University of Amsterdam
The stability of the dynamic Cournot equilibrium for a finite number of
firms with heterogeneous heuristics
Friso Cuijpers (10216995)
Bachelor thesis econometrics, supervised by Marius Ochea 25-06-2014
Abstract
This thesis tries to answer the question: How does the number of firms affect the stability of an oligopoly market consisting of naïve - and rational firms? In order to answer this question a model has been programmed to compute all outcomes for a dynamical Cournot model with naïve – and rational firms for any number of firms, starting values, adaption period and memory. The results show that two kinds of stable cycles arise. The amount of firms only affects the location of these cycles, but does not affect the stability of the markets.
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Introduction
Firms in an oligopoly compete with each other every day, month, year or period. How each firm decides what they are going to do the next period is determined by their strategy or heuristic. In the standard model it is assumed that firms know everything of their competitors and all of the firms use the same heuristic. In the real world firms do not know everything and do not use the same strategy every period. One of the first to come with a dynamical model to study firms switching between heuristics and having imperfect information was Sidney (1964), he
introduces the term “routine” which are the strategic characteristics of a firm. Firms using certain routines do better than firms with other routines, the firms using the bad routine will switch to the good routine and the bad routines die out just like natural selection.
This thesis is about dynamical Cournot models. These dynamical Cournot models are dynamical models to model competition between firms by letting the firms decide how much they will produce the next period. Many researchers like Droste (2002) have studied the stability of the Cournot-Nash equilibrium for a dynamic or evolutionary Cournot model. Most of these models assume a large number of firms, all firms can use several heuristics and each period two firms are drawn to compete in a Cournot duopoly. The two firms do not know which heuristic is used by the competitor. The reason to use this model for dynamical oligopolies is that the duopoly forms a dynamical system which can be analytically computed. This model might however not be very realistic for oligopolies, because even in the most basic Cournot model the results for an oligopoly are very different from a duopoly. That is why in this article all of the firms will compete against each other every period. This leads to more realistic results, but the dynamical systems become too extended to analytically compute.
The model which is used in this thesis is a dynamical Cournot model in which a finite number of firms compete against each other and can choose between a finite number of heuristics. For a prisoner’s dilemma Rubinstein (1986) argues that operating behavioral rules is costly. It also seems logical that using a more sophisticated strategy that uses more information would be more costly than simply imitating the other firms, for the firm needs better employees to be able to predict what the other firms will do and do the calculations needed. Therefore in the researched model a strategy using more information and computations will be more costly to use.
Each firm chooses a strategy from a finite set of behavioral rules. The different
behavioral rules and costs associating with the rules are known by all firms. The amount of firms using a certain behavioral rule and all profits made by these firms are known and can be used
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by firms to decide which behavioral rule they will use next period. Because firms use the past performance of a behavioral rule to decide whether they will switch and to which rule they will switch, rules with lower payoffs in all situations will eventually die out. Over time the model can converge to an equilibrium with all firms using the same behavioral rule, but the model may also come in a cycle where firms keep switching in the same pattern or the model will not converge at all.
This thesis addresses the following questions: How does the number of firms affect the stability of a market consisting of naïve - and rational firms? For different amounts of firms the model may have different stability characteristics. The stability characteristics being the point or cycle to which the model converges or whether it does not converge at all. This thesis is
interested to see whether the stability characteristics change with the amount of firms and how they change. This is researched by programming a model in which the amount of firms can be determined by the user. The output of the model will be vectors containing the produced quantities, the amount of firms using a certain strategy and the gained profits by each firm. The stability characteristics are analyzed using this model output.
The rest of the thesis is organized as followed. In section two the Cournot model and used model will be discussed. Section three will analyze model output. Section four will be the conclusion. The discussion will be in section five.
2 The Cournot Model
In paragraph two the standard Cournot model will be discussed and extended to the dynamical Cournot model with learning rules and eventually the used model.
Consider a homogeneous oligopoly Cournot model with f firms1. The inverse demand function P(Q) is a twice continuously differentiable function with ( ) and ( ) for every Q. Q is the aggregate production, ∑ , qi is the production of firm i. C(qi) is the
twice continuously differentiable cost function for every firm, with ( ) and ( ) for every qi. Each firm has profit function ( ) ( ), with ∑ .
