July  2018, 5(3): 223-230. doi: 10.3934/jdg.2018014

Critical transitions and Early Warning Signals in repeated Cooperation Games

Institute of Systems Sciences, Innovation and Sustainability Research, University of Graz, Graz, Austria

* Corresponding author: georg.jaeger@uni-graz.at

Received  November 2017 Revised  February 2018 Published  June 2018

Scanning a system's dynamics for critical transitions, i.e. sudden shifts from one system state to another, with the methodology of Early Warning Signals has been shown to yield promising results in many scientific fields. So far however, such investigations focus on aggregated system dynamics modeled with equation-based methods. In this paper the methodology of Early Warning Signals is applied to critical transitions found in the context of Cooperation Games. Since equation-based methods are not well suited to account for interactions in game theoretic settings, an agent-based model of a repeated Cooperation Game is used to generate data. We find that Early Warning Signals can be detected in agent-based simulations of such systems.

Citation: Christian Hofer, Georg Jäger, Manfred Füllsack. Critical transitions and Early Warning Signals in repeated Cooperation Games. Journal of Dynamics & Games, 2018, 5 (3) : 223-230. doi: 10.3934/jdg.2018014
References:
[1]

G. E. Bolton and A. Ockenfels, Self-centered fairness in games with more than two players, Handbook of Experimental Economics Results, 1 (2008), 531-540.  doi: 10.1016/S1574-0722(07)00059-5.  Google Scholar

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V. DakosS. R. CarpenterW. A. BrockA. M. EllisonV. GuttalA. R. IvesS. KéfiV. LivinaD. A. SeekellE. H. van Nes and M. Scheffer, Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data, PLOS ONE, 7 (2012), 1-20.  doi: 10.1371/journal.pone.0041010.  Google Scholar

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E. Fehr and S. Gachter, Cooperation and punishment in public goods experiments, American Economic Review, 90 (2000), 980-994.  doi: 10.1257/aer.90.4.980.  Google Scholar

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E. Fehr and K. M. Schmidt, Fairness, incentives, and contractual choices, European Economic Review, 44 (2000), 1057-1068.   Google Scholar

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U. FischbacherS. Gächter and E. Fehr, Are people conditionally cooperative? evidence from a public goods experiment, Economics Letters, 71 (2001), 397-404.   Google Scholar

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H. v. Foerster, Objects: Tokens for (eigen-)behaviors, ASC Cybernetics Forum, 8 (1976), 91–96. Reprinted in: Foerster H. von (1981) Observing systems. Intersystems Publications, Seaside CA: 274–285., Reprinted in: Foerster H. von (2003) Understanding understanding: Essays on cybernetics and cognition. Springer, New York: 261–271. Google Scholar

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G. Harras and D. Sornette, How to grow a bubble: A model of myopic adapting agents, Journal of Economic Behavior & Organization, 80 (2011), 137-152.  doi: 10.1016/j.jebo.2011.03.003.  Google Scholar

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Z.-Q. JiangW.-X. ZhouD. SornetteR. WoodardK. Bastiaensen and P. Cauwels, Bubble diagnosis and prediction of the 2005-2007 and 2008-2009 chinese stock market bubbles, Journal of Economic Behavior & Organization, 74 (2010), 149-162.   Google Scholar

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C. Kuehn, A mathematical framework for critical transitions: Bifurcations, fast-slow systems and stochastic dynamics, Physica D: Nonlinear Phenomena, 240 (2011), 1020-1035.   Google Scholar

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J. O. Ledyard, Public goods: A survey of experimental research, 1994. Google Scholar

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R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.  doi: 10.1038/269471a0.  Google Scholar

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P. E. McSharryL. A. Smith and L. Tarassenko, Prediction of epileptic seizures: Are nonlinear methods relevant?, Nat Med, 9 (2003), 241-242.  doi: 10.1038/nm0303-241.  Google Scholar

