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July  2019, 6(3): 241-257. doi: 10.3934/jdg.2019017

Spatial competitive games with disingenuously delayed positions

Marzieh Soltanolkottabi 1, , David Ben-Arieh 1, and  John (C-W) Wu 2,

1. 

Department of Industrial and Manufacturing Systems Engineering, Kansas State University, 2069 Rathbone Hall, 1701B Platt St., Manhattan, Kansas, Riley, USA
2. 

Department of Industrial and Manufacturing Systems Engineering, Kansas State University, 2061 Rathbone Hall, 1701A Platt St., Manhattan, Kansas, Riley, USA

Received  December 2018 Revised  May 2019 Published  July 2019

During the last decades, spatial games have received great attention from researchers showing the behavior of populations of players over time in a spatial structure. One of the main factors which can greatly affect the behavior of such populations is the updating scheme used to apprise new strategies of players. Synchronous updating is the most common updating strategy in which all players update their strategy at the same time. In order to be able to describe the behavior of populations more realistically several asynchronous updating schemes have been proposed. Asynchronous game does not use a universal clock and players can update their strategy at different time steps during the play.

In this paper, we introduce a new type of asynchronous strategy updating in which some of the players hide their updated strategy from their neighbors for several time steps. It is shown that this behavior can change the behavior of populations but does not necessarily lead to a higher payoff for the dishonest players. The paper also shows that with dishonest players, the average payoff of players is less than what they think they get, while they are not aware of their neighbors' true strategy.

Keywords: Spatial games, asynchronous updating, delayed positions.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Marzieh Soltanolkottabi, David Ben-Arieh, John (C-W) Wu. Spatial competitive games with disingenuously delayed positions. Journal of Dynamics & Games, 2019, 6 (3) : 241-257. doi: 10.3934/jdg.2019017
References:
[1]

K. M. Ariful Kabir, J. Tanimoto and Z. Wang, Influence of bolstering network reciprocity in the evolutionary spatial Prisoner's Dilemma game: A perspective, The European Physical Journal B, 91 (2018), Paper No. 312, 10 pp. doi: 10.1140/epjb/e2018-90214-6.  Google Scholar

[2]

J. M. Baetens, P. Van der Weeën and B. De Baets, Effect of asynchronous updating on the stability of cellular automata, Chaos, Solitons & Fractals, 45 (2012), 383-394.  doi: 10.1016/j.chaos.2012.01.002.  Google Scholar

[3]

S. Bandini, A. Bonomi and G. Vizzari, An analysis of different types and effects of asynchronicity in cellular automata update schemes, Natural Computing, 11 (2012), 277-287.  doi: 10.1007/s11047-012-9310-4.  Google Scholar

[4]

O. Bouré, N. Fates and V. Chevrier, First steps on asynchronous lattice-gas models with an application to a swarming rule, Natural Computing, 12 (2013), 551-560.  doi: 10.1007/s11047-013-9389-2.  Google Scholar

[5]

O. Bouré, N. Fates and V. Chevrier, Probing robustness of cellular automata through variations of asynchronous updating, Natural Computing, 11 (2012), 553-564.  doi: 10.1007/s11047-012-9340-y.  Google Scholar

[6]

O. Bouré, N. Fates and V. Chevrier, Robustness of cellular automata in the light of asynchronous information transmission, Unconventional Computation, 52–63, Lecture Notes in Comput. Sci., 6714, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-21341-0_11.  Google Scholar

[7]

N. W. H. Chan, C. Xu, S. K. Tey, Y. J. Yap and P. M. Hui, Evolutionary snowdrift game incorporating costly punishment in structured populations, Physica A: Statistical Mechanics and its Applications, 392 (2013), 168-176.  doi: 10.1016/j.physa.2012.07.078.  Google Scholar

[8]

H.-C. Chen and Y. Chow, Equilibrium selection in evolutionary games with imperfect monitoring, Journal of Applied Probability, 45 (2008), 388-402.  doi: 10.1239/jap/1214950355.  Google Scholar

[9]

