The best player rule : impact of imitation and memory in dynamic Cournot oligopolies

T H E B E S T P L AY E R R U L E

Impact of Imitation and Memory in Dynamic Cournot Oligopolies

ro b e rt v e r s c h u r e n 1 0 6 4 2 4 4 7

A Bachelor’s Thesis to obtain the degree in

e c o n o m e t r i c s & o p e r at i o n s r e s e a rc h

Supervisor: Dr. D. Kopányi June 26, 2016 – Final draft

S TAT E M E N T O F O R I G I N A L I T Y

This document is written by Student Robert Verschuren 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.

A B S T R A C T

In this thesis, we explore the impact of imitation with memory in the context of Cournot oligopolies. We model imitation with memory accord-ing to the best player rule by adjustaccord-ing the model developed by Vega-Redondo (1997) and Alós-Ferrer (2004). We run computer simulations with this model to determine how the best player rule affects the market outcome in the long-run in a Cournot oligopoly and how this outcome is influenced by memory length. We show that this model has in fact only one stochastically stable state, namely the Walrasian equilibrium. Moreover, this result is obtained regardless of the number of firms in the market or the memory length and is not affected by changes in the model parameters, such as the slope and intercept of the inverse-demand function. It also seems that relative payoff considerations remain more important than absolute ones when memory is introduced to this model, explaining why all other quantities lack stability. We therefore conclude that the best player rule leads to the Walrasian equilibrium in the long-run in a Cournot oligopoly, regardless of the number of firms in the market or the memory length, and that this result is robust with respect to the model parameters.

Keywords: Imitation, mutation, memory, stochastically stable state, Cournot oligopoly, Walrasian equilibrium.

C O N T E N T S

1 i n t ro d u c t i o n 1

2 m o d e l i n g t h e i m i tat i o n ru l e 5

2.1 Previous studies . . . 5

2.2 The imitation game . . . 7

2.2.1 Market structure . . . 8

2.2.2 The imitation model . . . 8

2.2.3 The best player rule . . . 9

2.3 Restrictions . . . 10

3 s i m u l at i n g t h e b e s t p l ay e r ru l e : m e t h o d o l o g y a n d data 13 3.1 Simulating the imitation process . . . 13

3.2 Simulation procedure . . . 14 4 s i m u l at i o n r e s u lt s 15 4.1 Simulation analysis . . . 15 4.2 Robustness . . . 20 5 c o n c l u s i o n 25 a wa l r a s i a n v e r s u s c o u r n o t e q u i l i b r i u m 27 b a d d i t i o n a l f i g u r e s a n d ta b l e s 29 r e f e r e n c e s 35 v

1

In the real, often complex world of oligopolies, firms have to operate without complete knowledge of the market and the price dynamics. Yet these firms still have to make difficult decisions without the capability of computing best replies, i.e., setting their production or price level whilst taking into account the level competing firms will settle at. One way firms cope with these uncertainties, is to use simple rules of thumb in their decision-making.

An example of such a rule of thumb is imitation with trial and error. Intuitively, imitation seems rather plausible. Since, in an oligopoly, firms are assumed to be aware of all decisions made in the market, including those of their competitors, the most profitable strategies observed are, as demonstrated byVega-Redondo(1997), immediately imitated and no firm is going to deviate from this strategy. He argues that the underlying principle is that firms who manage to obtain higher payoffs thrive at the expense of their competitors. Imitation can therefore be determined by observed success, where “success” can be measured in terms of achieving the highest profit. From this point of view, an oligopoly can be seen as a game in which firms tend to achieve the highest payoffs by imitating actions from the past, the imitation game. Trial and error is used in this game to model the (small) probability of a firm trying something new, through an experiment, also called mutation. So together, imitation and trial and error can account for the behavior of firms in such situations.

The use of this rather simple rule of thumb has major implications for the study of the market outcome. Without the use of imitation and trial and error, we would have to follow the classical approach to study the Nash equilibrium.Vega-Redondo(1997) points out that this approach is based upon absolute payoff considerations, where no firm has any con-cern for its position relative to its competitors. In reality however, this concern for relative payoff considerations is very real and has a huge im-pact on price dynamics. We therefore favor a more dynamic approach that reckons with this concern, and the use of imitation and trial and

2 i n t ro d u c t i o n

error enables such an evolutionary approach.Vega-Redondo (1997), for example, shows that this evolutionary approach provides an alternative basis for Walrasian behavior. In an oligopoly, this concretely means that evolutionary forces will drive firms to behave in a Walrasian-like man-ner, resulting in behavior “as if” they confronted prices as price takers.

Alós-Ferrer(2004) takes this approach even one step further by introduc-ing finite memory. He shows that, in contrast to Vega-Redondo (1997), the Walrasian equilibrium is no longer the only long-run equilibrium, as long as memory takes at least one period into consideration. On the con-trary, there appears to be a clear tension between the Walrasian and the Cournot equilibrium, resulting in the whole range of quantities be-tween the two equilibria being stabilized. Hence, it is apparent to see that memory has substantial consequences for the market outcome.

While the above shows that the use of imitation and trial and error has certain benefits, several challenges still remain. When we use this rule of thumb, it is for example not immediately clear how this will affect the long-run outcome in a Cournot oligopoly and whether the results will differ from those derived from the classical approach.Alós-Ferrer(2004), for instance, has made it apparent that the answers to the previous ques-tions depend on the memory length, but makes no further remark about how often the different equilibria are reached. It is also still unclear how the outcome is affected when we consider alternative ways to incorporate memory in the model, or imitation rules. The best player rule, for exam-ple, is such an alternative rule, where firms choose the most recent action of the player that earned the highest profit on average in the past. We therefore incorporate this rule in this thesis to investigate whether such an alternative imitation rule will yield similar results, or that it will lead to a different outcome. But more importantly, we investigate why this, possibly different, outcome arises to gain a more detailed insight into the effects of memory in a Cournot oligopoly. To achieve this, this thesis aims to answer the following research questions:

How does the best player rule affect the market outcome in the long-run in a Cournot oligopoly, and how is this outcome influenced by memory length?

