doi: 10.3934/jimo.2021071
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Evolution of revenue preference for competing firms with nonlinear inverse demand

1. 

School of Management Science and Engineering, Nanjing University of Finance and Economics, Nanjing 210023 China

2. 

Center for Behavioral Decision and Control, School of Management and Engineering, Nanjing University, Nanjing 210093, China

3. 

School of Business, Ningbo University, Ningbo 315211, China

* Corresponding author: Caichun Chai

Received  July 2020 Revised  February 2021 Early access April 2021

Fund Project: This study is funded by the National Natural Science Foundation of China (71871112, 71601098), and Natural Science Foundation of the Jiangsu Province (BK20190791)

This paper studies evolutionarily stable preferences of competing firms across independent markets. Two models are considered according to whether firms' preferences are discrete or continuous. When preferences are discrete, firms have two marketing strategies: profit maximization and revenue maximization. We find that, whether pure and mixed strategies are evolutionarily stable depends on the spectrum of pricing capability. When the pricing capability is moderate, the mixed strategy is an evolutionarily stable strategy. Revenue maximization is evolutionarily stable under relatively high pricing capability, whereas, in case of low pricing capability, firms opt to maximize their profits. Further, the stability of revenue preference is also examined under continuous preferences. We derive the conditions, under which a unique evolutionarily stable revenue preference appears as well as it is continuously stable. Our main results still hold when we extend our model to a general framework.

Citation: Caichun Chai, Tiaojun Xiao, Zhangwei Feng. Evolution of revenue preference for competing firms with nonlinear inverse demand. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021071
References:
[1]

P. A. Abrams, Modelling the adaptive dynamics of traits involved in inter-and intraspecific interactions: An assessment of three methods, Ecology Letters, 4 (2001), 166-175.  doi: 10.1046/j.1461-0248.2001.00199.x.

[2]

A. BaruaD. Chakraborty and H. CG, Entry, competitiveness and exports: Evidence from the indian firm data, Journal of Industry Competition and Trade, 12 (2012), 325-347.  doi: 10.1007/s10842-011-0096-3.

[3]

H. Bester and W. Güth, Is altruism evolutionarily stable?, Journal of Economic Behavior and Organization, 34 (1998), 193-209.  doi: 10.1016/S0167-2681(97)00060-7.

[4]

T. Boyacı and G. Gallego, Coordinating pricing and inventory replenishment policies for one wholesaler and one or more geographically dispersed retailers, International Journal of Production Economics, 77 (2002), 95-111.  doi: 10.1016/S0925-5273(01)00229-8.

[5]

C. ChaiT. Xiao and E. Francis, Is social responsibility for firms competing on quantity evolutionary stable?, Journal of Industrial and Management Optimization, 14 (2018), 325-347.  doi: 10.3934/jimo.2017049.

[6]

C. ChaiE. Francis and T. Xiao, Supply chain dynamics with assortative matching, Journal of Evolutionary Economics, 31 (2021), 179-206.  doi: 10.1007/s00191-020-00687-3.

[7]

C. Chai and T. Xiao, Wholesale pricing and evolutionarily stable strategy in duopoly supply chains with social responsibility, Journal of Systems Science and Systems Engineering, 28 (2019), 110-125.  doi: 10.1007/s11518-018-5392-6.

[8]

N. ChampagnatR. Ferričre and G. Ben Arous4, The canonical equation of adaptive dynamics: A mathematical view, Selection, 2 (2002), 73-83.  doi: 10.1556/Select.2.2001.1-2.6.

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R. Cressman, The Stability Concept of Evolutionary Game Theory: A Dynamic Approach, vol. 94, Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1992 doi: 10.1007/978-3-642-49981-4.

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E. DekelJ. Ely and O. Yilankaya, Evolution of preferences, Review of Economic Studies, 74 (2007), 685-704.  doi: 10.1093/restud/74.3.685.

[11]

U. Dieckmann, Can adaptive dynamics invade?, Trends in Ecology and Evolution, 12 (1997), 128-131.  doi: 10.1016/S0169-5347(97)01004-5.

[12]

U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, Journal of Mathematical Biology, 34 (1996), 579-612.  doi: 10.1007/BF02409751.