N. Fatès, Critical phenomena in cellular automata: Perturbing the update, the transitions, the topology, Acta Physica Polonica B, 2010. Google Scholar

[10]

N. Fatès, Quick convergence to a fixed point: A note on asynchronous elementary cellular automata, In Cellular Automata, 586–595, Springer International Publishing, 2014. Google Scholar

[11]

C. Grilo and L. Correia, Effects of asynchronism on evolutionary games, Journal of Theoretical Biology, 269 (2011), 109-122.  doi: 10.1016/j.jtbi.2010.10.022.  Google Scholar

[12]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1.  Google Scholar

[13]

Ito et al., Scaling the phase-planes of social dilemma strengths shows game-class changes in the five rules governing the evolution of cooperation, Royal Society Open Science, 2018, 181085. Google Scholar

[14]

T. Killingback and M. Doebeli, Spatial evolutionary game theory: Hawks and Doves revisited, Proceedings of the Royal Society of London. Series B: Biological Sciences, 263 (1996), 1135-1144.   Google Scholar

[15]

J. Lee, S. Adachi and F. Peper, A partitioned cellular automaton approach for efficient implementation of asynchronous circuits, The Computer Journal, 54 (2011), 1211-1220.  doi: 10.1093/comjnl/bxq089.  Google Scholar

[16]

R. Myerson, Game Theory: Analysis of Conflict Harvard Univ, Press, Cambridge, 1991.  Google Scholar

[17]

D. Newth and D. Cornforth, Asynchronous spatial evolutionary games, BioSystems, 95 (2009), 120-129.  doi: 10.1016/j.biosystems.2008.09.003.  Google Scholar

[18] M. A. Nowak, Evolutionary Dynamics, Harvard University Press, 2006.   Google Scholar
[19]

F. Peper, S. Adachi and J. Lee, Variations on the game of life, In Game of Life Cellular Automata, Springer London, 2010,235–255. Google Scholar

[20]

W. Radax and B. Rengs, Timing matters: Lessons from the CA literature on updating, arXiv preprint, arXiv: 1008.0941, 2010. Google Scholar

[21]

C. P. Roca, J. A. Cuesta and A. Sánchez, Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics, Physics of Life Reviews, 6 (2009), 208-249.  doi: 10.1016/j.plrev.2009.08.001.  Google Scholar

[22]

J. Tanimoto, Correlated asynchronous behavior updating with a mixed strategy system in spatial prisoner's dilemma games enhances cooperation, Chaos, Solitons & Fractals, 80 (2015), 39-46.  doi: 10.1016/j.chaos.2015.03.021.  Google Scholar

[23]

J. Tanimoto and H. Sagara, Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-player symmetric game, BioSystems, 90 (2007), 105-114.   Google Scholar

[24]

A. Valsecchi, L. Vanneschi and G. Mauri, A study on the automatic generation of asynchronous cellular automata rules by means of genetic algorithms, In Cellular Automata, 429–438, Springer Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-15979-4_45.  Google Scholar

[25]

X.-W. Wang, Z. Wang, S. Nie, L.-L. Jiang and B.-H. Wang, Impact of keeping silence on spatial reciprocity in spatial games, Applied Mathematics and Computation, 250 (2015), 848-853.  doi: 10.1016/j.amc.2014.11.023.  Google Scholar

[26]

Z. Wang, et al., Insight into the so-called spatial reciprocity, Physical Review E, 88 (2013), 042145. doi: 10.1103/PhysRevE.88.042145.  Google Scholar

[27]

Z. Wang, Universal scaling for the dilemma strength in evolutionary games, Physics of Life Reviews, 14 (2015), 1-30.   Google Scholar

[28]

A. Yamauchi, J. Tanimoto and A. Hagishima, An analysis of network reciprocity in Prisoner's Dilemma games using full factorial designs of experiment, BioSystems, 103 (2011), 85-92.  doi: 10.1016/j.biosystems.2010.10.006.  Google Scholar

[29]

J. Zhang and Z. Chen, Contact-based model for strategy updating and evolution of cooperation, Physica D: Nonlinear Phenomena, 323/324 (2016), 27-34.  doi: 10.1016/j.physd.2015.11.003.  Google Scholar

show all references

References:
[1]

