Advertisement

Cooperative Differential Games with Transferable Payoffs

  • Leon A. Petrosyan
  • Georges Zaccour
Reference work entry

Abstract

In many instances, players find it individually and collectively rational to sign a long-term cooperative agreement. A major concern in such a setting is how to ensure that each player will abide by her commitment as time goes by. This will occur if each player still finds it individually rational at any intermediate instant of time to continue to implement her cooperative control rather than switch to a noncooperative control. If this condition is satisfied for all players, then we say that the agreement is time consistent. This chapter deals with the design of schemes that guarantee time consistency in deterministic differential games with transferable payoffs.

Keywords

Cooperative differential games Time consistency Strong time consistency Imputation distribution procedure Shapley value Core 

References

  1. Angelova V, Bruttel LV, Güth W, Kamecke U (2013) Can subgame perfect equilibrium threats foster cooperation? An experimental test of finite-horizon folk theorems. Econ Inq 51: 1345–1356Google Scholar
  2. Baranova EM, Petrosjan LA (2006) Cooperative stochasyic games in stationary strategies. Game Theory Appl XI:1–7Google Scholar
  3. Başar T (1983) Performance bounds for hierarchical systems under partial dynamic information. J Optim Theory Appl 39:67–87Google Scholar
  4. Başar T (1984) Affine incentive schemes for stochastic systems with dynamic information. SIAM J Control Optim 22(2):199–210Google Scholar
  5. Başar T (1985) Dynamic games and incentives. In: Bagchi A, Jongen HT (eds) Systems and optimization. Lecture notes in control and information sciences, vol 66. Springer, Berlin/ New York, pp 1–13Google Scholar
  6. Başar T (1989) Stochastic incentive problems with partial dynamic information and multiple levels of hierarchy. Eur J Polit Econ V:203–217. (Special issue on “Economic Design”)Google Scholar
  7. Başar T, Olsder GJ (1980) Team-optimal closed-loop Stackelberg strategies in hierarchical control problems. Automatica 16(4):409–414Google Scholar
  8. Başar T, Olsder GJ (1995) Dynamic noncooperative game theory. Academic Press, New YorkGoogle Scholar
  9. Bauso D, Basar T (2016) Strategic thinking under social influence: scalability, stability and robustness of allocations. Eur J Control 32:1–15. https://doi.org/10.1016/j.ejcon.2016.04.006i
  10. Bayens E, Bitar Y, Khargonekar PP, Poolla K (2013) Coalitional aggregation of wind power. IEEE Trans Power Syst 28(4):3774–3784Google Scholar
  11. Benoit JP, Krishna V (1985) Finitely repeated games. Econometrica 53(4):905–922Google Scholar
  12. Boukas EK, Haurie A, Michel P (1990) An optimal control problem with a random stopping time. J Optim Theory Appl 64(3):471–480Google Scholar
  13. Breton M, Sokri A, Zaccour G (2008) Incentive equilibrium in an overlapping-generations environmental game. Eur J Oper Res 185:687–699Google Scholar
  14. Buratto A, Zaccour G (2009) Coordination of advertising strategies in a fashion licensing contract. J Optim Theory Appl 142:31–53Google Scholar
  15. Cansever DH, Başar T (1985) Optimum/near optimum incentive policies for stochastic decision problems involving parametric uncertainty. Automatica 21(5):575–584Google Scholar
  16. Chander P, Tulkens H (1997) The core of an economy with multilateral environmental externalities. Int J Game Theory 26:379–401Google Scholar
  17. Chang FR (2004) Stochastic optimization in continuous time. Cambridge University Press, CambridgeGoogle Scholar
