profit, but where one of them earns at least their Nash profit is called medium set M :

M = { s ∈ I2 | π(s) << (πl

i, πlj) } \ W (3.5)

Note that the Nash equilibrium lies in this subset n ∈ M . In the worst case W both players earn strictly less than the profit in the Nash equilibrium:

W = { s ∈ I2 | π(s) << π(n) } (3.6) The isoprofit curves for the profit in the Nash equilibrium, Stackelberg leader and follower are defined by Ini_{, I}li_{, I}fi _{respectively for player i = 1, 2. When an outcome lies on at least}

one of the isoprofit curves, this can be written as In= Ini_{∩ I}nj _{and likewise for the leader and}

follower isoprofit curves.

3.2.1 Quantity Competition

When quantity is the strategic variable of the duopoly model as described by Cournot, the reaction curves of both firms are downward sloping. This means that it is most profitable for both players to become the leader as already described in Section 2.1. We refer to these models as LL-models since both players want to become the leader. It is important to distinguish between the two varieties of the model for the analysis of the stability of the equilibrium outcomes, namely whether the better set B is empty or not. When B 6= ∅ a cartel solution exists where both players earn at least their leader profit. We call the associated model the LL(+) - model as shown in Figure 3.1. Figure 3.2 shows an example of a LL(-) - model, where B = ∅.

The market demand is defined as follows; with a, b, d > 0 and d a parameter defining the shape of the reaction function D(p) = a−p_b d as in Anderson & Engers (1992). When the demand curve is convex (d < 1) the subset B is empty leading to the LL(-)-model. In the case of a concave (d > 1) or linear (d = 1) demand curve the subset B is non-empty which makes the associated model a LL(+)-model. When demand is linear, the best set B consists of one point in which both players earn exactly the leader payoff. For this specification marginal costs can easily be implemented. The interest of this paper lies particularly in the collusive cartel outcome as a solution of the game, therefore it is important to take the value of d into account. We assume zero marginal costs as with costs there is not always an explicit solution when demand is non-linear. The profit function for player i = 1, 2 is defined as:

πi = max {0, (ai− b(qi+ qj))dqi}

In this model goods are homogeneous as there is only one value for b, but basic demand can differ across firms as there are two values for a (ai, aj). The best response for firm i without

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CHAPTER 3. THE BASIC MODEL 13

qi=

ai− bqj

b(1 + d)

The quantities asked by each firm in the Nash and Stackelberg equilibrium with leader i and follower j are added in Section B.1 of the Appendix. We find the accompanying profits of the equilibrium outcomes by rewriting the profit functions for the different quantities in the equilibrium. These profit functions can be found in the same section of the Appendix.

3.2.2 Price Competition

In this model we assume heterogeneous products as this satisfies the assumption of continuous and differentiable profit functions. The demand function is specified as follows: Di(p) = ai−

αipdi+ βipej with ai, αi, βi > 0. In general a change in the own price i will have a larger effect on

demand for good i than a change in the price of the other good j, therefore we specify α > β. We assume equal, constant marginal costs in every stage of the game, hence we can rewrite prices to net off marginal costs as in Singh & Vives (1984). The profit function of player i is specified as πi = (ai− αipdi + βipej)pi − ci(ai− αipdi + βipje). When we take a∗i = ai(1 − ci),

α_i∗= αi(1 − ci) and β∗i = βi(1 − ci), the profit function can be rewritten to:

πi = max {0, (a∗i − α ∗ ipdi + β

∗ ipej)pi}

The reaction functions are upward sloping as the derivative is always positive for βi > 0,

d > 0 and e > 0. Thus the price competition model is an example of an FF-model where both firms prefer to be the follower as πf ≥ πl_{. In this case the subset B is never empty, because it}

always contains the point si and sj. To facilitate the analysis we introduce a subset F :

F = { si ∈ I2 | π(s) ≥ πf } ∈ B ⇐⇒ πf ≥ πl (3.7)

The parameters d and e define the shape of the reaction functions, when d = e = 1 the relationship is linear as drawn in Figure 3.3. The reaction curve of firm i in the price competition model is defined as:

pi=

_a

i+ βipej

(d + 1)αi

1_d

Note that all the equations in this subsection hold for both players i = 1, 2. To make sure the Nash equilibrium is unique the following inequality must hold: e ≤ d ≤ 1. When d, e 6= 1 the model is non-linear and there is no closed form expression for the Nash and sequential equilibria, therefore in the following analysis we take d = e = 1. This can be done without loss of generality, because d and e only change the shape of the reaction functions, not the existence of the subsets or the order of profitability of the various equilibria. An example of a non-linear model can be found in Figure A.1 of the appendix.

