Developing a Non-Discrete Dynamic Game Model and Corresponding Monthly Collocation Solution Considering Variability in Reservoir Inflow

Abstract

So far a huge number of dynamic game models, both discrete and continuous, have been developed for reservoir operation. Most of them seek to overcome potential conflicts while allocating limited water among various users in reservoir systems. A number of them have successfully provided efficient approaches to tackle the problem of optimal water allocation. There are, however, still weaknesses in determining optimal policies. By taking account of the random nature of the inflow while producing monthly operating policies for a reservoir in a dynamic non-discrete framework, many of these deficiencies can be eliminated. In this study, considering the randomness in reservoir inflow, a continuous dynamic game model for water allocation in a reservoir system was developed. A corresponding monthly-basis solution based on collocation was then structured. This collocation method does not rely on first and second degree approximations. Instead, in order to evaluate the uncertainty triggered by the random variable, a discrete approximant was applied to quantify the random variable in the state transition function. A case study was carried out at the Zayandeh-Rud river basin in central Iran to identify the efficiency of the proposed method and evaluate the effect of uncertainty on decision variables. The results of the presented model (i.e., monthly-basis operating rules and value functions) proved to be more practical and reliable than similar continuous dynamic game models working on an annual basis. Moreover, the results show that bringing the stochasticity associated with inflow into the equation has an impact on the value functions and operating polices. Indeed, applying the mean of the random variable of inflow (deterministic form) reduce consistancy in monthly reliability indices throughout the year.

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References

  1. Carraro C, Marchiori C, Sgobbi A (2007) Negotiating on water: insights from noncooperative bargaining theory. Environ Dev Econ 12:329–349

    Article  Google Scholar 

  2. Dufournaud CM (1982) On the mutually beneficial cooperative scheme: dynamic change in the payoff matrix of international river basin schemes. Water Resour Res 18(4):764–772

    Article  Google Scholar 

  3. Dunn JC, Bertsekas DP (1989) Efficient dynamic programming implementations of Newton’s method for unconstrained optimal control problems. J Optim Theor Appl 63(1):23–38

    Article  Google Scholar 

  4. Fackler PL, Miranda MJ (1999) Hybrid methods for continuous state dynamic programming, Presented at computing in economics and finance, Boston College, June 1999, Online Available HTTP: http://www4.ncsu.edu/unity/users/p/pfackler/

  5. Fisvold GB, Caswell MF (2000) Transboundary water management: gametheoretic lessons for projects on the US–Mexico border. Agric Econ 24:101–111

    Google Scholar 

  6. Fundenberg D, Tirole J (1996) Game theory. MIT Press, Cambridge

    Google Scholar 

  7. Ganji A, Karamouz M, Khalili D (2007a) Development of stochastic conflict resolution models for reservoir operation, II. The value of players’ information availability and cooperative behavior. Adv Water Resour 30:157–168

    Article  Google Scholar 

  8. Ganji A, Khalili D, Karamouz M (2007b) Development of stochastic conflict resolution models for reservoir operation. I. The perfect symmetric stochastic model. Adv Water Resour 30:528–542

    Article  Google Scholar 

  9. Ganji A, Khalili D, Karamoz M, Ponnambalam K (2008) A fuzzy stochastic dynamic Nash game analysis of policies for managing water allocation in a reservoir system. J Water Resour Manage 22(1):51–66

    Article  Google Scholar 

  10. Gibbons R (1992) Game theory for applied economists. Princeton University Press, Princeton

    Google Scholar 

  11. Golubtsov PV, Lyubetsky VA (2003) Stochastic dynamic games with various types of information. Problems Info Trans 39(3):266–293

    Article  Google Scholar 

  12. Grafton RQ, Kompas T, Lindenmayer D (2005) Marine reserves with ecological uncertainty. Bull Math Biol 67:957–971

    Article  Google Scholar 

  13. Han Y, Huang YF, Wang GQ, Maqsood I (2010) A multi-objective linear programming model with interval parameters for water resources allocation in Dalian City. Water Resour Manage 25(2):449–463

    Article  Google Scholar 

  14. Homayounfar M, Ganji A, Martinez CJ (2011) A novel solution for stochastic dynamic game of water allocation from a reservoir using collocation method. Water Resour Manag 25(13):3427–3444

    Article  Google Scholar 

  15. Homayounfar M, Lai SH, Zomorodian M, Sepaskhah AR, Ganji A (2014) Optimal crop water allocation in case of drought occurrence, imposing deficit irrigation with proportional cutback constraint. Water Resour Manag 28(10):3207–3225

  16. Homayoun-Far M, Ganji A, Khalili D, Harris J (2010) Two solution methods ForDynamic game in reservoir operation. Adv Water Resour 33(7):752–761

    Article  Google Scholar 

  17. Jin L, Huang G, Fan Y, Nie X, Cheng G (2012) A hybrid dynamic dual interval programming for irrigation water allocation under uncertainty. Water Resour Manage 26(5):1183–1200

    Article  Google Scholar 

  18. Kaviani S, Hassanli AM, Homayounfar H  (2014) Optimal crop water allocation based on constrain-state method and nonnormal stochastic Variable. Water Resour Manag. doi:10.1007/s11269-014-0856-z

  19. Kerachian R, Karamouz M (2007) A stochastic conflict resolution model for water quality management in reservoir-river systems. Adv Water Resour 30:866–882

    Article  Google Scholar 

  20. Kilgour DM, Dinar A (2001) Flexible water sharing within an international river Basin. Environ Resour Econ 18(1):43–60