  18. Chiarella C, Kemp MC, Long NV, Okuguchi K (1984) On the economics of international fisheries. Int Econ Rev 25:85–92Google Scholar
  19. De Frutos J, Martín-Herrán G (2015) Does flexibility facilitate sustainability of cooperation over time? A case study from environmental economics. J Optim Theory Appl 165:657–677Google Scholar
  20. De Giovanni P, Reddy PV, Zaccour G (2016) Incentive strategies for an optimal recovery program in a closed-loop supply chain. Eur J Oper Res 249:605–617Google Scholar
  21. Dockner EJ, Jorgensen S, Van Long N, Sorger G (2000) Differential games in economics and management science. Cambridge University Press, CambridgeGoogle Scholar
  22. Dutta PK (1995) A folk theorem for stochastic games. J Econ Theory 66:1–32Google Scholar
  23. Ehtamo H, Hämäläinen RP (1986) On affine incentives for dynamic decision problems. In: Basar T (ed) Dynamic games and applications in economics. Springer, Berlin, pp 47–63Google Scholar
  24. Ehtamo H, Hämäläinen RP (1989) Incentive strategies and equilibria for dynamic games with delayed information. J Optim Theory Appl 63:355–370Google Scholar
  25. Ehtamo H, Hämäläinen RP (1993) A cooperative incentive equilibrium for a resource management problem. J Econ Dyn Control 17:659–678Google Scholar
  26. Engwerda J (2005) Linear-quadratic dynamic optimization and differential games. Wiley, New YorkGoogle Scholar
  27. Eswaran M, Lewis T (1986) Collusive behaviour in finite repeated games with bonding. Econ Lett 20:213–216Google Scholar
  28. Flesch J, Kuipers J, Mashiah-Yaakovi A, Shmaya E, Shoenmakers G, Solan E, Vrieze K (2014) Nonexistene of subgame-perfect-equilibrium in perfect information games with infinite horizon. Int J Game Theory 43:945–951Google Scholar
  29. Flesch J, Predtetchinski A (2015) On refinements of subgame perfect ε-equilibrium. Int J Game Theory. https://doi.org/10.1007/s00182-015-0468-8
  30. Friedman JW (1986) Game theory with applications to economics. Oxford University Press, OxfordGoogle Scholar
  31. Gao L, Jakubowski A, Klompstra MB, Olsder GJ (1989) Time-dependent cooperation in games. In: Başar TS, Bernhard P (eds) Differential games and applications. Springer, BerlinGoogle Scholar
  32. Gillies DB (1953) Some theorems on N-person games. Ph.D. Thesis, Princeton UniversityGoogle Scholar
  33. Gromov D, Gromova E (2014) Differential games with random duration: a hybrid systems formulation. Contrib Game Theory Manag 7:104–119Google Scholar
  34. Gromov D, Gromova E (2016) On a class of hybrid differential games. Dyn Games Appl. https://doi.org/10.1007/s13235-016-0185-3
  35. Gromova EV, Lopez-Barrientos JD (2015) A differential game model for the extraction of non-renewable resources with random initial times. Contrib Game Theory Manag 8:58–63Google Scholar
  36. Haurie A (1976) A note on nonzero-sum differential games with bargaining solution. J Optim Theory Appl 18:31–39Google Scholar
  37. Haurie A (2005) A multigenerational game model to analyze sustainable development. Ann Oper Res 137(1):369–386Google Scholar
  38. Haurie A, Krawczyk JB, Roche M (1994) Monitoring cooperative equilibria in a stochastic differential game. J Optim Theory Appl 81:79–95Google Scholar
  39. Haurie A, Krawczyk JB, Zaccour G (2012) Games and dynamic games. Scientific World, SingaporeGoogle Scholar
  40. Haurie A, Pohjola M (1987) Efficient equilibria in a differential game of capitalism. J Econ Dyn Control 11:65–78Google Scholar