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R. O. Murphy and K. A. Ackermann, Social value orientation: Theoretical and measurement issues in the study of social preferences, Personality and Social Psychology Review, 18 (2014), 13-41.   Google Scholar

[22]

J. F. Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences, 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48.  Google Scholar

[23]

H. H. Nax and M. Perc, Directional learning and the provisioning of public goods, Scientific Reports, 5 (2015), P8010.  doi: 10.1038/srep08010.  Google Scholar

[24]

M. SchefferJ. BascompteW. A. BrockV. BrovkinS. R. CarpenterV. DakosH. HeldE. H. Van NesM. Rietkerk and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.  doi: 10.1038/nature08227.  Google Scholar

[25]

M. SchefferS. CarpenterJ. A. FoleyC. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.  doi: 10.1038/35098000.  Google Scholar

[26]

D. Sornette, Critical market crashes, Physics Reports, 378 (2003), 1-98.  doi: 10.1016/S0370-1573(02)00634-8.  Google Scholar

[27]

D. Sornette, Physics and financial economics (1776–2014): Puzzles, ising and agent-based models, Reports on Progress in Physics, 77 (2014), 062001, 28pp. doi: 10.1088/0034-4885/77/6/062001.  Google Scholar

show all references

References:
[1]

G. E. Bolton and A. Ockenfels, Self-centered fairness in games with more than two players, Handbook of Experimental Economics Results, 1 (2008), 531-540.  doi: 10.1016/S1574-0722(07)00059-5.  Google Scholar

[2]

M. N. Burton-ChellewH. H. Nax and S. A. West, Payoff-based learning explains the decline in cooperation in public goods games, Proceedings of the Royal Society of London B: Biological Sciences, 282 (2015), 20142678.  doi: 10.1098/rspb.2014.2678.  Google Scholar

[3]

G. Charness and M. Rabin, Understanding social preferences with simple tests, The Quarterly Journal of Economics, 117 (2002), 817-869.   Google Scholar

[4]

A. Chaudhuri, Sustaining cooperation in laboratory public goods experiments: A selective survey of the literature, Experimental Economics, 14 (2011), 47-83.  doi: 10.1007/s10683-010-9257-1.  Google Scholar

[5]

L. DaiD. VorselenK. S. Korolev and J. Gore, Generic indicators for loss of resilience before a tipping point leading to population collapse, Science, 336 (2012), 1175-1177.  doi: 10.1126/science.1219805.  Google Scholar

[6]

V. Dakos and J. Bascompte, Critical slowing down as early warning for the onset of collapse in mutualistic communities, Proceedings of the National Academy of Sciences, 111 (2014), 17546-17551.  doi: 10.1073/pnas.1406326111.  Google Scholar

[7]

V. DakosS. R. CarpenterW. A. BrockA. M. EllisonV. GuttalA. R. IvesS. KéfiV. LivinaD. A. SeekellE. H. van Nes and M. Scheffer, Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data, PLOS ONE, 7 (2012), 1-20.  doi: 10.1371/journal.pone.0041010.  Google Scholar

[8]

E. Fehr and S. Gachter, Cooperation and punishment in public goods experiments, American Economic Review, 90 (2000), 980-994.  doi: 10.1257/aer.90.4.980.  Google Scholar

[9]

E. Fehr and K. M. Schmidt, Fairness, incentives, and contractual choices, European Economic Review, 44 (2000), 1057-1068.   Google Scholar

[10]

U. FischbacherS. Gächter and E. Fehr, Are people conditionally cooperative? evidence from a public goods experiment, Economics Letters, 71 (2001), 397-404.   Google Scholar

[11]

H. v. Foerster, Objects: Tokens for (eigen-)behaviors, ASC Cybernetics Forum, 8 (1976), 91–96. Reprinted in: Foerster H. von (1981) Observing systems. Intersystems Publications, Seaside CA: 274–285., Reprinted in: Foerster H. von (2003) Understanding understanding: Essays on cybernetics and cognition. Springer, New York: 261–271. Google Scholar

