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Sequencing grey games
Game theoretical modelling of a dynamically evolving network Ⅱ: Target sequences of score 1
1. | School of Mathematics and Statistics, The University of Sheffield, Hounsfield Road, Sheffield, S3 7RH, UK |
2. | Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, UK |
In previous work we considered a model of a population where individuals have an optimum level of social interaction, governed by a graph representing social connections between the individuals, who formed or broke those links to achieve their target number of contacts. In the original work an improvement in the number of links was carried out by breaking or joining to a randomly selected individual. In the most recent work, however, these actions were often not random, but chosen strategically, and this led to significant complications. One of these was that in any state, multiple individuals might wish to change their number of links. In this paper we consider a systematic analysis of the structure of the simplest class of non-trivial cases, where in general only a single individual has reason to make a change, and prove some general results. We then consider in detail an example game, and introduce a method of analysis for our chosen class based upon cycles on a graph. We see that whilst we can gain significant insight into the general structure of the state space, the analysis for specific games remains difficult.
References:
[1] |
B. Allen and M. A. Nowak,
Games on graphs, EMS Surveys in Mathematical Sciences, 1 (2014), 113-151.
doi: 10.4171/EMSS/3. |
[2] |
V. Bala and S. Goyal,
A model of non-cooperative network formation, Econometrica, 68 (2000), 1181-1230.
doi: 10.1111/1468-0262.00155. |
[3] |
M. Broom and C. Cannings,
A dynamic network population model with strategic link formation governed by individual preferences, J. Theor. Biol., 335 (2013), 160-168.
doi: 10.1016/j.jtbi.2013.06.024. |
[4] |
M. Broom and C. Cannings,
Graphic deviation, Discrete Mathematics, 338 (2015), 701-711.
doi: 10.1016/j.disc.2014.12.011. |
[5] |
M. Broom and C. Cannings,
Game theoretical modelling of a dynamically evolving network Ⅰ: General target sequences, Journal of Dynamic Games, 4 (2017), 285-318.
doi: 10.3934/jdg.2017016. |
[6] |
C. Capitanio, Sociability and response to video playback in adult male rhesus monkeys (macac mulatta), Primates, 43 (2002), 169-177. Google Scholar |
[7] |
R. C. Connor, M. R. Helthaus and L. M. Barre,
Superalliances of bottlenose dolphins, Nature, 397 (1999), 571-572.
doi: 10.1038/17501. |
[8] |
P. I. M. Dunbar,
Neocortex size as a constraint on group size in primates, J. Human Evoluion, 22 (1992), 469-493.
doi: 10.1016/0047-2484(92)90081-J. |
[9] |
B. Dutta and M. O. Jackson, The stability and effeciency of directed communication networks, Rev. Econ. Design, 5 (2015), 251-272. Google Scholar |
[10] |
C. S. Elton, Animal Ecology, Sidgwick & Jackson, London, 1927. Google Scholar |
[11] |
P. Erdos and P. T. Gallai, Grafok eloirt fokszamu pontokkal, Matematikai Lapo, 11 (1960), 264-274. Google Scholar |
[12] |
S. L. Hakimi,
On the realizability of a set of integers as degrees of the vertices of a graph. Ⅰ, J. Soc. Indust. Appl. Math., 10 (1962), 496-506.
doi: 10.1137/0110037. |
[13] |
W. Hässelbarth,
Die Verzweightheit von Graphen, Comm. in Math. and Computer Chem. (MATCH), 16 (1984), 3-17.
|
[14] |
V. Havel, A remark on the existence of finite graphs, Časopis Pěst. Mat., 81 (1956), 477–480. |
[15] |
M. O. Jackson, The stability and efficiency of economic and social networks, Networks and Groups, Springer, (2003). Google Scholar |
[16] |
M. O. Jackson, Social and Economic Networks, Princeton University Press, Princeton, NJ,
2008. |
[17] |
M. O. Jackson, An overview of social networks and economic applications, Handbook of Social Economics, Elsevier, (2011), 512–585 Google Scholar |
[18] |
M. O. Jackson and A. Wolinsky,
A strategic model of social and economic networks, J. Econ. Theory, 71 (1996), 44-74.
doi: 10.1006/jeth.1996.0108. |
[19] |
R. Merris and T. Roby, The lattice of threshold graphs, J. Inequal. Pure and Appl. Math., 6 (2005), Art. 2, 21 pp. |
[20] |
R. Noë, Biological markets: Partner choice as the driving force behind the evolution of cooperation, Economics in Nature. Social Dilemmas, Mate Choice and Biological Markets, (2001), 93–118. Google Scholar |
[21] |
R. Noë and P. Hammerstein, Biological markets: Supply and demand determine the effect of partner choice in cooperation, mutualism and mating, Behav. Ecol. Sociobio., 35 (1994), 1-11. Google Scholar |
[22] |
J. M. Pacheco, A. Traulsen and M. A. Nowak,
Active linking in evolutionary games, J. Theor. Biol., 243 (2006), 437-443.
