Evolutionary dynamics of gradient
learning in quantity-setting oligopolies
Rick van der Peet 5886805 University of Amsterdam Bachelor thesis: assignment 2 Supervisor: Rob van Hemert Instructor: Marius Ochea June 27, 2014
2
Index
1. Introduction ... 3
2. Model for two competing heuristics ... 6
2.1 Implementing the model ... 10
2.2 Example Jacobian ... 11
3. Simulations ... 12
3.1 Best-reply versus gradient learning... 12
3.1.1 Simulation series with linear costs ... 12
3.1.2 Simulation series with quadratic costs ... 13
3.2 Rational versus gradient learning ... 14
3.2.1 Simulation series with linear costs ... 14
3.2.2 Simulation series with quadratic costs ... 15
4. Three heuristic oligopoly ... 16
4.1 Implementing the model ... 17
4.2 Simulation results ... 19
4.2.1 Simulation series with linear costs ... 19
4.2.2 Simulation series with quadratic costs ... 20
5. Conclusion and discussion ... 22
3 1. Introduction
In competitive markets firms are trying to stay ahead of the game to ensure their success in the market. Important decisions have to be made on all kinds of levels and different divisions within the company. There are multiple ways to gather the information needed to make these decisions. These different ways will be called heuristics in this thesis. For firms it can be difficult to figure out which heuristic under which circumstances performs best. Plus, heuristics may be costly to implement since the needed information may be hard to gather
This thesis focusses on the question how firms should determine their output. Cournot (1838) introduced on this topic a duopoly where firms produce under the assumption that each firm believes that their competition would produce the same in the new period as they did in the previous period. This way of determining the output is the so called best-reply heuristic.
Other research focussed on different heuristics, some also naïve like the best-reply heuristic, others more rational. These heuristics have been researched facing different circumstances, e.g. a change from a Cournot to a Bertrand competition. The introduction of more than one heuristics in the same oligopoly market lead to a new field of interest: the evolutionary dynamics with heterogeneous heuristics; where at least two heuristics are being investigated against each other. Some heuristics may perform better than others, and the importance of the better performing heuristics is expected to grow in the market as the proportions of the heuristics change: heuristics that turn out to give less satisfying results could ‘die out’ when firms switch to the better performing heuristics. A central topic in the evolutionary dynamics is whether or not stability will be reached; and the question if that result would be robust if e.g. parameters or the cost function would be changed.
Hommes et al. (2011) researched an evolutionary version of Theocharis (1960)’s work, showing that the introduction of more sophisticated heuristics tend to stabilize the dynamics. This research shows that the use of different heuristics can lead to different results, and that introducing heterogeneous heuristics can lead to interesting results. Droste et al. (2002) introduced a general evolutionary dynamical system, in which a large finite number of heuristics can be investigated . Huang (2010) however state that the strategy switching dynamics for more than two heuristics are more difficult to analyse then models with two strategies, since various possible dynamics could intervene the process and affect the results leading to biased results and chaotic-like behaviour.
4 So although a market with multiple heuristics may be more interesting; the study on the interaction between two heuristics gives a clearer and a more informative insight on the evolutionary dynamics of the used heuristics.
Droste et al. (2002) researched the evolutionary performance of the best-reply heuristic versus a rational heuristic as an example for the model they introduced. They found that if the best-reply function is unstable, this could possibly lead to complicated dynamics. Therefor they expect similar or even more complicated phenomena for more complicated games. This thesis will research the two heuristics used by Droste et al. (2002) in addition to a third heuristic; the gradient learning heuristic.
Whereas the best-reply heuristic has been criticized by its naïve assumption to expect that other firms keep the same output as the last period (Seade, 1980), the rational heuristic is defined by its knowledge of future output of all other companies and is because of the
information needed costly to operate for firms. The gradient learning heuristic is a more naïve one compared to the rational heuristic. With gradient learning firms adjust their strategic variable in the direction of the optimum choice based on their own performance in the last period; in a quantity-setting competition firms change their last period’s quantity by the partial derivative of the profit function with respect to quantity. This means that firms base their new output on how they could have adjusted their performance in the last period to gain a higher profit. Firms using the gradient learning heuristic only take into account their own output, without looking at the market structure or the effect that changes in output of the competition could have on the firm’s own profit. It is therefore possible that a company gets the incentive to produce more in the new period, but ends up facing a lower demand than last period, resulting in a smaller profit.
