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Convexity In Stochastic Cooperative Situations

Author

Listed:
  • JUDITH TIMMER

    (Center and Department of Econometrics and Operations Research, Tilburg University, P. O. Box 90153, 5000 LE Tilburg, The Netherlands)

  • PETER BORM

    (Center and Department of Econometrics and Operations Research, Tilburg University, P. O. Box 90153, 5000 LE Tilburg, The Netherlands)

  • STEF TIJS

    (Center and Department of Econometrics and Operations Research, Tilburg University, P. O. Box 90153, 5000 LE Tilburg, The Netherlands)

Abstract

This paper introduces a new model concerning cooperative situations in which the payoffs are modeled by random variables. We analyze these situations by means of cooperative games with random payoffs. Special attention is paid to three types of convexity, namely coalitional-merge, individual-merge and marginal convexity. The relations between these types are studied and in particular, as opposed to their deterministic counterparts for TU games, we show that these three types of convexity are not equivalent. However, all types imply that the core of the game is nonempty. Sufficient conditions on the preferences are derived such that the Shapley value, defined as the average of the marginal vectors, is an element of the core of a convex game.

Suggested Citation

  • Judith Timmer & Peter Borm & Stef Tijs, 2005. "Convexity In Stochastic Cooperative Situations," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 7(01), pages 25-42.
    • Handle: RePEc:wsi:igtrxx:v:07:y:2005:i:01:n:s0219198905000387
    DOI: 10.1142/S0219198905000387
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    1. Suijs, Jeroen & Borm, Peter & De Waegenaere, Anja & Tijs, Stef, 1999. "Cooperative games with stochastic payoffs," European Journal of Operational Research, Elsevier, vol. 113(1), pages 193-205, February.
    2. Ruud Hendrickx & Judith Timmer & Peter Borm, 2002. "A note on NTU convexity," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 29-37.
    3. Suijs, Jeroen & Borm, Peter, 1999. "Stochastic Cooperative Games: Superadditivity, Convexity, and Certainty Equivalents," Games and Economic Behavior, Elsevier, vol. 27(2), pages 331-345, May.
    4. Suijs, Jeroen & De Waegenaere, Anja & Borm, Peter, 1998. "Stochastic cooperative games in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 209-228, July.
    5. Ichiishi, Tatsuro, 1981. "Super-modularity: Applications to convex games and to the greedy algorithm for LP," Journal of Economic Theory, Elsevier, vol. 25(2), pages 283-286, October.
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    Cited by:

    1. Ruud Hendrickx & Judith Timmer & Peter Borm, 2002. "A note on NTU convexity," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 29-37.
    2. R. Branzei & O. Branzei & S. Alparslan Gök & S. Tijs, 2010. "Cooperative interval games: a survey," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 18(3), pages 397-411, September.
    3. Hendrickx, R.L.P. & Borm, P.E.M. & Timmer, J.B., 2000. "On Convexity for NTU-Games," Other publications TiSEM ef12f1e8-87f5-41b4-97e4-7, Tilburg University, School of Economics and Management.
    4. Junnosuke Shino & Shinichi Ishihara & Shimpei Yamauchi, 2022. "Shapley Mapping and Its Axiomatizations in n -Person Cooperative Interval Games," Mathematics, MDPI, vol. 10(21), pages 1-14, October.
    5. Judith Timmer & Peter Borm & Stef Tijs, 2004. "On three Shapley-like solutions for cooperative games with random payoffs," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(4), pages 595-613, August.
    6. Hendrickx, R.L.P., 2004. "Cooperation and allocation," Other publications TiSEM ab33e762-204c-46e2-86b1-0, Tilburg University, School of Economics and Management.
    7. Yanovskaya, E. & Brânzei, R. & Tijs, S.H., 2008. "Monotonicity Problems of Interval Solutions and the Dutta-Ray Solution for Convex Interval Games," Other publications TiSEM 22884fa9-68cc-4b67-8c27-2, Tilburg University, School of Economics and Management.
    8. Alparslan Gök, S.Z. & Branzei, O. & Branzei, R. & Tijs, S., 2011. "Set-valued solution concepts using interval-type payoffs for interval games," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 621-626.
    9. Judith Timmer, 2006. "The Compromise Value for Cooperative Games with Random Payoffs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 95-106, August.
    10. Tijs, S.H. & Timmer, J.B. & Brânzei, R., 2001. "Compensations in Information Collecting Situations," Discussion Paper 2001-2, Tilburg University, Center for Economic Research.
    11. Laszlo A. Koczy, 2019. "The risk-based core for cooperative games with uncertainty," CERS-IE WORKING PAPERS 1906, Institute of Economics, Centre for Economic and Regional Studies.
    12. Timmer, J.B., 2000. "The Compromise Value for Cooperative Games with Random Payoffs," Discussion Paper 2000-98, Tilburg University, Center for Economic Research.
    13. Benati, Stefano & López-Blázquez, Fernando & Puerto, Justo, 2019. "A stochastic approach to approximate values in cooperative games," European Journal of Operational Research, Elsevier, vol. 279(1), pages 93-106.
    14. Tijs, S.H. & Timmer, J.B. & Brânzei, R., 2001. "Compensations in Information Collecting Situations," Other publications TiSEM c180b734-2a6e-4461-979d-5, Tilburg University, School of Economics and Management.
    15. Xuan Vinh Doan & Tri-Dung Nguyen, 2019. "Technical Note—Robust Newsvendor Games with Ambiguity in Demand Distributions," Operations Research, INFORMS, vol. 68(4), pages 1047-1062, July.
    16. Donald Nganmegni Njoya & Issofa Moyouwou & Nicolas Gabriel Andjiga, 2021. "The equal-surplus Shapley value for chance-constrained games on finite sample spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(3), pages 463-499, June.
    17. D. Bauso & J. Timmer, 2009. "Robust dynamic cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 38(1), pages 23-36, March.
    18. Timmer, J.B., 2000. "The Compromise Value for Cooperative Games with Random Payoffs," Other publications TiSEM 08aefe4e-00a8-4e58-9e54-3, Tilburg University, School of Economics and Management.
    19. R. Branzei & S. Gök & O. Branzei, 2011. "Cooperative games under interval uncertainty: on the convexity of the interval undominated cores," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 19(4), pages 523-532, December.
    20. Francesco Passarelli, 2007. "Asymmetric Bargaining," ISLA Working Papers 26, ISLA, Centre for research on Latin American Studies and Transition Economies, Universita' Bocconi, Milano, Italy, revised Jan 2007.

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    More about this item

    Keywords

    Cooperative games; random variables; preferences; convexity; Subject Classification: 91A12;
    All these keywords.

    JEL classification:

    • B4 - Schools of Economic Thought and Methodology - - Economic Methodology
    • C0 - Mathematical and Quantitative Methods - - General
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D5 - Microeconomics - - General Equilibrium and Disequilibrium
    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics

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