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August & September  2019, 12(4&5): 989-1003. doi: 10.3934/dcdss.2019067

Environmental game modeling with uncertainties

1. 

Business School, University of Shanghai for Science and Technology, Shanghai 200093, China

2. 

School of Science, Shandong University of Technology, Shandong 255049, China

* Corresponding author: Shaojian Qu

Received  July 2017 Revised  December 2017 Published  November 2018

Fund Project: The first author is supported by NSSF grant.

We model environmental games with stochastic data based on an imprecise distribution which is assumed to be attached to an a-priori known set. Our model is different from previous games where the probability distribution of the uncertain data is precisely given. Our model is also different from the robust games which presents a robust optimization approach to game models with the uncertain data in a compact convex set without probabilistic information which can lead to overly conservative solutions. A distributionally robust approach is used to cope with our setting in the games by combining the stochastic optimization and robust optimization approaches which can be termed as the distributionally robust environmental games. We show that the existence of an equilibrium for the distributionally robust environmental games under mild assumptions. The computation method for equilibrium, with the first- and second-information about the probability of uncertain data, can be reformulated as a semidefinite programming problem which can be tractably realized. Numerical tests are given to show the efficiency of the proposed methods.

Citation: Ying Ji, Shaojian Qu, Fuxing Chen. Environmental game modeling with uncertainties. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 989-1003. doi: 10.3934/dcdss.2019067
References:
[1]

M. Aghassi and D. Bertsimas, Robust game theory, Mathematical Programming, 107 (2006), 231-273.  doi: 10.1007/s10107-005-0686-0.

[2]

A. Ben Tal and A. Nemirovski, Robust convex optimization, Mathematical Methods of Operations Research, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769.

[3]

M. BretonG. Zaccour and M. Zahaf, A game-theoretic formulation of joint implementation of environmental projects, European Journal of Operational Research, 168 (2005), 221-239.  doi: 10.1016/j.ejor.2004.04.026.

[4]

S. Boyd and L. Vandenberghe, Convex Optimization Cambridge, U.K.: Cambridge University, Press, 2004. doi: 10.1017/CBO9780511804441s.

[5]

X. ChenM. SimP. Sun and J. W. Zhang, A linear-decision based approximation approach to stochastic programming, Operational Research, 56 (2008), 344-357.  doi: 10.1287/opre.1070.0457.

[6]

E. Delage and S. Mannor, Distributionally robust optimization under moment uncertainty with application data-driven problems, Operational Research, 58 (2010), 596-612.  doi: 10.1287/opre.1090.0741.

[7]

A. M. Fink, Equilibrium in a stochastic n-person game, Journal of Science in Hiroshima University, series A-I, 28 (1964), 89-93. 

[8]

R. W. FreundF. Jarre and C. H. Vogelbusch, Nonlinear semidefinite programming: Sensitivity, convergence, and an application in passive reduced-order modeling, Mathematical Programming, 109 (2007), 581-611.  doi: 10.1007/s10107-006-0028-x.

[9]

J. Goh and M. Sim, Distributionally robust optimization and its tractable approximation, Operation Reserch, 58 (2010), 902-917.  doi: 10.1287/opre.1090.0795.

[10]

J. Goh and M. Sim, Robust optimization made easy with ROME, Operation Reserch, 59 (2011), 973-985.  doi: 10.1287/opre.1110.0944.

[11]

M. Grant and S. Boyd, Graph implementations for nonsmooth convex programs, Lecture Notes in Control and Information Sciences, 371 (2008), 95-110.  doi: 10.1007/978-1-84800-155-8_7.

[12]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21.../../cvx.

[13]

J. C. Harsanyi, Games with incomplete information played by ’Bayesian’ players’, Parts Ⅰ-Ⅲ, Management Science, 14 (1968), 159-182, 320-334, 486-502. doi: 10.1287/mnsc.14.5.320.

[14]

Z. L. HuJ. Gao and L. J. Hong, Robust simulation of global warming policies using the DICE model, Management Science, 58 (2012), 2190-2206. 

[15]

J. B. Huang, Z. F. Li and S. O. Finance, Risk Hedging Strategies and Its Utility under Distributional Uncertainty Chinese Journal of Management Science, 01 (2017).

[16]

K. Isii, On the sharpness of Chebyshev-type inequalities, Annals of the Institute of Statistical Mathematics, 12 (1963), 185-197.  doi: 10.1007/BF02868641.

[17]

B. Jadamba and F. Raciti, RA variational inequality approach to a class of environmental equilibrium problems, Applied Mathematics, 3 (2012), 1723-1728. 