The prices asked in the Nash equilibrium by player i and in the Stackelberg equilibrium by leader i and follower j for this specification of the model are included in Section B.2 of the

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Figure 3.1: LL(+) model

Note. The values of the parameters are a1= a2= 1, b = 1, d = 2.

Figure 3.2: LL(-) model

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Figure 3.3: FF model

Note. The values of the parameters are a1= a2= 0.55, α1= α2= 0.9, β1= β2= 0.7 and d = e = 1.

appendix. We find the associated profits of these equilibria by plugging these equations into the earlier mentioned profit functions, which can be found in the same section of the appendix.

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Chapter 4

Timing of the Game

The model described in this paper distinguishes itself from usual repeated competition models by the timing of the game, which is therefore an important feature of this paper. We first motivate the timing of the game on the basis of an example, followed by the general time structure of the game. Section 4.2 explains how multiple games are linked together in the repeated multistage game Gk_(h).

4.1 Timing G(h)

In the game G(h) there are two firms in the market, let us say Bon Pain and Bakerworld who are the only companies selling baguettes in the Kalverstraat. At a certain moment the price of a baguettes at Bon Pain is e2.10 and at the Bakerworld e2, these are called the initial values from history h at the start of the game. Now both companies can choose to change their prices or wait for the other company to be able to react to their price change. When one of the companies chooses to change their price, we assume that the firm has to keep that promise, because otherwise it could hurt the brands equity by being untrustworthy. When for example Bon Pain chooses to decrease their price to e1.90 to attract more customers and Bakerworld chooses to wait, Bakerworld could react to this price cut by further decreasing its own price or keeping it the same. When both firms are waiting for each other, nothing happens and the initial prices of e2.10 and 2 euro’s hold for another period. The next chapter investigates whether a collusive solution in which both firms ask higher prices and have higher profits is stable or that the firms keep competing against each other with price cuts which leads to lower prices and a lower payoff for both firms.

The general time structure is as follows: starting with initial values from history h = (h1, h2) ∈ I2 both firms can adjust the value of their strategic variable in the setting period

S(1) and trade their products in the trading period T (1). In the setting period each firm has the choice to set their strategic variable s = (si, sj) ∈ I2 or to wait which is denoted by wi

or wj respectively. It is important that once a player has decided to set its strategic variable

equal to a certain value, it is not able to change it for that specific game. When one of the

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CHAPTER 4. TIMING OF THE GAME 17 firms decides to set and the other waits, the waiting firm is capable of giving its best response to this output with outcomes (s1, r2(s1)) or (r1(s2), s2). In the case both firms wait, the initial

values h remain the same for another trading period. After T (1) the initial values are updated and the one-shot game ends. The strategies for the players i and j are summarized in the table below. Table 4.1: Outcomes of G(h) Player 2 Player 1 s2 w2 s1 (s1, s2) (s1, r2(s1)) w1 (r1(s2), s2) (h1, h2)

In addition we can find the extended version of the game G(h) in Figure A.2 of the appendix. This shows that we can also see our model as an extensive form game with simultaneous moves as in Osborne & Rubinstein (1994).

4.2 Timing multistage G

k

(h)

The multistage version Gk(h) repeats the game described in the previous section k times. After the first trading period it continues with a setting period S(2) with the outcomes of T (1) as the initial values. Subsequently when players have chosen to wait or to set their amount the trading period T (2) starts where firms could potentially respond and the outcome of the game is revealed. The game continues in similar fashion for a predetermined number of k periods, where it is assumed that demand and profit functions remain the same in every succeeding stage. This assumption only holds when the time between the stages k stays sufficiently small so that the values of the trading period of the preceding period are valid initial values for the next game. The figure gives a visual interpretation of how the stages are linked together.

Figure 4.1: Timeline Gk_(h)

H S(1) T (1) S(2) T (2)

The initial proposals from history in H before the first trading period originate from previous periods or when the firms have competed against each other in the past, for example before a take over or in another establishment in the neighborhood. In the Kalverstraat example when one of the bakeries decides to open a shop in the street, the prices from another establishment can be taken as the initial value for the new firm. In this paper we choose different initial values from every disjoint subset and describe their influence on the resulting solution of the game. The model could easily be extended with a pre-bargaining stage before H, but we choose not to include this as there is no clear economic reasoning behind it.