Every firm wants to maximize its profit . This leads to the first order condition:
( ) ( ) ( ) (1)
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and to ensure a local maximum second order condition: ( ) (
)
( ) . The best-response or reaction curve is found by solving (1) for qi. The
best-response function gives that qi that maximizes the profit of firm i, given what the other f-1 firms
do,
( ) (2)
In the standard Cournot model it is assumed that all the firms have full information and thus know what the other firms do and can use this to solve the best-response function to maximize their profit. This thesis analyzes what happens when not all of the firms have all information and using more information will cost the firm more. Within a dynamical Cournot model it is possible to let some firms use less information or less up-to-date information and analyze the effects.
Within a dynamic setting the results found in (1) and (2) can still be used. The best-response function now computes the qi,t as a function of Q-1,t in which qi is the quantity for a
certain period t and Q-i,t is the aggregate quantity of all the competitors for a certain period t.
Assuming full information the response function can now be written as:
( ) (3)
In result (3) firms use the aggregate quantity of the competitors in a period t, to determine their own quantity for that same period t. Result (3) can be altered to have firms only use information from previous periods to determine their quantity for the future.
2.1 Learning Rules
Learning rules are strategies that firms can use to determine their future quantity. This thesis will focus on two of these strategies. The strategy defined by result (3) is the rational strategy, this strategy uses full information and thus will be costly to use. Strategies do not necessarily use full information. An example of a strategy that only uses information from previous periods, is the strategy where the firm makes his decision based on the competitors’ quantities from last period. It seems that it is not so smart to use this strategy, yet it is a very well possible strategy for firms in the real world. For markets that are near or in a Nash-equilibrium and will probably stay around that equilibrium it is much easier and cheaper for firms to react to the previous
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period and assume that their competitors will do the same next period, while not losing much income because they are near a Nash-equilibrium. From now on this strategy will be called the naïve strategy. For the naïve strategy Theocharis (1960) has shown that the Nash-equilibrium of a market consisting of players that are all using this strategy is unstable for three or more
players.
In this thesis firms with different strategies compete against each other and firms will get the possibility to switch between strategies based on the past performance of firms using that strategy. This way firms can switch to the strategy that seems better. There are infinite possible combinations of strategies that can be analyzed, but this thesis focusses on rational firms vs. naïve firms.
2.3 The Model
The model will be a dynamical Cournot model with both rational firms and naïve firms. The total number of firms is given by f and the number of firms using the Nash-strategy will be given by n. The firms using the same strategy have the same inverse demand function, cost function and best-response function, and therefore they will produce the same quantity. The quantity produced by a naïve firm will be denoted as xt and the quantity produced by a rational firm will
be denoted as yt. The used inverse demand function and cost functions are:
Inverse demand ( ) ( ( ) )
costs ( ) for the rational firms
costs ( ) for the naïve firms
With a, b, c, d ≥ 0.
( ) for every combination of xt and yt. ( ) for both xi and yi. The second
order condition: ( ) ( ) ( ) , so solving the first order
condition will give a local maximum. The inverse demand function and cost functions satisfy the conditions for Cournot models.
The profit functions are given by:
( ) ( ) ( ( ( ) )) ( ) ( ) ( ( ( ) ))
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Based on the average profits of the previous m periods a firm change its strategy every ap periods. m is the memory of the firms and ap is the adaption period, the adaption period is added because firms in the real world will change their quantity more often than their strategy. The rational firms know about this mechanic to predict nt+1. Computing the derivatives of the
profit functions and solving them to xt and yt will give the response functions. The
best-response functions are:
( ) (( ) ( ) ) (4) ( ) (( ) ( ) ) (5)
xt+1 only depends on x, y and n from period t, because they assume that the output from their
competitors will stay the same in period t+1. yt+1 depends on x, y and n from period t+1 and thus
predicts the future by using more information.
For any market the Nash-equilibrium can be computed with the following equation:
( ) (6)
The used model is in the appendix, together with an explanation on how to use it.
3 Results
In this section the results gained by using the programmed model are analyzed, in order to find the effect of the amount of firms on the stability of the market. For a given market the amount of rational firms in a market will be analyzed first and secondly the produced quantities will be analyzed.
