doi: 10.3934/dcdss.2022046
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Social norms for the stability of international enviromental agreements

1. 

Department of Economics and Finance, University of Bari, Largo Abbazia S. Scolastica, 53 70124 Bari, Italy

2. 

Department of Economics, Management and Territory, University of Foggia, Via Alberto da Zara 11, 71121 Foggia, Italy

* Corresponding author: Lucia Maddalena

Received  August 2021 Early access March 2022

This paper is devoted to study the stability of international environmental agreements (IEAs) in a pollution abatement context. Countries can decide to cooperate or to defect. Defector countries decide on their abatement levels by minimizing their own total cost whereas, signatory countries decide on their abatement levels by minimizing the aggregate of all cooperators.

In the model, all countries have the same environmental damage instead, respect to the non-environmental cost, we assume that each signatory country has to punish a non-signatory for its behaviour, at some cost to itself (see [17]). We propose two different cases in which we have that punishment is directly proportional to the level of pollution (see [6] or not (see [5]). Punishments can be in the form of trade sanctions or import tariffs, as a measure to encourage cooperation.

We model a differential game in order to determine both the optimal path of the abatement levels and stock pollutant as results of feedback Nash equilibria. Stability conditions, such as internal and external stability, are applied showing that different answers about the size of a stable IEA can be obtained.

Citation: Marta Biancardi, Lucia Maddalena, Giovanni Villani. Social norms for the stability of international enviromental agreements. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022046
References:
[1] S. Barrett, Environment and Statecraft: The Strategy of Environmental Treaty-Making, Oxford University Press, Oxford, 2003. 
[2]

S. Barrett, Self-enforcing international environmental agreements, Oxford Economic Papers, 46 (1994), 878-894. 

[3]

M. E. Biancardi, International environmental agreement: A dynamical model of emissions reduction, Nonlinear Dynamics in Economics, Finance and Social Sciences, Eds. Springer, (2010), 73–93. doi: 10.1007/978-3-642-04023-8_5.

[4]

M. Biancardi and G. Villani, International environmental agreements with asymmetric countries, Computational Economics, 36 (2010), 69-92. 

[5]

G. I. Bischi, F. Lamantia and L. Sbragia, Competition and cooperation in natural resources exploitation: An evolutionary game approach, In: C. Carraro and V. Fragnelli (Eds.) Game Practice and the Environment. Northampton: Edward Elgar Publishing, (2004), 187–211.

[6]

M. BretonL. Sbragia and G. Zaccour, A dynamic model for international environmental agreements, Environmental and Resource Economics, 45 (2010), 25-48. 

[7]

E. Calvio and S. J. Rubio, Dynamic Models of International Environmental Agreements: A Differential Game Approach., The International Review of Environmental and Resource Economics, Vol. 6, 2012.

[8]

C. Carraro and D. Siniscalco, Strategies for the international protection of the environment, Journal of Public Economics, 52 (1993), 309-328. 

[9]

C. d'AspremontA. JacqueminJ. Gabszewicz and J. A. Weymark, On the stability of collusive price leadership, Canadian Journal of Economics, 16 (1983), 17-25. 

[10]

E. Diamantoudi and E. Sartzetakis, Stable international environmental agreements: An analytical approach, Journal of Public Economic Theory, 8 (2006), 247-263. 

[11]

M. Hoel, International environmental conventions: The case of uniform reductions of emissions, Environmental and Resource Economics, 2 (1992), 141-159. 

[12]

M. Hoel and K. Schneider, Incentives to participate in a international environmental agreement, Environmental and Resource Economics, 9 (1997), 153-170. 

[13]

M. Labriet and R. Loulou, Coupling climate damages and GHG abatement costs in a linear programming framework, Environmental Modeling and Assessment, 8 (2003), 261-274. 

[14]

S. J. Rubio and B. Casino, Self-enforcing international environmental agreements with a stock pollutant, Spanish Economic Review, 7 (2005), 89-109. 

