Abstract
The family of discounted Shapley values is analyzed for cooperative games in coalitional form. We consider the bargaining protocol of the alternating random proposer introduced in Hart and Mas-Colell (Econometrica 64:357–380, 1996). We demonstrate that the discounted Shapley values arise as the expected payoffs associated with the bargaining equilibria when a time discount factor is considered. In a second model, we replace the time cost with the probability that the game ends without agreements. This model also implements these values in transferable utility games, moreover, the model implements the \(\alpha \)-consistent values in the nontransferable utility setting.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
There are at most \(n\) rounds.
A game is zero-monotonic if \(v(S)\le v(T)\) whenever \(S\subset T\), and \( v(i)=0\) for all player \(i\) in the game.
Here \(s\) stands for the cardinality of the active coalition \(S\) at this moment.
That is, for every period \(t=0,1,2 \ldots \), the utility of amount \(x\) obtained at period \(t\) can be represented by \(\delta ^{t}x\)
A particular case of games without transferable utility, in which the boundary of the feasible sets is defined by a hyperplane.
See Hart and Mas-Colell (1996).
We normalize it so that \(\sum _{i\in S}\lambda ^{i}=1\).
The order in which the respondents are asked does not affect the result of the bargaining.
References
Binmore, K., & Dasgupta, P. (1987). The economics of bargaining. Oxford: Blackwell.
Binmore, K., Rubinstein, A., & Wolinsky, A. (1986). The nash bargaining solution in economic modelling. The RAND Journal of Economics, 17, 176–188.
Driessen, T. H., & Radzik, T. (2002). A weighted pseudo-potential approach to values fot TU-games. International Transactions in Operational Research, 9, 303–320.
Hart, S. (1994). On prize games. In N. Meggido (Ed.), Essays in game theory (pp. 111–121). New York: Springer.
Hart, S., & Mas-Colell, A. (1989). Potential, value, and consistency. Econometrica, 57, 589–614.
Hart, S., & Mas-Colell, A. (1996). Bargaining and value. Econometrica, 64, 357–380.
Joosten, R. (1996). Dynamics, equilibria and values dissertation. Maastricht: Maastricht University.
Ju, Y., & Wettstein, D. (2009). Implementing cooperative solution concepts: A generalized bidding approach. Economic Theory, 39, 307–330.
Kawamori, T. (2014). Hart-Mas-Colell lmplementation of the discounted Shapley value. Osaka University of Economics Working Paper Series No. 2014–2.
Maschler, M., & Owen, G. (1989). The consistent Shapley value for hyperplane games. International Journal of Game Theory, 18, 389–407.
Maschler, M., & Owen, G. (1992). The consistent Shapley value for ames without side payments. In R. Selten (Ed.), Rational interaction (pp. 5–12). New York: Springer.
Myerson, R. B. (1980). Conference structures and fair allocation rules. International Journal of Game Theory, 9, 169–182.
Nash, J. F. (1953). Two-person cooperative games. Econometrica, 21, 128–140.
Osborne, M., & Rubinstein, A. (1990). Bargaining and markets. San Diego: Academic Press.
Pérez-Castrillo, D., & Wettstein, D. (2001). Bidding for the surplus: A non-cooperative approach to the Shapley value. Journal of Economic Theory, 100, 274–294.
Roth, A. (1989). Risk aversion and the relationship between Nash’s solution and subgame perfect equilibrium of sequential bargaining. Journal of Risk and Uncertainty, 2, 353–365.
Rubinstein, A. (1982). Perfect equilibrium in a bargaining model. Econometrica, 50, 97–109.
Shapley, L. S. (1953). A value for n-person games. In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the theory of games II (annals of mathematics studies 28) (pp. 307–317). Princeton: Princeton University Press.
Sobolev, A. I. (1973). A characterization of optimality principles in cooperative games by functional equations (Russian). Mathematical Methods of Social Sciences, 6, 94–151.
Stähl, I. (1972). Bargaining theory. Stockholm: Stockholm School of Economics.
van den Brink, R., & Funaki, Y. (2010). Axiomatization and implementation of discounted Shapley values. SSRN Electronic Journal 01/2010. doi:10.2139/ssrn.1636201.
van den Brink, R., Funaki, Y., & Ju, Y. (2011). Reconciling marginalism with egalitarianism: Consistency, monotonicity, and implementation of egalitarian Shapley values. Social Choice and Welfare, 40(3), 693–714. doi:10.1007/s00355-011-0634-2.
Young, P. (1985). Monotonic solutions for cooperative games. International Journal of Game Theory, 14, 65–72.
Acknowledgments
We wish to thank Francesc Carreras for several comments and suggestions. Emilio Calvo thanks the Ministry of Science and Technology, the European Feder Funds under project ECO2010-20584, and the Generalitat Valenciana under the Excellence Programs Prometeo 2009/068 and ISIC2012/021 for their financial support. Esther Gutiérrez-López wishes to thank financial support from the Spanish Ministry of Science and Technology and the European Regional Development Funds under project ECO2012-33618, and from the Universidad del País Vasco / Euskal Herriko Unibertsitatea (UFI 11/51).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Calvo, E., Gutiérrez-López, E. A strategic approach for the discounted Shapley values. Theory Decis 80, 271–293 (2016). https://doi.org/10.1007/s11238-015-9500-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11238-015-9500-5