The gradient heuristic also knows advantages; Arrow and Hurwicz (1960) state that the gradient learning heuristic is a plausible heuristic for firms in games which are played multiple times. They also state that it could be very difficult or even impossible to reach the maximum profit by a direct calculation; so it is a reasonable hypothesis that firms approach the maximum profit they can achieve by adjusting their output using the gradient learning heuristic. The gradient learning heuristic can thus be seen as an easy heuristic to implement and a possible cheap one, since it does not require knowledge of output of other firms nor insight in e.g. the market demand.
The research in this thesis focusses on three parts: in the first part the companies in the market can choose and switch between the gradient learning heuristic and the best-reply heuristic; in the second part the companies in the market can choose and switch between the
5 gradient learning heuristic and a rational heuristic; in the third part the companies in the market can choose and switch between the gradient learning heuristic, a rational heuristic and the best-reply heuristic. In all three parts two different sub-researches will be held: one where the market is characterized by a linear demand function and a linear cost function (as in: Hommes et al. (2011) and Anufriev et al. (2013)); the other where the market is characterized by a linear demand function and a quadratic cost function (as in: Droste et al. (2002)).
The model used in the research of this thesis is the same as the evolutionary dynamical model Droste et al. (2002) introduced. In this model every discrete-time period a large
population of firms is matched in pairs randomly to play a symmetric Cournot duopoly game. The goods are perfect substitutes, which according to Singh and Vives (1984) ensures that quantity competition is the dominant competition compared to price competition. Each period, firms compare expected profits which they would achieve using one of the two heuristics and choose optimally. This means that each time-period firms have the option to switch between the available heuristics. Droste et al. (2002) found that when evolutionary pressure increases complicated quantity fluctuations arise around un unstable equilibrium, with firms switching between the heuristics.
Anufriev et al. (2013) considered a Bertrand oligopoly where firms used either the least squares heuristic or the gradient learning heuristic. Both the least squares heuristic and the gradient learning heuristic can be considered naïve heuristics compared to the rational heuristic. However, Anufriev et al. (2013) found a similar result as Droste et al. (2002); concluding that in most cases cyclical switching between the two heuristics occurred.
The goal of the research of this thesis is to study the evolutionary performance of the gradient learning heuristic in respect to the best reply heuristic (another naïve heuristic) and in respect to the rational heuristic; and to see whether or not the results are robust for a change in cost function from a linear to a quadratic function. Comparing the results with the results of Droste et al. (2002) could then give more insight in the underlying dynamics of the three heuristics. Ultimately a competitive market where all three heuristics are available will be simulated to find out if the results of Droste et al. (2002) and the results of the other simulations of this thesis are visible.
In Section 2 the model is further explained and clarified with the introduction of the model specifications. Section 3 follows with the simulation results of best-reply versus gradient learning and rational versus gradient learning. Section 4 then presents the model and simulation results of gradient learning versus rational versus best-reply. Section 5 concludes this thesis.
6 2. Model for two competing heuristics
Before any simulations can be carried out a complete and working model is needed. The basis is the general evolutionary dynamical model of Droste et al. (2002); where each
discrete-time period a large number of firms is matched in pairs to play a symmetric Cournot duopoly. Firms determine their quantity each time period based on one of the three heuristics, depending on which two heuristics will be available in that particular simulation. The three heuristics studied in this thesis are the gradient learning heuristic, the rational heuristic and the best-reply heuristic. This research is carried out in two kinds of simulations, one where firms can choose and switch between the gradient learning heuristic and the best-reply heuristic and the other one where firms can choose and switch between the gradient learning heuristic and the rational heuristic.
The linear inverse demand function is given by
( ) ( ),
with Q the total output in that particular time period.
Both simulation series (gradient learning versus best-reply and gradient learning versus rational) will be carried out with a linear cost function; and after that the same simulation series will be carried out, only this time with a quadratic cost function; these functions are given by
( ) and
( )
respectively.
The inverse demand function and cost function result in the reaction function given by
( )
7 with the corresponding Cournot-Nash equilibrium quantity
for the simulation series with the linear cost function.
The reaction function for the simulation series with the quadratic cost function is given by
( )
with the corresponding Cournot-Nash equilibrium quantity
Each discrete-time period firms make expectations about the profit they would receive for both heuristics and choose optimally. The resulting profit function is given by
( ) ( )
With T an additional payment representing the costs when using the rational heuristic. The use of the best-reply heuristic and the gradient learning heuristic are free from costs, the rational heuristic requires information and knowledge which are not freely available.