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Chapter 5

Solutions of the Game

As already alluded to at the end of the previous chapter the starting values from history h have a large influence on the outcome of the game. This paragraph describes for various starting values the possible outcomes of the game, using the concept of dominance. It starts with the one-shot version of the game in Section 5.1, which we will further extend to a multistage game in Section 5.2. This section also lists under which conditions the implicit cartel outcome is a solution of the game. The last subsection shows whether the system returns to the collusive outcome under small shocks on the market.

5.1 One-stage game

This section lists the possible equilibrium outcomes for the starting values from the five possible disjoint subsets as described in Section 3.2. In the possible equilibria outcomes none of the players have an incentive to deviate, so they are all potential solutions of the multistage game. Some strategies however are weakly dominated by other strategies, therefore we assume that they do not occur in the one-stage game.

The first important thing to note is that when profits for the initial value are non-negative there is always a situation for which a player is better off waiting wi instead of setting himself

to an amount si. In the case the other player sets an amount sj, player i has the opportunity

of reacting with his best response r(sj) and πj(r(sj), sj) ≥ πj(si, sj), because by definition

the best response of player i is the output that optimizes his profit given the output of player j. If the other player waits wj, the outcome of the previous value (hi, hj) holds for another

period if player i also waits. When h has a non-negative profit, this is never dominated by setting an amount because for every si there is an amount the player j can produce that leads

to zero-prices caused by overproduction and thus non-positive profits in the quantity model or asking prices equal to the marginal cots in the price competition model. Note that in this case it is not assumed that player j necessarily plays his best-response. If the initial values leads to a negative payoff for player i, waiting is dominated by setting si = 0, but we assume that

this never occurs as profits are non-negative as described Section 3.1. This summarizes to the

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CHAPTER 5. SOLUTIONS OF THE GAME 19 following:

Lemma 1. Playing wait wi is never dominated by setting an amount si before the elimination

of dominated strategies.

Irrespective of the value of h the choice to set an amount si = xi never dominates another

value si = yi with xi 6= yi. For example in the quantity model if a player can choose between

producing a small amount xi and a large amount yi with xi < yi, it depends on the strategy of

the other player what the optimal amount is. If the other player also sets sj and he produces

a large quantity, it is better for the other player to produce the small amount xi, because

otherwise the price will decrease which damages the profitability of player i. On the other hand if player j produces a small amount it is better for the player to set to the large amount yi.

Thus it always depends on the value of the other player sj which quantity is more profitable

for the other player to produce and two different setting values can never dominate each other. A similar reasoning holds for other strategic variables as the optimal value si always depends

on what the other player does.

Lemma 2. Setting an amount si = xi never dominates setting on another amount si= yi with

xi 6= yi before the elimination of dominated strategies.

As already described by Hamilton & Slutsky (1990) the profit in the Nash equilibrium never exceeds the leader profit. If player i is the leader he chooses his most profitable point on player j’s reaction function. This includes the point of the Nash equilibrium and the leader profit is therefore never less than the profit in the Nash equilibrium.

Lemma 3. Setting si= li dominates si = ni when player j chooses wj.

The strategies c1 = (l1, w2), c2 = (w1, l2) and n = (n1, n2) are equilibria for every

ini-tial subset of this game because none of the players make more profit by only changing their own strategy; however, depending on the starting initial values these strategies may be dom-inated in the one-period game. The non-domdom-inated equilibrium outcomes are described for h in every disjoint subset. It is possible a certain subset is empty for certain specification of the model, for example B in the LL(-)-model. Which subsets are non-empty for the different model specifications is already described in Section 3.2.

• Start with the assumption that h ∈ B: When player j waits and player i chooses si, the

highest possible profit for player i is the leader profit. This is less than the profit player i has when he chooses wi, because in this case the original values hold for another period

and their profit is at least as high as the leader profit when h ∈ B. Following the same reasoning it is never beneficial for player j to choose a certain amount sj as this will never

give him more than the leader profit. So the dominant strategy for both players is waiting, with strict dominance if h >> πl. Thus the strategy is w which leads to the equilibrium outcome h.