All outputs are generated with the same parameter values for the inverse demand functions and cost functions. These values are , , and .
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For various amounts of firms has the model been run to gain an overview of the different markets and the different kinds of cycles to which the dynamical system converges.
(Graph 1) The amount of rational firms over 100 periods for various amounts of firms
All the results are gained by running the programmed model for the number of firms (f) 2 up to and including 20, starting values ⌈ ⌉, and , adaption period and memory for 100 periods. If a result is gained with a different input, then only the part which is different from this input will be given.
In graph 1 there are three kinds of stable cycles and two stable points. The stable points are the markets that converge to a point where all the firms use one of the strategies. The market with four firms converges to a stable point with only rational firms. This market reaches that stable point in which all firms are rational firms because in the first few periods where other markets slowly converge to a stable cycle, this market at some point reaches a point where all firms are rational firms and within the programmed model it is impossible to get out of that state.
The other markets with a stable point converge to a point where all firms are naïve firms. These markets start producing the Nash-quantity or something very close to it, when there are both naïve and rational firms. When producing the Nash-quantity both the naïve and rational strategy will produce the Nash-quantity again in the next period, making it more profitable to use the naïve strategy.
The other markets either converge to a stable cycle in which the amount of rational firms creates a triangular shape with the t-axis (cycle 1), a stable cycle in which the amount of rational firms creates a triangular shape with the t-axis with a dent in the top (cycle 2) or stable cycle in
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which the amount of rational firms creates a triangular shape with the t-axis with a dent in the bottom (cycle 3). For larger amounts of firms the number of rational firms within these cycles is always greater than zero or only hits zero for one period, for example the markets with eighteen and twenty firms. For smaller amounts of firms will the number of rational firms become zero for several periods, this happens because a negative amount of rational firms is not realistic.
Another thing is that the maximum amount of rational firms in these cycles does rise when the number of firms rises, but the maximum amount of rational firms stays relatively low. The maximum amount of rational firms in these cycles is always approximately 1/3 of the total number of firms.
Graph 1 also shows that there is no connection between the amount of firms and the length of the stable cycle. The stable cycle for eighteen firms is twelve periods long, the stable cycle for fourteen firms is fourteen periods long and the stable cycle for six firms is ten periods long.
One of the markets which converges to cycle 1 is the market with twenty firms. All markets that converge to cycle 1 are very comparable, but note that for a smaller amount of firms there will be periods in which the amount of rational firms becomes zero.
(Graph 2) The amount of rational firms over 100 periods for 20 firms
After eighteen periods has this market converged to a stable sixteen period cycle. The Nash-equilibrium for this market is .
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(Graph 3) The produced quantities over 100 periods for 20 firms
The produced quantities also converge to a stable cycle of sixteen periods. The quantities start to fluctuate when the amount of rational firms hits zero, this probably happens because a
market consisting solely out of naïve firms is not stable, like Theocharis (1960) has shown. Then every period one more firms starts to use the rational strategy until the market gets close to the part where the quantities are approximately 1/3. 1/3 is the Nash-equilibrium for this market, so when the quantities start to get closer to 1/3 it becomes more profitable to use the naïve strategy. It is more profitable because the naïve strategy is cheaper to use and near the Nash-equilibrium will the benefits not change that much.
Besides stable cycle 1, two other cycles can be seen in graph 1. The two cycles are very similar to cycle 1. When looking at the amount of rational firms is stable cycle 2 the same as stable cycle 1 but with a period dent in the top and stable cycle 3 is the same but with a 1-period dent in the bottom. Two examples can be seen graph 4. For these stable cycles the produces quantities start to diverge from the Nash-equilibrium when the amount rational firms hits the minimum of the cycle. The diversion from the Nash-equilibrium makes it more profitable to switch to the rational strategy and as the amount of rational firms goes up, will the produced quantities converge towards the Nash-equilibrium again. When close to the Nash-equilibrium is becomes more profitable to switch to the naïve strategy and this keeps repeating itself.
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(Graph 4) The amount of rational firms over 100 periods for 12 and 16 firms
The starting values of the produced quantities dot not matter that much because the market tends to converge towards the Nash-equilibrium. However the convergence of the markets is highly dependent on the starting value for the number of rational firms.
