Mathematical Programming

, Volume 163, Issue 1–2, pp 25–56 | Cite as

Dynamic linear programming games with risk-averse players

  • Alejandro Toriello
  • Nelson A. Uhan
Full Length Paper Series A


Motivated by situations in which independent agents wish to cooperate in some uncertain endeavor over time, we study dynamic linear programming games, which generalize classical linear production games to multi-period settings under uncertainty. We specifically consider that players may have risk-averse attitudes towards uncertainty, and model this risk aversion using coherent conditional risk measures. For this setting, we study the strong sequential core, a natural extension of the core to dynamic settings. We characterize the strong sequential core as the set of allocations that satisfy a particular finite set of inequalities that depend on an auxiliary optimization model, and then leverage this characterization to establish sufficient conditions for emptiness and non-emptiness. Qualitatively, whereas the strong sequential core is always non-empty when players are risk-neutral, our results indicate that cooperation in the presence of risk aversion is much more difficult. We illustrate this with an application to cooperative newsvendor games, where we find that cooperation is possible when it least benefits players, and may be impossible when it offers more benefit.


Cooperative game Stochastic linear program Risk measure 



The authors’ work was partially supported by the National Science Foundation via Grant CMMI 1265616.


  1. 1.
    Ahmed, S., Çakmak, U., Shapiro, A.: Coherent risk measures in inventory problems. Eur. J. Oper. Res. 182, 226–238 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alparslan-Gök, S.Z., Miquel, S., Tijs, S.: Cooperation under interval uncertainty. Math. Methods Oper. Res. 69, 99–109 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Avrachenkov, K., Cottatellucci, L., Maggi, L.: Cooperative Markov decision processes: time consistency, greedy players satisfaction, and cooperation maintenance. Int. J. Game Theory 42, 239–262 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bertsimas, D., Brown, D.B.: Constructing uncertainty sets for robust linear optimization. Oper. Res. 57, 1483–1495 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cachon, G.P., Netessine, S.: Game theory in supply chain analysis. In: Tutorials in Operations Research: Models, Methods, and Applications for Innovative Decision Making, pp. 200–233. INFORMS (2006)Google Scholar
  7. 7.
    Chalkiadakis, G., Markakis, V., Boutilier, C.: Coalition formation under uncertainty: bargaining equilibria and the Bayesian core stability concept. In: Proceedings of the 6th International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2007), pp. 412–419 (2007)Google Scholar
  8. 8.
    Chan, T.C.Y., Mahmoudzadeh, H., Purdie, T.G.: A robust-CVaR optimization approach with application to breast cancer therapy. Eur. J. Oper. Res. 238(3), 876–885 (2014)Google Scholar
  9. 9.
    Charnes, A., Granot, D.: Prior solutions: extensions of convex nucleolus solutions to chance-constrained games. In: Proceedings of the Computer Science and Statistics Seventh Symposium at Iowa State University, pp. 323–332 (1973)Google Scholar
  10. 10.
    Chen, X.: Inventory centralization games with price-dependent demand and quantity discount. Oper. Res. 57, 1394–1406 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, X., Zhang, J.: Duality approaches to economic lot sizing games. IOMS: Operations Management Working Paper OM-2006-01, Stern School of Business, New York University (2006)Google Scholar
  12. 12.
    Chen, X., Zhang, J.: A stochastic programming duality approach to inventory centralization games. Oper. Res. 57, 840–851 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, Y., Xu, M., Zhang, Z.: A risk-averse newsvendor model under the CVaR criterion. Oper. Res. 57, 1040–1044 (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Choi, S., Ruszczyński, A.: A risk-averse newsvendor with law invariant coherent measures of risk. Oper. Res. Lett. 36, 77–82 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Choi, S., Ruszczyński, A.: A multi-product risk-averse newsvendor with exponential utility function. Eur. J. Oper. Res. 214, 78–84 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Choi, S., Ruszczyński, A., Zhao, Y.: A multi-product risk-averse newsvendor with law invariant coherent measures of risk. Oper. Res. 59, 346–364 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Choquet, G.: Theory of capacities. Ann de l’Institut Fourier 5, 131–295 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Deng, X., Ibaraki, T., Nagamochi, H.: Algorithmic aspects of the core of combinatorial optimization games. Math. Oper. Res. 24, 751–766 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and Their Applications (Calgary International Conference on Combinatorial Structures and Their Applications), pp. 69–87. Gordon and Breach, New York (1970)Google Scholar
  20. 20.
    Elkind, E., Pasechnik, D., Zick, Y.: Dynamic weighted voting games. In: Proceedings of the 12th International Conference on Autonomous Agents and Multiagent Systems (AAMAS) (2013)Google Scholar
  21. 21.