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CHAPTER 5. SOLUTIONS OF THE GAME 20 • Now assume h ∈ Uj_{\B: In this subset player j has a profit at least as high as his leader}

profit, so he never benefits from setting an amount and his dominant strategy is wi. By

elimination of dominated strategies together with the fact that the other player’s profit from the initial value is lower than his leader profit, he gains by setting sj = lj. Thus

the strategies are (wi, lj) which results in player i playing his best-response which is the

follower’s profit and this leads to the outcome ci = (fi, lj). He will not choose to set

si = ni, because πin ≤ πli as described in Lemma 3 and therefore choosing the leader

amount dominates playing the Nash amount when player i assumes player j waits, so the Nash equilibrium is eliminated by iterated dominance. Note that this describes the outcomes of the two distinct subsets U1\B and U2_{\B with strategies (w}

1, l2) and (l1, w2)

and outcomes (f1, l2) and (l1, f2) respectively.

• Next start with h ∈ M : In this case in h both players have lower payoffs than their leader payoffs, but one of the players earns at least their Nash payoff. Both players benefit from setting si = li when the other player waits wj, where in turn player j will use his

best response resulting in the follower profit. Waiting w is not an equilibrium outcome because for each player it is dominated by setting si = li. As one of the players always

has a payoff at least as high as the Nash payoff for this player waiting dominates si= ni.

Because si = ni is never played, sj = nj is eliminated by iterated dominance, as it never

yields a higher payoff than setting sj = lj. Thus n is not an outcome in undominated

strategies and the possible strategies are (l1, w2) and (w1, l2) with outcomes c1 and c2.

Which firms moves first, the so called Stackelberg warfare, can not be derived theoretically (Stackelberg, 1934). Without extra information it is therefore not possible to draw any conclusions on which of the two equilibria occurs.

• Last consider h ∈ W : All players have a lower profit in h than their leader profit and the profit in the Nash equilibrium. Therefore both players have an incentive to deviate and one of them sets si = li or si = ni. These strategies do not dominate each other

according to Lemma 2 as no strategies can be eliminated by waiting. When player i sets, it is more profitable for player j to wait as described in Lemma 1. Player j now uses his best response which results in the equilibrium outcomes (li, fi) and (ni, nj). Following

this reasoning for both players the possible equilibria outcomes are s1, s2, n.

Proposition 1. The undominated equilibrium outcomes of the one-stage game G(h) depending on the subset of the starting values h ∈ I2 are listed in the table below.

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CHAPTER 5. SOLUTIONS OF THE GAME 21

Table 5.1: Non-Dominated equilibrium outcomes G(h)

Location of h Strategies Equilibrium outcomes B w = (w1, w2) h = (h1, h2) a b

Uj\B ci = (li, wj) si = (li, fj) b

M c1, c2 s1, s2 b

W c1, c2, n s1, s2, n

a

s1, s2are dominated. bn is dominated.

5.2 Two- and Multistage game

In the previous section the possible equilibrium outcomes of the one-stage game are given for each of the five disjoint subsets, which holds for both the LL and FF - models. This section extends the analysis first to the two-stage game followed by a multistage game with a predetermined number of repetitions k. It considers both dominated and non-dominated strategies of the one-stage game and defines under which conditions which solution is an outcome of the game starting with the analysis for the LL - models followed by the FF - model. To facilitate the analysis the subsets of the different equilibrium outcomes are described in Table 5.2.

Table 5.2: Subsets of Equilibrium outcomes G(h)

Equilibrium outcome ∈ Subset LL ∈ Subset F F

h = (h1, h2) B B

s1= (l1, f2) U1\B B

s2= (f1, l2) U2\B B

n = (n1, n2) M M

5.2.1 Two stage LL - model

First, we list all possible equilibrium outcomes of the LL - models, to find which equilibria satisfy the subgame perfect refinement we use backward induction. Starting with all possible equilibrium outcomes in the second stage, we list what the possibilities are in the preliminary stage and select the optimal strategy in every situation.

• The equilibrium outcome h ∈ B during trading period T (2) is only possible when the initial value for the second stage and thus the outcome of T (1) lies in subset B as well. Table 5.2 shows that the only outcome that lies in B is the outcome h, which in turn only arises when the values in S(1) are in the subset B. This equilibrium outcome gives both players in each stage of G2(h) a payoff at least as high as their leader payoff.

