Neighborhood strong superiority and evolutionary stability of polymorphic profiles in asymmetric games

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July 2022, 9(3): 253-266. doi: 10.3934/jdg.2022012

Neighborhood strong superiority and evolutionary stability of polymorphic profiles in asymmetric games

Aradhana Narang ^, and A. J. Shaiju

Department of Mathematics, Indian Institute of Technology Madras, Chennai-600036, Tamil Nadu, India

^* Corresponding author: Aradhana Narang

Received August 2021 Revised March 2022 Published July 2022 Early access June 2022

Fund Project: The second named author would like to acknowledge financial support from SERB, Department of Science and Technology, Govt. of India, through the project MTR/2017/000674 titled "Evolutionary Stability in Asymmetric Games with Continuous Strategy Space"

In symmetric evolutionary games with continuous strategy spaces, Cressman [6] has proved an interesting stability result for the associated replicator dynamics relating the concepts of neighborhood superiority and neighborhood attracting for polymorphic states with respect to the weak topology. Similar stability results are also established for monomorphic profiles in 2-player asymmetric games [8]. In the present paper, we use the model of asymmetric evolutionary games introduced by Mendoza-Palacios and Hernández-Lerma [17] and obtain a stability result for polymorphic profiles in $ n $-player asymmetric evolutionary games with continuous action spaces using the concept of neighborhood strong superiority (Definition 2.3). In particular, we prove that neighborhood strong superior polymorphic profiles are neighborhood attracting. It is also shown that a polymorphic neighborhood strong superior profile is in fact a vector of Dirac measures. Moreover, we establish that the notion of neighborhood strong superiority does not imply strong uninvadability and vice-versa.

Keywords: Asymmetric evolutionary games, replicator dynamics, games with continuous strategy spaces, neighborhood superiority, neighborhood attracting, polymorphic profiles, monomorphisms, strong uninvadable profiles.

Mathematics Subject Classification: Primary: 91A22, 91A10, 34A34, 34G20, 34D20; Secondary: 92D25.

Citation: Aradhana Narang, A. J. Shaiju. Neighborhood strong superiority and evolutionary stability of polymorphic profiles in asymmetric games. Journal of Dynamics and Games, 2022, 9 (3) : 253-266. doi: 10.3934/jdg.2022012

References:

[1]	D. Balkenborg and K. H. Schlag, Evolutionarily stable sets, Internat. J. Game Theory, 29 (2001), 571-595. doi: 10.1007/s001820100059.
[2]	D. Balkenborg and K. H. Schlag, On the evolutionary selection of sets of Nash equilibria, J. Econom. Theory, 133 (2007), 295-315. doi: 10.1016/j.jet.2005.08.008.
[3]	I. M. Bomze, Cross entropy minimization in uninvadable states of complex populations, Journal of Mathematical Biology, 30 (1991), 73-87.
[4]	I. M. Bomze and B. M. Pötscher, Game Theoretical Foundations of Evolutionary Stability, Lecture Notes in Economics and Mathematical Systems, 324, Springer, Berlin, 1989. doi: 10.1007/978-3-642-45660-2.
[5]	I. M. Bomze and J. W. Weibull, Does neutral stability imply Lyapunov stability?, Games and Economic Behavior, 11 (1995), 173-192. doi: 10.1006/game.1995.1048.
[6]	R. Cressman, Stability of the replicator equation with continuous strategy space, Math. Social Sci., 50 (2005), 127-147. doi: 10.1016/j.mathsocsci.2005.03.001.
[7]	R. Cressman, Continuously stable strategies, neighborhood superiority and two-player games with continuous strategy space, Internat. J. Game Theory, 38 (2009), 221-247. doi: 10.1007/s00182-008-0148-z.
[8]	R. Cressman, CSS, NIS and dynamic stability for two-species behavioral models with continuous trait spaces, J. Theoret. Biol., 262 (2010), 80-89. doi: 10.1016/j.jtbi.2009.09.019.
[9]	R. Cressman, Beyond the symmetric normal form: Extensive form games, asymmetric games and games with continuous strategy spaces, Evolutionary Game Dynamics, Proceedings of Symposia in Applied Mathematics, 69 (2011), 27-59.
[10]	R. Cressman and J. Apaloo, Evolutionary game theory, in Handbook of Dynamic Game Theory, Springer, Cham, Switzerland, 2018,461-510. doi: 10.1007/978-3-319-44374-4_6.
[11]	R. Cressman and J. Hofbauer, Measure dynamics on a one-dimensional continuous trait space: Theoretical foundations for adaptive dynamics, Theoretical Population Biology, 67 (2005), 47-59.
[12]	R. Cressman, J. Hofbauer and F. Riedel, Stability of the replicator equation for a single species with a multi-dimensional continuous trait space, J. Theoret. Biol., 239 (2006), 273-288. doi: 10.1016/j.jtbi.2005.07.022.
[13]	I. Eshel and E. Sansone, Evolutionary and dynamic stability in continuous population games, J. Math. Biol., 46 (2003), 445-459. doi: 10.1007/s00285-002-0194-2.
[14]	D. Gulick, Encounters with Chaos and Fractals, CRC Press, United States, 2012.
[15]	D. Hingu, Asymptotic stability of strongly uninvadable sets, Ann. Oper. Res., 287 (2020), 737-749. doi: 10.1007/s10479-017-2695-9.
[16]	D. Hingu, K. S. Mallikarjuna Rao and A. J. Shaiju, Evolutionary stability of polymorphic population states in continuous games, Dyn. Games Appl., 8 (2018), 141-156. doi: 10.1007/s13235-016-0207-1.
[17]	S. Mendoza-Palacios and O. Hernández-Lerma, Evolutionary dynamics on measurable strategy spaces: Asymmetric games, J. Differential Equations, 259 (2015), 5709-5733. doi: 10.1016/j.jde.2015.07.005.
[18]	S. Mendoza-Palacios and O. Hernández-Lerma, Stability of the replicator dynamics for games in metric spaces, J. Dyn. Games, 4 (2017), 319-333. doi: 10.3934/jdg.2017017.
[19]	A. Narang and A. J. Shaiju, Evolutionary stability of polymorphic profiles in asymmetric games, Dyn. Games Appl., 9 (2019), 1126-1142. doi: 10.1007/s13235-019-00302-6.
[20]	A. Narang and A. J. Shaiju, Globally strong uninvadable sets of profiles in asymmetric games, International Game Theory Review, 22 (2020), 1950014.
[21]	A. Narang and A. J. Shaiju, Stability of faces in asymmetric evolutionary games, Ann. Oper. Res., 304 (2021), 343-359. doi: 10.1007/s10479-021-04157-2.
[22]	T. W. L. Norman, Dynamically stable sets in infinite strategy spaces, Games Econom. Behav., 62 (2008), 610-627. doi: 10.1016/j.geb.2007.05.005.
[23]	J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces, Econom. Theory, 17 (2001), 141-162. doi: 10.1007/PL00004092.
[24]	J. Oechssler and F. Riedel, On the dynamic foundation of evolutionary stability in continuous models, J. Econom. Theory, 107 (2002), 223-252. doi: 10.1006/jeth.2001.2950.
[25]	K. Ritzberger and J. W. Weibull, Evolutionary selection in normal-form games, Econometrica: Journal of the Econometric Society, 63 (1995), 1371-1399. doi: 10.2307/2171774.
[26]	H. L. Royden, Real Analysis, 3rd ed, Macmillan, New York, 1988.
[27]	L. Samuelson, Limit evolutionarily stable strategies in two-player, normal form games, Games Econom. Behav., 3 (1991), 110-128. doi: 10.1016/0899-8256(91)90008-3.
[28]	L. Samuelson and J. Zhang, Evolutionary stability in asymmetric games, J. Econom. Theory, 57 (1992), 363-391. doi: 10.1016/0022-0531(92)90041-F.
[29]	W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, 2010.
[30]	R. Selten, A note on evolutionarily stable strategies in asymmetric animal conflicts, J. Theoret. Biol., 84 (1980), 93-101. doi: 10.1016/S0022-5193(80)81038-1.
[31]	R. Selten, Evolutionary stability in extensive two-person games, Math. Social Sci., 5 (1983), 269-363. doi: 10.1016/0165-4896(83)90012-4.
[32]	A. J. Shaiju and P. Bernhard, Evolutionarily robust strategies: Two nontrivial examples and a theorem, Advances in Dynamic Games and their Applications, "Annals of the International Society of Dynamic Games", (2009), 377-395.
[33]	A. N. Shiryaev, Probability, vol. 2, Springer, Berlin, 1996. doi: 10.1007/978-1-4757-2539-1.
[34]	J. M. Smith, Evolution and the Theory of Games, Cambridge university press, Cambridge, 1982.
[35]	P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9.
[36]	B. Thomas, On evolutionarily stable sets, J. Math. Biol., 22 (1985), 105-115. doi: 10.1007/BF00276549.
[37]	M. van Veelen and P. Spreij, Evolution in games with a continuous action space, Econom. Theory, 39 (2009), 355-376. doi: 10.1007/s00199-008-0338-8.