K. M. Ariful Kabir, J. Tanimoto and Z. Wang, Influence of bolstering network reciprocity in the evolutionary spatial Prisoner's Dilemma game: A perspective, The European Physical Journal B, 91 (2018), Paper No. 312, 10 pp. doi: 10.1140/epjb/e2018-90214-6.  Google Scholar

[2]

J. M. Baetens, P. Van der Weeën and B. De Baets, Effect of asynchronous updating on the stability of cellular automata, Chaos, Solitons & Fractals, 45 (2012), 383-394.  doi: 10.1016/j.chaos.2012.01.002.  Google Scholar

[3]

S. Bandini, A. Bonomi and G. Vizzari, An analysis of different types and effects of asynchronicity in cellular automata update schemes, Natural Computing, 11 (2012), 277-287.  doi: 10.1007/s11047-012-9310-4.  Google Scholar

[4]

O. Bouré, N. Fates and V. Chevrier, First steps on asynchronous lattice-gas models with an application to a swarming rule, Natural Computing, 12 (2013), 551-560.  doi: 10.1007/s11047-013-9389-2.  Google Scholar

[5]

O. Bouré, N. Fates and V. Chevrier, Probing robustness of cellular automata through variations of asynchronous updating, Natural Computing, 11 (2012), 553-564.  doi: 10.1007/s11047-012-9340-y.  Google Scholar

[6]

O. Bouré, N. Fates and V. Chevrier, Robustness of cellular automata in the light of asynchronous information transmission, Unconventional Computation, 52–63, Lecture Notes in Comput. Sci., 6714, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-21341-0_11.  Google Scholar

[7]

N. W. H. Chan, C. Xu, S. K. Tey, Y. J. Yap and P. M. Hui, Evolutionary snowdrift game incorporating costly punishment in structured populations, Physica A: Statistical Mechanics and its Applications, 392 (2013), 168-176.  doi: 10.1016/j.physa.2012.07.078.  Google Scholar

[8]

H.-C. Chen and Y. Chow, Equilibrium selection in evolutionary games with imperfect monitoring, Journal of Applied Probability, 45 (2008), 388-402.  doi: 10.1239/jap/1214950355.  Google Scholar

[9]

N. Fatès, Critical phenomena in cellular automata: Perturbing the update, the transitions, the topology, Acta Physica Polonica B, 2010. Google Scholar

[10]

N. Fatès, Quick convergence to a fixed point: A note on asynchronous elementary cellular automata, In Cellular Automata, 586–595, Springer International Publishing, 2014. Google Scholar

[11]

C. Grilo and L. Correia, Effects of asynchronism on evolutionary games, Journal of Theoretical Biology, 269 (2011), 109-122.  doi: 10.1016/j.jtbi.2010.10.022.  Google Scholar

[12]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1.  Google Scholar

[13]

Ito et al., Scaling the phase-planes of social dilemma strengths shows game-class changes in the five rules governing the evolution of cooperation, Royal Society Open Science, 2018, 181085. Google Scholar

[14]

T. Killingback and M. Doebeli, Spatial evolutionary game theory: Hawks and Doves revisited, Proceedings of the Royal Society of London. Series B: Biological Sciences, 263 (1996), 1135-1144.   Google Scholar

[15]

J. Lee, S. Adachi and F. Peper, A partitioned cellular automaton approach for efficient implementation of asynchronous circuits, The Computer Journal, 54 (2011), 1211-1220.  doi: 10.1093/comjnl/bxq089.  Google Scholar

[16]

R. Myerson, Game Theory: Analysis of Conflict Harvard Univ, Press, Cambridge, 1991.  Google Scholar

[17]

D. Newth and D. Cornforth, Asynchronous spatial evolutionary games, BioSystems, 95 (2009), 120-129.  doi: 10.1016/j.biosystems.2008.09.003.  Google Scholar

[18] M. A. Nowak, Evolutionary Dynamics, Harvard University Press, 2006.   Google Scholar
[19]