[15]

S. J. Rubio and A. Ulph, Self-enforcing international environmental agreements revisited, Oxford Economic Papers, 58 (2006), 233-263. 

[16]

S. J. Rubio and A. Ulph, An infinite-horizon model of dynamic membership of international environmental agreements, Journal of Environmental Economics and Management, 54 (2007), 296-310. 

[17]

R. Sethi and E. Somanathan, The evolution of social norms in common property resource use, The American Economic Review, 86 (1996), 766-788. 

[18]

A. de. Zeeuw, Dynamic effects on the stability of international environmental agreements, Journal of Environmental Economics and Management, 55 (2008), 163-174. 

show all references

References:
[1] S. Barrett, Environment and Statecraft: The Strategy of Environmental Treaty-Making, Oxford University Press, Oxford, 2003. 
[2]

S. Barrett, Self-enforcing international environmental agreements, Oxford Economic Papers, 46 (1994), 878-894. 

[3]

M. E. Biancardi, International environmental agreement: A dynamical model of emissions reduction, Nonlinear Dynamics in Economics, Finance and Social Sciences, Eds. Springer, (2010), 73–93. doi: 10.1007/978-3-642-04023-8_5.

[4]

M. Biancardi and G. Villani, International environmental agreements with asymmetric countries, Computational Economics, 36 (2010), 69-92. 

[5]

G. I. Bischi, F. Lamantia and L. Sbragia, Competition and cooperation in natural resources exploitation: An evolutionary game approach, In: C. Carraro and V. Fragnelli (Eds.) Game Practice and the Environment. Northampton: Edward Elgar Publishing, (2004), 187–211.

[6]

M. BretonL. Sbragia and G. Zaccour, A dynamic model for international environmental agreements, Environmental and Resource Economics, 45 (2010), 25-48. 

[7]

E. Calvio and S. J. Rubio, Dynamic Models of International Environmental Agreements: A Differential Game Approach., The International Review of Environmental and Resource Economics, Vol. 6, 2012.

[8]

C. Carraro and D. Siniscalco, Strategies for the international protection of the environment, Journal of Public Economics, 52 (1993), 309-328. 

[9]

C. d'AspremontA. JacqueminJ. Gabszewicz and J. A. Weymark, On the stability of collusive price leadership, Canadian Journal of Economics, 16 (1983), 17-25. 

[10]

E. Diamantoudi and E. Sartzetakis, Stable international environmental agreements: An analytical approach, Journal of Public Economic Theory, 8 (2006), 247-263. 

[11]

M. Hoel, International environmental conventions: The case of uniform reductions of emissions, Environmental and Resource Economics, 2 (1992), 141-159. 

[12]

M. Hoel and K. Schneider, Incentives to participate in a international environmental agreement, Environmental and Resource Economics, 9 (1997), 153-170. 

[13]

M. Labriet and R. Loulou, Coupling climate damages and GHG abatement costs in a linear programming framework, Environmental Modeling and Assessment, 8 (2003), 261-274. 

[14]

S. J. Rubio and B. Casino, Self-enforcing international environmental agreements with a stock pollutant, Spanish Economic Review, 7 (2005), 89-109. 

[15]

S. J. Rubio and A. Ulph, Self-enforcing international environmental agreements revisited, Oxford Economic Papers, 58 (2006), 233-263. 

[16]

S. J. Rubio and A. Ulph, An infinite-horizon model of dynamic membership of international environmental agreements, Journal of Environmental Economics and Management, 54 (2007), 296-310. 

[17]

R. Sethi and E. Somanathan, The evolution of social norms in common property resource use, The American Economic Review, 86 (1996), 766-788. 

[18]

A. de. Zeeuw, Dynamic effects on the stability of international environmental agreements, Journal of Environmental Economics and Management, 55 (2008), 163-174. 