[13]

A. A. Elsadany and A. E. Matouk, Dynamical behaviors of fractional-order Lotka–Volterra predator–prey model and its discretization, Journal of Applied Mathematics and Computing, 49 (2015), 269-283.  doi: 10.1007/s12190-014-0838-6.

[14]

D. Friedman and B. Sinervo, Evolutionary Games in Natural, Social, and Virtual Worlds, Oxford University Press, 2016. doi: 10.1093/acprof:oso/9780199981151.001.0001.

[15]

J. GaleK. G. Binmore and L. Samuelson, Learning to be imperfect: The ultimatum game, Games and Economic Behavior, 8 (1995), 56-90.  doi: 10.1016/S0899-8256(05)80017-X.

[16]

S. A. H. GeritzE'. KisdiG. Mesze'NA and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evolutionary Ecology, 12 (1998), 35-57.  doi: 10.1023/A:1006554906681.

[17]

W. Güth and M. Yaari, An evolutionary approach to explain reciprocal behavior in a simple strategic game, U. Witt. Explaining Process and Change–Approaches to Evolutionary Economics. Ann Arbor, 23–34.

[18]

J. Hofbauer and K. Sigmund, Adaptive dynamics and evolutionary stability, Applied Mathematics Letters, 3 (1990), 75-79.  doi: 10.1016/0893-9659(90)90051-C.

[19]

J. HuQ. Hu and Y. Xia, Who should invest in cost reduction in supply chains?, International Journal of Production Economics, 207 (2019), 1-18.  doi: 10.1016/j.ijpe.2018.10.002.

[20]

S. Huck and J. Oechssler, The indirect evolutionary approach to explaining fair allocations, Games and Economic Behavior, 28 (1999), 13-24.  doi: 10.1006/game.1998.0691.

[21]

M. Konigstein and W. Müller, Combining rational choice and evolutionary dynamics: The indirect evolutionary approach, Metroeconomica, 51 (2000), 235-256.  doi: 10.1111/1467-999X.00090.

[22]

M. KopelF. Lamantia and F. Szidarovszky, Evolutionary competition in a mixed market with socially concerned firms, Journal of Economic Dynamics and Control, 48 (2014), 394-409.  doi: 10.1016/j.jedc.2014.06.001.

[23]

J.-F. Le GalliardR. Ferrière and U. Dieckmann, The adaptive dynamics of altruism in spatially heterogeneous populations, Evolution, 57 (2003), 1-17.  doi: 10.1111/j.0014-3820.2003.tb00211.x.

[24]

B. J. McGill and J. S. Brown, Evolutionary game theory and adaptive dynamics of continuous traits, Annual Review of Ecology, Evolution, and Systematics, 38 (2007), 403-435.  doi: 10.1146/annurev.ecolsys.36.091704.175517.

[25]

X. Meng and L. Zhang, Evolutionary dynamics in a lotka-volterra competition model with impulsive periodic disturbance, Mathematical Methods in the Applied Sciences, 39 (2016), 177-188.  doi: 10.1002/mma.3467.

[26]

L. MuJ. Ma and L. Chen, A 3-dimensional discrete model of housing price and its inherent complexity analysis, Journal of Systems Science and Complexity, 22 (2009), 415-421.  doi: 10.1007/s11424-009-9174-6.

[27]

A. K. Naimzada and L. Sbragia, Oligopoly games with nonlinear demand and cost functions: two boundedly rational adjustment processes, Chaos, Solitons & Fractals, 29 (2006), 707-722.  doi: 10.1016/j.chaos.2005.08.103.

[28]

S. Nicoleta, The theory of the firm and the evolutionary games, The Annals of the University of Oradea, 22 (2013), 533-542. 

[29]

T. OffermanJ. Potters and J. Sonnemans, Imitation and belief learning in an oligopoly experiment, The Review of Economic Studies, 69 (2002), 973-997.  doi: 10.1111/1467-937X.00233.

[30]

K. M. Page and M. A. Nowak, A generalized adaptive dynamics framework can describe the evolutionary ultimatum game, Journal of Theoretical Biology, 209 (2001), 173-179.  doi: 10.1006/jtbi.2000.2251.

[31]

P. Rhode and M. Stegeman, Non-Nash equilibria of Darwinian dynamics with applications to duopoly, International Journal of Industrial Organization, 19 (2001), 415-453.  doi: 10.1016/S0167-7187(99)00025-9.