doi: 10.1016/j.jtbi.2006.06.027. |
[23] |
J. M. Pacheco, A. Traulsen and M. A. Nowak, Coevolution of strategy and structure in complex networks with dynamical linking, Phys. Rev. Lett., 97 (2006), 258103.
doi: 10.1103/PhysRevLett.97.258103. |
[24] |
J. Pepper, J. Mitani and D. Watts, General gregariousness and specific social preferences among wild chimpanzees, Int. J. Primatol., 20 (1999), 613-632. Google Scholar |
[25] |
M. Perc and A. Szolnoki, Coevolutionarygames - A mini review, BioSystems, 99 (2010), 109-125. Google Scholar |
[26] |
E. Ruch and I. Gutman,
The branching extent of graphs, J. Combin. Inform. Systems Sci., 4 (1979), 285-295.
|
[27] |
L. S. Shapley,
Stochastic games, Proc. Nat. Acad. Sci. U.S.A., 39 (1979), 1095-1100.
doi: 10.1073/pnas.39.10.1953. |
[28] |
A. M. Sibbald and R. J. Hooper,
Sociability and willingness of individual sheep to move away from their companions in order to graze, Applied Animal Behaviour, 86 (2004), 51-62.
doi: 10.1016/j.applanim.2003.11.010. |
[29] |
R. Southwell and C. Cannings,
Some models of reproducing graphs: Ⅰ pure reproduction, Applied Mathematics, 1 (2010), 137-145.
doi: 10.4236/am.2010.13018. |
[30] |
R. Southwell and C. Cannings, Some models of reproducing graphs. Ⅱ. Age capped reproduction, Applied Mathematics, 1 (2010), 251-259. Google Scholar |
[31] |
R. Southwell and C. Cannings,
Some models of reproducing graphs. Ⅲ. Game based reproduction, Applied Mathematics, 1 (2010), 335-343.
doi: 10.4236/am.2010.15044. |
[32] |
B. Voelkl and C. Kasper,
Social structure of primate interaction networks facilitates the emergence of cooperation, Biology Letters, 5 (2009), 462-464.
doi: 10.1098/rsbl.2009.0204. |
[33] |
B. Voelkl and R. Noë,
The influence of social structure on the propagation of social information in artificial primate groups: A graph-based simulation approach, J. Theor. Biol., 252 (2008), 77-86.
doi: 10.1016/j.jtbi.2008.02.002. |
[34] |
J. Wiszniewski, C. Brown and L. M. Möller,
Complex patterns of male alliance formation in dolphin social networks, Journal of Mammalogy, 93 (2012), 239-250.
doi: 10.1644/10-MAMM-A-366.1. |
show all references
References:
[1] |
B. Allen and M. A. Nowak,
Games on graphs, EMS Surveys in Mathematical Sciences, 1 (2014), 113-151.
doi: 10.4171/EMSS/3. |
[2] |
V. Bala and S. Goyal,
A model of non-cooperative network formation, Econometrica, 68 (2000), 1181-1230.
doi: 10.1111/1468-0262.00155. |
[3] |
M. Broom and C. Cannings,
A dynamic network population model with strategic link formation governed by individual preferences, J. Theor. Biol., 335 (2013), 160-168.
doi: 10.1016/j.jtbi.2013.06.024. |
[4] |
M. Broom and C. Cannings,
Graphic deviation, Discrete Mathematics, 338 (2015), 701-711.
doi: 10.1016/j.disc.2014.12.011. |
[5] |
M. Broom and C. Cannings,
Game theoretical modelling of a dynamically evolving network Ⅰ: General target sequences, Journal of Dynamic Games, 4 (2017), 285-318.
doi: 10.3934/jdg.2017016. |
[6] |
C. Capitanio, Sociability and response to video playback in adult male rhesus monkeys (macac mulatta), Primates, 43 (2002), 169-177. Google Scholar |
[7] |
R. C. Connor, M. R. Helthaus and L. M. Barre,
Superalliances of bottlenose dolphins, Nature, 397 (1999), 571-572.
doi: 10.1038/17501. |
[8] |
P. I. M. Dunbar,
Neocortex size as a constraint on group size in primates, J. Human Evoluion, 22 (1992), 469-493.