The gradient learning heuristic uses the partial derivative of the profit function with respect to output of time period t to conduct the output for time period t+1 based on last period’s output; the function is given by
The rational heuristic knows which fractions play which heuristic and what their output will be in the next time period and base their own output on that knowledge; the function is given by
8 ( ( ) )
with nt the proportion of rational firms at time-period t.
The best-reply heuristic uses the output of time period t to conduct the output for time period t+1; the function is therefore given by
( ( ) )
here with nt the proportion of best-reply firms in time-period t.
The xt, yt, and zt represent the output supplied by respectively the gradient learning
heuristic, the rational heuristic and the best-reply heuristic. This leads to the following profit functions
( ( ( ) ))
( ( ( ) ))
and
( ( ( ) ))
The fractions are calculated using the replicator dynamics formula used by Droste et al. (2002):
( )
( )
for the simulation series where companies can choose and switch between the rational and the gradient learning heuristic; and
( )
( )
for the simulation series where companies can choose and switch between the best-reply and the gradient learning heuristic. With , and the realized profits for companies using
9 the gradient learning heuristic, the rational heuristic and the best-reply heuristic respectively at time period t. The parameter corresponds to the noise level, representing the stability of the market: each period the fraction companies are replaced by new companies. This results in both heuristic being chosen by 0.5 of the new firms (Droste et al. (2002)).
The Cournot-Nash equilibrium fraction with costs T for using the rational heuristic is given by
( ( ) √ ( ) ( )
The Cournot-Nash equilibrium fraction for the series with the choice between the gradient learning heuristic and the best-reply heuristic (no costs T for using an heuristic) is given by
This model requires several restrictions; quantities and prices cannot be negative for all discrete time-periods, so , , and ( ) .
The costs however cannot be positive, so
( ) .
Furthermore the assumption is made that companies will not chose a quantity that results in a negative profit; in that case a company will chose the closest quantity which leads to a profit equal to zero.
10
2.1 Implementing the model
In all simulation series the parameters are fixed as in Droste et al. (2002):
This results in a Cournot-Nash equilibrium quantity and profit given by
and
if the costs are linear. With the Cournot-Nash equilibrium fraction if the gradient learning heuristic and the best-reply heuristic are competing (see Section 2). For gradient versus rational the Cournot-Nash equilibrium fraction is given by
If the costs are quadratic the Cournot-Nash equilibrium quantity and profit are given by
and
Again with the Cournot-Nash equilibrium fraction if the gradient learning heuristic and the best-reply heuristic are competing (see Section 2). For gradient versus rational the Cournot-Nash equilibrium fraction is given by
11
2.2 Example Jacobian
For the competition of gradient learning versus best-reply with linear costs a 3x3 Jacobian matrix is calculated: ( )
With resulting eigenvalues
( √ ) and
( √ )
The eigenvalues and are between -1 and 1 for . The equilibrium with the Cournot-Nash quantities and fraction is locally stable if ; if
.
For quadratic costs the 3x3 Jacobian matrix is given by
(
)
With resulting eigenvalues
( √ ) and
( √ )
The eigenvalues and are between -1 and 1 for . The equilibrium with the Cournot-Nash quantities and fraction is locally stable if, for ; if .
12 3. Simulations
In the first part of this section the best-reply heuristic will be competing against the gradient learning heuristic. In the second part the rational heuristic will be competing against the gradient learning heuristic. Both parts for linear and for quadratic costs.
All simulation series will be reviewed for varying parameters; with en the main parameters. The evolutionary outcomes shown in this chapter are robust for the initial stages of the quantities and the fractions.
3.1 Best-reply versus gradient learning
In these simulation series the best-reply heuristic competes with the gradient learning heuristic. In section 3.1.1 the companies face linear costs ( ) and in section 3.1.2 the companies face quadratic costs ( )
3.1.1 Simulation series with linear costs
Figure 1.I shows the ‘neutralizing’ effect of the δ parameter; the bigger the value of δ is on its interval [0,0.5], the closer the evolutionary outcome of is to a value of 0.5. The parameter has a big influence (see figure 1.II), however, the best-reply heuristic performs better evolutionary than the gradient learning heuristic regardless the value of .