  41. Haurie A, Tolwinski B (1985) Definition and properties of cooperative equilibria in a two-player game of infinite duration. J Optim Theory Appl 46(4):525–534Google Scholar
  42. Haurie A, Zaccour G (1986) A differential game model of power exchange between interconnected utilities. In: Proceedings of the 25th IEEE conference on decision and control, Athens, pp 262–266Google Scholar
  43. Haurie A, Zaccour G (1991) A game programming approach to efficient management of interconnected power networks. In: Hämäläinen RP, Ehtamo H (eds) Differential games – developments in modelling and computation. Springer, BerlinGoogle Scholar
  44. Jørgensen S, Martín-Herrán G, Zaccour G (2003) Agreeability and time-consistency in linear-state differential games. J Optim Theory Appl 119:49–63Google Scholar
  45. Jørgensen S, Martín-Herrán G, Zaccour G (2005) Sustainability of cooperation overtime in linear-quadratic differential game. Int Game Theory Rev 7:395–406Google Scholar
  46. Jørgensen S, Zaccour G (2001) Time consistent side payments in a dynamic game of downstream pollution. J Econ Dyn Control 25:1973–1987Google Scholar
  47. Jørgensen S, Zaccour G (2002a) Time consistency in cooperative differential game. In: Zaccour G (ed) Decision and control in management science: in honor of professor Alain Haurie. Kluwer, Boston, pp 349–366Google Scholar
  48. Jørgensen S, Zaccour G (2002b) Channel coordination over time: incentive equilibria and credibility. J Econ Dyn Control 27:801–822Google Scholar
  49. Kaitala V, Pohjola M (1990) Economic development and agreeable redistribution in capitalism: efficient game equilibria in a two-class neoclassical growth model. Int Econ Rev 31:421–437Google Scholar
  50. Kaitala V, Pohjola M (1995) Sustainable international agreements on greenhouse warming: a game theory study. Ann Int Soc Dyn Games 2:67–87Google Scholar
  51. Kostyunin S, Palestini A, Shevkoplyas E (2014) On a nonrenewable resource extraction game played by asymmetric firms. J Optim Theory Appl 163(2):660–673Google Scholar
  52. Kostyunin SY, Shevkoplyas EV (2011) On the simplification of the integral payoff in differential games with random duration. Vestnik St Petersburg Univ Ser 10(4):47–56Google Scholar
  53. Kozlovskaya NV, Petrosyan LA, Zenkevich NA (2010) Coalitional solution of a game-theoretic emission reduction model. Int Game Theory Rev 12(3):275–286Google Scholar
  54. Lehrer E, Scarsini M (2013) On the core of dynamic cooperative games. Dyn Games Appl 3: 359–373Google Scholar
  55. Mailath GJ, Postlewaite A, Samuelson L (2005) Contemporaneous perfect epsilon-equilibria. Games Econ Behav 53:126–140Google Scholar
  56. Marín-Solano J, Shevkoplyas EV (2011) Non-constant discounting and differential games with random time horizon. Automatica 47(12):2626–2638Google Scholar
  57. Martín-Herrán G, Rincón-Zapatero JP (2005) Efficient Markov perfect Nash equilibria: theory and application to dynamic fishery games. J Econ Dyn Control 29:1073–1096Google Scholar
  58. Martín-Herrán G, Taboubi S, Zaccour G (2005) A time-consistent open-loop Stackelberg equilibrium of shelf-space allocation. Automatica 41:971–982Google Scholar
  59. Martín-Herrán G, Zaccour G (2005) Credibility of incentive equilibrium strategies in linear-state differential games. J Optim Theory Appl 126:1–23Google Scholar
  60. Martín-Herrán G, Zaccour G (2009) Credible linear-incentive equilibrium strategies in linear-quadratic differential games. Advances in dynamic games and their applications. Ann Int Soci Dyn Games 10:1–31Google Scholar
  61. Opathella C, Venkatesh B (2013) Managing uncertainty of wind energy with wind generators cooperative. IEEE Trans Power Syst 28(3):2918–2928Google Scholar