Figure 1.  The solid line is the evolution of pollution s(t) in the the first model while, the dashodot line is the evolution in the second model. The gold surface is the evolution of s(t) in the first model while the violet in the second model.
Table 1.  Coalition Stability with $ p = 0.20 $
Coalition $ \psi < 0 $ $ \psi=0 $ $ \psi=0 $
$ \xi > 0 $ $ \xi > 0 $ $ \xi=0 $
First Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.00; 6.94] $ $ \xi\,\in\,[0.00; 5.55] $ Stable
$ m=4 $ $ \xi\,\in\,[6.94; 12.69] $ $ \xi\,\in\,[5.55; 11.11] $ Never Stable
$ m=5 $ $ \xi\,\in\,[12.69; 18.11] $ $ \xi\,\in\,[11.11; 16.66] $ Never Stable
$ m=6 $ $ \xi\,\in\,[18.11; 23.39] $ $ \xi\,\in\,[16.66; 22.22] $ Never Stable
$ m=7 $ $ \xi\,\in\,[23.39; 28.59] $ $ \xi\,\in\,[22.22; 27.77] $ Never Stable
$ m=8 $ $ \xi\,\in\,[28.59; 33.75] $ $ \xi\,\in\,[27.77; 33.33] $ Never Stable
$ m=9 $ $ \xi\,\in\,[33.75; 38.88] $ $ \xi\,\in\,[33.33; 38.88] $ Never Stable
$ m=10 $ $ \xi >38.88 $ $ \xi >38.88 $ Never Stable
Second Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.0000; 0.0005] $ $ \xi\,\in\,[0.0000; 0.0004] $ Stable
$ m=4 $ $ \xi\,\in\,[0.0005;0.0010] $ $ \xi\,\in\,[0.0004; 0.0009] $ Never Stable
$ m=5 $ $ \xi\,\in\,[0.0010; 0.0015] $ $ \xi\,\in\,[0.0009; 0.0014] $ Never Stable
$ m=6 $ $ \xi\,\in\,[0.0015; 0.0021] $ $ \xi\,\in\,[0.0014; 0.0019] $ Never Stable
$ m=7 $ $ \xi\,\in\,[0.0021; 0.0027] $ $ \xi\,\in\,[0.0019; 0.0026] $ Never Stable
$ m=8 $ $ \xi\,\in\,[0.0027; 0.0035] $ $ \xi\,\in\,[0.0026; 0.0034] $ Never Stable
$ m=9 $ $ \xi\,\in\,[0.0035; 0.0044] $ $ \xi\,\in\,[0.0034; 0.0044] $ Never Stable
$ m=10 $ $ \xi\,\in\,[0.0044; 2.4800] $ $ \xi\,\in\,[0.0044; 7.0100] $ Never Stable
Coalition $ \psi < 0 $ $ \psi=0 $ $ \psi=0 $
$ \xi > 0 $ $ \xi > 0 $ $ \xi=0 $
First Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.00; 6.94] $ $ \xi\,\in\,[0.00; 5.55] $ Stable
$ m=4 $ $ \xi\,\in\,[6.94; 12.69] $ $ \xi\,\in\,[5.55; 11.11] $ Never Stable
$ m=5 $ $ \xi\,\in\,[12.69; 18.11] $ $ \xi\,\in\,[11.11; 16.66] $ Never Stable
$ m=6 $ $ \xi\,\in\,[18.11; 23.39] $ $ \xi\,\in\,[16.66; 22.22] $ Never Stable
$ m=7 $ $ \xi\,\in\,[23.39; 28.59] $ $ \xi\,\in\,[22.22; 27.77] $ Never Stable
$ m=8 $ $ \xi\,\in\,[28.59; 33.75] $ $ \xi\,\in\,[27.77; 33.33] $ Never Stable
$ m=9 $ $ \xi\,\in\,[33.75; 38.88] $ $ \xi\,\in\,[33.33; 38.88] $ Never Stable
$ m=10 $ $ \xi >38.88 $ $ \xi >38.88 $ Never Stable
