Pongou, Roland and Tondji, JeanBaptiste (2016): Valuing Inputs Under Supply Uncertainty : The Bayesian Shapley Value.

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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 posteriorithat 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 noncooperative 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 reallife situations.
Item Type:  MPRA Paper 

Original Title:  Valuing Inputs Under Supply Uncertainty : The Bayesian Shapley Value 
Language:  English 
Keywords:  Input valuation, uncertainty, a priori Shapley value, Bayesian Shapley value, rationalizability 
Subjects:  C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C70  General D  Microeconomics > D2  Production and Organizations > D20  General D  Microeconomics > D8  Information, Knowledge, and Uncertainty > D80  General J  Labor and Demographic Economics > J3  Wages, Compensation, and Labor Costs > J30  General 
Item ID:  74747 
Depositing User:  M. JeanBaptiste Tondji 
Date Deposited:  27 Oct 2016 00:10 
Last Modified:  27 Oct 2016 00:11 
References:  Aguiar, V. H., Pongou, R., and Tondji, JB. (2016). Measuring and decomposing the distance to the Shapley wage function with limited data. Mimeo. Allen, F., and Gale, D. (2000). Financial contagion. Journal of political economy, 108(1), 133. Battiston, S., Gatti, D. D., Gallegati, M., Greenwald, B., and Stiglitz, J. E. (2012). Liaisons dangereuses: Increasing connectivity, risk sharing, and systemic risk. Journal of Economic Dynamics and Control, 36(8), 11211141. Freixas, J. (2005). The ShapleyShubik power index for games with several levels of approval in the input and output. Decision Support Systems, 39, 185195. Gale, D., and Shapley, L., (1962). College admissions and the stability of marriage. American Mathematical Monthly 69, 915. Hsiao, ChihRu, and Raghavan T.E.S. (1993). Shapley value for multichoice cooperative games, I. Games and Economics Behavior, 5, 240256. Lin, J. (2016). Using Weighted Shapley Values to measure the systemic Risk of interconnected banks. Pacific Economic Review. Nash, F. J. (1951). Noncooperative games. The Annals of Mathematics, 54(2), 286295. Pongou, R. (2010). The economics of fidelity in network formation. PhD Dissertation, Brown University. Pongou, R., and Serrano, R. (2013). Fidelity networks and longrun trends in HIV/AIDS gender gaps. The American Economic Review, 103(3), 298302. Pongou, R., and Serrano, R. (2016). Volume of trade and dynamic network formation in twosided economies. Journal of Mathematical Economics, 63, 147163. Pongou, R., and Tondji, JB. (2016). Noncooperative games with Shapley payoffs. Working paper. Roth, A. (1988). The Shapley value: essays in honor of Lloyd S. Shapley, Cambridge University Press. Serrano, R. (2013). Lloyd Shapley's matching and game theory. Scandinavian Journal of Economics, 115, 599618. Shapley, L. (1953). A Value for nperson games. In Contributions to the Theory of Games, 2 , 307317. Shapley, L., and Shubik, M. (1967). Ownership and the production function. The Quarterly Journal of Economics, 88111. Young, H. P. (1985). Monotonic solutions of cooperative games. International Journal of Game Theory, 14 (2), 6572. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/74747 