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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 & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 989-1003. doi: 10.3934/dcdss.2019067
##### References:
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Mannor, Distributionally robust optimization under moment uncertainty with application data-driven problems, Operational Research, 58 (2010), 596-612. doi: 10.1287/opre.1090.0741. Google Scholar [7] A. M. Fink, Equilibrium in a stochastic n-person game, Journal of Science in Hiroshima University, series A-I, 28 (1964), 89-93. Google Scholar [8] R. W. Freund, F. 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. Google Scholar [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. Google Scholar [10] J. Goh and M. Sim, Robust optimization made easy with ROME, Operation Reserch, 59 (2011), 973-985. doi: 10.1287/opre.1110.0944. Google Scholar [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. Google Scholar [12] M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21.../../cvx.Google Scholar [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. Google Scholar [14] Z. L. Hu, J. Gao and L. J. Hong, Robust simulation of global warming policies using the DICE model, Management Science, 58 (2012), 2190-2206. Google Scholar [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).Google Scholar [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. Google Scholar [17] B. Jadamba and F. Raciti, RA variational inequality approach to a class of environmental equilibrium problems, Applied Mathematics, 3 (2012), 1723-1728. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [21] E. Kardes, F. 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. Google Scholar [22] S. W. Lam, T. S. Ng, M. Sim and J. H. Song, Multiple objectives satisficing under uncertainty, Operation Reserch, 61 (2013), 214-227. doi: 10.1287/opre.1120.1132. Google Scholar [23] G. Lan, Z. 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. Google Scholar [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. Google Scholar [25] A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, 77 (1997), 301-320. doi: 10.1007/BF02614439. Google Scholar [26] A. Shapiro, Worst-case distribution analysis of stochastic programs, Mathematical Programming, 107 (2006), 91-96. doi: 10.1007/s10107-005-0680-6. Google Scholar [27] M. Tidball and G. Zaccour, An environmental game with coupling constraints, Environmental Modeling & Assessment, 10 (2005), 153-158. Google Scholar [28] Y. Zhang, S. Song, Z. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE Transactions on Intelligent Transportation Systems, 99 (2017), 1-11. Google Scholar

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. Google Scholar [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. Google Scholar [3] M. Breton, G. 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. Google Scholar [4] S. Boyd and L. Vandenberghe, Convex Optimization Cambridge, U.K.: Cambridge University, Press, 2004. doi: 10.1017/CBO9780511804441s. Google Scholar [5] X. Chen, M. Sim, P. 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. Google Scholar [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. Google Scholar [7] A. M. Fink, Equilibrium in a stochastic n-person game, Journal of Science in Hiroshima University, series A-I, 28 (1964), 89-93. Google Scholar [8] R. W. Freund, F. 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. Google Scholar [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. Google Scholar [10] J. Goh and M. Sim, Robust optimization made easy with ROME, Operation Reserch, 59 (2011), 973-985. doi: 10.1287/opre.1110.0944. Google Scholar [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. Google Scholar [12] M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21.../../cvx.Google Scholar [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. Google Scholar [14] Z. L. Hu, J. Gao and L. J. Hong, Robust simulation of global warming policies using the DICE model, Management Science, 58 (2012), 2190-2206. Google Scholar [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).Google Scholar [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. Google Scholar [17] B. Jadamba and F. Raciti, RA variational inequality approach to a class of environmental equilibrium problems, Applied Mathematics, 3 (2012), 1723-1728. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [21] E. Kardes, F. 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. Google Scholar [22] S. W. Lam, T. S. Ng, M. Sim and J. H. Song, Multiple objectives satisficing under uncertainty, Operation Reserch, 61 (2013), 214-227. doi: 10.1287/opre.1120.1132. Google Scholar [23] G. Lan, Z. 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. Google Scholar [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. Google Scholar [25] A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, 77 (1997), 301-320. doi: 10.1007/BF02614439. Google Scholar [26] A. Shapiro, Worst-case distribution analysis of stochastic programs, Mathematical Programming, 107 (2006), 91-96. doi: 10.1007/s10107-005-0680-6. Google Scholar [27] M. Tidball and G. Zaccour, An environmental game with coupling constraints, Environmental Modeling & Assessment, 10 (2005), 153-158. Google Scholar [28] Y. Zhang, S. Song, Z. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE Transactions on Intelligent Transportation Systems, 99 (2017), 1-11. Google Scholar
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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