F. Peper, S. Adachi and J. Lee, Variations on the game of life, In Game of Life Cellular Automata, Springer London, 2010,235–255. Google Scholar

[20]

W. Radax and B. Rengs, Timing matters: Lessons from the CA literature on updating, arXiv preprint, arXiv: 1008.0941, 2010. Google Scholar

[21]

C. P. Roca, J. A. Cuesta and A. Sánchez, Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics, Physics of Life Reviews, 6 (2009), 208-249.  doi: 10.1016/j.plrev.2009.08.001.  Google Scholar

[22]

J. Tanimoto, Correlated asynchronous behavior updating with a mixed strategy system in spatial prisoner's dilemma games enhances cooperation, Chaos, Solitons & Fractals, 80 (2015), 39-46.  doi: 10.1016/j.chaos.2015.03.021.  Google Scholar

[23]

J. Tanimoto and H. Sagara, Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-player symmetric game, BioSystems, 90 (2007), 105-114.   Google Scholar

[24]

A. Valsecchi, L. Vanneschi and G. Mauri, A study on the automatic generation of asynchronous cellular automata rules by means of genetic algorithms, In Cellular Automata, 429–438, Springer Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-15979-4_45.  Google Scholar

[25]

X.-W. Wang, Z. Wang, S. Nie, L.-L. Jiang and B.-H. Wang, Impact of keeping silence on spatial reciprocity in spatial games, Applied Mathematics and Computation, 250 (2015), 848-853.  doi: 10.1016/j.amc.2014.11.023.  Google Scholar

[26]

Z. Wang, et al., Insight into the so-called spatial reciprocity, Physical Review E, 88 (2013), 042145. doi: 10.1103/PhysRevE.88.042145.  Google Scholar

[27]

Z. Wang, Universal scaling for the dilemma strength in evolutionary games, Physics of Life Reviews, 14 (2015), 1-30.   Google Scholar

[28]

A. Yamauchi, J. Tanimoto and A. Hagishima, An analysis of network reciprocity in Prisoner's Dilemma games using full factorial designs of experiment, BioSystems, 103 (2011), 85-92.  doi: 10.1016/j.biosystems.2010.10.006.  Google Scholar

[29]

J. Zhang and Z. Chen, Contact-based model for strategy updating and evolution of cooperation, Physica D: Nonlinear Phenomena, 323/324 (2016), 27-34.  doi: 10.1016/j.physd.2015.11.003.  Google Scholar

Figure 1.  Illustrating Hiding Strategy
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Figure 2.  Percentage of Hawks in the final lattice
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Figure 3.  Final distribution of players in the lattice in Figure 2
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Figure 4.  Average of percentage of Hawks in the final lattice for different values of $ b $
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Figure 5.  Average payoff of players in the final lattice using different $ b $ values
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Figure 6.  Average of payoff for various player types using different $ b $ values
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Figure 7.  Percentage of Hawks using different time steps
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Figure 8.  Percentage of Hawks using different percentage of dishonest players
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Figure 9.  Percentage of Hawks for 20 different randomly generated initial lattices for $ b $ equal to 3.2, 3.3, 3.4 and 3.5
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Table 1.  Payoff matrix for chicken game
Hawk Dove
Hawk $ (b-C)/2 $ $ b $
Dove 0 $ b/2 $
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Hawk Dove
Hawk $ (b-C)/2 $ $ b $
Dove 0 $ b/2 $
Table 2.  Average of percentage of Hawks in the final lattice for different values of $ b $
b value 1 3 5 7 9
Synchronous updating 0.04922 0.19522 0.27369 0.73168 1
Asynchronous updating (displayed strategy) 0.01184 0.12838 0.41834 0.82762 1
Asynchronous updating (true strategy) 0.0121 0.12589 0.42632 0.82852 1
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b value 1 3 5 7 9
Synchronous updating 0.04922 0.19522 0.27369 0.73168 1
Asynchronous updating (displayed strategy) 0.01184 0.12838 0.41834 0.82762 1
Asynchronous updating (true strategy) 0.0121 0.12589 0.42632 0.82852 1
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