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doi: 10.3934/jimo.2021071

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 Published  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

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

[39]

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

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

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

[39]

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

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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