    Faigle, U., Kern, W.: On some approximately balanced combinatorial cooperative games. Math. Methods Oper. Res. 38, 141–152 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Filar, J., Petrosjan, L.A.: Dynamic cooperative games. Int. Game Theory Rev. 2, 47–65 (2000)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Flåm, S.D.: Stochastic programming, cooperation, and risk exchange. Optim. Methods Softw. 17, 493–504 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gale, D.: The core of a monetary economy without trust. J. Econ. Theory 19, 456–491 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gillies, D.B.: Solutions to general non-zero-sum games. In: Tucker, A.W., Luce, R.D. (eds.) Contributions to the Theory of Games volume 40 of Annals of Mathematics Studies, vol. IV, pp. 47–85. Princeton University Press, Princeton (1959)Google Scholar
  26. 26.
    Glashoff, K., Gustafson, S.: Linear Optimization and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-infinite Programs, vol. 45 of Applied Mathematical Sciences, English edition. Springer, New York (1983)Google Scholar
  27. 27.
    Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley Series in Mathematical Methods in Practice. Wiley, Chichester (1998)Google Scholar
  28. 28.
    Goemans, M.X., Skutella, M.: Cooperative facility location games. J. Algorithms 50, 194–214 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gopaladesikan, M., Uhan, N.A., Zou, J.: A primal-dual algorithm for computing a cost allocation in the core of economic lot-sizing games. Oper. Res. Lett. 40, 453–458 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Gotoh, J., Takano, Y.: Newsvendor solutions via conditional value-at-risk minimization. Eur. J. Oper. Res. 179, 80–96 (2007)CrossRefzbMATHGoogle Scholar
  31. 31.
    Granot, D.: A generalized linear production model: a unifying model. Math. Program. 34, 212–222 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Granot, D., Huberman, G.: Minimum cost spanning tree games. Math. Program. 21, 1–18 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Guardiola, L.A., Meca, A., Puerto, J.: Production-inventory games and PMAS-games: characterization of the Owen point. Math. Soc. Sci. 57, 96–108 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Guardiola, L.A., Meca, A., Puerto, J.: Production-inventory games: a new class of totally balanced combinatorial optimization games. Games Econ. Behav. 65, 205–219 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Habis, H., Herings, P.J.-J.: A note on the weak sequential core of dynamic TU games. Int. Game Theory Rev. 12(4), 407–416 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Habis, H., Herings, P.J.-J.: Core concepts for incomplete market economics. J. Math. Econ. 47(4–5), 595–609 (2011a)CrossRefzbMATHGoogle Scholar
  37. 37.
    Habis, H., Herings, P.J.-J.: Transferable utility games with uncertainty. J. Econ. Theory 146, 2126–2139 (2011b)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Hartman, B.C., Dror, M.: Cost allocation in continuous-review inventory models. Nav. Res. Logist. 43, 449–561 (1996)CrossRefzbMATHGoogle Scholar
  39. 39.
    Hartman, B.C., Dror, M.: Allocation of gains from inventory centralization in newsvendor environments. IIE Trans. 37, 93–107 (2005)CrossRefGoogle Scholar
  40. 40.
    Hartman, B.C., Dror, M., Shaked, M.: Cores of inventory centralization games. Games Econ. Behav. 31, 26–49 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    He, S., Zhang, J., Zhang, S.: Polymatroid optimization, submodularity, and joint replenishment games. Oper. Res. 60, 128–137 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Huber, P.: Robust Statistics. Wiley, New York (1981)CrossRefzbMATHGoogle Scholar
  43. 43.
    Iancu, D.A., Petrik, M., Subramanian, D.: Tight approximations of dynamic risk measures. Math. Oper. Res. (2014). ForthcomingGoogle Scholar
  44. 44.
    Ieong, S., Shoham, Y.: Bayesian coalitional games. In: Proceedings of the 23rd AAAI Conference on Artificial Intelligence, pp. 95–100 (2008)Google Scholar
  45. 45.
    Kalai, E., Zemel, E.: Totally balanced games and games of flow. Math. Oper. Res. 7, 476–478 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Kranich, L., Perea, A., Peters, H.: Core concepts for dynamic TU games. Int. Game Theory Rev. 7, 43–61 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Lehrer, E., Scarsini, M.: On the core of dynamic cooperative games. Dyn. Games Appl. 3, 359–373 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Maschler, M., Peleg, B., Shapley, L.S.: Geometric properties of the kernel, nucleolus, and related solution concepts. Math. Oper. Res. 4, 303–338 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Montrucchio, L., Scarsini, M.: Large newsvendor games. Games Econ. Behav. 58, 316–337 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Müller, A., Scarsini, M., Shaked, M.: The newsvendor game has a non-empty core. Games Econ. Behav. 38, 118–126 (2002)CrossRefzbMATHGoogle Scholar
  51. 51.