Several studies other thanVega-Redondo(1997) andAlós-Ferrer(2004) have previously investigated this topic.Alchian (1950), for example, ar-gues why the behavior of firms in an oligopoly should be modeled in a non-rational way.Schlag (1999), on the other hand, makes an extensive study of all kinds of imitation rules and on what grounds these rules can be criticized. The overall conclusion is that the answers to these questions are not straightforward. In order to give a meaningful answer, we first have to select an imitation model. In the articles ofVega-Redondo(1997) and Alós-Ferrer (2004), imitate the best is used as model, where simply the quantity that led to the highest profit in memory is imitated. In this thesis on the contrary, the best player rule is being used, for it provides a better formulation of imitation with memory with more applications for the real world. Secondly, we can note that although Alós-Ferrer (2004)

i n t ro d u c t i o n 3

provides information on the effect of memory on the market outcome in the long-run, he does not mention how often the different stochastically

stable states are reached, or which memory length firms will choose when

they have the opportunity to do so. In this thesis, we investigate the possible market outcome in a Cournot oligopoly, especially the stochas-tically stable states, under the best player rule and further analyze how this outcome is affected by memory. To do so, we present a theoretical framework and scrutinize the model developed byVega-Redondo (1997) andAlós-Ferrer(2004) to adjust it in such a way that it incorporates the best player rule. The analysis furthermore involves computer simulations and a formal analysis. Through deduction, evidence-based answers are provided for the research questions posed earlier.

The remainder of this thesis is concerned with taking the steps de-scribed previously. We provide a theoretical background in the first part of Chapter2, whereas in the second part we give an explanation of how the concepts of imitation and memory can be incorporated. Subsequently, we present a model which takes these concepts into account, and at the end of this chapter we impose several restrictions on this model for the sake of simplicity. In Chapter 3, we explain the simulation process, dis-cussing which methods are used and why, with what purpose. Afterwards, we present the results and subsequently interpret them with regard to the research questions posed earlier, in Chapter4. Finally, the conclusion of this thesis follows in Chapter5.

2

This thesis aims to explain how the best player rule affects the market outcome in the long-run in a Cournot oligopoly, and how this outcome is influenced by memory length. To achieve this, we adjust the formal model developed byVega-Redondo(1997) andAlós-Ferrer(2004), which fully describes the behavior of firms in a dynamic Cournot oligopoly, to incorporate the best player rule. This model is established in such a way that, under several assumptions, a well-behaved Cournot oligopoly is ensured.

In this chapter, we describe the model for the imitation game. We give an overview of the main findings in the related literature in the next section, explaining the intuition behind the effects of imitation and memory in dynamic oligopolies.

2.1 p r e v i o u s s t u d i e s

A wide range of literature has been published on the use and implications of imitation and trial and error in a Cournot oligopoly. These studies vary with imitation rules on how to explain non-rational behavior, but also with assumptions and constraints applied to the models in question.

Alchian (1950), for example, argues that it is not realistic to assume that firms behave rationally in an oligopoly but that their behavior should in fact be described in a non-rational way. He, as well asArmstrong and Huck(2010), explains that the standard assumption of profit maximiza-tion no longer makes sense when there is uncertainty present in the mar-ket. Firms may, for instance, have no complete knowledge of the market and the price dynamics, or have no clue on how to anticipate the decisions of their competitors. Instead of the standard assumption of profit maxi-mization,Alchian(1950) recommends using imitation and trial and error to cope with this non-rational behavior and makes some suggestions on how to implement this. This recommendation is supported by Matthey

6 m o d e l i n g t h e i m i tat i o n ru l e

(2010), who obtains clear experimental evidence of firms’ intention to imitate.

One of these suggestions is modeled byVega-Redondo(1997), who con-siders an N -firm Cournot oligopoly in which imitation and trial and error play an important role. He investigates whether the central hypothesis of the Walrasian theory, which states that firms are price takers, is necessary for explaining Walrasian behavior, or that an alternative explanation can be provided which is not based on the absence of this so-called “monopoly power”. The introduction of imitation and trial and error enables him to use a more evolutionary approach for his research.

Since spite plays an important role in this evolutionary approach, it is useful to define spite in this context more formally for the remaining part of this thesis. Whilst taking imitation of the highest profit into account, spite implies that only relative payoffs matter.Vega-Redondo(1997), for instance, shows that the Walrasian equilibrium, in which aggregate pro-duction leads to a price equal to marginal cost, is favored above the Cournot-Nash equilibrium rather quickly because of the effects of spite. In other words, a firm can earn higher profits by deviating from the Walrasian equilibrium, but the competitors who decided to remain in the Walrasian equilibrium will, according toVega-Redondo (1997), earn even higher profits. Deviating will therefore result in a bad relative posi-tion. Likewise, a firm will earn lower profits if it decides to produce the Walrasian quantity instead of the Cournot one, but the competitors who decided to remain in the Cournot-Nash equilibrium will earn even lower profits. In this case, deviating will actually result in a good relative posi-tion. From an evolutionary perspective, this means that relative payoffs are considered to be more important than absolute payoffs.

But more importantly, this evolutionary approach allowsVega-Redondo

(1997) to develop an alternative basis for Walrasian behavior, in which we can avoid any consideration to the absence of monopoly power al-together. As the probability of experimenting by trying something new goes to zero, and trial and error becomes irrelevant, this limiting behav-ior shows that the distribution of the induced stochastic process ends up concentrated around the unique symmetric Walrasian equilibrium in the long-run. In other words,Vega-Redondo (1997) points out that evo-lutionary forces will drive firms to behave in a Walrasian-like manner in a Cournot oligopoly, resulting in behavior “as if” they confronted prices as given.Possajennikov (2003) obtains a similar result by a decrease in rationality, where firms can only observe the realized output levels and profits but not the intended strategies.

Alós-Ferrer (2004) considers the same Cournot oligopoly model, but comes to a very different conclusion when he introduces finite memory. He shows that, in contrast toVega-Redondo (1997), the Walrasian equi-librium is no longer the only long-run equiequi-librium, as long as memory takes at least one period into consideration. Actually, he draws the con-clusion that there appears to be a clear tension between the Walrasian and the Cournot equilibrium, resulting in the whole range of quantities

2.2 the imitation game 7 between the two equilibria being stabilized. Intuitively, this difference is due to the fact that, in the case firms remember profits earned in the past, destabilization of the Cournot equilibrium is less likely to oc-cur. A single firm can, for example, decide to mutate from the Cournot equilibrium and consequently may earn more than the competing firms who decided not to mutate, but the Cournot equilibrium will still be the largest profit remembered. Therefore, the mutating firm will “correct its mistake”, which clarifies the name trial and error of the experimentation process. In other words, the quantities above the Walrasian and below the Cournot one lack stability and so it is highly unlikely that these will be observed. This implies that it is very much likely that, in the long-run, all quantities within the interval limited by the Walrasian and the Cournot quantities will be observed. It is interesting to note that the same pre-diction arises in some oligopoly models with highly sophisticated firms, such as inFerreira(2003) and in Delgado and Moreno (2004).