Second Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.0000; 0.0005] $ $ \xi\,\in\,[0.0000; 0.0004] $ Stable
$ m=4 $ $ \xi\,\in\,[0.0005;0.0010] $ $ \xi\,\in\,[0.0004; 0.0009] $ Never Stable
$ m=5 $ $ \xi\,\in\,[0.0010; 0.0015] $ $ \xi\,\in\,[0.0009; 0.0014] $ Never Stable
$ m=6 $ $ \xi\,\in\,[0.0015; 0.0021] $ $ \xi\,\in\,[0.0014; 0.0019] $ Never Stable
$ m=7 $ $ \xi\,\in\,[0.0021; 0.0027] $ $ \xi\,\in\,[0.0019; 0.0026] $ Never Stable
$ m=8 $ $ \xi\,\in\,[0.0027; 0.0035] $ $ \xi\,\in\,[0.0026; 0.0034] $ Never Stable
$ m=9 $ $ \xi\,\in\,[0.0035; 0.0044] $ $ \xi\,\in\,[0.0034; 0.0044] $ Never Stable
$ m=10 $ $ \xi\,\in\,[0.0044; 2.4800] $ $ \xi\,\in\,[0.0044; 7.0100] $ Never Stable
Table 2.  Coalition Stability with $ p = 0.30 $
Coalition $ \psi > 0 $ $ \psi=0 $ $ \psi=0 $
$ \xi > 0 $ $ \xi > 0 $ $ \xi=0 $
First Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.00; 15.62] $ $ \xi\,\in\,[0.00; 12.50] $ Stable
$ m=4 $ $ \xi\,\in\,[15.62; 28.57] $ $ \xi\,\in\,[12.50; 25.00] $ Never Stable
$ m=5 $ $ \xi\,\in\,[28.57; 40.76] $ $ \xi\,\in\,[25.00; 37.50] $ Never Stable
$ m=6 $ $ \xi\,\in\,[40.76; 52.63] $ $ \xi\,\in\,[37.50; 50.00] $ Never Stable
$ m=7 $ $ \xi\,\in\,[52.63; 64.33] $ $ \xi\,\in\,[50.00; 62.50] $ Never Stable
$ m=8 $ $ \xi\,\in\,[64.33; 75.94] $ $ \xi\,\in\,[62.50; 75.00] $ Never Stable
$ m=9 $ $ \xi\,\in\,[75.94; 87.50] $ $ \xi\,\in\,[75.00; 87.50] $ Never Stable
$ m=10 $ Never Stable ($ \Omega <0 $) Never Stable ($ \Omega <0 $) Never Stable ($ \Omega <0 $)
Second Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.0000; 0.0012] $ $ \xi\,\in\,[0.0000; 0.0010] $ Stable
$ m=4 $ $ \xi\,\in\,[0.0012;0.0024] $ $ \xi\,\in\,[0.0010; 0.0021] $ Never Stable
$ m=5 $ $ \xi\,\in\,[0.0024; 0.0038] $ $ \xi\,\in\,[0.0021; 0.0034] $ Never Stable
$ m=6 $ $ \xi\,\in\,[0.0038; 0.0053] $ $ \xi\,\in\,[0.0034; 0.0049] $ Never Stable
$ m=7 $ $ \xi\,\in\,[0.0053; 0.0073] $ $ \xi\,\in\,[0.0049; 0.0069] $ Never Stable
$ m=8 $ $ \xi\,\in\,[0.0073; 0.0099] $ $ \xi\,\in\,[0.0069; 0.0096] $ Never Stable
$ m=9 $ $ \xi\,\in\,[0.0099; 0.0137] $ $ \xi\,\in\,[0.0096; 0.0134] $ Never Stable
$ m=10 $ Never Stable ($ \Omega' <0 $) Never Stable ($ \Omega' <0 $) Never Stable
Coalition $ \psi > 0 $ $ \psi=0 $ $ \psi=0 $
$ \xi > 0 $ $ \xi > 0 $ $ \xi=0 $
First Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.00; 15.62] $ $ \xi\,\in\,[0.00; 12.50] $ Stable
