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Linear Transformation of Products: Games and Economies

Author

Listed:
  • J. Timmer

    (Tilburg University)

  • P. Borm

    (Tilburg University)

  • J. Suijs

    (Tilburg University)

Abstract

In this paper, we introduce situations involving the linear transformation of products (LTP). LTP situations are production situations where each producer has a single linear transformation technique. First, we approach LTP situations from a (cooperative) game theoretical point of view. We show that the corresponding LTP games are totally balanced. By extending an LTP situation to one where a producer may have more than one linear transformation technique, we derive a new characterization of (nonnegative) totally balanced games: each totally balanced game with nonnegative values is a game corresponding to such an extended LTP situation. The second approach to LTP situations is based on a more economic point of view. We relate (standard) LTP situations to economies in two ways and we prove that the economies are standard exchange economies (with production). Relations between the equilibria of these economies and the cores of cooperative LTP games are investigated.

Suggested Citation

  • J. Timmer & P. Borm & J. Suijs, 2000. "Linear Transformation of Products: Games and Economies," Journal of Optimization Theory and Applications, Springer, vol. 105(3), pages 677-706, June.
    • Handle: RePEc:spr:joptap:v:105:y:2000:i:3:d:10.1023_a:1004601509292
    DOI: 10.1023/A:1004601509292
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    1. Ehud Kalai & Eitan Zemel, 1982. "Totally Balanced Games and Games of Flow," Mathematics of Operations Research, INFORMS, vol. 7(3), pages 476-478, August.
    2. Timmer, J.B. & Llorca, N. & Tijs, S.H., 1999. "Games Arising from Infinite Production Situations," Other publications TiSEM 4573bb7e-7b84-4792-a330-b, Tilburg University, School of Economics and Management.
    3. Vito Fragnelli & Fioravante Patrone & Enrico Sideri & Stef Tijs, 1999. "Balanced games arising from infinite linear models," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(3), pages 385-397, December.
    4. Shapley, Lloyd S. & Shubik, Martin, 1969. "On market games," Journal of Economic Theory, Elsevier, vol. 1(1), pages 9-25, June.
    5. Timmer, J.B. & Llorca, N. & Tijs, S.H., 1999. "Games Arising from Infinite Production Situations," Discussion Paper 1999-57, Tilburg University, Center for Economic Research.
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    Cited by:

    1. Ichiro Nishizaki & Tomohiro Hayashida & Shinya Sekizaki & Kenta Tanaka, 2023. "Averaged dual solution for linear production games and its characterization," 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. 31(2), pages 523-555, June.
    2. Timmer, J.B. & Llorca, N. & Tijs, S.H., 1999. "Games Arising from Infinite Production Situations," Discussion Paper 1999-57, Tilburg University, Center for Economic Research.
    3. Peter Borm & Herbert Hamers & Ruud Hendrickx, 2001. "Operations research games: A survey," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(2), pages 139-199, December.
    4. Aparicio, Juan & Pastor, Jesús T. & Vidal, Fernando & Zofío, José L., 2017. "Evaluating productive performance: A new approach based on the product-mix problem consistent with Data Envelopment Analysis," Omega, Elsevier, vol. 67(C), pages 134-144.
    5. Yea, Minyoung & Kim, Daeki & Cheong, Taesu & Moon, Joon & Kang, Sungho, 2022. "Baking and slicing the pie: An application to the airline alliance's profit-sharing based on cooperative game theory," Journal of Air Transport Management, Elsevier, vol. 102(C).
    6. Tijs, S.H. & Timmer, J.B. & Llorca, N. & Sánchez-Soriano, J., 2000. "The Owen Set and the Core of Semi-Infinite Linear Production Situations," Other publications TiSEM c28286c5-deeb-4e8a-b8dd-c, Tilburg University, School of Economics and Management.
    7. Ichiro Nishizaki & Tomohiro Hayashida & Shinya Sekizaki & Kojiro Furumi, 2023. "A two-stage linear production planning model with partial cooperation under stochastic demands," Annals of Operations Research, Springer, vol. 320(1), pages 293-324, January.
    8. Herrera-Restrepo, Oscar & Triantis, Konstantinos, 2019. "Enterprise design through complex adaptive systems and efficiency measurement," European Journal of Operational Research, Elsevier, vol. 278(2), pages 481-497.
    9. Tijs, S.H. & Timmer, J.B. & Llorca, N. & Sánchez-Soriano, J., 2000. "The Owen Set and the Core of Semi-Infinite Linear Production Situations," Discussion Paper 2000-49, Tilburg University, Center for Economic Research.
    10. Borrero, D.V. & Hinojosa, M.A. & Mármol, A.M., 2016. "DEA production games and Owen allocations," European Journal of Operational Research, Elsevier, vol. 252(3), pages 921-930.
    11. Yu, Anyu & You, Jianxin & Rudkin, Simon & Zhang, Hao, 2019. "Industrial carbon abatement allocations and regional collaboration: Re-evaluating China through a modified data envelopment analysis," Applied Energy, Elsevier, vol. 233, pages 232-243.
    12. Lozano, S., 2013. "DEA production games," European Journal of Operational Research, Elsevier, vol. 231(2), pages 405-413.
    13. Lozano, S. & Hinojosa, M.A. & Mármol, A.M., 2015. "Set-valued DEA production games," Omega, Elsevier, vol. 52(C), pages 92-100.
    14. Gutiérrez, E. & Llorca, N. & Sánchez-Soriano, J. & Mosquera, M., 2018. "Sustainable allocation of greenhouse gas emission permits for firms with Leontief technologies," European Journal of Operational Research, Elsevier, vol. 269(1), pages 5-15.
    15. Timmer, J.B. & Llorca, N. & Tijs, S.H., 1999. "Games Arising from Infinite Production Situations," Other publications TiSEM 4573bb7e-7b84-4792-a330-b, Tilburg University, School of Economics and Management.

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    More about this item

    Keywords

    linear transformation; cooperative games; economies; price equilibria;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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