Alós-Ferrer (2004) furthermore concludes that the result obtained by

Vega-Redondo (1997), since it is based on the relative payoff considera-tions mentioned earlier, is not robust with respect to the absolute pay-off considerations introduced by the model including memory. We can completely ascribe this conclusion to the introduction of memory, for the only difference between the model used by Alós-Ferrer (2004) and

Vega-Redondo(1997) is the implementation of finite memory. Empirical motivation for this implementation is delivered byMatthey (2010), who demonstrates in an experiment that the majority of firms considers the performance of its competitors over more than one period.

Among other things,Alós-Ferrer(2004) demonstrates that memory al-lows for the comparison of the success of a new strategy with that of the previous one. Naturally, if the mutation results in lower payoffs, the mu-tating firm can correct its mistake by going back to the previous strategy. It is therefore evident that for models of bounded rationality this intro-duces a better framework for response considerations without explicitly having to assume that firms compute each best reply. It is this interplay between relative success, doing relatively better than competitors, and better response, improving themselves relative to the past, which influ-ence the two properties of the long-run equilibria. The first property is intertwined with Nash equilibria, namely the intertemporal comparison of own payoffs, whereas the second property is intertwined with global stability and the effects of spite.

Now that we have provided a theoretical framework, we can model imitation and trial and error formally under the best player rule.

2.2 t h e i m i tat i o n g a m e

In the previous section, an overview has been given of the main findings in the existing literature on imitation and trial and error, and the intuition behind the most important theoretical results has been described. In this section, we use this overview to modify the model developed by

Vega-8 m o d e l i n g t h e i m i tat i o n ru l e

Redondo (1997) and Alós-Ferrer (2004) to incorporate the best player rule in the context of a Cournot oligopoly.

To do so, we first have to explain how the market structure should look like for the sake of tractability. Several assumptions on this structure are imposed and explained in the following subsection.

2.2.1 Market structure

To ensure that the model represents a well-behaved Cournot oligopoly, we have to make several assumptions. We first consider a market in which

N firms are active and all produce a homogeneous good. Suppose that

the demand for this good is given by the inverse-demand function P : R+→R+. We now assume that this function P(·)has a strictly positive

intercept when no quantity is produced at all and that it is non-negative, decreasing and concave in the interval [0, Qmax]. The inverse-demand function then equals zero outside of this interval. We further assume that all firms have an identical cost function C : R+ → R+, which is

increasing and strictly convex. For the market to be meaningful, it is also necessary to assume that the marginal costs lie below the intercept of the inverse-demand function.

We can now define the Walrasian and the Cournot quantity as fol-lows. The Walrasian quantity qW is such that P(N qW)qW − C(qW) ≥

P(N qW)q − C(q)for all q ≥ 0. Similarly, the Cournot quantity qC is such

that P(N qC)qC− C(qC)≥ P(N − 1)qC+qq − C(q)for all q ≥ 0. It is interesting to note that the Walrasian quantity yields the optimal out-come when firms are price takers, that is they get the highest profit at a price of P(N qW). In contrast, firms do take into account that their quantity choice affects the price for the Cournot equilibrium, settling at a price of P(N qC). This intuition is explained in more mathematical detail in Appendix A, under the restrictions imposed in Section 2.3. To-gether, these assumptions ensure us that the Walrasian and the Cournot quantities exist, and more specifically that they are unique and strictly positive.

We make the model for the Cournot oligopoly described above explicit in the next subsection, incorporating the concept of memory.

2.2.2 The imitation model

We can now further explain the characteristics of the Cournot oligopoly model under the assumptions stated previously. Time, for instance, is assumed to be discrete in this model. We also assume that all output levels that firms can announce belong to a common finite grid Γ =

{λδ,(λ+1)δ, ..., νδ} with δ > 0, which is thought to be small, and

λ, ν ∈ N, where λ < ν, under the condition that qW and qC are in

this grid. At each time period, firms play the Cournot game by simulta-neously choosing their output levels from this gridΓ. They are assumed to have no knowledge of either the demand or the cost function. The only

2.2 the imitation game 9 information firms have at their disposal when choosing an output level for period t+1 is which quantities were produced and the correspond-ing profits realized by all firms in the last K+1 periods with K ≥ 0, including the current period. We define q_i(t) as the output level of firm

i in period t, and q−i(t)as the vector of output levels of its competitors

in that same period. The profit function of each firm i individually in period t is then given by

πi(t) =Π  qi(t), q−i(t)  =P N X j=1 qj(t) ! qi(t)− C  qi(t)  . (1)

Note that the model without memory, that is with K = 0, corre-sponds to the model used by Vega-Redondo (1997). He uses the con-cept of stochastically stable states, which are those states that are in the support of the limit invariant distribution of the imitation process as the probability ε of experimentation approaches zero, to show that the Walrasian equilibrium is the only possible stochastically stable state. As explained in Schenk-Hoppé (2000) orNewton (2015), the interpretation of stochastically stable states is that, for small but positive ε, the process can be found in these states for most of the time.

We specify the imitation rule used in this model more formally in the following subsection, where we present a mathematical notion of imita-tion and trial and error.

2.2.3 The best player rule

We complete this model for the imitation game by specifying the imita-tion and experimentaimita-tion process. The behavior of firms can be catego-rized into two types in this model, imitation or trial and error. Firms can, for example, simply imitate the quantity that led to the highest profit in the last K+1 periods, but, instead of imitating, a firm can also experi-ment by choosing a random quantity. This experiexperi-mentation happens with a (small) probability of ε ≥ 0 independent across time and firms, where all quantities in the output grid Γ have a positive probability of being selected. This implies that several firms might simultaneously decide to experiment in the same period, but that this occurrence is unlikely for small ε, since the probability of any two given firms experimenting in the same period is proportional to ε2.

We can also formalize imitation and trial and error mathematically, giving more insight in the implications of this rule of thumb. First, we fix a probability ε ∈ [0, 1) and define qi(t) and q−i(t) as described in the previous section. The behavior of each individual firm i can then be categorized as one of the following two types.

The first type of behavior to be considered is “imitate the best player”. First, assume that imitation occurs with probability 1 − ε. Next, define

B_tK as the set of observed output levels produced by the player with the highest profit on average in memory. More formally,

10 m o d e l i n g t h e i m i tat i o n ru l e B_tK =  qj(t)|j ∈ {1, ..., N } and PK i=0πj(t − i) K+1 ≥ PK i=0πj0(t − i) K+1 ∀j0 ∈ {1, ..., N }  . (2)

From this definition, it can be stated that qi(t+1) =q∗, where q∗ ∈ BtK, selected at random if B_tK is not a singleton, according to an uniform probability distribution with full support on B_tK.