$ m=4 $ $ \xi\,\in\,[15.62; 28.57] $ $ \xi\,\in\,[12.50; 25.00] $ Never Stable
$ m=5 $ $ \xi\,\in\,[28.57; 40.76] $ $ \xi\,\in\,[25.00; 37.50] $ Never Stable
$ m=6 $ $ \xi\,\in\,[40.76; 52.63] $ $ \xi\,\in\,[37.50; 50.00] $ Never Stable
$ m=7 $ $ \xi\,\in\,[52.63; 64.33] $ $ \xi\,\in\,[50.00; 62.50] $ Never Stable
$ m=8 $ $ \xi\,\in\,[64.33; 75.94] $ $ \xi\,\in\,[62.50; 75.00] $ Never Stable
$ m=9 $ $ \xi\,\in\,[75.94; 87.50] $ $ \xi\,\in\,[75.00; 87.50] $ Never Stable
$ m=10 $ Never Stable ($ \Omega <0 $) Never Stable ($ \Omega <0 $) Never Stable ($ \Omega <0 $)
Second Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.0000; 0.0012] $ $ \xi\,\in\,[0.0000; 0.0010] $ Stable
$ m=4 $ $ \xi\,\in\,[0.0012;0.0024] $ $ \xi\,\in\,[0.0010; 0.0021] $ Never Stable
$ m=5 $ $ \xi\,\in\,[0.0024; 0.0038] $ $ \xi\,\in\,[0.0021; 0.0034] $ Never Stable
$ m=6 $ $ \xi\,\in\,[0.0038; 0.0053] $ $ \xi\,\in\,[0.0034; 0.0049] $ Never Stable
$ m=7 $ $ \xi\,\in\,[0.0053; 0.0073] $ $ \xi\,\in\,[0.0049; 0.0069] $ Never Stable
$ m=8 $ $ \xi\,\in\,[0.0073; 0.0099] $ $ \xi\,\in\,[0.0069; 0.0096] $ Never Stable
$ m=9 $ $ \xi\,\in\,[0.0099; 0.0137] $ $ \xi\,\in\,[0.0096; 0.0134] $ Never Stable
$ m=10 $ Never Stable ($ \Omega' <0 $) Never Stable ($ \Omega' <0 $) Never Stable
Table 3.  Coalition Stability with $ p = 0.40 $
Coalition $ \psi < 0 $ $ \psi=0 $ $ \psi=0 $
$ \xi > 0 $ $ \xi > 0 $ $ \xi=0 $
First Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.00; 27.77] $ $ \xi\,\in\,[0.00; 22.22] $ Stable
$ m=4 $ $ \xi\,\in\,[27.77; 50.79] $ $ \xi\,\in\,[22.22; 44.44] $ Never Stable
$ m=5 $ $ \xi\,\in\,[50.79; 72.46] $ $ \xi\,\in\,[44.44; 66.66] $ Never Stable
$ m=6 $ $ \xi\,\in\,[ 72.46; 93.56] $ $ \xi\,\in\,[66.66; 88.88] $ Never Stable
$ m=7 $ $ \xi\,\in\,[93.56; 114.37] $ $ \xi\,\in\,[88.88; 111.11] $ Never Stable
$ m=8 $ $ \xi\,\in\,[114.37; 135.02] $ $ \xi\,\in\,[111.11; 133.33] $ Never Stable
$ m=9 $ Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $) Never Stable
$ m=10 $ Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $)
Second Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.0000; 0.0023] $ $ \xi\,\in\,[0.0000; 0.0018] $ Stable
$ m=4 $ $ \xi\,\in\,[0.0023; 0.0040] $ $ \xi\,\in\,[0.0018; 0.0040] $ Never Stable
$ m=5 $ $ \xi\,\in\,[0.0040; 0.0073] $ $ \xi\,\in\,[0.0040; 0.0066] $ Never Stable
$ m=6 $ $ \xi\,\in\,[0.0073; 0.0100] $ $ \xi\,\in\,[0.0066; 0.0100] $ Never Stable