  62. Ordeshook PC (1986) Game theory and political theory. Cambridge University Press, Cambridge, UKGoogle Scholar
  63. Osborne MJ, Rubinstein A (1994) A course in game theory. MIT Press, Cambridge, MAGoogle Scholar
  64. Parilina E (2014) Strategic stability of one-point optimality principles in cooperative stochastic games. Math Game Theory Appl 6(1):56–72 (in Russian)Google Scholar
  65. Parilina EM (2015) Stable cooperation in stochastic games. Autom Remote Control 76(6): 1111–1122Google Scholar
  66. Parilina E, Zaccour G (2015a) Approximated cooperative equilibria for games played over event trees. Oper Res Lett 43:507–513Google Scholar
  67. Parilina E, Zaccour G (2015b) Node-consistent core for games played over event trees. Automatica 55:304–311Google Scholar
  68. Petrosyan LA (1977) Stability of solutions of differential games with many participants. Vestnik Leningrad State Univ Ser 1 4(19):46–52Google Scholar
  69. Petrosyan LA (1993) Strongly time-consistent differential optimality principles. Vestnik St Petersburg Univ Ser 1 4:35–40Google Scholar
  70. Petrosyan LA (1995) Characteristic function of cooperative differential games. Vestnik St Petersburg Univ Ser 1 1:48–52Google Scholar
  71. Petrosyan LA, Baranova EM, Shevkoplyas EV (2004) Multistage cooperative games with random duration, mathematical control theory, differential games. Trudy Inst Mat i Mekh UrO RAN 10(2):116–130 (in Russian)Google Scholar
  72. Petrosjan L, Danilov NN (1979) Stability of solutions in nonzero sum differential games with transferable payoffs. J Leningrad Univ N1:52–59 (in Russian)Google Scholar
  73. Petrosjan L, Danilov NN (1982) Cooperative differential games and their applications. Tomsk University Press, TomskGoogle Scholar
  74. Petrosjan L, Danilov NN (1986) Classification of dynamically stable solutions in cooperative differential games. Isvestia of high school 7:24–35 (in Russian)Google Scholar
  75. Petrosyan LA, Gromova EV (2014) Two-level cooperation in coalitional differential games. Trudy Inst Mat i Mekh UrO RAN 20(3):193–203Google Scholar
  76. Petrosjan LA, Mamkina SI (2006) Dynamic games with coalitional structures. Int Game Theory Rev 8(2):295–307Google Scholar
  77. Petrosjan LA, Murzov NV (1966) Game-theoretic problems of mechanics. Litovsk Mat Sb 6:423–433 (in Russian)Google Scholar
  78. Petrosyan LA, Shevkoplyas EV (2000) Cooperative differential games with random duration Vestnik Sankt-Peterburgskogo Universiteta. Ser 1. Matematika Mekhanika Astronomiya 4: 18–23Google Scholar
  79. Petrosyan LA, Shevkoplyas EV (2003) Cooperative solutions for games with random duration. Game theory and applications, vol IX. Nova Science Publishers, New York, pp 125–139Google Scholar
  80. Petrosjan L, Zaccour G (2003) Time-consistent Shapley value of pollution cost reduction. J Econ Dyn Control 27:381–398Google Scholar
  81. Petrosjan L, Zenkevich NA (1996) Game theory. World Scientific, SingaporeGoogle Scholar
  82. Radner R (1980) Collusive behavior in noncooperative epsilon-equilibria of oligopolies with long but finite lives. J Econ Theory 22:136–154Google Scholar
  83. Reddy PV, Shevkoplyas E, Zaccour G (2013) Time-consistent Shapley value for games played over event trees. Automatica 49(6):1521–1527Google Scholar
  84. Reddy PV, Zaccour G (2016) A friendly computable characteristic function. Math Soc Sci 82: 18–25Google Scholar
  85. Rincón-Zapatero JP, Martín-Herrán G, Martinez J (2000) Identification of efficient Subgame-perfect Nash equilibria in a class of differential games. J Optim Theory Appl 104:235–242Google Scholar