(Graph 5) The amount of rational firms over 100 periods for 15 firms with different n0
The market with fifteen firms converges to a stable point where all firms use the naïve strategy, but when the starting value n0 is changed to six instead of eight the market converges to a
stable cycle 1. This same effect has been researched for all the markets and it is still unclear when and why exactly the convergence of a market changes its stable point or cycle. However
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even when looking at all the possible values for n0, all the markets still converge to one of the
two stable points or one of the three stable cycles.
Based on infinite population models one could expect multiple attractors or chaos to arise when changing the starting value n0. What the exact reason is why this kind of behavior
does not arise now is not known, but there are a few things different from the standard infinite population models that might explain this. First thing is that this model only seeks for solutions within the economically feasible solutions, so positive prices and positive quantities. Second thing is that this model uses a very simple rule to determine the next number of rational firms nt+1, while the standard infinite population models use more difficult dynamics like the replicator
dynamics or logit dynamics.
Although n0 affects to which cycle or point the market converges, it will not change the
conclusions because only the effect of the amount of firms is researched and the markets still converge to one of the stable points or cycles.
Previous results were for ap=m=1, but do the results still hold for different ap and m?
(Graph 6) The amount of rational firms over 200 periods for 16 firms with ap=m=2
The amount of rational firms for this market with ap=m=1 can be seen in graph 4. By changing ap and m to two the stable cycle has switched from a stable cycle 3 to a stable cycle 1. Also the minimum amount of rational firms has changed from one to zero. For all markets the effects of higher ap and m, with ap=m, has been researched. Just like for n0 it seems that ap and m do
affect to which stable point or cycle the market converges, but still all markets converge to one of the two stable points or one of the three stable cycles. Because all of the markets with ap=m
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still converge to the same stable points or cycles, is the effect of the amount of firms on the stability of the market unchanged.
The results show that the markets always converges to a stable point where all firms use one of the strategies or to a stable cycle. The amount of firms does not affect the length of these stable cycles, but does affect the location of the cycles. The maximum number of rational firms is always approximately 1/3 of the amount of firms. The same five stable points or cycles arise independent of the amount of firms is and thus the amount of firms does not have an effect on the stability of the market.
Both kinds of cycles have one thing in common. The number of rational firms only goes up at some point because having more naïve firms makes the market less stable, like
Theocharis (1960) has shown. Because the market starts to diverge from the Nash-equilibrium when there are more naïve firms, there always arises a point where the rational firms can start making more profit than the naïve firms, making the number of rational firms go up.
4 Conclusion
This thesis tries to answer the question: How does the number of firms affect the stability of an oligopoly market consisting of naïve - and rational firms? In order to answer this question a model has been programmed to compute all outcomes for a dynamical Cournot model with naïve – and rational firms for any number of firms, starting values, adaption period and memory.
This model is computed for the numbers of firms two up to and including twenty. These results show that for a small number of firms the market converges to a stable point, while for a large number of firms the market converges to either a stable cycle 1 or a stable cycle 2. The number of firms does not affect whether the market converges to a stable point or cycle or whether it diverges. Because all the markets converge to either a stable point or cycle. For the stable cycles there is no relation between the length of the stable cycles and the number of firms. The number of firms does affect the location of the stable cycles, the maximum of the stable cycles is always approximately 1/3 of the amount of firms.
The number of firms does not affect the stability of the market, because all the markets converge to the same stable points or cycles without a clear relation with the amount of firms.
This thesis also confirms Theocharis’ (1960) result. Theocharis has shown that a market consisting solely out of naïve firms that are not producing the Nash-quantity is not stable. For both large and small number of firms the results show that the produced quantities in a market with few or no rational firms diverges from the Nash-equilibrium.
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5 DiscussionIn this section assumptions and flaws are listed and explained.
Due to a lack of time the effect of the amount of firms on the stability of the market with an adaption period, longer memory or both (with ap≠m) has not been analyzed. The
programmed model does the ability to compute these results.