• When the outcome in T (2) is s1_{, the initial value must lie in U}2_{\B, M or W in}

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CHAPTER 5. SOLUTIONS OF THE GAME 22 Table 5.2 indicates that the equilibrium outcome of T (1) must be s2, n or h. When subset B 6= ∅ and dominated strategies are also considered the profit cπ_ilin the first period is also possible for player i = 1, 2. The same reasoning applies when the outcome in T (2) is s2, but with reversed strategies, outcomes and payoffs for the players. Table 5.3 summarizes these results for player i and j as they all hold for both players.

• The Nash equilibrium outcome n is only an undominated solution in the second trading period if the initial value lies in subset W , which is not a subset of one of the equilibrium outcomes. When dominated strategies are also considered the Nash equilibrium is a possible strategy in all of five disjoint subsets, which means that in the preceding period T (1) all of the strategies of Table 5.2 can be played. This leads to the payoffs as described in Table 5.3.

Proposition 2. The possible payoffs for the LL - model of the two-stage game G2(h) with the accompanying subsets for h ∈ H to make this outcome feasible are listed in the table below.

Table 5.3: Candidates of equilibrium outcomes G2_{(h) in the LL-model}

Location of h S(1) T (1) S(2) T (2) Profit i Profit j

B w h w h 2cπl 2cπl a _B _w _h ci _si cπl +πl cπl +πf I2\Uj b _cj _sj _πf_{+ π}l _πl_{+ π}f W b n n πn+ πl πn+ πf a _B _wb _h n n cπl +πn cπl +πn a _I2_\Uj b _cj _sj _πf_{+ π}n _πl_{+ π}n a _W b _n _n _2πn _2πn Note. c ≥ 1. a

The strategy in the second stage S(2) is dominated. b The whole strategy set I2 _{when dominated strategies are considered.}

Equilibrium outcome LL(+) - model

When B 6= ∅ as in the LL(+) - model both players prefer the payoff resulting from the equilib-rium outcome from this subset as they earn at least two times the leader payoff, which is most profitable for both of them among all possible payoffs. The question is under which conditions this is an undominated equilibrium outcome in the two-stage game. Therefore two cases are considered, first with h ∈ B and in the second case h 6∈ B.

1. When h ∈ B: The profit of the initial values is at least as high as the leader profit, so setting an amount is never more profitable than waiting as this will never lead to a higher payoff. In this case, for each player waiting dominates any other strategy for both stages of the game.

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CHAPTER 5. SOLUTIONS OF THE GAME 23 2. When h 6∈ B: The minimum profit of the collusive solution is 2π_il. This solution is an outcome of the game when neither of the two players obtain a higher payoff by deviating from this strategy. When player j sets his strategic variable equal to the collusive solution bj, as defined in 3.4, the other player could benefit by deviating and selling more products,

for example, by increasing his quantity in the quantity model. However, in the next round of the two-stage model, player i is punished for this behavior. In this round the initial value h ∈ Ui\B as player i obtained a higher payoff than the leader payoff by deviating from the cartel solution. Thus, for player i the dominant strategy is to wait, wi, and for

player j, who receives less than his leader payoff, it is most profitable to set sj = lj. Player

i reacts using his best response ri(lj) = fi and the solution in the stage after deviation

equals sj. Thus for the implicit cartel solution to be stable the following must hold:

πi(ri(bj), bj) + πi(fi, lj) ≤ 2πil ∀i = 1, 2 (5.1)

When this condition holds not every point b ∈ B is necessarily an outcome of the game. To illustrate this take a point x1 ∈ B as in Figure 5.1. This point is an equilibrium

outcome as none of the players has an incentive to deviate. When Condition 5.1 holds none of the players has an incentive to deviate to a solution outside subset B, but what happens when deviations to points inside B are also considered? Player 1 could benefit by increasing the value of his strategic variable and when player 2 does not change his output this results in the outcome x2. Now player 1 cannot increase his profit without

leaving subset B, but player 2, who gets exactly the leader profit in x2 can by increasing

the value of his strategic variable which results in outcome x3. This continues until both

players get precisely their leader profit in the point bu or x1 and is stable when:

πi(x2) + πi(x3) ≤ 2πi(x1) ∀i = 1, 2 (5.2)

So for h 6∈ B all cartel solutions b ∈ B are equilibrium outcomes when Condition 5.1 and 5.2 hold. If only the first condition hold the only outcome is bu.