  86. Saad W, Han Z, Debbah M, Hjorungnes A, Başar T (2009) Coalitional game theory for communication networks [A tutorial]. IEEE Signal Process Mag Spec Issue Game Theory 26(5):77–97Google Scholar
  87. Saad W, Han Z, Zheng R, Hjorungnes A, Başar T, Poor HV (2012) Coalitional games in partition form for joint spectrum sensing and access in cognitive radio networks. IEEE J Sel Top Sign Process 6(2):195–209Google Scholar
  88. Shevkoplyas EV (2009) The Hamilton-Jacobi-Bellman equation for differential games with random duration. Upravlenie bolshimi systemami 26.1, M.:IPU RAN, pp 385–408 (in Russian)Google Scholar
  89. Shevkoplyas EV (2011) The Shapley value in cooperative differential games with random duration. In: Breton M, Szajowski K (eds) Advances in dynamic games. Ann Int Soci Dyn Games 11(part4):359–373Google Scholar
  90. Tolwinski B, Haurie A, Leitmann G (1986) Cooperative equilibria in differential games. J Math Anal Appl 119:182–202Google Scholar
  91. Von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, PrincetonGoogle Scholar
  92. Wrzaczek S, Shevkoplyas E, Kostyunin S (2014) A differential game of pollution control with overlapping generations. Int Game Theory Rev 16(3):1450005–1450019Google Scholar
  93. Yeung DWK, Petrosjan L (2001) Proportional time-consistent solutions in differential games. In: Proceedings of international conference on logic, game theory and applications, St. Petersburg, pp 254–256Google Scholar
  94. Yeung DWK, Petrosjan L (2004) Subgame consistent cooperative solutions in stochastic differential games. J Optim Theory Appl 120:651–666MathSciNetCrossRefGoogle Scholar
  95. Yeung DWK, Petrosjan L (2005a) Cooperative stochastic differential games. Springer, New YorkGoogle Scholar
  96. Yeung DWK, Petrosjan L (2005b) Consistent solution of a cooperative stochastic differential game with nontransferable payoffs. J Optim Theory Appl 124:701–724MathSciNetCrossRefGoogle Scholar
  97. Yeung DWK, Petrosjan L (2006) Dynamically stable corporate joint ventures. Automatica 42: 365–370MathSciNetCrossRefGoogle Scholar
  98. Yeung DWK, Petrosjan L, Yeung PM (2007) Subgame consistent solutions for a class of cooperative stochastic differential games with nontransferable payoffs. Ann Int Soc Dyn Games 9:153–170MathSciNetCrossRefGoogle Scholar
  99. Yeung DWK (2006) An irrational-behavior-proof condition in cooperative differential games. Int Game Theory Rev 8(4):739–744MathSciNetCrossRefGoogle Scholar
  100. Yeung DWK, Petrosyan LA (2012) Subgame consistent economic optimization. Birkhauser, New YorkCrossRefGoogle Scholar
  101. Zaccour G (2003) Computation of characteristic function values for linear-state differential games. J Optim Theory Appl 117(1):183–194MathSciNetCrossRefGoogle Scholar
  102. Zaccour G (2008) Time consistency in cooperative differential games: a tutorial. INFOR 46(1): 81–92MathSciNetGoogle Scholar
  103. Zhang B, Johari R, Rajagopal R (2015) Competition and coalition formation of renewable power producers. IEEE Trans Power Syst 30(3):1624–1632CrossRefGoogle Scholar
  104. Zheng YP, Başar T, Cruz JB (1982) Incentive Stackelberg strategies for deterministic multi-stage decision processes. IEEE Trans Syst Man Cybern SMC-14(1):10–20CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Applied Mathematics-Control ProcessesSt Petersburg State UniversitySt PetersburgRussia
  2. 2.Department of Decision SciencesGERAD, HEC MontréalMontrealCanada

Personalised recommendations