I: on the x-axis and parameter =0.05 II: on the x-axis and parameter δ=0.02
Fig 1. Bifurcation diagrams with on the y axis. Initial values and =0.5
As seen in figure 1.II the fraction displays a chaotic-like behaviour with a small range
13 The Cournot-Nash equilibrium fraction for this simulation series, with and
, is . It is clear that this fraction is different than the evolutionary outcome of the simulations, even though according to section 2.2 a stable Cournot-Nash equilibrium fraction was expected.
As stated earlier, the best-reply heuristic outperforms the gradient learning heuristic, regardless the values of parameters and δ.
3.1.2 Simulation series with quadratic costs
In these quadratic costs series a different evolutionary dynamics are displayed than in the linear costs series (see Figure 2).
I: parameter δ=0.0006. II: parameter δ=0.06.
Fig 2. Bifurcation diagram with on the x-axis and on the y axis. Initial values
and =0.5.
In this simulation series the best-reply heuristic outperforms the gradient learning heuristic for < 0.7723, as shown in Figure 2. This threshold value is perhaps close to the calculated , however the evolutionary outcome of the fraction best-reply companies is not equal to the Cournot-Nash equilibrium fraction ( ).
For larger a chaotic behaviour of the fraction appears. In Figure 2 the difference is displaced for two different values for the parameter δ (I: δ=0.0006 and II: parameter δ=0.06). A larger δ pulls the evolutionary outcome of the fractions towards 0.5.
As stated above; for all ⟨ and all δ ⟩ the fraction best-reply companies is bigger than the fraction gradient learning companies.
14
3.2 Rational versus gradient learning
In these simulation series the rational heuristic competes with the gradient learning heuristic. In section 3.2.1 the companies face linear costs ( ) and in section 3.2.2 the companies face quadratic costs ( )
3.2.1 Simulation series with linear costs
Increasing δ on its interval ⟩ moves the fraction towards =0.5 (see Figure 3.I), while the effect of an increasing T is a smaller fraction using the rational heuristic. Figure 3.II shows the bifurcation diagram with on the x-axis.
I: on the x-axis and parameter =0.5 II: on the x-axis and parameter δ=0.06
Fig 3. Bifurcation diagrams with on the y axis. Initial values and =0.5, with parameter T=1.
The value of has a big influence on the evolutionary outcome of the process. The interval from 0 to close to 0.6 results in a somewhat chaotic behaviour of , however the range of varies little. After the period doubling bifurcation close after ≈0.62 a seemingly linear increase of occurs in . Revealing that the outcome of the evolutionary process depends greatly on the value of the parameter ; since for smaller values the majority of companies are using the gradient learning heuristic, and for values closer to 1 the majority of companies are using the rational heuristic, with practically all companies using the rational heuristic if =1. Figure 3.II is calculated with a relatively large δ; for smaller δ the difference between the evolutionary outcomes of using a small or a large ⟨ is even bigger.
For and the evolutionary outcome of the rational companies is 0.01433; which means that the Cournot-Nash equilibrium fraction is not
15 reached. Without giving proof that is based on the eigenvalues of the Jacobian, this is
evidence that the simulation series is not stable using these values for the parameters.
Concluding; the value of the parameter has a decisive influence on the evolutionary outcome of the process. Small values of closer to 0 give the gradient learning companies a good ability to adapt while values of closer to 1 results in the gradient learning heuristic unable to compete with the rational heuristic.
3.2.2 Simulation series with quadratic costs
As shown in Figure 4 the results for these simulation series depend on the value of the parameters T and . For T=1 the evolutionary outcome of the fraction rational companies lies within a range close to 0.5.
I: on the x-axis and parameter =0.5 II: on the x-axis and parameter T=1
Fig 4. Bifurcation diagrams with on the y axis; initial values and =0.5. Parameter δ=0.0006.
To study the performance more accurate a time series is carried out (see Figure 5.I). In this time series it is clear that for the biggest part the gradient learning heuristic slightly outperforms the rational heuristic; but generally the fractions are very close to each other; very different than the calculated Cournot-Nash equilibrium fraction .
In Figure 5.II an example is shown for different initial stages . In the first periods it is clearly depending on the initial value which heuristic performs better. Ultimately the fractions converge to the same value however.
I:initial value =0.5 II: initial { ( ) ( ) ( )}
Fig. 5. Time series of the fraction for different initial values . Parameters =0.5,
δ=0.0006 and T=1. Initial values 1.