doi: 10.1016/0047-2484(92)90081-J. |
[9] |
B. Dutta and M. O. Jackson, The stability and effeciency of directed communication networks, Rev. Econ. Design, 5 (2015), 251-272. Google Scholar |
[10] |
C. S. Elton, Animal Ecology, Sidgwick & Jackson, London, 1927. Google Scholar |
[11] |
P. Erdos and P. T. Gallai, Grafok eloirt fokszamu pontokkal, Matematikai Lapo, 11 (1960), 264-274. Google Scholar |
[12] |
S. L. Hakimi,
On the realizability of a set of integers as degrees of the vertices of a graph. Ⅰ, J. Soc. Indust. Appl. Math., 10 (1962), 496-506.
doi: 10.1137/0110037. |
[13] |
W. Hässelbarth,
Die Verzweightheit von Graphen, Comm. in Math. and Computer Chem. (MATCH), 16 (1984), 3-17.
|
[14] |
V. Havel, A remark on the existence of finite graphs, Časopis Pěst. Mat., 81 (1956), 477–480. |
[15] |
M. O. Jackson, The stability and efficiency of economic and social networks, Networks and Groups, Springer, (2003). Google Scholar |
[16] |
M. O. Jackson, Social and Economic Networks, Princeton University Press, Princeton, NJ,
2008. |
[17] |
M. O. Jackson, An overview of social networks and economic applications, Handbook of Social Economics, Elsevier, (2011), 512–585 Google Scholar |
[18] |
M. O. Jackson and A. Wolinsky,
A strategic model of social and economic networks, J. Econ. Theory, 71 (1996), 44-74.
doi: 10.1006/jeth.1996.0108. |
[19] |
R. Merris and T. Roby, The lattice of threshold graphs, J. Inequal. Pure and Appl. Math., 6 (2005), Art. 2, 21 pp. |
[20] |
R. Noë, Biological markets: Partner choice as the driving force behind the evolution of cooperation, Economics in Nature. Social Dilemmas, Mate Choice and Biological Markets, (2001), 93–118. Google Scholar |
[21] |
R. Noë and P. Hammerstein, Biological markets: Supply and demand determine the effect of partner choice in cooperation, mutualism and mating, Behav. Ecol. Sociobio., 35 (1994), 1-11. Google Scholar |
[22] |
J. M. Pacheco, A. Traulsen and M. A. Nowak,
Active linking in evolutionary games, J. Theor. Biol., 243 (2006), 437-443.
doi: 10.1016/j.jtbi.2006.06.027. |
[23] |
J. M. Pacheco, A. Traulsen and M. A. Nowak, Coevolution of strategy and structure in complex networks with dynamical linking, Phys. Rev. Lett., 97 (2006), 258103.
doi: 10.1103/PhysRevLett.97.258103. |
[24] |
J. Pepper, J. Mitani and D. Watts, General gregariousness and specific social preferences among wild chimpanzees, Int. J. Primatol., 20 (1999), 613-632. Google Scholar |
[25] |
M. Perc and A. Szolnoki, Coevolutionarygames - A mini review, BioSystems, 99 (2010), 109-125. Google Scholar |
[26] |
E. Ruch and I. Gutman,
The branching extent of graphs, J. Combin. Inform. Systems Sci., 4 (1979), 285-295.
|
[27] |
L. S. Shapley,
Stochastic games, Proc. Nat. Acad. Sci. U.S.A., 39 (1979), 1095-1100.
doi: 10.1073/pnas.39.10.1953. |
[28] |
A. M. Sibbald and R. J. Hooper,
Sociability and willingness of individual sheep to move away from their companions in order to graze, Applied Animal Behaviour, 86 (2004), 51-62.
doi: 10.1016/j.applanim.2003.11.010. |
[29] |
R. Southwell and C. Cannings,
Some models of reproducing graphs: Ⅰ pure reproduction, Applied Mathematics, 1 (2010), 137-145.
doi: 10.4236/am.2010.13018. |
[30] |
R. Southwell and C. Cannings, Some models of reproducing graphs. Ⅱ. Age capped reproduction, Applied Mathematics, 1 (2010), 251-259. Google Scholar |
[31] |
R. Southwell and C. Cannings,
Some models of reproducing graphs. Ⅲ. Game based reproduction, Applied Mathematics, 1 (2010), 335-343.
doi: 10.4236/am.2010.15044. |
[32] |
B. Voelkl and C. Kasper,
Social structure of primate interaction networks facilitates the emergence of cooperation, Biology Letters, 5 (2009), 462-464.
doi: 10.1098/rsbl.2009.0204. |
[33] |
B. Voelkl and R. Noë,
The influence of social structure on the propagation of social information in artificial primate groups: A graph-based simulation approach, J. Theor. Biol., 252 (2008), 77-86.
doi: 10.1016/j.jtbi.2008.02.002. |
[34] |
J. Wiszniewski, C. Brown and L. M. Möller,
Complex patterns of male alliance formation in dolphin social networks, Journal of Mammalogy, 93 (2012), 239-250.
doi: 10.1644/10-MAMM-A-366.1. |







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