As seen in graph 1, once a market only has rational firms there will never be any naïve firm. A market with only rational firms will produce the Nash-equilibrium and within this
equilibrium it is cheaper to switch to the naïve strategy. The rational firms never change their strategy to the naïve strategy because within the model firms only change their strategy based on the previous periods. In the previous periods there were no naïve firms so the profits gained by the naïve strategy in the previous periods is zero, any profit gained by producing the Nash-equilibrium as a rational firm is greater than zero so no rational firm will ever change its strategy. The model can be extended to make it possible for rational firms to compute what their profits would have been when they change their strategy, but the rational will have to do more
computations which should be more costly. The decision has been made not to add this feature because there only was one market in which this occurred.
6 References
- Droste E., Hommes C., Tuinstra J. (2002). Endogenous fluctuations under evolutionary
pressure in Cournot competition. Games and Economic Behavior 40, pp. 232-269.
- Rubinstein A. (1986). Finite Automata Play the Repeated Prisoner’s Dilemma. Journal of Economic Theory 39, pp. 83-96.
- Sidney G. & Winter JR. (1964). Economic “Natural Selection” and the Theory of the
Firm. Yale Economic Essays Vol. 4, No. 1, pp. 225-272.
- Theocharis, R. D. (1960). On the stability of the Cournot solution on the oligopoly
Appendix
The used model is programmed, using Wolfram Mathematica.
For any number of firms f, starting values n0, x0 and y0, adaption period ap, memory m
and periods p this program is able to generate vectors with the quantities produced by the rational firms and the naïve firms, the amount of firms using the rational strategy and the profits gained by the rational firms and the Naïve-firms for each period. Using the quantity vectors it is possible to analyze whether this market converges to a Nash-equilibrium and whether this equilibrium is stable. The vector containing the amount of firms using the Nash-strategy is to analyze whether the market reaches an equilibrium where both strategies are used and the amount of firms using them remains constant, one strategy completely dies out or they keep switching between strategies. The profit vectors are used to analyze how the strategies perform and to see if this market does better or worse than the standard (non-dynamical) Cournot model.
Bachelor Thesis Program
Friso Cuijpers
The stability of the dynamic Cournot equilibrium for a finite number of firms with
heterogeneous heuristics
Program
DynamicEquilibrium[f_,nash_, x1_, y1_,ap_, m_, p_]:=Module[{a,b,c,d, i,x,y,n,pX,pY,count,countx,county, z}, Off[Solve::svars]; a=17; b=1; c=1/10; d=10; X=Array[x[#] &,p]; Y=Array[y[#] &,p]; Nash=Array[n[#] &, p]; VectorprofitX=Array[pX[#] &,p]; VectorprofitY=Array[pY[#] &, p]; X[[1]]=x1; Y[[1]]=y1; Nash[[1]]=nash;
(* x1 en y1 zijn de startwaarden voor de hoeveelheden vectoren X en Y *)
i=1;
While [i< p, (* start van while-loop 1 *)
If[f-Nash[[i]] 0,profitX=0,profitX=X[[i]]*(a-b*(Nash[[i]]*Y[[i]]+(f-Nash[[i]])*X[[i]]))-d*X[[i]] ;];
If[Nash[[i]] 0,profitY=0,profitY=Y[[i]]*(a-b*(Nash[[i]]*Y[[i]]+(f-Nash[[i]])*X[[i]]))-d*Y[[i]]-c ;];