show all references

References:

[1]	D. Balkenborg and K. H. Schlag, Evolutionarily stable sets, Internat. J. Game Theory, 29 (2001), 571-595. doi: 10.1007/s001820100059.
[2]	D. Balkenborg and K. H. Schlag, On the evolutionary selection of sets of Nash equilibria, J. Econom. Theory, 133 (2007), 295-315. doi: 10.1016/j.jet.2005.08.008.
[3]	I. M. Bomze, Cross entropy minimization in uninvadable states of complex populations, Journal of Mathematical Biology, 30 (1991), 73-87.
[4]	I. M. Bomze and B. M. Pötscher, Game Theoretical Foundations of Evolutionary Stability, Lecture Notes in Economics and Mathematical Systems, 324, Springer, Berlin, 1989. doi: 10.1007/978-3-642-45660-2.
[5]	I. M. Bomze and J. W. Weibull, Does neutral stability imply Lyapunov stability?, Games and Economic Behavior, 11 (1995), 173-192. doi: 10.1006/game.1995.1048.
[6]	R. Cressman, Stability of the replicator equation with continuous strategy space, Math. Social Sci., 50 (2005), 127-147. doi: 10.1016/j.mathsocsci.2005.03.001.
[7]	R. Cressman, Continuously stable strategies, neighborhood superiority and two-player games with continuous strategy space, Internat. J. Game Theory, 38 (2009), 221-247. doi: 10.1007/s00182-008-0148-z.
[8]	R. Cressman, CSS, NIS and dynamic stability for two-species behavioral models with continuous trait spaces, J. Theoret. Biol., 262 (2010), 80-89. doi: 10.1016/j.jtbi.2009.09.019.
[9]	R. Cressman, Beyond the symmetric normal form: Extensive form games, asymmetric games and games with continuous strategy spaces, Evolutionary Game Dynamics, Proceedings of Symposia in Applied Mathematics, 69 (2011), 27-59.
[10]	R. Cressman and J. Apaloo, Evolutionary game theory, in Handbook of Dynamic Game Theory, Springer, Cham, Switzerland, 2018,461-510. doi: 10.1007/978-3-319-44374-4_6.
[11]	R. Cressman and J. Hofbauer, Measure dynamics on a one-dimensional continuous trait space: Theoretical foundations for adaptive dynamics, Theoretical Population Biology, 67 (2005), 47-59.
[12]	R. Cressman, J. Hofbauer and F. Riedel, Stability of the replicator equation for a single species with a multi-dimensional continuous trait space, J. Theoret. Biol., 239 (2006), 273-288. doi: 10.1016/j.jtbi.2005.07.022.
[13]	I. Eshel and E. Sansone, Evolutionary and dynamic stability in continuous population games, J. Math. Biol., 46 (2003), 445-459. doi: 10.1007/s00285-002-0194-2.
[14]	D. Gulick, Encounters with Chaos and Fractals, CRC Press, United States, 2012.
[15]	D. Hingu, Asymptotic stability of strongly uninvadable sets, Ann. Oper. Res., 287 (2020), 737-749. doi: 10.1007/s10479-017-2695-9.
[16]	D. Hingu, K. S. Mallikarjuna Rao and A. J. Shaiju, Evolutionary stability of polymorphic population states in continuous games, Dyn. Games Appl., 8 (2018), 141-156. doi: 10.1007/s13235-016-0207-1.
[17]	S. Mendoza-Palacios and O. Hernández-Lerma, Evolutionary dynamics on measurable strategy spaces: Asymmetric games, J. Differential Equations, 259 (2015), 5709-5733. doi: 10.1016/j.jde.2015.07.005.