Equilibrium outcome LL(-) - model

When B = ∅ as in the LL(-) - case player i prefers the payoff πn+ πl resulting from playing si = ni in the first stage and si = li in the second stage; however, this never constitutes an

equilibrium, as for player j this leads to payoffs πn_{+ π}f _{which is an even lower payoff than 2π}n_,

so player j has an incentive to deviate.

The remaining outcomes of the two stage game are {si, sj} and {n, n} for trading period 1 and 2 respectively with corresponding payoffs πf + πl and 2πn. Which of these outcomes has a higher payoff depends on the specification of the inverse demand and costs functions. In the basic Cournot duopoly game with linear demand and no costs 2πn≥ πf_{+ π}l_{. The equilibrium}

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CHAPTER 5. SOLUTIONS OF THE GAME 24

Figure 5.1: Equilibrium outcome b in LL(+) - model

outcome n is most profitable for both players as the price in the Nash equilibrium is higher than the price in the Stackelberg equilibrium. This inequality always holds for concave, convex and linear inverse demand functions with a1 = a2 and no costs (Anderson & Engers, 1992),

which results from the fact that the aggregate output in the Nash equilibrium is lower than in the Stackelberg equilibrium and closer to the optimal monopoly output. Because the aggregate output in the Stackelberg equilibrium is higher than in the Nash equilibrium, the introduction of costs never changes this result, as producing less never has higher costs than a higher production level. In the more general specification of the model as described in Subsection 3.2.1, the profits of the follower and leader should be compared with two times the Nash profit, this simplifies to the following comparison:

π_il+ π_if ≶ 2πin ⇐⇒ ( 2 d+ 1) d₍2 d+ 1) ≶ ( 1 d2 + 1 + 2 d) d₍2 d+ 2) (5.3)

The first and second term of the right-hand side of the equation are larger as _d12 > 0 for

d > 0 and (2_d+ 2) > (2_d+ 1), thus we can conclude that two times the Nash payoff is larger than the leader plus follower payoff, see Appendix B.3 for the full derivation. In the LL(-) - model players prefer n over a two-cycle {si, sj}; however, this equilibrium is not supported by weak dominance as after playing Nash once, the initial values lie in subset M and both players prefer si = li over si = ni as shown in Table 5.1. So the equilibrium outcome in the LL(-) - model is

the two-cycle {si, sj}. Note that only in the first period is it possible to play Nash if h ∈ W which leads to slightly higher aggregate payoff.

For the one-period LL(-) - model there is no collusive equilibrium, but when we look at the repeated game and also consider mixed strategies, there could exist a subgame perfect solution for which both players earn a higher aggregate payoff. Consider the collusive outcome where

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CHAPTER 5. SOLUTIONS OF THE GAME 25 both players earn half the monopoly profit, which we will call πm. A deviation would give

profit πl when the other decides to wait and 0 when the other also plays the leader amount.

This last fact can be proven with the value of the parameter in the LL(-) - model (d < 1) and the profit function. Take for simplicity ai = aj = 1 and bi = 1, profits are zero when

ai≤ b(qi+ qj) =⇒ 1 ≤ (qi+ qj). In this example qi = _1+d1 , so the equation becomes 1 ≤ (_1+d2 ),

which simplifies to (1 + d) ≤ 2 and further to d ≤ 1, which is in definition true for the LL(-) model. Note that we assume that profits can never be negative, so in this case both players earn the lowest amount: 0. We can summarise this game with the following payoff matrix:

Table 5.4: Cooperation in the LL(-) - model

Player i

Player j

cooperate defect cooperate πm, πm πf, πl

defect πl, πf 0

For the one shot version of the game there are two equilibria with one player producing the leader amount and the other the follower. However when we repeat this game and allow for mixed strategies, there is a subgame perfect equilibrium where both players cooperate in the first round and play a mixed strategy between cooperating and defecting in the second round, when πm+ u ∗ v(πm) + u ∗ (1 − v)(πf) + (1 − u) ∗ v(πl) + (1 − u) ∗ (1 − v) ∗ 0 > πl+ πf with u and v

the chance of player i and j cooperating. When a player deviates from cooperation in the first round, he will earn the leader amount. In the second round he waits, because there is no amount he can produce for which he is better off and the other player becomes the leader, so that in the second round he will become the follower. The existence of this equilibrium strongly depends on the value of d and the cooperation parameters u and v. We find that for d smaller than 0.55 cooperation is never an outcome. When we assume two symmetric players, cooperation is an outcome for values of d between 0.55 and 1, in particular we find that for d = 0.84 players are indifferent between cooperating and defecting such that u = v = 0.5.