The second type of behavior to be considered is experimentation. First, assume that experimentation, or trial and error, occurs with probability

ε. Here qi(t+1) is a randomly chosen quantity from the output grid Γ,

according to an uniform probability distribution with full support on Γ. Furthermore, notice that for each fixed ε, the model described is in fact a finite Markov chain. In such a context, monomorphic states can be defined as mon(q, K) = ((q, ..., q), ...,(q, ..., q)) ∈ ΓN(K+1)_{, for a given}

quantity q ∈Γ. In the model with memory, that is K > 0, this refers to the state wherein all firms have produced the same quantity q ∈Γ for the last K+1 periods. The singletons formed by monomorphic states are in this case the recurrent classes of the process without mutations (ε=0). This implies that, in the occurrence of no mutation, a monomorphic state cannot be left because of the imitation process, and the probability that all firms will be imitating the same quantity will always be strictly posi-tive, given any non-monomorphic state.Young (1993) orEllison (2000), for example, shows that in this case only these states might be stochas-tically stable and that we can therefore restrict our analysis to these monomorphic states only.

Now that we have presented a formal model which incorporates the best player rule, all that remains is to impose several restrictions for the sake of simplicity.

2.3 r e s t r i c t i o n s

In the previous sections, an outline of imitation and memory in a Cournot oligopoly has been given and afterwards a model has been presented, which incorporates the best player rule in this context. To prevent the analysis of this imitation rule of getting overly complex, we impose cer-tain restrictions on this oligopoly model. First of all, a linear inverse-demand function is used, given by P(Q) = 400 − 0.8Q where Q is the aggregate output. Secondly, all firms are assumed to have constant and equal marginal costs, with C(qi) = 4qi for each firm i. These restric-tions remain essentially the same in the second part of the analysis, but the corresponding parameters of the inverse-demand and marginal cost functions may vary. Also, the number of firms is set to N = 5 to begin with, and is extended to 2, 3, 4, 9, 10 and 11 later on in the analysis. Under these restrictions, the Walrasian and the Cournot quantities are

qW =82.5 and qC =99 respectively. A mathematical proof of the value of these quantities is given in AppendixA, under the assumptions of a

2.3 restrictions 11 linear inverse-demand function and constant marginal costs. Since the Walrasian and the Cournot output level are assumed to be included in the output gridΓ, we consider the values δ=0.25, λ=320 and ν=408, or equivalentlyΓ={80, 80.25, ..., 102}.

In this chapter, the model for the best player rule has been presented. The importance and significance of imitation with trial and error in a Cournot oligopoly has been discussed at first, and we have afterwards adjusted the model developed byVega-Redondo (1997) and Alós-Ferrer

(2004) to incorporate the best player rule. In the next chapter, we de-scribe the imitation process of this model, elaborating on the simulation methodology.

3

M E T H O D O L O G Y A N D D ATA

In the previous chapter, the model for the best player rule has been presented. We can run computer simulations with this model to determine how this rule affects the long-run market outcome in a Cournot oligopoly and how this outcome is influenced by memory length. But to do so, it first has to be clear what exactly should be simulated, how we should do this and, more specifically, how many rounds we should consider in each initialization for the oligopoly to end up in a stochastically stable state. In this chapter, we describe the simulation process of the best player rule, providing an explanation on how imitation and trial and error are simulated in this thesis and elaborating on the simulation procedure in matlab. In the next section, we first explain how the imitation process is simulated.

3.1 s i m u l at i n g t h e i m i tat i o n p ro c e s s

Before running any simulations with the model presented in Chapter 2, it has to be clear how we should simulate the best player rule. From the specification of the imitation and experimentation process in Section2.2, it is straightforward to see that the production level for every next time period, or round, should be simulated according to the best player rule or chosen at random from the output grid Γ. So first of all, we have to determine whether a firm is going to imitate or mutate, with probabilities of 1 − ε and ε respectively. If we set this probability of mutation to, for example, five percent, or equivalently ε = 0.05, then each firm in every round has a five percent chance of producing a quantity chosen at random from all possible output levels in the grid Γ, independent of the output levels of its competitors. Naturally, if a firm is not mutating, it is imitating and it has to be determined which quantity will be imitated in each round according to the best player rule, or Equation (2) to be more specific.

14 s i m u l at i n g t h e b e s t p l ay e r ru l e : m e t h o d o l o g y a n d data

But to determine which quantity should be imitated, this rule requires knowledge of all the production levels and corresponding realized profits in the last K+1 rounds. We therefore have to store, for each round in memory, all these decisions together with the corresponding profits and have to update each time the firms progress to a next round. This means that in, for example, round t the output levels and profits of rounds t, t − 1, ..., t − K have to be remembered, but in round t+1 only the output levels and profits of rounds t+1, t, ..., t − K+1 matter and it is no longer relevant to remember what happened in round t − K. Consequently, we first calculate which player has the highest profit on average in the last K+1 rounds, and afterwards set the quantity which will be imitated to the production level of this player in the previous round.

Now that it is clear what we should simulate and how, it remains to present the simulation procedure in matlab.

3.2 s i m u l at i o n p ro c e d u r e

From the previous section, it has become apparent how to simulate the imitation process, but several issues still have to be addressed before it is fully clear what the exact simulation procedure is in matlab. The previous section does not, for example, cover how we should determine which quantity should be imitated in the first round of a run, for there is no memory yet to base this computation upon. We therefore generate, at the beginning of each initialization, a “starting memory”, where for each of the N firms K+1 quantities are chosen at random from the output grid Γ. Before initiating this procedure in matlab, we set the random number generator to rng(1) to ensure that subsequent research can replicate the results of this thesis. Furthermore, note that all quantities in the grid Γ have an equal probability of being chosen, making the market outcome in the long-run independent of the initial state of the Cournot oligopoly.

Randomly choosing this starting memory is, however, not enough to ensure that the imitation process eventually ends up in a stochastically stable state. For the imitation process to converge to a stable state, it is also necessary to consider enough rounds in each run. In other words, there have to be so many rounds that eventually, it does not matter anymore if a firm happens to mutate, all other firms will still imitate the same quantity as prior to this mutation. For that reason, we consider at most 1.000.000 rounds for each of the 1.000 initializations with the mutation parameter ε set to 1%. When the imitation process has indeed converged, we then, as shown in Section2.2, only need to determine which quantity was imitated in the last round for each initialization.

In conclusion, it has been made clear in this chapter what exactly should be simulated and how we should do this in matlab. Using this methodology, we simulate the model described in Section2.2and discuss the results and their interpretation in the analysis of the next chapter.