VectorprofitX[[i]]=profitX; VectorprofitY[[i]]=profitY;
If[IntegerPart[i/ap]i/ap, (* start van if-conditie 1 *) If[i<m, AvgVectorprofitX=Sum[VectorprofitX[[j]],{j,1,i}]/i; AvgVectorprofitY=Sum[VectorprofitY[[j]],{j,1,i}]/i; , AvgVectorprofitX=Sum[VectorprofitX[[j]],{j,i-m+1,i}]/m; AvgVectorprofitY=Sum[VectorprofitY[[j]],{j,i-m+1,i}]/m; ];
If[AvgVectorprofitX>AvgVectorprofitY, (* start van if-conditie 2 *) If[Nash[[i]]-1 0, Nash[[i+1]]=Nash[[i]]-1, Nash[[i+1]]=0];,
If[Nash[[i]]+1 f, Nash[[i+1]]=Nash[[i]]+1, Nash[[i+1]]=f];, Nash[[i+1]]=Nash[[i]]]; (* einde van if-conditie 2 *)
If[f-Nash[[i+1]]==0, X[[i+1]]=0; maxy=Maximize[{Y[[i+1]]*(b*((Nash[[i+1]]-1)*z+Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]))-d*Y[[i+1]]-c, Y[[i+1]] 0, a-b*((Nash[[i+1]])*Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]) 0, z 0},Y[[i+1]]]; z=Y[[i+1]]; vectormax=Array[maxy[[2,1,2,1,#]] &,Length[maxy[[2,1,2,1]]]]; vectorsol=Array[Solve[vectormax[[#,1]]Y[[i+1]]&&vectormax[[#,2]],Y[[i+1]]] &,Length[vectormax]]; vectorsol=Select[vectorsol,UnsameQ[#,{}]&]; Y[[i+1]]=vectorsol[[1,1,1,2]] , If[Nash[[i+1]] 0, Y[[i+1]]=0;
maxx=Maximize[{X[[i+1]]*(a-b*(Nash[[i]]*Y[[i]]+(f-Nash[[i]]-1)*X[[i]]+X[[i+1]]))-d*X[[i+1]], X[[i+1]] 0, a-b*(Nash[[i+1]]*Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]) 0},X[[i+1]]];
X[[i+1]]=maxx[[2,1,2]]; ,
maxx=Maximize[{X[[i+1]]*(a-b*(Nash[[i]]*Y[[i]]+(f-Nash[[i]]-1)*X[[i]]+X[[i+1]]))-d*X[[i+1]], X[[i+1]] 0, a-b*(Nash[[i+1]]*Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]) 0},X[[i+1]]]; maxy=Maximize[{Y[[i+1]]*(b*((Nash[[i+1]]-1)*z+Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]))-d*Y[[i+1]]-c, Y[[i+1]] 0, a-b*((Nash[[i+1]])*Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]) 0, z 0,X[[i+1]] 0},Y[[i+1]]]; z=Y[[i+1]]; vectormax=Array[maxy[[2,1,2,1,#]] &,Length[maxy[[2,1,2,1]]]]; vectorsol=Array[Solve[vectormax[[#,1]]Y[[i+1]]&&vectormax[[#,2]],{Y[[i+1]],X [[i+1]]}] &,Length[vectormax]]; vectorsol=Select[vectorsol,UnsameQ[#,{}]&]; countx=1;
While[countx Length[maxx[[2,1,2]]-1], county=1; While[county Length[vectorsol], If[Length[vectorsol[[county,1]]]1, If[Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]maxx[[2,1,2,1,countx,1 ]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]],Y[[i+1]]}]{}, sol=Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]maxx[[2,1,2,1,countx, 1]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]],Y[[i+1]]}]; county=county+1, sol=Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]maxx[[2,1,2,1,countx, 1]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]],Y[[i+1]]}]; county=Length[vectorsol]+1; countx=Length[maxx[[2,1,2]]]; ]; , If[Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]==vectorsol[[county,1,2 ,2]]&&X[[i+1]]maxx[[2,1,2,1,countx,1]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]],Y [[i+1]]}]{}, sol=Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]==vectorsol[[county,1, 2,2]]&&X[[i+1]]maxx[[2,1,2,1,countx,1]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]], Y[[i+1]]}]; county=county+1, sol=Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]==vectorsol[[county,1, 2,2]]&&X[[i+1]]maxx[[2,1,2,1,countx,1]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]], Y[[i+1]]}]; county=Length[vectorsol]+1; countx=Length[maxx[[2,1,2]]]; ]; ]; ]; countx=countx+1; ]; If[sol {}, i=p;, X[[i+1]]=sol[[1,1,2]]; Y[[i+1]]=sol[[1,2,2]];]; ]; ];