4. Three heuristic oligopoly
In this section the competition where companies can choose and switch between the three analysed heuristics will be studied and simulated. Again, both for linear and for quadratic costs. The basis of the model remains unchanged, but the important formulas are different for the three heuristic simulation series.
The formulas for the output are given by
( ( ) ) and ( ( ) )
for respectively the gradient learning, the rational and the best-reply heuristic. With the fraction rational companies and the fraction best-reply companies at time period t. The fractions are calculated by
( )
( ) and
( )
17 With the fraction gradient learning companies at time period t.
The profits are calculated by
( ( ( ) ))
( ( ( ) ))
and
( ( ( ) ))
The Cournot-Nash equilibrium quantities and profits remains unchanged, the Cournot-Nash equilibrium fractions are given by
√ ( ) and or ( √ ( ) ) √ ( )
The Cournot-Nash equilibrium fraction gradient learning companies is equal to .
4.1 Implementing the model
As in section 2 the following parameters are fixed:
18 The Cournot-Nash equilibrium quantities and profits remain unchanged;
and
If ; the linear costs case; and
and
for ; the quadratic costs case.
Using in the linear costs case the Cournot-Nash equilibrium fractions are given by
and
With the fraction gradient learning companies
For the quadratic costs case the Cournot-Nash equilibrium fractions are given by
and
With the fraction gradient learning companies
19
4.2 Simulation results
In these simulation series the parameter has again an equalizing effect; this time pulling the fraction outcomes of the simulation series towards . Lowering the parameter T, the costs of using the rational heuristic, has a negligible effect; ultimately for all T > 0 it is more profitable for companies to use one of the two naïve heuristics.
For the simulation results shown in this section the parameter will be set to 0.0006 and T will remain fixed on 1.
4.2.1 Simulation series with linear costs
First the simulations are run with ; the linear costs case.
I: on the x- and on the y-axis II: on the y- and on the x-axis III:1- - on y- and on the x-axis
Figure 6. Bifurcation diagrams with fraction rational (I), best-reply (II) and gradient (III) companies on the y-axis and with (I) and (II and III) on the x-axis. rational companies
with and fixed on 0.5 (I) and with fixed on 0.0006 (II and III). Initial values and .
As seen in Figure 6, the parameter has only influence for , but for all in the interval the outcome is that the majority of companies uses the best-reply heuristic.
The fraction rational companies (0.00327) is close to the Cournot-Nash equilibrium fraction (0.003249). The fraction outcomes of the gradient learning and the best-reply companies are very different as compared to the calculated Cournot-Nash equilibrium fractions.
The initial values of the quantities influence the initial courses of the fractions, but the evolutionary outcome however is unaffected; this is shown in Figure 7.
20
I: =0.05; =0.05 II: =1/3; =1/3
Fig. 7. Time series of the fractions (red), (blue) and (green) for different initial values . Parameters =0.5, δ=0.0006 and T=1. Initial values 1.
4.2.2 Simulation series with quadratic costs
Simulation series with ; with quadratic costs.
I: on the y-axis II: on the y-axis III:1- - on the y-axis
Figure 8. Bifurcation diagrams with fraction rational (I), best-reply (II) and gradient (III) companies on the y-axis and with on the x-axis. Parameters fixed on 0.0006 and T on 1.
Initial values and .
The parameter has again influence on the evolutionary outcome. For large values of in the interval [0,1] this leads to chaotic-like behavior. For two values of the process will be investigated; and .
21
I: time period 0 to 100 II: time period 0 to 2500
Fig. 9. Time series of the fractions (red), (blue) and (green) for is 0.75,
δ=0.0006 and T=1. Initial values 1.
I: time period 0 to 390 II: time period 0 to 1000
Fig. 10. Time series of the fractions (red), (blue) and (green) for different initial values . Parameters =0.87, δ=0.0006 and T=1. Initial values 1.
For smaller than 0,837 it is clear that evolutionary wise the best-reply heuristic performs best (see Figure 9). The value of the resulting Cournot-Nash equilibrium fraction rational companies (0.003685) is close to the calculated . For an interesting pattern exists, with all three fractions fluctuating. However, for all the best-reply heuristic still performs best on average (see Figure 10).