4

In the previous chapter, we have presented the methodology for simulat-ing the model discussed in Section2.2. In combination with this method-ology, we have simulated this model to determine how the best player rule affects the long-run market outcome in a Cournot oligopoly. Special attention is being paid during these simulations to the question whether a different outcome will result if the number of firms or the memory length changes.

In this chapter, we present and interpret the results of these simulations. Furthermore, the robustness of these results and the limitations of this research are also addressed accordingly. We first present these results in the next section, followed by an interpretation.

4.1 s i m u l at i o n a n a ly s i s

We have simulated the best player rule using the model and methodology presented in Section2.2and Chapter3 respectively. The restrictions im-posed in Section2.3have been taken into account throughout this entire analysis. All production levels reported represent the imitated quantity in the last round for each initialization, where we have set the mutation parameter ε to 1%, and are expressed in units. The results of this research are divided into two separate cases, no memory versus short memory and short memory versus long memory.

In the first case, we compare no memory with short memory to de-termine if the best player rule selects a different imitating quantity if memory is introduced to the model. We show a visual representation of this comparison in Figure1. From this figure we can conclude that intro-ducing finite memory yields qualitatively the same outcome for the best player rule as before this introduction. It turns out that the imitated quantity is the Walrasian one in almost every run and that all other quantities lack stability. This implies that it is very much likely that, in the long-run, the only quantity observed will be the Walrasian one.

16 s i m u l at i o n r e s u lt s Imitated quantity 90 94 98 102 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW Imitated quantity 90 94 98 102 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW (a) K =0 (b) K=1

Figure 1. No memory versus short memory. The likelihood of imitating

a quantity after 10.000 rounds with N=5 firms is shown for K=0 on the left and for K=1 on the right. These probabilities are based on 1.000 initializations with the mutation parameter ε set to 1%. The red line illustrates the Walrasian output level. Imitated quantity 90 94 98 102 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW Imitated quantity 90 94 98 102 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW (a) K =5 (b) K=100

Figure 2. Short memory versus long memory. The likelihood of imitating

a quantity with N=5 firms is shown for K=5 after 10.000 rounds on the left and for K = 100 after 1.000.000 rounds on the right. These probabilities are based on 1.000 initializations with the mutation parameter ε set to 1%. The red line illustrates the Walrasian output level.

4.1 simulation analysis 17 In other words, the Walrasian equilibrium turns out to be the only stochastically stable state. It seems that, when using the best player rule, relative payoff considerations remain more important than absolute ones when memory is introduced to the model. The effects of spite causes us to again favor the Walrasian equilibrium above the Cournot-Nash equi-librium rather quickly. This means that when a firm decides to deviate from the Walrasian equilibrium, it will consequently earn a higher profit, but the competitors who decided to remain in the Walrasian equilibrium will earn even higher profits. Spite therefore implies that, under the best player rule, deviating will result in a bad relative position. The remark can be made that Alós-Ferrer (2004) finds the opposite when using im-itate the best as imitation rule. This explains why the introduction of memory has, in contrast toAlós-Ferrer (2004), no effect on the long-run outcome, since the same reasoning concerning the payoff considerations applies when no memory is present. Moreover, because the two different imitation rules result in exactly the same selection procedure when there is no memory in the model, the same outcome arises as in the research ofVega-Redondo (1997), namely the Walrasian equilibrium.

We find similar results in the second case where we compare short memory with long memory to determine if the memory length affects the selection procedure of which quantity should be imitated under the best player rule. This comparison is visually represented in Figure2where we have set the memory length to a high value to represent long memory, namely K=100. From this figure the conclusion can be drawn that the results remain qualitatively the same for the best player rule no matter what the memory length is. We should, however, point out that the Wal-rasian quantity is imitated a lot less in the long-run for K =100. But we also notice that the probability of imitating the Walrasian quantity in-creases as we consider more and more rounds in each initialization. This means that we expect that the Walrasian quantity will be imitated at the end of almost every initialization regardless of the memory length if we consider enough rounds, and that any other quantity will lack stability. These results therefore indicate that the long-run market outcome in this Cournot oligopoly is independent of the memory length and, more specif-ically, that the Walrasian equilibrium is always the only stochastically stable state in this oligopoly. Moreover, it seems that relative payoff con-siderations are of far greater importance than absolute ones for any choice of memory length when we use the best player rule. Consequently, the memory length has no effect on the outcome under the best player rule and, again, the same outcome arises as in the research ofVega-Redondo

(1997), namely the Walrasian equilibrium.

But the memory length could, nonetheless, still have an effect on the profit that firms earn on average in the long-run. To investigate this possible effect, we have considered the profit per firm earned on average over the last 1.000 rounds for each initialization. The results of this study are presented for different choices of memory length K in Figure 3 and Table1.

18 s i m u l at i o n r e s u lt s

Memory length

0 5 10 15 20 30 40 50 75 100

Profit earned per firm on average

0 50 100 150 200 250 300 350 400 Insignificant Significant

Figure 3. Profit per firm earned on average for each K. The profit

per firm earned on average in the last 1.000 rounds ¯πk is shown for N = 5 firms for each value of K =k. These payoffs are based on 1.000 initializations

with the mutation parameter ε set to 1%, and with 10.000 rounds if K < 100 and 1.000.000 rounds if K =100. The blue bars indicate that the profit is not significantly larger than ¯π0 at a significance level of 1%, whereas the red bars do represent significantly larger profits. Further information on the statistical significance can be found in Table1.

Table 1

Statistical significance of the profit per firm for each K The table presents the estimates, standard errors and the corresponding t-Statistics and p-values of the profits per firm earned on average shown in Fig-ure3. These values are used to test the null hypothesis of ¯πk= ¯π0 when K=k against the alternative hypothesis of ¯πk> ¯π0with a Student’s t-test.

Parameter Estimate Standard Error t-Statistic Probability

¯π0 81.54 143.69 0.000 0.500 ¯π5 67.28 132.53 −0.108 1.000 ¯π10 69.58 141.33 −0.085 0.996 ¯π15 73.06 143.36 −0.059 0.969 ¯π20 79.31 146.28 −0.015 0.686 ¯π30 83.19 164.51 0.010 0.376 ¯π40 102.59 191.41 0.110 0.000 ¯π50 137.57 235.31 0.238 0.000 ¯π75 217.44 315.36 0.431 0.000 ¯π100 359.13 403.61 0.688 0.000

4.1 simulation analysis 19 From this figure and this table it is apparent to see that the profit per firm earned on average over the last 1.000 rounds increases as the memory length increases and that this increase is actually statistically significant for larger values of K with respect to a Student’s t-test with a 1% sig-nificance level. This profit increases since, if the memory length K is extended, we take more output levels into account when we select which quantity we should imitate. But, consequently, this will also mean that we will then take more mutations into account in this selection proce-dure, for if the memory length increases, more time is required before these mutations have disappeared from memory and the selection pro-cedure. Moreover, the differences between the profits of firms earned on average in memory tend to grow bigger as the memory length K further increases, since the profits of non-mutating firms increase the most in the occurrence of a mutation. If the mutating firm then by chance also hap-pens to be the best player in that round, it could occur that this firm will remain the best player after its mutation, leading the market away from the Walrasian equilibrium and resulting in much higher payoffs. Such an occurrence will happen more frequently as the memory length K in-creases since the selection procedure will then take more mutations into account. Firms can therefore increase their profits earned on average in the long-run under the best player rule by selecting, as a group, a longer memory length.