5.2.2 Two stage FF - model

As in the previous subsection we first list all possible equilibrium outcomes in the two-stage game and investigate which outcomes are undominated but now for the model where both players prefer to be the follower. The results of this analysis are listed in Table 5.5.

• Start with w played in the second stage S(2), Table 5.1 shows that this is only an equi-librium outcome if the initial values h ∈ B. The outcome of the previous period in the FF - model must be h or si as shown in Table 5.2. Note that the subset B is split into F and B\F as the preferences for the outcomes from these subsets differ. When h ∈ F every player has a profit at least as high as the follower profit in both stages of the game and this payoff dominates all other outcomes. When h ∈ B\F they both earn an amount

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CHAPTER 5. SOLUTIONS OF THE GAME 26 between their leader and follower payoff, so each player would rather be the follower. • When si _{is the outcome of the second stage T (2), the initial value must lie in U}j_{\B, M or}

W in undominated strategies or in B when we also consider dominated strategies. Table 5.2 shows that the only equilibrium outcome in undominated strategies in the first stage is n as all other outcomes lie in B which is only part of the dominated strategies. These results together with their payoffs are summarised in Table 5.5.

• The outcome n is only a played in undominated strategies when the initial value h ∈ W , but because none of the equilibrium outcomes lies in W it is never part of undominated strategies. When we also examine undominated strategies, n can be a solution of all the subsets in I2.

Proposition 3. The possible payoffs for the FF - model of the two-stage game G2(h) with the accompanying subsets for h ∈ H to make this outcome feasible are listed in the table below.

Table 5.5: Candidates of equilibrium outcomes G2_{(h) in the FF - model}

Location of H S(1) T (1) S(2) T (2) Profit i Profit j F w h w h 2cπf _2cπf B\F 2dπl 2dπl I2\Uj b _cj _sj _2πf _2πl a _B _w _h ci si cπl +πl cπl +πf a _I2_\Uj b _cj _sj _πf _{+ π}l _πl_{+ π}f W b n n πn+ πl πn+ πf a _B _w _h n n cπl +πn cπl +πn a _I2_\Uj b _cj _sj _πf _{+ π}n _πl_{+ π}n a _W b _n _n _2πn _2πn Note. c, d ≥ 1. dπl< πf.

a_{The strategy in the second stage S(2) is dominated.} b_{The whole strategy set I}2

when dominated strategies are considered.

Equilibrium outcome FF - model

In the FF - model the subset B is nonempty and thus it is always possible for both players to earn at least their leader payoff in every stage of the game. The question is under which conditions the collusive solutions b ∈ B are outcomes of the game. We start our analysis with the initial values h ∈ B followed by h 6∈ B where we split B into B\F and the Pareto superior subset F .

1. When h ∈ B: Neither of the players has an incentive to deviate and become the leader as in this subset they both earn at least their leader profit. Table 5.5 shows that it is most profitable for both players to earn the payoff from values in the subset F ∈ B, where they

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CHAPTER 5. SOLUTIONS OF THE GAME 27 both earn more than two times the follower profit. When h ∈ B\F each player i would prefer to the follower, which can be obtained by playing si = wiif the other player chooses

sj = lj. However, lj is never played as the payoff from wj dominates the leader payoff for

player j when h ∈ B. So all h ∈ B are outcomes of the game with both players playing wait in every stage of the game.

2. When h 6∈ B:

• First consider h ∈ Ui_{\B: In this case player j earns less than his leader profit and}

has an incentive to set sj = lj. Player i reacts with his best response and earns the

higher follower profit, the solution lies in the subset B\F . As we described before, solutions with h ∈ B are outcomes of the game with each player choosing to wait in every succeeding stage of the game.

• Now consider h 6∈ (Ui_{∪ U}j_{): In this set each player has a strictly lower payoff than}

their leader payoff. i profits from setting si = li and j benefits even more by sj = fj.