    Article  Google Scholar 

  21. LaFrance JT, Barney LD (1991) The envelope theorem in dynamic optimization. J Econ Dyn Cont 15:355–385

    Article  Google Scholar 

  22. Li YP, Huang GH (2008) Interval-parameter two-stage stochastic nonlinear programming for water resources management under uncertainty. Water Resour Manage 22(6):681–698

    Article  Google Scholar 

  23. Loucks DP, Stedinger JR, Haith DA (1988) Water resources system planning and analysis. Prentice-Hall, England Cliffs

    Google Scholar 

  24. Madani K (2010) Game theory and water resources. J Hydrol 381:225–238

    Article  Google Scholar 

  25. Madani K, Dinar A (2012a) Non-cooperative institutions for sustainable common pool resource management: application to groundwater. Ecol Econ 74:34–45

    Article  Google Scholar 

  26. Madani K, Dinar A (2012b) Cooperative institutions for sustainable common pool resource management: application to groundwater. Water Resour Res 48(9):W09553

    Google Scholar 

  27. Madani K, Hipel KW (2007) Strategic insights into the Jordan River conflict. In: Kabbes KC (ed) Proceeding of the 2007 world environmental and water resources congress. American Society of Civil Engineers, Florida, pp 1–10

    Google Scholar 

  28. Madani K, Hipel KW (2011) Non-cooperative stability definitions for strategic analysis of generic water resources conflicts. Water Resour Manag 25:1949–1977. doi:10.1007/s11269-011-9783-4

    Article  Google Scholar 

  29. Madani K, Lund JR (2011) A Monte-Carlo game theoretic approach for multi-criteria decision making under uncertainty. Adv Water Resour 34(5):607–616

    Article  Google Scholar 

  30. Maskin E, Tirole J (1994) Markov perfect equilibrium. Harvard institute of economic research. Harvard Univ, Cambridge

    Google Scholar 

  31. Miranda MJ, Fackler PL (2002) Applied computational economics and finance. MIT press

  32. Murray DM, Yakowitz SJ (1984) Differential dynamic programming and Newton’s method for discrete optimal control problems. J Optim Theor Appl 43(3):359–414

    Article  Google Scholar 

  33. Parrachino I, Dinar A, Patrone F (2006) Cooperative game theory and its application to natural, environmental, and water resource issues: 3. Application to water resources. World bank policy research working paper No. 4074, WPS4074. World Bank Policy Research Working Paper 4074, Washington

    Google Scholar 

  34. Petit ML (1990) Control theory and dynamic games in economic policy analysis. Cambridge Univ Press, Cambridge

    Google Scholar 

  35. Provencher B, Bishop RC (1997) An estimable dynamic model of recreation behavior with an application to great lakes angling. J Environ Econ Manage 33(2):107–127

    Article  Google Scholar 

  36. Rogers P (1969) A game theory approach to the problems of international river basins. Water Resour Res 5(4):749–760

    Article  Google Scholar 

  37. Rubio SJ, Casino B (2003) Strategic behavior and efficiency in the common property extraction of groundwater. Environ Resour Econ 26(1):73–87

    Article  Google Scholar 

  38. Sadegh M, Mahjouri N, Kerachian R (2010) Optimal inter-basin water allocation using crisp and fuzzy shapley games. Water Resour Manage 24(10):2291–2310

    Article  Google Scholar 

  39. Sechi GM, Zucca R, Zuddas P (2013) Water costs allocation in complex systems using a cooperative game theory approach. Water Resour Manage 27(6):1781–1796

    Article  Google Scholar 

  40. Sheikhmohammady M, Madani K (2008) Bargaining over the Caspian Sea–the largest Lake on the earth. In: Babcock RW, Walton R (eds) Proceeding of the 2008 world environmental and water resources congress. American Society of Civil Engineers, Honolulu, pp 1–9

    Google Scholar 

  41. Supalla R, Klaus B, Yeboah O, Bruins R (2002) A game theory approach to deciding who will supply instream flow water. J Am Water Resour Assoc 38(4):959–966

    Article  Google Scholar 

  42. Tolwinski B (1989) Newton-type methods for stochastic games. Diff Games Appl 119:128–144

    Article  Google Scholar 

  43. Wu X, Whittington D (2006) Incentive compatibility and conflict resolution in international river basins: a case study of the Nile Basin. Water Resources Research 42, W02417. doi:10.1029/2005WR004238

  44. Zara S, Dinar A, Patrone F (2006) Cooperative game theory and its application to natural, environmental, and water resource issues: 2. Application to Natural and Environmental Resources. World Bank Policy Research Working Paper 4073

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Acknowledgments

The authors express the gratitude to Dr. Kaveh Madani at the Centre for Environmental Policy at the Imperial College London for his valuable comments and helpful discussions. Authors thank the Editor and anonymous reviewers for their constructive comments. The first author is grateful to the Government of Malaysia and the University of Malaya (UM) for financial support of the first author through the Bright Spark Fellowship Scheme. The first author also would like to thanks the Faculty of Engineering, University of Malaya for facilitating this study.

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Correspondence to Sai Hin Lai.

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Homayounfar, M., Lai, S.H., Zommorodian, M. et al. Developing a Non-Discrete Dynamic Game Model and Corresponding Monthly Collocation Solution Considering Variability in Reservoir Inflow. Water Resour Manage 29, 2599–2618 (2015). https://doi.org/10.1007/s11269-015-0959-1

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Keywords

  • Monthly collocation
  • Conflict resolution
  • Reservoir operation
  • Dynamic continous game