    Myerson, R.B.: Virtual utility and the core for games with incomplete information. J. Econ. Theory 136, 260–285 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Nagarajan, M., Sošić, G.: Game-theoretic analysis of cooperation among supply chain agents: review and extensions. Eur. J. Oper. Res. 187, 719–745 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Owen, G.: On the core of linear production games. Math. Program. 9(1), 358–370 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Özen, U., Fransoo, J., Norde, H., Slikker, M.: Cooperation between multiple newsvendors with warehouses. Manuf. Serv. Oper. Manag. 10, 311–324 (2008)Google Scholar
  55. 55.
    Özener, O.Ö., Ergun, Ö., Savelsbergh, M.: Allocating cost of service to customers in inventory routing. Oper. Res. 61, 112–125 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Petrosjan, L.A.: Cooperative stochastic games. In: Proceedings of the 10th International Symposium on Dynamic Games and Applications, vol. 2 (2002)Google Scholar
  57. 57.
    Predtetchinski, A.: The strong sequential core for stationary cooperative games. Games Econ. Behav. 61, 50–66 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Predtetchinski, A., Herings, P.J.-J., Peters, H.: The strong sequential core for two-period economies. J. Math. Econ. 38, 465–482 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Predtetchinski, A., Herings, P.J.-J., Peters, H.: The strong sequential core in a dynamic exchange economy. Econ. Theory 24, 147–162 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2(3), 21–41 (2000)CrossRefGoogle Scholar
  61. 61.
    Ruszczyński, A., Shapiro, A.: Conditional risk mappings. Math. Oper. Res. 31(3), 544–561 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Samet, D., Zemel, E.: On the core and dual set of linear programming games. Math. Oper. Res. 9, 309–316 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Schmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17, 1163–1170 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory, Second Edition. SIAM and Mathematical Optimization Society, Philadelphia, Pennsylvania (2014)Google Scholar
  66. 66.
    Shapley, L.S.: A value for \(n\)-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games, Volume II, vol. 28 of Annals of Mathematics Studies, pp. 307–317. Princeton University Press, Princeton (1953)Google Scholar
  67. 67.
    Shapley, L.S., Shubik, M.: The assignment game I: the core. Int. J. Game Theory 1, 111–130 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Slikker, M., Fransoo, J., Wouters, M.: Cooperation between multiple news-vendors with transshipments. Eur. J. Oper. Res. 167, 370–380 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Suijs, J., Borm, P.: Stochastic cooperative games: superadditivity, convexity, and certainty equivalents. Games Econ. Behav. 27(2), 331–345 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Suijs, J., Borm, P., de Waegenaere, A., Tijs, S.: Cooperative games with stochastic payoffs. Eur. J. Oper. Res. 113, 193–205 (1999a)CrossRefzbMATHGoogle Scholar
  71. 71.
    Suijs, J., Borm, P., de Waegenare, A., Tijs, S.: Cooperative games with stochastic payoffs. Eur. J. Oper. Res. 113, 193–205 (1999b)CrossRefzbMATHGoogle Scholar
  72. 72.
    Tamir, A.: On the core of network synthesis games. Math. Program. 50, 123–135 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Timmer, J., Borm, P., Tijs, S.: On three Shapley-like solutions for cooperative games with random payoffs. Int. J. Game Theory 32, 595–613 (2003)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Toriello, A., Uhan, N.A.: On traveling salesman games with asymmetric costs. Oper. Res. 61, 1429–1434 (2013). Technical noteGoogle Scholar
  75. 75.
    Toriello, A., Uhan, N.A.: Dynamic cost allocation for economic lot sizing games. Oper. Res. Lett. 42, 82–84 (2014)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Uhan, N.A.: Stochastic linear programming games with concave preferences. Eur. J. Oper. Res. 243(2), 637–646 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    van den Heuvel, W., Borm, P., Hamers, H.: Economic lot-sizing games. Eur. J. Oper. Res. 176, 1117–1130 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    van Gellekom, J.R.G., Potters, J.A.M., Reijnierse, J.H., Engel, M.C., Tijs, S.H.: Characterization of the Owen set of linear production processes. Games Econ. Behav. 32, 139–156 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)zbMATHGoogle Scholar
  80. 80.
    Wu, M., Zhu, S.X., Teunter, R.H.: The risk-averse newsvendor problem with random capacity. Eur. J. Oper. Res. 231(2), 328–336 (2013)Google Scholar
  81. 81.
    Xin, L., Goldberg, D.A., Shapiro, A.: Distributionally robust multistage inventory models with moment constraints. arXiv:1304.3074 (2013)
  82. 82.
    Xu, N., Veinott, A.F.: Sequential stochastic core of a cooperative stochastic programming game. Oper. Res. Lett. 41(5), 430–435 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    Yaari, M.E.: The dual theory of choice under risk. Econometrica 55(1), 95–115 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  84. 84.
    Yu, G.: Min-max optimization of several classical discrete optimization problems. J. Optim. Theory Appl. 98(1), 221–242 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    Zhang, J.: Cost allocation for joint replenishment models. Oper. Res. 57, 146–156 (2009)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Mathematics DepartmentUnited States Naval AcademyAnnapolisUSA

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