[32]

E. Rudis, CEO challenge 2006: Perspectives and analysis, Conference Board, 2006.

[33]

M. E. Schaffer, Are profit-maximisers the best survivors?: A Darwinian model of economic natural selection, Journal of Economic Behavior and Organization, 12 (1989), 29-45.  doi: 10.1016/0167-2681(89)90075-9.

[34]

Y. Shirata, The evolution of fairness under an assortative matching rule in the ultimatum game, International Journal of Game Theory, 41 (2012), 1-21.  doi: 10.1007/s00182-011-0271-0.

[35]

J. M. Smith, The theory of games and the evolution of animal conflicts, Journal of Theoretical Biology, 47 (1974), 209-221.  doi: 10.1016/0022-5193(74)90110-6.

[36]

J. M. Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18.  doi: 10.1038/246015a0.

[37] J. M. Smith, Evolution and the theory of games, Cambridge University Press, Cambridge, 1982. 
[38]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Mathematical Biosciences, 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9.

[39]

A. Traulsen and C. Hauert, Stochastic evolutionary game dynamics, Reviews of Nonlinear Dynamics and Complexity, 2 (2009), 25-61. 

[40]

A. A. Tsay and N. Agrawal, Channel dynamics under price and service competition, Manufacturing & Service Operations Management, 2 (2000), 372-391.  doi: 10.1287/msom.2.4.372.12342.

[41]

J. W. Weibull, Evolutionary Games Theory, The MIT Press, Cambridge, Massachusetts, London, England, 1995.

[42]

T. Xiao and G. Yu, Marketing objectives of retailers with differentiated goods: An evolutionary perspective, Journal of Systems Science and Systems Engineering, 15 (2006), 359-374.  doi: 10.1007/s11518-006-5013-7.

[43]

Y. Yi and H. Yang, An evolutionary stable strategy for retailers selling complementary goods subject to indirect network externalities, Economic Modelling, 62 (2017), 184-193.  doi: 10.1016/j.econmod.2016.12.021.

[44]

Y. Yi and H. Yang, Wholesale pricing and evolutionary stable strategies of retailers under network externality, European Journal of Operational Research, 259 (2017), 37-47.  doi: 10.1016/j.ejor.2016.09.014.

show all references

References:
[1]

P. A. Abrams, Modelling the adaptive dynamics of traits involved in inter-and intraspecific interactions: An assessment of three methods, Ecology Letters, 4 (2001), 166-175.  doi: 10.1046/j.1461-0248.2001.00199.x.

[2]

A. BaruaD. Chakraborty and H. CG, Entry, competitiveness and exports: Evidence from the indian firm data, Journal of Industry Competition and Trade, 12 (2012), 325-347.  doi: 10.1007/s10842-011-0096-3.

[3]

H. Bester and W. Güth, Is altruism evolutionarily stable?, Journal of Economic Behavior and Organization, 34 (1998), 193-209.  doi: 10.1016/S0167-2681(97)00060-7.

[4]

T. Boyacı and G. Gallego, Coordinating pricing and inventory replenishment policies for one wholesaler and one or more geographically dispersed retailers, International Journal of Production Economics, 77 (2002), 95-111.  doi: 10.1016/S0925-5273(01)00229-8.

[5]

C. ChaiT. Xiao and E. Francis, Is social responsibility for firms competing on quantity evolutionary stable?, Journal of Industrial and Management Optimization, 14 (2018), 325-347.  doi: 10.3934/jimo.2017049.

[6]

C. ChaiE. Francis and T. Xiao, Supply chain dynamics with assortative matching, Journal of Evolutionary Economics, 31 (2021), 179-206.  doi: 10.1007/s00191-020-00687-3.

[7]

C. Chai and T. Xiao, Wholesale pricing and evolutionarily stable strategy in duopoly supply chains with social responsibility, Journal of Systems Science and Systems Engineering, 28 (2019), 110-125.  doi: 10.1007/s11518-018-5392-6.

[8]

N. ChampagnatR. Ferričre and G. Ben Arous4, The canonical equation of adaptive dynamics: A mathematical view, Selection, 2 (2002), 73-83.  doi: 10.1556/Select.2.2001.1-2.6.

