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Valuing Inputs Under Supply Uncertainty: The Bayesian Shapley Value

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  • Roland Pongou

    (Department of Economics, University of Ottawa, Ottawa, ON)

  • Jean-Baptiste Tondji

    (Department of Economics, University of Ottawa, Ottawa, ON)

Abstract

We consider the problem of valuing inputs in a production environment in which input supply is uncertain. Inputs can be workers in a firm, risk factors for a disease, securities in a financial market, or nodes in a networked economy. Each input takes its values from a finite set, and uncertainty is modeled as a probability distribution over this set. First, we provide an axiomatic solution to our valuation problem, defining three intuitive axioms which we use to uniquely characterize a valuation scheme that we call the a priori Shapley value. Second, we solve the problem of valuing inputs a posteriori - that is, after observing output. This leads to the Bayesian Shapley value. Third, we consider the problem of rationalizing uncertainty when the inputs are rational workers supplying labor in a non-cooperative production game in which payoffs are given by the Shapley wage function. We find that probability distributions over labor supply that can be supported as mixed strategy Nash equilibria always exist. We also provide an intuitive condition under which we prove the existence of a pure strategy Nash equilibrium. We present several applications of our theory to real-life situations.

Suggested Citation

  • Roland Pongou & Jean-Baptiste Tondji, 2016. "Valuing Inputs Under Supply Uncertainty: The Bayesian Shapley Value," Working Papers 1617E, University of Ottawa, Department of Economics.
  • Handle: RePEc:ott:wpaper:1617e
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    References listed on IDEAS

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    1. Franklin Allen & Douglas Gale, 2000. "Financial Contagion," Journal of Political Economy, University of Chicago Press, vol. 108(1), pages 1-33, February.
    2. Battiston, Stefano & Delli Gatti, Domenico & Gallegati, Mauro & Greenwald, Bruce & Stiglitz, Joseph E., 2012. "Liaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk," Journal of Economic Dynamics and Control, Elsevier, vol. 36(8), pages 1121-1141.
    3. Pongou, Roland & Serrano, Roberto, 2016. "Volume of trade and dynamic network formation in two-sided economies," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 147-163.
    4. Roberto Serrano, 2013. "Lloyd Shapley's Matching and Game Theory," Scandinavian Journal of Economics, Wiley Blackwell, vol. 115(3), pages 599-618, July.
    5. Victor Aguiar & Roland Pongou & Jean-Baptiste Tondji, 2016. "Measuring and Decomposing the Distance to the Shapley Wage Function with Limited Data," Working Papers 1613e, University of Ottawa, Department of Economics.
    6. Stefan Krasa & Nicholas C. Yannelis, 2005. "The value allocation of an economy with differential information," Studies in Economic Theory, in: Dionysius Glycopantis & Nicholas C. Yannelis (ed.), Differential Information Economies, pages 507-526, Springer.
    7. Roland Pongou & Roberto Serrano, 2013. "Fidelity Networks and Long-Run Trends in HIV/AIDS Gender Gaps," American Economic Review, American Economic Association, vol. 103(3), pages 298-302, May.
    8. Hsiao Chih-Ru & Raghavan T. E. S., 1993. "Shapley Value for Multichoice Cooperative Games, I," Games and Economic Behavior, Elsevier, vol. 5(2), pages 240-256, April.
    9. L. S. Shapley & Martin Shubik, 1967. "Ownership and the Production Function," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 81(1), pages 88-111.
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    Cited by:

    1. Josep Freixas & Montserrat Pons, 2022. "A critical analysis on the notion of power," Annals of Operations Research, Springer, vol. 318(2), pages 911-933, November.
    2. Ghislain H. Demeze-Jouatsa & Roland Pongou & Jean-Baptiste Tondji, 2021. "A Free and Fair Economy: A Game of Justice and Inclusion," Papers 2107.12870, arXiv.org.
    3. Pongou, Roland & Sidie, Ghislain Junior & Tchuente, Guy & Tondji, Jean-Baptiste, 2022. "Profits, Pandemics, and Lockdown Effectiveness in Nursing Home Networks," National Institute of Economic and Social Research (NIESR) Discussion Papers 540, National Institute of Economic and Social Research.
    4. Roland Pongou & Guy Tchuente & Jean-Baptiste Tondji, 2020. "An Economic Model of Health-vs-Wealth Prioritization During COVID-19: Optimal Lockdown, Network Centrality, and Segregation," Working Papers 2009E Classification-E61,, University of Ottawa, Department of Economics.
    5. Tido Takeng, Rodrigue, 2022. "Uncertain production environment and communication structure," Journal of Mathematical Economics, Elsevier, vol. 102(C).
    6. Friedman, Jane & Parker, Cameron, 2018. "The conditional Shapley–Shubik measure for ternary voting games," Games and Economic Behavior, Elsevier, vol. 108(C), pages 379-390.
    7. Afodjo, Nabil & Pongou, Roland, 2024. "Efficiency and maximality in anonymous two-sided economies," Games and Economic Behavior, Elsevier, vol. 146(C), pages 184-195.
    8. Demeze-Jouatsa, Ghislain-Herman & Pongou, Roland & Tondji, Jean-Baptiste, 2021. "A Free and Fair Economy: A Game of Justice and Inclusion," Center for Mathematical Economics Working Papers 653, Center for Mathematical Economics, Bielefeld University.
    9. Pongou, Roland & Tchantcho, Bertrand, 2021. "Round-robin political tournaments: Abstention, truthful equilibria, and effective power," Games and Economic Behavior, Elsevier, vol. 130(C), pages 331-351.
    10. Ghislain Herman Demeze-Jouatsa & Roland Pongou & Jean-Baptiste Tondji, 2024. "Justice, inclusion, and incentives," Journal of Theoretical Politics, , vol. 36(2), pages 101-131, April.
    11. Roland Pongou & Guy Tchuente & Jean-Baptiste Tondji, 2023. "Optimal interventions in networks during a pandemic," Journal of Population Economics, Springer;European Society for Population Economics, vol. 36(2), pages 847-883, April.
    12. Pongou, Roland & Tchuente, Guy & Tondji, Jean-Baptiste, 2021. "Optimally Targeting Interventions in Networks during a Pandemic: Theory and Evidence from the Networks of Nursing Homes in the United States," GLO Discussion Paper Series 957, Global Labor Organization (GLO).
    13. Victor H. Aguiar & Roland Pongou & Roberto Serrano & Jean-Baptiste Tondji, 2018. "An Index of Unfairness," Working Papers 2018-9, Brown University, Department of Economics.
    14. Roland Pongou & Guy Tchuente & Jean-Baptiste Tondji, 2021. "Optimally Targeting Interventions in Networks during a Pandemic: Theory and Evidence from the Networks of Nursing Homes in the United States," Papers 2110.10230, arXiv.org.

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    More about this item

    Keywords

    Input valuation; uncertainty; a priori Shapley value; Bayesian Shapley value; rationalizability;
    All these keywords.

    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • D20 - Microeconomics - - Production and Organizations - - - General
    • D80 - Microeconomics - - Information, Knowledge, and Uncertainty - - - General
    • J30 - Labor and Demographic Economics - - Wages, Compensation, and Labor Costs - - - General

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