[18]

B. Jadamba and F. Raciti, On the modeling of some environmental games with uncertain data, Journal of Optimization Theory} & \emph{Applications,, 167 (2015), 959-968.  doi: 10.1007/s10957-013-0389-2.

[19]

J. Janssen, Does international emissions trading jeopardize joint implemention? Distinguishing the Kyoto mechanisms for economic perspectives, In: Abele, H., Heller, T.C., Schleicher, S.P. (Eds.), Designing Climate Policy, Austrian Council of Climate Change, Graz, (2000), 247-277, Abele H, Heller T C, Schleicher S P. Designing Climate Policy: The Challenge of the Kyoto Protocol[J]. International Journal of Infectious Diseases, 6 (2001), S54.

[20]

S. Kakutani, RA generalization of Brouwer's fixed point theorem, Duke Mathematical Journal, 8 (1941), 457-459.  doi: 10.1215/S0012-7094-41-00838-4.

[21]

E. KardesF. Ord$\acute{o}\bar{n}$ez and R. W. Hall, Discounted robust stochastic games and an application to aueueing control, Operation Reserch, 59 (2011), 365-382.  doi: 10.1287/opre.1110.0931.

[22]

S. W. LamT. S. NgM. Sim and J. H. Song, Multiple objectives satisficing under uncertainty, Operation Reserch, 61 (2013), 214-227.  doi: 10.1287/opre.1120.1132.

[23]

G. LanZ. Lu and R. D. C. Monteiro, Primal-dual first-order methods with iteration-complexity for cone programming, Mathematical Programming, 126 (2011), 1-29.  doi: 10.1007/s10107-008-0261-6.

[24]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained problems, Society for Industrial and Applied Mathematics, 17 (2006), 969-996.  doi: 10.1137/050622328.

[25]

A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, 77 (1997), 301-320.  doi: 10.1007/BF02614439.

[26]

A. Shapiro, Worst-case distribution analysis of stochastic programs, Mathematical Programming, 107 (2006), 91-96.  doi: 10.1007/s10107-005-0680-6.

[27]

M. Tidball and G. Zaccour, An environmental game with coupling constraints, Environmental Modeling & Assessment, 10 (2005), 153-158. 

[28]

Y. ZhangS. SongZ. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE Transactions on Intelligent Transportation Systems, 99 (2017), 1-11. 

show all references

References:
[1]

M. Aghassi and D. Bertsimas, Robust game theory, Mathematical Programming, 107 (2006), 231-273.  doi: 10.1007/s10107-005-0686-0.

[2]

A. Ben Tal and A. Nemirovski, Robust convex optimization, Mathematical Methods of Operations Research, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769.

[3]

M. BretonG. Zaccour and M. Zahaf, A game-theoretic formulation of joint implementation of environmental projects, European Journal of Operational Research, 168 (2005), 221-239.  doi: 10.1016/j.ejor.2004.04.026.

[4]

S. Boyd and L. Vandenberghe, Convex Optimization Cambridge, U.K.: Cambridge University, Press, 2004. doi: 10.1017/CBO9780511804441s.

[5]

X. ChenM. SimP. Sun and J. W. Zhang, A linear-decision based approximation approach to stochastic programming, Operational Research, 56 (2008), 344-357.  doi: 10.1287/opre.1070.0457.

[6]

E. Delage and S. Mannor, Distributionally robust optimization under moment uncertainty with application data-driven problems, Operational Research, 58 (2010), 596-612.  doi: 10.1287/opre.1090.0741.

[7]

A. M. Fink, Equilibrium in a stochastic n-person game, Journal of Science in Hiroshima University, series A-I, 28 (1964), 89-93. 

[8]

R. W. FreundF. Jarre and C. H. Vogelbusch, Nonlinear semidefinite programming: Sensitivity, convergence, and an application in passive reduced-order modeling, Mathematical Programming, 109 (2007), 581-611.  doi: 10.1007/s10107-006-0028-x.

[9]

J. Goh and M. Sim, Distributionally robust optimization and its tractable approximation, Operation Reserch, 58 (2010), 902-917.  doi: 10.1287/opre.1090.0795.

[10]

J. Goh and M. Sim, Robust optimization made easy with ROME, Operation Reserch, 59 (2011), 973-985.  doi: 10.1287/opre.1110.0944.

[11]

M. Grant and S. Boyd, Graph implementations for nonsmooth convex programs, Lecture Notes in Control and Information Sciences, 371 (2008), 95-110.  doi: 10.1007/978-1-84800-155-8_7.