The question now arises whether a firm can also strengthen its position

relative to its competitors by individually selecting a different memory

length. If we for instance consider N =5 firms active in the market and a single firm decides to select a different memory length, we would then have two groups who each select a different memory length, one with 4 firms and another with just one firm. As we can see from FigureB.1 in Appendix B, no group would in this case earn a higher profit than its competing group regardless of the memory length a group adopts, since relative payoffs are not only dominant between groups but also remain so between firms individually, and the Walrasian equilibrium would remain to be the only stochastically stable state in this oligopoly. Selecting a different memory length as an individual firm will therefore not gain any relative advantage. It actually turns out that this result remains qualitatively the same no matter the size of each group or the number of groups in total. Even if firms can decide on the memory length they adopt individually as shown in FigureB.2, they will still not gain any relative advantage. We can therefore conclude that selecting a different memory length, as an individual or as a subgroup, will not, under the best player rule, strengthen the position of a firm relative to its competitors and will still lead to the Walrasian equilibrium in the long-run.

It remains of interest to determine how robust these results are with respect to certain parameter changes, such as the number of firms or the slope and intercept of the inverse-demand function for example. We out-line the outcome of this investigation in the second part of this analysis, followed by a discussion of the limitations of this research.

20 s i m u l at i o n r e s u lt s

4.2 ro b u s t n e s s

We have made clear in the previous section that, under the restrictions imposed in Section 2.3, the Walrasian equilibrium is the only stochasti-cally stable state regardless of the memory length. But it is still unclear whether the same outcome arises for different values of N , that is the number of firms in the Cournot oligopoly.

In the first part of the analysis, we fixed the number of firms active in the market at a single value, namely N = 5. We have afterwards expanded this analysis by also evaluating the market with 2, 3, 4, 9, 10 and 11 firms and visualized this for N = 3 and N = 9 in Figure B.3 and Figure B.4 respectively. Each of these additional values of N yields qualitatively the same results as when we consider N =5, meaning that the results are robust with respect to the number of firms N .

The number of firms active in the market does have an effect, however, on the profit that firms earn on average in the long-run. We can see from Figure4and Table 2that the profit per firm earned on average over the last 1.000 rounds no longer increases significantly as the memory length increases, but that this profit is relatively high and actually remains more or less the same for each value of K. Intuitively, this difference is due to the fact that the number of firms active in the market has increased and that the profits of firms earned on average in memory vary less for a longer memory length. Naturally, an increase in the number of firms coincides with taking more output levels into account, and consequently also more mutations, when determining which quantity should be imi-tated. This explains why the overall profit is, regardless of the value of

K, relatively high. The likelihood that a mutating firm also happens to

be precisely that one firm that is the best player in that round now de-creases since simply more firms could be mutating. Moreover, the memory length K will barely matter anymore for the probability that this occur-rence takes place. A mutation will therefore lead the market away from the Walrasian equilibrium less often when the number of firms active in the market increases. The same reasoning can be followed in the oppo-site direction when the number of firms active in the market decreases, resulting in qualitatively the same outcome as for N = 5 as we can see from FigureB.5 and Table3 in AppendixB.

Besides the value of N , we are also particularly interested to deter-mine whether the analyses we made previously will lead to more or less the same results under less strict restrictions. The two most important restrictions imposed in Section2.3are the use of a linear inverse-demand function P(Q) =a − bQ, with a=400 and b=0.8, and the assumption of constant and equal marginal costs for each firm, or C(qi) =cqi with

c=4. The main purpose of these restrictions is to ensure that the inverse-demand function is linear and that the marginal costs are constant and equal for each firm. At first, we only considered unilateral changes in the parameters, allowing for any strictly positive value of a, b and c while still taking the purpose of these restrictions into account.

4.2 robustness 21

Memory length

0 5 10 15 20 30 40 50 75 100

Profit earned per firm on average

0 20 40 60 80 100 120 140 160 180

Figure 4. Profit per firm earned on average for each K with more firms. The profit per firm earned on average in the last 1.000 rounds ¯πk is shown for N = 10 firms for each value of K =k. These payoffs are based on

1.000 initializations with the mutation parameter ε set to 1%, and with 10.000 rounds if K < 100 and 1.000.000 rounds if K = 100. The blue bars indicate that the profit is not significantly larger than ¯π0 at a significance level of 1%. Further information on the statistical significance can be found in Table2.

Table 2

Statistical significance of the profit per firm with more firms The table presents the estimates, standard errors and the corresponding t-Statistics and p-values of the profits per firm earned on average shown in Fig-ure4. These values are used to test the null hypothesis of ¯πk = ¯π0when K =k against the alternative hypothesis of ¯πk> ¯π0with a Student’s t-test.

Parameter Estimate Standard Error t-Statistic Probability

¯π0 174.32 117.89 0.000 0.500 ¯π5 177.91 116.78 0.031 0.166 ¯π10 164.922 116.97 −0.080 0.994 ¯π15 166.99 115.45 −0.064 0.978 ¯π20 165.74 117.11 −0.073 0.990 ¯π30 164.52 122.90 −0.080 0.994 ¯π40 153.39 130.44 −0.161 1.000 ¯π50 157.57 135.14 −0.124 1.000 ¯π75 152.96 150.92 −0.142 1.000 ¯π100 173.36 165.43 −0.006 0.573

22 s i m u l at i o n r e s u lt s

Qualitatively the same outcome results from these unilateral deviations as before these deviations, making it interesting to further investigate bi-and trilateral changes in these parameters. Figure B.6 and Figure B.7 show two of such trilateral changes in the parameters. From these figures we can conclude that these simultaneous changes turn out to also yield qualitatively the same outcome, for every combination of the three pa-rameters actually. This means that the results are robust with respect to changes in the parameters a, b and c.