, (* als het aantal Nash-firms niet hoeft te veranderen gebeurt het onderstaande *) Nash[[i+1]]=Nash[[i]]; If[f-Nash[[i+1]]==0, X[[i+1]]=0; maxy=Maximize[{Y[[i+1]]*(b*((Nash[[i+1]]-1)*z+Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]))-d*Y[[i+1]]-c, Y[[i+1]] 0,
a-b*((Nash[[i+1]])*Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]) 0, z 0},Y[[i+1]]]; z=Y[[i+1]]; vectormax=Array[maxy[[2,1,2,1,#]] &,Length[maxy[[2,1,2,1]]]]; vectorsol=Array[Solve[vectormax[[#,1]]Y[[i+1]]&&vectormax[[#,2]],Y[[i+1]]] &,Length[vectormax]]; vectorsol=Select[vectorsol,UnsameQ[#,{}]&]; Y[[i+1]]=vectorsol[[1,1,1,2]] , If[Nash[[i+1]]==0, Y[[i+1]]=0;
maxx=Maximize[{X[[i+1]]*(a-b*(Nash[[i]]*Y[[i]]+(f-Nash[[i]]-1)*X[[i]]+X[[i+1]]))-d*X[[i+1]], X[[i+1]] 0, a-b*(Nash[[i+1]]*Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]) 0},X[[i+1]]];
X[[i+1]]=maxx[[2,1,2]]; ,
maxx=Maximize[{X[[i+1]]*(a-b*(Nash[[i]]*Y[[i]]+(f-Nash[[i]]-1)*X[[i]]+X[[i+1]]))-d*X[[i+1]], X[[i+1]] 0, a-b*(Nash[[i+1]]*Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]) 0},X[[i+1]]]; maxy=Maximize[{Y[[i+1]]*(b*((Nash[[i+1]]-1)*z+Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]))-d*Y[[i+1]]-c, Y[[i+1]] 0, a-b*((Nash[[i+1]])*Y[[i+1]]+(f-Nash[[i+1]])*X[[i+1]]) 0, z 0,X[[i+1]] 0},Y[[i+1]]]; z=Y[[i+1]]; vectormax=Array[maxy[[2,1,2,1,#]] &,Length[maxy[[2,1,2,1]]]]; vectorsol=Array[Solve[vectormax[[#,1]]Y[[i+1]]&&vectormax[[#,2]],{Y[[i+1]],X [[i+1]]}] &,Length[vectormax]]; vectorsol=Select[vectorsol,UnsameQ[#,{}]&]; countx=1; While[countx Length[maxx[[2,1,2]]-1], county=1; While[county Length[vectorsol], If[Length[vectorsol[[county,1]]]1, If[Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]maxx[[2,1,2,1,countx,1 ]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]],Y[[i+1]]}]{}, sol=Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]maxx[[2,1,2,1,countx, 1]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]],Y[[i+1]]}]; county=county+1, sol=Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]maxx[[2,1,2,1,countx, 1]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]],Y[[i+1]]}]; county=Length[vectorsol]+1; countx=Length[maxx[[2,1,2]]]; ]; , If[Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]==vectorsol[[county,1,2 ,2]]&&X[[i+1]]maxx[[2,1,2,1,countx,1]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]],Y
[[i+1]]}]{}, sol=Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]==vectorsol[[county,1, 2,2]]&&X[[i+1]]maxx[[2,1,2,1,countx,1]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]], Y[[i+1]]}]; county=county+1, sol=Solve[Y[[i+1]]==vectorsol[[county,1,1,2]]&&X[[i+1]]==vectorsol[[county,1, 2,2]]&&X[[i+1]]maxx[[2,1,2,1,countx,1]]&&maxx[[2,1,2,1,countx,2]],{X[[i+1]], Y[[i+1]]}]; county=Length[vectorsol]+1; countx=Length[maxx[[2,1,2]]]; ]; ]; ]; countx=countx+1; ]; If[sol {}, i=p;, X[[i+1]]=sol[[1,1,2]]; Y[[i+1]]=sol[[1,2,2]];]; ]; ];
]; (* einde van if-conditie 1 *) i=i+1;
Clear[z];
]; (* einde van while-loop 1 *)
If[f-Nash[[p]] 0,profitX=0,profitX=X[[p]]*(a-b*(Nash[[p]]*Y[[p]]+(f-Nash[[p]])*X[[p]]))-d*X[[p]] ;]; If[Nash[[p]] 0,profitY=0,profitY=Y[[p]]*(a-b*(Nash[[p]]*Y[[p]]+(f-Nash[[p]])*X[[p]]))-d*Y[[p]]-c ;]; VectorprofitX[[p]]=profitX; VectorprofitY[[p]]=profitY; Matrix=({
{X, Y, Nash, VectorprofitX, VectorprofitY} });
Return[Matrix]; ];