[12]

G. Harras and D. Sornette, How to grow a bubble: A model of myopic adapting agents, Journal of Economic Behavior & Organization, 80 (2011), 137-152.  doi: 10.1016/j.jebo.2011.03.003.  Google Scholar

[13]

Z.-Q. JiangW.-X. ZhouD. SornetteR. WoodardK. Bastiaensen and P. Cauwels, Bubble diagnosis and prediction of the 2005-2007 and 2008-2009 chinese stock market bubbles, Journal of Economic Behavior & Organization, 74 (2010), 149-162.   Google Scholar

[14]

C. Kuehn, A mathematical framework for critical transitions: Bifurcations, fast-slow systems and stochastic dynamics, Physica D: Nonlinear Phenomena, 240 (2011), 1020-1035.   Google Scholar

[15]

J. O. Ledyard, Public goods: A survey of experimental research, 1994. Google Scholar

[16]

T. M. Lenton, Early warning of climate tipping points, Nature Clim. Change, 1 (2011), 201-209.  doi: 10.1038/nclimate1143.  Google Scholar

[17]

T. M. LentonH. HeldE. KrieglerJ. W. HallW. LuchtS. Rahmstorf and H. J. Schellnhuber, Tipping elements in the earth's climate system, Proceedings of the National Academy of Sciences, 105 (2008), 1786-1793.  doi: 10.1073/pnas.0705414105.  Google Scholar

[18]

B. LittR. EstellerJ. EchauzM. D'AlessandroR. ShorT. HenryP. PennellC. EpsteinR. BakayM. Dichter and G. Vachtsevanos, Epileptic seizures may begin hours in advance of clinical onset: A report of five patients, Neuron, 30 (2001), 51-64.  doi: 10.1007/978-90-481-3018-4_9.  Google Scholar

[19]

R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.  doi: 10.1038/269471a0.  Google Scholar

[20]

P. E. McSharryL. A. Smith and L. Tarassenko, Prediction of epileptic seizures: Are nonlinear methods relevant?, Nat Med, 9 (2003), 241-242.  doi: 10.1038/nm0303-241.  Google Scholar

[21]

R. O. Murphy and K. A. Ackermann, Social value orientation: Theoretical and measurement issues in the study of social preferences, Personality and Social Psychology Review, 18 (2014), 13-41.   Google Scholar

[22]

J. F. Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences, 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48.  Google Scholar

[23]

H. H. Nax and M. Perc, Directional learning and the provisioning of public goods, Scientific Reports, 5 (2015), P8010.  doi: 10.1038/srep08010.  Google Scholar

[24]

M. SchefferJ. BascompteW. A. BrockV. BrovkinS. R. CarpenterV. DakosH. HeldE. H. Van NesM. Rietkerk and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.  doi: 10.1038/nature08227.  Google Scholar

[25]

M. SchefferS. CarpenterJ. A. FoleyC. Folke and B. Walker, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.  doi: 10.1038/35098000.  Google Scholar

[26]

D. Sornette, Critical market crashes, Physics Reports, 378 (2003), 1-98.  doi: 10.1016/S0370-1573(02)00634-8.  Google Scholar

[27]

D. Sornette, Physics and financial economics (1776–2014): Puzzles, ising and agent-based models, Reports on Progress in Physics, 77 (2014), 062001, 28pp. doi: 10.1088/0034-4885/77/6/062001.  Google Scholar

Figure 1.  Time development of the rate of cooperators in a simulated Cooperation Game with a population of 1000 agents. The highlighted fraction of the time series (orange) was considered for EWS analysis.
Figure 2.  Critical transition of cooperation in a Cooperation Game. Simulated data (orange) is compared to a Hill function (blue) that is known to replicate critical transitions on an aggregated level.
Figure 3.  Results of EWS analysis for pure (left column) as well as for detrended data (right column). Apart from the kurtosis, all indicators show EWSs.
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