In line with these additional analyses, we have also investigated if the number of rounds in each initialization was sufficient for the imitation process to converge and if we had considered enough initializations. For this purpose, all results were simulated with both 10.000 and 100.000 rounds in each initialization and subsequently compared to determine whether the imitation process has already converged after 10.000 rounds. It turns out that for every value of the model parameters N , K, a, b and

c both simulations yield qualitatively the same outcome, meaning that

10.000 rounds are enough to consider for the imitation process to con-verge. For illustrative purposes, we have included FigureB.8to visualize this comparison. The only exception is when K = 100, in which case 10.000 rounds are not enough for the imitation process to converge since it requires more time before we favor one firm above another. Intuitively, this is due to the fact that the profit of a firm earned on average in the last 100 rounds does not alter very much after one successful output choice in a round. We should therefore consider at least 1.000.000 rounds when

K=100. We can note though, that if the mutation parameter ε is set to a lower rate of 0.1% for example, the imitation process converges faster and that 10.000 rounds would already suffice for this convergence even for

K=100. The resulting state would, however, be no longer stochastically stable if the mutation parameter ε was set to a higher rate. Additionally, we have made a comparison of simulations considering 1.000 initializa-tions with those considering 10.000 initializainitializa-tions, and both simulainitializa-tions yield, as we can see in for example Figure B.9, qualitatively the same outcome for every value of the model parameters. In other words, the results are robust with respect to the number of rounds we consider in each initialization and with respect to the amount of initializations.

Even though we have made an extensive analysis of the best player rule, there are still some limitations in the research in this thesis. The main focus of this thesis was, for example, on a linear inverse-demand function and constant and equal marginal costs. It would therefore be interesting for future research to investigate whether the use of a non-linear inverse-demand function or non-constant and perhaps even unequal marginal costs will lead to qualitatively the same outcome. It would also be worth investigating if the same outcome will result in a Bertrand or a Stackelberg competition. Another possible subject of investigation could be if different imitation rules will lead to different behavior of firms.

In conclusion, we have investigated how the best player rule affects the long-run market outcome in a Cournot oligopoly and how this is

influ-4.2 robustness 23 enced by memory length. We present a summary of the most important findings as well as some concluding remarks in the next, and also final, chapter of this thesis.

5

The main objective of this thesis was to determine how the best player rule affects the market outcome in the long-run in a Cournot oligopoly and how this outcome is influenced by memory length. This is partic-ularly interesting since in the real world of oligopolies, firms have to operate without complete knowledge of the market and the price dynam-ics. It therefore makes sense for firms to use simple rules of thumb in their decision-making to cope with these uncertainties, such as imitation with trial and error. Combining imitation with trial and error leads al-most automatically to the best player rule. To analyze such an imitation rule, we have adjusted a pre-existing model for a Cournot oligopoly to incorporate this rule for simulation purposes.

After an extensive study, we have found that the Walrasian equilibrium is the only stochastically stable state in this oligopoly regardless of the memory length a firm adopts. In other words, we expect that the Wal-rasian quantity is the only quantity that will be observed in this market in the long-run, and that all other output levels lack stability and will therefore not be observed in the long-run. Intuitively, this is due to the fact that relative payoffs remain of greater importance than absolute ones when memory is introduced to the best player rule. This study was ex-tended with some additional research, investigating the robustness of this outcome with respect to certain parameter changes such as the number of firms. It has turned out that changing the number of firms or the param-eters of the inverse-demand and cost functions leads to qualitatively the same results, and that 10.000 rounds suffice for the imitation process to converge. We can therefore conclude that the best player rule yields the Walrasian outcome in the long-run in a Cournot oligopoly, regardless of the number of firms or the memory length, and that this result is robust with respect to the model parameters.

We were primarily focused on a linear inverse-demand function and constant and equal marginal costs in our research in this thesis. But, naturally, a non-linear inverse-demand function or non-constant marginal

26 c o n c l u s i o n

costs could also be of interest. This makes it particularly interesting for future research to investigate whether the results remain qualitatively the same when we impose less strict restrictions. Future research should also focus on the type of competition to investigate whether the same outcome will result in a Bertrand or a Stackelberg oligopoly. Another possible subject of interest could be to explore how the behavior of firms responds to other imitation rules.

The results presented in this thesis are not only theoretically relevant, but also empirically. Real firms in an oligopoly could base their strategic decisions on this simple rule of thumb without the need of computing best replies. But if these firms wish to maximize their profits, this would not be the right course of action for them, since they will most prob-ably end up in the highly competitive and barely profitable Walrasian equilibrium in the long-run. Real firms in an oligopoly should therefore not incorporate the best player rule and are advised to follow a different imitation strategy.

A

Recall the profit function given in Equation (1) in Section 2.2. A more general version of this function, invariant of time, is given by

πi =P(Q)qi− C(qi).

Now consider the restrictions imposed in Section2.3. In line with these restrictions, we assume that the inverse-demand function is of the linear form P(Q) =a − bQ, with a, b > 0, and that all firms have constant and

equal marginal costs of the form C(qi) =cqi, with c > 0. The approach for deriving the Walrasian and the Cournot equilibria is very similar, but differs on one point substantially. In the Walrasian equilibrium, firms act as price takers, assuming that P(Q) =p is given, whereas firms do take

into account that their quantity choice affects the price for the Cournot equilibrium.

In the case of the Walrasian equilibrium, the maximization problem of each individual firm i is given by

max qi∈R+ πi = max qi∈R+ {pq_i− C(qi)}= max qi∈R+ {(p − c)qi}. (3)

The corresponding supply function S_i(p)of firm i is then defined as

Si(p) =    ∞ if p > c q if p=c 0 if p < c (4)

where q ∈ R₊. Next, recall the condition that in a market equilibrium demand should equal supply, or

D(p) =S(p) (5)

where D(p) =P−1(p) = a−p_b and S(p) =PN

i=1Si(p)are the demand and aggregate supply functions respectively. We can now note that such an equilibrium only exists if p=c, and also that this production decision is

28 wa l r a s i a n v e r s u s c o u r n o t e q u i l i b r i u m

exactly the same for each individual firm, since they are assumed to have equal and constant marginal costs. Each firm will therefore produce the same quantity in the symmetric Walrasian equilibrium, and S(p) =N q.

Together with Equation (5), this implies that the production level is given by

qW = a − c

N b . (6)

From this equation, we can verify that under the restrictions imposed in Section2.3, the Walrasian quantity is in fact qW =99.

The computations involved in the derivation of the Cournot equilib-rium are slightly more complicated. First of all, the maximization prob-lem of each individual firm i is given by

max qi∈R+ πi = max qi∈R+ {P(Q)qi− C(qi)}= max qi∈R+ {(a − bQ − c)qi}. (7)

Following the standard procedure of profit maximization requires the use of the first and second order conditions. The following constraint must be met to satisfy the first order condition of this maximization problem.