[18]	S. Mendoza-Palacios and O. Hernández-Lerma, Stability of the replicator dynamics for games in metric spaces, J. Dyn. Games, 4 (2017), 319-333. doi: 10.3934/jdg.2017017.
[19]	A. Narang and A. J. Shaiju, Evolutionary stability of polymorphic profiles in asymmetric games, Dyn. Games Appl., 9 (2019), 1126-1142. doi: 10.1007/s13235-019-00302-6.
[20]	A. Narang and A. J. Shaiju, Globally strong uninvadable sets of profiles in asymmetric games, International Game Theory Review, 22 (2020), 1950014.
[21]	A. Narang and A. J. Shaiju, Stability of faces in asymmetric evolutionary games, Ann. Oper. Res., 304 (2021), 343-359. doi: 10.1007/s10479-021-04157-2.
[22]	T. W. L. Norman, Dynamically stable sets in infinite strategy spaces, Games Econom. Behav., 62 (2008), 610-627. doi: 10.1016/j.geb.2007.05.005.
[23]	J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces, Econom. Theory, 17 (2001), 141-162. doi: 10.1007/PL00004092.
[24]	J. Oechssler and F. Riedel, On the dynamic foundation of evolutionary stability in continuous models, J. Econom. Theory, 107 (2002), 223-252. doi: 10.1006/jeth.2001.2950.
[25]	K. Ritzberger and J. W. Weibull, Evolutionary selection in normal-form games, Econometrica: Journal of the Econometric Society, 63 (1995), 1371-1399. doi: 10.2307/2171774.
[26]	H. L. Royden, Real Analysis, 3rd ed, Macmillan, New York, 1988.
[27]	L. Samuelson, Limit evolutionarily stable strategies in two-player, normal form games, Games Econom. Behav., 3 (1991), 110-128. doi: 10.1016/0899-8256(91)90008-3.
[28]	L. Samuelson and J. Zhang, Evolutionary stability in asymmetric games, J. Econom. Theory, 57 (1992), 363-391. doi: 10.1016/0022-0531(92)90041-F.
[29]	W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, 2010.
[30]	R. Selten, A note on evolutionarily stable strategies in asymmetric animal conflicts, J. Theoret. Biol., 84 (1980), 93-101. doi: 10.1016/S0022-5193(80)81038-1.
[31]	R. Selten, Evolutionary stability in extensive two-person games, Math. Social Sci., 5 (1983), 269-363. doi: 10.1016/0165-4896(83)90012-4.
[32]	A. J. Shaiju and P. Bernhard, Evolutionarily robust strategies: Two nontrivial examples and a theorem, Advances in Dynamic Games and their Applications, "Annals of the International Society of Dynamic Games", (2009), 377-395.
[33]	A. N. Shiryaev, Probability, vol. 2, Springer, Berlin, 1996. doi: 10.1007/978-1-4757-2539-1.
[34]	J. M. Smith, Evolution and the Theory of Games, Cambridge university press, Cambridge, 1982.
[35]	P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9.
[36]	B. Thomas, On evolutionarily stable sets, J. Math. Biol., 22 (1985), 105-115. doi: 10.1007/BF00276549.
[37]	M. van Veelen and P. Spreij, Evolution in games with a continuous action space, Econom. Theory, 39 (2009), 355-376. doi: 10.1007/s00199-008-0338-8.

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