This solution lies in B and after this one player gets the leader and the other the follower payoff in every stage of the game as described in row 3 of Table 5.5. There are no further conditions needed for this solution to be an outcome of the game. • Note that the solutions s1 _{and s}2 _{lie in B\F and each player would be better or}

at least as good off with a solution in F . Could this under certain conditions be a undominated equilibrium outcome? The minimum payoff in this subset is 2πf_i, which lies on the isoprofit curve IF of the follower profit. Let f ∈ If be a point on one of these isoprofit curves, a one time deviation of player i would give him profit πi(ri(fj), fj). The solution lies in Ui\B so player j earns less than his leader payoff.

In the next round player j sets sj = lj and player i who has waited responds with fi

resulting in profit π_if. For a point on the boundary of F to be stable the following inequality must hold: πi(ri(fj), fj) + πif ≤ 2π

f

i, which simplifies to πi(ri(fj), fj) ≤ πif

∀i = 1, 2. However, the left hand side of this inequality is never smaller, because player i can always choose fi leading to the follower profit. A best response yields an

even higher payoff and therefore f ∈ If is never an outcome of the game. When we consider all points within F the following equation should hold to make sure none of the players has an incentive to deviate:

πi(ri(fj), fj) + πf_i ≤ 2cπi(f) (5.4)

When this condition holds, a point f ∈ F is stable when the initial value h 6∈ B.

5.2.3 Multistage constant game

When players are aware they are competing against each other for multiple stages they can take this information into account and try to maximize their total profit over multiple stages.

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CHAPTER 5. SOLUTIONS OF THE GAME 28 The collusive solution might be more often an outcome when the game G(h) is played multiple times, because punishment after deviation has a larger impact when it last for more than one game. For each of the models the conditions under which a solution is an outcome of the game are listed in the table below, with k a predetermined number of stages of the game.

Proposition 4. The equilibrium outcomes depending on the position of the initial values h and a predetermined number of stages k for the LL(+), LL(-) and FF model are listed in Table 5.6.

Table 5.6: Steady States Multistage Gk(h)

h Model Outcome Conditions

∈ B LL(+) h -FF h -6∈ B LL(+) bu πi(ri(bj), bj) + +k₂πif+ (k2 − 1)π l i≤ kπil a b πi(ri(bj), bj) +k₂πif + (k2 − 1)πli ≤ kcπil a, πi(x2) + πi(x3) + ... ≤ kπi(x1) b {si_{, s}j_} c _else LL(-) {si_{, s}j_} c -FF f πi(ri(fj), fj) + (k − 1)π f i ≤ kcπi(f) si else and h 6∈ Ui

Note. All conditions should hold for player i = 1, 2. c ≥ 1.

a

If k mod 2 = 0, else πi(ri(bj), bj) + k−1₂ πf_i +k−1₂ πil ≤ kcπil. b With x1 ∈ B,

x2∈ Ili, x3∈ Ilj. c When h ∈ W n is possible in the first period.

After how many periods collusive solutions are an outcome of the game is of particular interest for the different specification of the model. As it is not possible to obtain a general solution, we calculate the number of periods k needed for the price and quantity models men-tioned in the model section. When we consider the model with linear demand (d = 1) the intersection point of the two leader isoprofit costs is: bu_i = _4ba, which is at the same time the only point in the subset B. It is stable when a one time deviation is less profitable than earning the payoff in this point for all k periods as specified in the table above. The best response to the collusive solution bu_i is: ri(bui) = ri(_4ba) = 3a_8b and the profit from a one time

devia-tion equals: π_idev = (a − b(3a_8b + 2a_8b)3a_8b = 3a_8b2 + 15a_64b2. Comparing this with the profit from the two-cycle and leader profit, the following equation must hold for an even number of periods:

3a2 8b + 15a2 64b + k 2 a2 16b+ ( k 2 − 1) a2 8b ≤ k a2

8b. The equation does not depend upon the values of a and b

as each side can be divided by a_b2 when a, b > 0. This inequality holds for k ≥ 2, so deviation is less profitable even when the game only lasts for two stages. As the values of a and b have no influence on the number of stages needed for the collusive solution to be an outcome of the game, we take a = b = 1 in the upcoming analysis.

The model with concave demand (d > 1) has a larger set B and as an example we consider d = 2 as in Figure 3.1. The quantities belonging to the upper corner intersection point bu of the two isoprofit curves are approximately 0.2358. When a player deviates, he plays his

Implicit cartels : cooperation in a duopoly with endogenous timing of decisions

Master’s Thesis