$ m=7 $ $ \xi\,\in\,[0.0100; 0.0150] $ $ \xi\,\in\,[0.0100; 0.0147] $ Never Stable
$ m=8 $ $ \xi\,\in\,[0.0150; 0.0230] $ $ \xi\,\in\,[0.0147; 0.0224] $ Never Stable
$ m=9 $ Never Stable ($ \Omega' > 0 $) Never Stable ($ \Omega' > 0 $) Never Stable
$ m=10 $ Never Stable ($ \Omega' > 0 $) Never Stable ($ \Omega' > 0 $) Never Stable
Coalition $ \psi < 0 $ $ \psi=0 $ $ \psi=0 $
$ \xi > 0 $ $ \xi > 0 $ $ \xi=0 $
First Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.00; 27.77] $ $ \xi\,\in\,[0.00; 22.22] $ Stable
$ m=4 $ $ \xi\,\in\,[27.77; 50.79] $ $ \xi\,\in\,[22.22; 44.44] $ Never Stable
$ m=5 $ $ \xi\,\in\,[50.79; 72.46] $ $ \xi\,\in\,[44.44; 66.66] $ Never Stable
$ m=6 $ $ \xi\,\in\,[ 72.46; 93.56] $ $ \xi\,\in\,[66.66; 88.88] $ Never Stable
$ m=7 $ $ \xi\,\in\,[93.56; 114.37] $ $ \xi\,\in\,[88.88; 111.11] $ Never Stable
$ m=8 $ $ \xi\,\in\,[114.37; 135.02] $ $ \xi\,\in\,[111.11; 133.33] $ Never Stable
$ m=9 $ Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $) Never Stable
$ m=10 $ Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $)
Second Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.0000; 0.0023] $ $ \xi\,\in\,[0.0000; 0.0018] $ Stable
$ m=4 $ $ \xi\,\in\,[0.0023; 0.0040] $ $ \xi\,\in\,[0.0018; 0.0040] $ Never Stable
$ m=5 $ $ \xi\,\in\,[0.0040; 0.0073] $ $ \xi\,\in\,[0.0040; 0.0066] $ Never Stable
$ m=6 $ $ \xi\,\in\,[0.0073; 0.0100] $ $ \xi\,\in\,[0.0066; 0.0100] $ Never Stable
$ m=7 $ $ \xi\,\in\,[0.0100; 0.0150] $ $ \xi\,\in\,[0.0100; 0.0147] $ Never Stable
$ m=8 $ $ \xi\,\in\,[0.0150; 0.0230] $ $ \xi\,\in\,[0.0147; 0.0224] $ Never Stable
$ m=9 $ Never Stable ($ \Omega' > 0 $) Never Stable ($ \Omega' > 0 $) Never Stable
$ m=10 $ Never Stable ($ \Omega' > 0 $) Never Stable ($ \Omega' > 0 $) Never Stable
Table 4.  Coalition Stability with $ p = 0.50 $
Coalition $ \psi < 0 $ $ \psi=0 $ $ \psi=0 $
$ \xi > 0 $ $ \xi > 0 $ $ \xi=0 $
First Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.00; 43.40] $ $ \xi\,\in\, [0.00; 34.72] $ Stable
$ m=4 $ $ \xi\,\in\, [43.40; 79.36] $ $ \xi\,\in\, [34.72; 69.44] $ Never Stable
$ m=5 $ $ \xi\,\in\, [79.36; 113.22] $ $ \xi\,\in\, [ 69.44 ;104.16] $ Never Stable
$ m=6 $ $ \xi\,\in\, [113.22; 146.19] $ $ \xi\,\in\, [104.16; 138.88] $ Never Stable
$ m=7 $ $ \xi\,\in\, [146.19; 178.71] $ $ \xi\,\in\, [138.88; 173.61] $ Never Stable
$ m=8 $ Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $) Never Stable