FOC: ∂πi

∂qi

=0

a − bQ − bqi− c=0

It follows from this condition that the best response function of each firm

i is then defined as

qbr_i (q−i) =

a − bq−i− c

2b . (8)

We can now note that this decision is exactly the same for each individ-ual firm, since we have assumed that all firms have eqindivid-ual and constant marginal costs. Each firm will therefore produce the same quantity in the symmetric Cournot equilibrium, and qC₁ =q₂C =...=qC_N =qC. Together with Equation (8), this implies that the production level is given by

qC = a − c

(N+1)b. (9)

It now remains to verify the second order condition, to ensure that this is actually a maximum. SOC: ∂ 2_π i ∂q2 i   _q i=qC ≤ 0 −2 ≤ 0

The Cournot quantity given in Equation (9) satisfies the first and second order conditions, so from this equation we can verify that under the restrictions imposed in Section2.3, the Cournot quantity is in fact qC = 82.5.

B

30 a d d i t i o n a l f i g u r e s a n d ta b l e s

Profit per group on average

-200 0 200 400 600 800 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Group 1 Group 2

Figure B.1. Two groups adopting a different memory length. The

like-lihood of earning a profit per group on average in the last 1.000 rounds is shown for 4 firms in the first group, who consider a memory length of 5, and only 1 firm in the second group, who incorporates a memory length of 20. These prob-abilities are based on 1.000 initializations with 10.000 rounds and the mutation parameter ε set to 1%.

Profit per group on average

-200 0 200 400 600 800 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Group 1 Group 2 Group 3 Group 4 Group 5

Figure B.2. Firms adopting a different memory length individually.

The likelihood of earning a profit per individual firm on average in the last 1.000 rounds is shown for a total of 5 firms who consider a memory length of 0, 5, 10, 20 and 30 respectively. These probabilities are based on 1.000 initializations with 10.000 rounds and the mutation parameter ε set to 1%.

a d d i t i o n a l f i g u r e s a n d ta b l e s 31 Imitated quantity 156 160 164 168 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Simulations qW Imitated quantity 156 160 164 168 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Simulations qW (a) K =0 (b) K=5

Figure B.3. Imitated quantities involving less firms. The likelihood of

imitating a quantity after 10.000 rounds with N =3 firms is shown for K=0 on the left and for K =5 on the right. These probabilities are based on 1.000 initializations with the mutation parameter ε set to 1%. The red line illustrates the Walrasian output level.

Imitated quantity 46 50 54 58 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Simulations qW Imitated quantity 46 50 54 58 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Simulations qW (a) K =0 (b) K=5

Figure B.4. Imitated quantities involving more firms. The likelihood of

imitating a quantity after 10.000 rounds with N =9 firms is shown for K=0 on the left and for K =5 on the right. These probabilities are based on 1.000 initializations with the mutation parameter ε set to 1%. The red line illustrates the Walrasian output level.

32 a d d i t i o n a l f i g u r e s a n d ta b l e s

Memory length

0 5 10 15 20 30 40 50 75 100

Profit earned per firm on average

0 1000 2000 3000 4000 5000 6000 7000 8000 Insignificant Significant

Figure B.5. Profit per firm earned on average for each K with less firms. The profit per firm earned on average in the last 1.000 rounds ¯πk is shown for N = 2 firms for each value of K = k. These payoffs are based on

1.000 initializations with the mutation parameter ε set to 1%, and with 10.000 rounds if K < 100 and 1.000.000 rounds if K=100. The blue bars indicate that the profit is not significantly larger than ¯π0at a significance level of 1%, whereas the red bars do represent significantly larger profits. Further information on the statistical significance can be found in Table3.

Table 3

Statistical significance of the profit per firm with less firms The table presents the estimates, standard errors and the corresponding t-Statistics and p-values of the profits per firm earned on average shown in Fig-ure B.5. These values are used to test the null hypothesis of ¯πk = ¯π0 when

K=k against the alternative hypothesis of ¯πk> ¯π0 with a Student’s t-test. Parameter Estimate Standard Error t-Statistic Probability

¯π0 436.45 971.87 0.000 0.500 ¯π5 1392.98 1834.71 0.521 0.000 ¯π10 2386.48 2261.51 0.862 0.000 ¯π15 3005.08 2396.80 1.072 0.000 ¯π20 3516.70 2619.92 1.176 0.000 ¯π30 4446.74 2736.78 1.465 0.000 ¯π40 5182.00 2725.47 1.741 0.000 ¯π50 5770.34 2935.11 1.817 0.000 ¯π75 6704.16 2806.15 2.234 0.000 ¯π100 7546.72 2794.68 2.544 0.000

a d d i t i o n a l f i g u r e s a n d ta b l e s 33 Imitated quantity 81 85 89 93 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW Imitated quantity 81 85 89 93 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW (a) K =0 (b) K=5

Figure B.6. Imitated quantities with parameter combination 1. The

likelihood of imitating a quantity after 10.000 rounds with(a, b, c) = (189, 0.4, 9) and N=5 firms is shown for K=0 on the left and for K=5 on the right. These probabilities are based on 1.000 initializations with the mutation parameter ε set to 1%. The red line illustrates the Walrasian output level.

Imitated quantity 81 85 89 92 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW Imitated quantity 81 85 89 92 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW (a) K =0 (b) K=5

Figure B.7. Imitated quantities with parameter combination 2. The

likelihood of imitating a quantity after 10.000 rounds with(a, b, c) = (901, 2, 1) and N=5 firms is shown for K=0 on the left and for K=5 on the right. These probabilities are based on 1.000 initializations with the mutation parameter ε set to 1%. The red line illustrates the Walrasian output level.

34 a d d i t i o n a l f i g u r e s a n d ta b l e s Imitated quantity 90 94 98 102 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW Imitated quantity 90 94 98 102 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW (a) K =0 (b) K=5

Figure B.8. Imitated quantities based on more rounds. The likelihood of

imitating a quantity after 100.000 rounds with N=5 firms is shown for K=0 on the left and for K =5 on the right. These probabilities are based on 1.000 initializations with the mutation parameter ε set to 1%. The red line illustrates the Walrasian output level.

Imitated quantity 90 94 98 102 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW Imitated quantity 90 94 98 102 Probability 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Simulations qW (a) K =0 (b) K=5

Figure B.9. Imitated quantities based on more initializations. The

like-lihood of imitating a quantity after 10.000 rounds with N = 5 firms is shown for K=0 on the left and for K =5 on the right. These probabilities are based on 10.000 initializations with the mutation parameter ε set to 1%. The red line illustrates the Walrasian output level.

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