[9]

R. Cressman, The Stability Concept of Evolutionary Game Theory: A Dynamic Approach, vol. 94, Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1992 doi: 10.1007/978-3-642-49981-4.

[10]

E. DekelJ. Ely and O. Yilankaya, Evolution of preferences, Review of Economic Studies, 74 (2007), 685-704.  doi: 10.1093/restud/74.3.685.

[11]

U. Dieckmann, Can adaptive dynamics invade?, Trends in Ecology and Evolution, 12 (1997), 128-131.  doi: 10.1016/S0169-5347(97)01004-5.

[12]

U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, Journal of Mathematical Biology, 34 (1996), 579-612.  doi: 10.1007/BF02409751.

[13]

A. A. Elsadany and A. E. Matouk, Dynamical behaviors of fractional-order Lotka–Volterra predator–prey model and its discretization, Journal of Applied Mathematics and Computing, 49 (2015), 269-283.  doi: 10.1007/s12190-014-0838-6.

[14]

D. Friedman and B. Sinervo, Evolutionary Games in Natural, Social, and Virtual Worlds, Oxford University Press, 2016. doi: 10.1093/acprof:oso/9780199981151.001.0001.

[15]

J. GaleK. G. Binmore and L. Samuelson, Learning to be imperfect: The ultimatum game, Games and Economic Behavior, 8 (1995), 56-90.  doi: 10.1016/S0899-8256(05)80017-X.

[16]

S. A. H. GeritzE'. KisdiG. Mesze'NA and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evolutionary Ecology, 12 (1998), 35-57.  doi: 10.1023/A:1006554906681.

[17]

W. Güth and M. Yaari, An evolutionary approach to explain reciprocal behavior in a simple strategic game, U. Witt. Explaining Process and Change–Approaches to Evolutionary Economics. Ann Arbor, 23–34.

[18]

J. Hofbauer and K. Sigmund, Adaptive dynamics and evolutionary stability, Applied Mathematics Letters, 3 (1990), 75-79.  doi: 10.1016/0893-9659(90)90051-C.

[19]

J. HuQ. Hu and Y. Xia, Who should invest in cost reduction in supply chains?, International Journal of Production Economics, 207 (2019), 1-18.  doi: 10.1016/j.ijpe.2018.10.002.

[20]

S. Huck and J. Oechssler, The indirect evolutionary approach to explaining fair allocations, Games and Economic Behavior, 28 (1999), 13-24.  doi: 10.1006/game.1998.0691.

[21]

M. Konigstein and W. Müller, Combining rational choice and evolutionary dynamics: The indirect evolutionary approach, Metroeconomica, 51 (2000), 235-256.  doi: 10.1111/1467-999X.00090.

[22]

M. KopelF. Lamantia and F. Szidarovszky, Evolutionary competition in a mixed market with socially concerned firms, Journal of Economic Dynamics and Control, 48 (2014), 394-409.  doi: 10.1016/j.jedc.2014.06.001.

[23]

J.-F. Le GalliardR. Ferrière and U. Dieckmann, The adaptive dynamics of altruism in spatially heterogeneous populations, Evolution, 57 (2003), 1-17.  doi: 10.1111/j.0014-3820.2003.tb00211.x.

[24]

B. J. McGill and J. S. Brown, Evolutionary game theory and adaptive dynamics of continuous traits, Annual Review of Ecology, Evolution, and Systematics, 38 (2007), 403-435.  doi: 10.1146/annurev.ecolsys.36.091704.175517.

[25]

X. Meng and L. Zhang, Evolutionary dynamics in a lotka-volterra competition model with impulsive periodic disturbance, Mathematical Methods in the Applied Sciences, 39 (2016), 177-188.  doi: 10.1002/mma.3467.

[26]

L. MuJ. Ma and L. Chen, A 3-dimensional discrete model of housing price and its inherent complexity analysis, Journal of Systems Science and Complexity, 22 (2009), 415-421.  doi: 10.1007/s11424-009-9174-6.

[27]

A. K. Naimzada and L. Sbragia, Oligopoly games with nonlinear demand and cost functions: two boundedly rational adjustment processes, Chaos, Solitons & Fractals, 29 (2006), 707-722.  doi: 10.1016/j.chaos.2005.08.103.