$ m=9 $ Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $) Never Stable
$ m=10 $ Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $) Never Stable
Second Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\, [0.0000; 0.0038] $ $ \xi\,\in\, [0.0000; 0.0030] $ Stable
$ m=4 $ $ \xi\,\in\, [0.0038; 0.0077] $ $ \xi\,\in\, [0.0030; 0.0066] $ Never Stable
$ m=5 $ $ \xi\,\in\, [0.0077; 0.0127] $ $ \xi\,\in\, [0.0066;0.0113 ] $ Never Stable
$ m=6 $ $ \xi\,\in\, [0.0127; 0.0197] $ $ \xi\,\in\, [0.0113; 0.0179] $ Never Stable
$ m=7 $ $ \xi\,\in\, [ 0.0197; 0.0317] $ $ \xi\,\in\, [ 0.0179; 0.0288] $ Never Stable
$ m=8 $ Never Stable ($ \Omega' > 0 $) Never Stable ($ \Omega' > 0 $) Never Stable
$ m=9 $ Never Stable ($ \Omega' > 0 $) Never Stable ($ \Omega' > 0 $) Never Stable
$ m=10 $ Never Stable ($ \Omega' > 0 $) Never Stable ($ \Omega' > 0 $) Never Stable
Coalition $ \psi < 0 $ $ \psi=0 $ $ \psi=0 $
$ \xi > 0 $ $ \xi > 0 $ $ \xi=0 $
First Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\,[0.00; 43.40] $ $ \xi\,\in\, [0.00; 34.72] $ Stable
$ m=4 $ $ \xi\,\in\, [43.40; 79.36] $ $ \xi\,\in\, [34.72; 69.44] $ Never Stable
$ m=5 $ $ \xi\,\in\, [79.36; 113.22] $ $ \xi\,\in\, [ 69.44 ;104.16] $ Never Stable
$ m=6 $ $ \xi\,\in\, [113.22; 146.19] $ $ \xi\,\in\, [104.16; 138.88] $ Never Stable
$ m=7 $ $ \xi\,\in\, [146.19; 178.71] $ $ \xi\,\in\, [138.88; 173.61] $ Never Stable
$ m=8 $ Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $) Never Stable
$ m=9 $ Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $) Never Stable
$ m=10 $ Never Stable ($ \Omega > 0 $) Never Stable ($ \Omega > 0 $) Never Stable
Second Model
$ m=1 $
$ m=2 $ Stable
$ m=3 $ $ \xi\,\in\, [0.0000; 0.0038] $ $ \xi\,\in\, [0.0000; 0.0030] $ Stable
$ m=4 $ $ \xi\,\in\, [0.0038; 0.0077] $ $ \xi\,\in\, [0.0030; 0.0066] $ Never Stable
$ m=5 $ $ \xi\,\in\, [0.0077; 0.0127] $ $ \xi\,\in\, [0.0066;0.0113 ] $ Never Stable
$ m=6 $ $ \xi\,\in\, [0.0127; 0.0197] $ $ \xi\,\in\, [0.0113; 0.0179] $ Never Stable
$ m=7 $ $ \xi\,\in\, [ 0.0197; 0.0317] $ $ \xi\,\in\, [ 0.0179; 0.0288] $ Never Stable
$ m=8 $ Never Stable ($ \Omega' > 0 $) Never Stable ($ \Omega' > 0 $) Never Stable
$ m=9 $ Never Stable ($ \Omega' > 0 $) Never Stable ($ \Omega' > 0 $) Never Stable
$ m=10 $ Never Stable ($ \Omega' > 0 $) Never Stable ($ \Omega' > 0 $) Never Stable
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