[12]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21.../../cvx.

[13]

J. C. Harsanyi, Games with incomplete information played by ’Bayesian’ players’, Parts Ⅰ-Ⅲ, Management Science, 14 (1968), 159-182, 320-334, 486-502. doi: 10.1287/mnsc.14.5.320.

[14]

Z. L. HuJ. Gao and L. J. Hong, Robust simulation of global warming policies using the DICE model, Management Science, 58 (2012), 2190-2206. 

[15]

J. B. Huang, Z. F. Li and S. O. Finance, Risk Hedging Strategies and Its Utility under Distributional Uncertainty Chinese Journal of Management Science, 01 (2017).

[16]

K. Isii, On the sharpness of Chebyshev-type inequalities, Annals of the Institute of Statistical Mathematics, 12 (1963), 185-197.  doi: 10.1007/BF02868641.

[17]

B. Jadamba and F. Raciti, RA variational inequality approach to a class of environmental equilibrium problems, Applied Mathematics, 3 (2012), 1723-1728. 

[18]

B. Jadamba and F. Raciti, On the modeling of some environmental games with uncertain data, Journal of Optimization Theory} & \emph{Applications,, 167 (2015), 959-968.  doi: 10.1007/s10957-013-0389-2.

[19]

J. Janssen, Does international emissions trading jeopardize joint implemention? Distinguishing the Kyoto mechanisms for economic perspectives, In: Abele, H., Heller, T.C., Schleicher, S.P. (Eds.), Designing Climate Policy, Austrian Council of Climate Change, Graz, (2000), 247-277, Abele H, Heller T C, Schleicher S P. Designing Climate Policy: The Challenge of the Kyoto Protocol[J]. International Journal of Infectious Diseases, 6 (2001), S54.

[20]

S. Kakutani, RA generalization of Brouwer's fixed point theorem, Duke Mathematical Journal, 8 (1941), 457-459.  doi: 10.1215/S0012-7094-41-00838-4.

[21]

E. KardesF. Ord$\acute{o}\bar{n}$ez and R. W. Hall, Discounted robust stochastic games and an application to aueueing control, Operation Reserch, 59 (2011), 365-382.  doi: 10.1287/opre.1110.0931.

[22]

S. W. LamT. S. NgM. Sim and J. H. Song, Multiple objectives satisficing under uncertainty, Operation Reserch, 61 (2013), 214-227.  doi: 10.1287/opre.1120.1132.

[23]

G. LanZ. Lu and R. D. C. Monteiro, Primal-dual first-order methods with iteration-complexity for cone programming, Mathematical Programming, 126 (2011), 1-29.  doi: 10.1007/s10107-008-0261-6.

[24]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained problems, Society for Industrial and Applied Mathematics, 17 (2006), 969-996.  doi: 10.1137/050622328.

[25]

A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, 77 (1997), 301-320.  doi: 10.1007/BF02614439.

[26]

A. Shapiro, Worst-case distribution analysis of stochastic programs, Mathematical Programming, 107 (2006), 91-96.  doi: 10.1007/s10107-005-0680-6.

[27]

M. Tidball and G. Zaccour, An environmental game with coupling constraints, Environmental Modeling & Assessment, 10 (2005), 153-158. 

[28]

Y. ZhangS. SongZ. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE Transactions on Intelligent Transportation Systems, 99 (2017), 1-11. 

Table 1.  Numerical results for the robustness of our method
$\delta$ NSCW CSCW USCW
TSC 0 2.0001 2.446 1.2269
TASC 0 1.8 2.1333 1.8
TS 0.01 2 2.4456 2
TASC 0.01 1.801 2.1565 1.801
TSC 0.03 1.9981 2.4448 1.9981
TASC 0.03 1.803 2.1602 1.803
TSC 0.05 1.9906 2.444 1.9906
TASC 0.05 1.805 2.1625 1.805
TSC 0.1 1.9811 2.4421 1.9811
TASC 0.1 1.8155 2.1681 1.8155
$\delta$ NSCW CSCW USCW
TSC 0 2.0001 2.446 1.2269
TASC 0 1.8 2.1333 1.8
TS 0.01 2 2.4456 2
TASC 0.01 1.801 2.1565 1.801
TSC 0.03 1.9981 2.4448 1.9981
TASC 0.03 1.803 2.1602 1.803
TSC 0.05 1.9906 2.444 1.9906
TASC 0.05 1.805 2.1625 1.805
TSC 0.1 1.9811 2.4421 1.9811
TASC 0.1 1.8155 2.1681 1.8155
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