[28]

S. Nicoleta, The theory of the firm and the evolutionary games, The Annals of the University of Oradea, 22 (2013), 533-542. 

[29]

T. OffermanJ. Potters and J. Sonnemans, Imitation and belief learning in an oligopoly experiment, The Review of Economic Studies, 69 (2002), 973-997.  doi: 10.1111/1467-937X.00233.

[30]

K. M. Page and M. A. Nowak, A generalized adaptive dynamics framework can describe the evolutionary ultimatum game, Journal of Theoretical Biology, 209 (2001), 173-179.  doi: 10.1006/jtbi.2000.2251.

[31]

P. Rhode and M. Stegeman, Non-Nash equilibria of Darwinian dynamics with applications to duopoly, International Journal of Industrial Organization, 19 (2001), 415-453.  doi: 10.1016/S0167-7187(99)00025-9.

[32]

E. Rudis, CEO challenge 2006: Perspectives and analysis, Conference Board, 2006.

[33]

M. E. Schaffer, Are profit-maximisers the best survivors?: A Darwinian model of economic natural selection, Journal of Economic Behavior and Organization, 12 (1989), 29-45.  doi: 10.1016/0167-2681(89)90075-9.

[34]

Y. Shirata, The evolution of fairness under an assortative matching rule in the ultimatum game, International Journal of Game Theory, 41 (2012), 1-21.  doi: 10.1007/s00182-011-0271-0.

[35]

J. M. Smith, The theory of games and the evolution of animal conflicts, Journal of Theoretical Biology, 47 (1974), 209-221.  doi: 10.1016/0022-5193(74)90110-6.

[36]

J. M. Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18.  doi: 10.1038/246015a0.

[37] J. M. Smith, Evolution and the theory of games, Cambridge University Press, Cambridge, 1982. 
[38]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Mathematical Biosciences, 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9.

[39]

A. Traulsen and C. Hauert, Stochastic evolutionary game dynamics, Reviews of Nonlinear Dynamics and Complexity, 2 (2009), 25-61. 

[40]

A. A. Tsay and N. Agrawal, Channel dynamics under price and service competition, Manufacturing & Service Operations Management, 2 (2000), 372-391.  doi: 10.1287/msom.2.4.372.12342.

[41]

J. W. Weibull, Evolutionary Games Theory, The MIT Press, Cambridge, Massachusetts, London, England, 1995.

[42]

T. Xiao and G. Yu, Marketing objectives of retailers with differentiated goods: An evolutionary perspective, Journal of Systems Science and Systems Engineering, 15 (2006), 359-374.  doi: 10.1007/s11518-006-5013-7.

[43]

Y. Yi and H. Yang, An evolutionary stable strategy for retailers selling complementary goods subject to indirect network externalities, Economic Modelling, 62 (2017), 184-193.  doi: 10.1016/j.econmod.2016.12.021.

[44]

Y. Yi and H. Yang, Wholesale pricing and evolutionary stable strategies of retailers under network externality, European Journal of Operational Research, 259 (2017), 37-47.  doi: 10.1016/j.ejor.2016.09.014.

Figure 1.  The overall flowchart for the proposed methodology
Figure 2.  Phase portrait when $ c(17+\sqrt{193})/6<a<c(4+\sqrt{13}) $
Figure 3.  Phase portrait when $ a\leq c(17+\sqrt{193})/6 $
Figure 4.  Phase portrait when $ a\geq c(4+\sqrt{13}) $
Figure 5.  Firms' payoffs change with the degree of revenue preference
Figure 6.  The dynamics of firms' revenue preference
Table 1.  The material payoff matrix
$R$$P$
$R$$2a^2(a-5c), $$(a-3c)(2a-c)(a+2c), $
$2a^2(a-5c)$$(a-3c)^2(2a-c)$
$P$$(a-3c)^2(2a-c), $$2(a-c)^3, $
$(a-3c)(2a-c)(a+2c)$$2(a-c)^3$
$R$$P$
$R$$2a^2(a-5c), $$(a-3c)(2a-c)(a+2c), $
$2a^2(a-5c)$$(a-3c)^2(2a-c)$
$P$$(a-3c)^2(2a-c), $$2(a-c)^3, $
$(a-3c)(2a-c)(a+2c)$$2(a-c)^3$
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