22 5. Conclusion and discussion
The goal of this research was to study the evolutionary performance of the gradient learning heuristic in respect to other heuristics; to see how this ‘more reasonable’ heuristic, which is cheap and easy to implement for companies, would perform evolutionary. This study of the evolutionary performance of the gradient learning heuristic in competition with either the best-reply heuristic, the rational heuristic or both has produced various and interesting results; displaying complicated dynamics in all simulation series, just as the results of Droste et al. (2002). First the simulation series of the gradient learning heuristic versus one other heuristic were run; and after the simulation series where all three heuristics are competing in the same market. This way the underlying effects that the heuristics have on each other evolutionary-wise can be studied, which would not have been clear just from studying the three-heuristic simulation series (Huang, 2010).
Figure 11 gives an overview on the evolutionary outcome of the simulation series. It shows a distinctive evolutionary winner; the best-reply heuristic. This heuristic performs best evolutionary wise according to the one-on-one simulation series. This result was also shown in the simulation series where all three heuristics are competing in the same market.
Linear costs Quadratic costs
Gradient vs. best-reply Best-reply Best-reply (until threshold)
Gradient vs. rational Depending on parameters Perform equally well
Best-reply vs. rational Best-reply Best-reply
Figure 11. Evolutionary outcome of competing heuristics accounted for linear or quadratic costs.
The results appeared to be different than the stable Cournot-Nash equilibrium fractions and threshold parameter values. However, only for the gradient learning versus best-reply heuristic the eigenvalues of the Jacobian are calculated, therefor it is possible that for the gradient learning versus the rational heuristic the outcome is not stable because the used parameters are outside the stable interval. Another possibility is that the used restrictions in the simulation models are the reason for the biased results; since the analytic calculations do not account for all used restrictions; this could be researched further in other studies. Clear is that, although the rational heuristic can have an advantage on the short term, on the long term the naïve and cheap best-reply heuristic outperforms the rational heuristic. Important to see is
23 that the best-reply heuristic performs best in both the linear and the quadratic costs series, although the outcomes of all series differ for using linear or quadratic costs.
The gradient heuristic performs somewhat poorly compared to the expectations based on the Cournot-Nash equilibrium fraction, although in Section 4 it is clear that the gradient learning heuristic performs better than the rational heuristic. Interesting to see are the different results when varying the parameter . Especially when the gradient learning heuristic
competes with the rational heuristic the value of decides the outcome of the evolutionary process; showing that for close to 1 the disadvantage of the gradient learning heuristic becomes apparent, since this heuristic only takes the firm’s own output into account, and fails to keep in mind that other companies also might adapt their output. It shows that using a lower value of is a way to counter this downside.
Even though the best-reply heuristic seems to be a better heuristic evolutionary-wise than the gradient learning heuristic, both these heuristics can an advantage over the expensive rational heuristic in the three-heuristic simulation series. For the best-reply heuristic this is also true in the series versus only the rational heuristic, for all values of the parameters. For the gradient heuristic in the value of can be deciding for the outcome when competing the rational heuristic, showing the importance of countering the downside of this heuristic.
24 6. References
Anufriev, M., D. Kopányi and J. Tuinstra (2013): “Learning in cycles in Bertrand competition with differentiated commodities and competing learning rules”, Journal of Economic
Dynamics & Control, Vol. 37(12), 2562-2581.
Arrow, K. J. and L. Hurwicz (1960): “Some Remarks on the Equilibria of Economic Systems”. Econometrica, Vol. 28(3), 640-646.
Cournot, A. (1838): “Recherches sur les Prindpes Mathe’matiques de la Theorie des
Richness”. English edition: “Researches into the Mathematical Principles of the Theory of
Wealth” Edited by N. Bacon. New-York: Macmillan, 1897.
Droste, E., C.H. Hommes and J. Tuinstra (2002): “Endogenous Fluctuations under
Evolutionary Pressure in Cournot competition”, Games and Economic Behavior, Vol. 40, 232-269.
Hommes, C. H., M. I. Ochea and J. Tuinstra (2011): “On the Stability of the Cournot Equilibrium: An Evolutionary Approach”, CeNDEF Working Paper 11-10.
Huang, W. (2010): “On the complexity of strategy-switching dynamics”, Journal of
Economic Behavior and Organisation, Vol. 75(3), 445-460.
Seade, J. (1980): “The stability of Cournot revisited. J. Econ. Theory, Vol. 23, 15-27.
Singh, N. and X. Vives (1984): “Price and quantity competition in a differentiated duopoly”.
The Rand Journal of Economics, Vol. 15(4), 546-554.
Theocharis, R. D. (1960): “On the stability of the Cournot solution on the oligopoly problem”,
