Abstract
The model of incomplete cooperative games incorporates uncertainty into the classical model of cooperative games by considering a partial characteristic function. Thus the values for some of the coalitions are not known. The main focus of this paper is 1-convexity under this framework. We are interested in two heavily intertwined questions. First, given an incomplete game, how can we fill in the missing values to obtain a complete 1-convex game? Second, how to determine in a rational, fair, and efficient way the payoffs of players based only on the known values of coalitions? We illustrate the analysis with two classes of incomplete games—minimal incomplete games and incomplete games with defined upper vector. To answer the first question, for both classes, we provide a description of the set of 1-convex extensions in terms of its extreme points and extreme rays. Based on the description of the set of 1-convex extensions, we introduce generalisations of three solution concepts for complete games, namely the \(\tau \)-value, the Shapley value and the nucleolus. For minimal incomplete games, we show that all of the generalised values coincide. We call it the average value and provide different axiomatisations. For incomplete games with defined upper vector, we show that the generalised values do not coincide in general. This highlights the importance and also the difficulty of considering more general classes of incomplete games.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alparslan Gök, S. Z. (2009). Cooperative interval games. Ph.D. thesis, Middle East Technical University.
Alparslan Gök, S. Z. (2014). Cooperative ellipsoidal games: A survey. Modeling, Dynamics, Optimization and Bioeconomics, I(73), 279–284.
Bilbao, J. M. (2012). Cooperative games on combinatorial structures (Vol. 26). Springer Science & Business Media.
Albizuri, M. J., Masuya, S., & Zarzuelo, J. M. (2022). Characterization of a value for games under restricted cooperation. Annals of Operations Research, 318, 773.
Bok, J., Černý, M., Hartman, D., & Hladík, M. (2020). Positivity and convexity in incomplete cooperative games. arXiv preprint arXiv:1812.05821.
Bok, J., & Hladík, M. (2015). Selection-based approach to cooperative interval games. In: Communications in Computer and Information Science, ICORES 2015—International Conference on Operations Research and Enterprise Systems, (vol. 577, pp. 40–53).
Branzei, R., Branzei, O., Alparslan Gök, S. Z., & Tijs, S. (2010). Cooperative interval games: A survey. Central European Journal of Operations Research, 18(3), 397–411.
Branzei, R., Dimitrov, D., & Tijs, S. (2008). Models in Cooperative Game Theory. Springer.
Calso, E., Tijs, S. H., Valenciano, F., & Zarzuelo, J. M. (1995). On axiomatization of the \(\tau \)-value. TOP, 3, 35–46.
Curiel, I. (2013). Cooperative Game Theory and Applications: Cooperative Games Arising from Combinatorial Optimization Problems. Springer.
Dehez, P. (2021). 1-convex transferable utility games, a reappraisal. Tech. rep.
Deng-Feng, L. (2016a). Models and methods for interval-valued cooperative games in economic management. Springer International Publishing.
Deng-Feng, L. (2016b). Stochastic cooperative games: Theory and applications (Vol. 31). Springer.
Driessen, T. (1988). Cooperative games, solutions and applications, theory and decision library C (Vol. 3). Kluwer.
Driessen, T. S. H. (1985a). Contributions to the theory of cooperative games: the\(\tau \)-value and k-convex games. Ph.D. thesis, Radboud University: Nijmegen.
Driessen, T. S. H. (1985b). Properties of \(1\)-convex \(n\)-person games. OR Spektrum, 7, 19–26.
Driessen, T. S. H. (2012). 1-concave basis for tu games and the library game. TOP, 20, 578–591.
Driessen, T. S. H., & Funaki, Y. (1991). Coincidence of and collinearity between game theoretic solutions. OR Spektrum, 13(1), 15–30.
Driessen, T. S. H., Muto, S., & Nakayama, M. (1992). A cooperative game of information trading: The core, the nucleolus and the kernel. Zeitschrift für Operations Research, 36(1), 55–72.
Fagile, U. (1989). Cores of games with restricted cooperation. ZOR Methods and models of Operations Research, 33, 405–422.
Grabisch, M. (2016). Set functions, games and capacities in decision making. Springer.
Kopelowitz, A. (1967). Computation of the kernels of simple games and the nucleolus of cooperative games as locuses in the strong \(\epsilon \)-core. Research Memo,31.
Lemaire, J. (1991). Cooperative Game Theory and its Insurance Applications. Center for Research on Risk and Insurance: Wharton School of the University of Pennsylvania.
Mareš, M. (2013). Fuzzy cooperative games: Cooperation with vague expectations Physica, 72.
Masuya, S. (2021). An approximated Shapley value for partially defined cooperative games. Procedia Computer Science, 192, 100–108.
Masuya, S., & Inuiguchi, M. (2016). A fundamental study for partially defined cooperative games. Fuzzy Optimization Decision Making, 15(1), 281–306.
Peleg, B., & Sudhölter, P. (2007). Introduction to the Theory of Cooperative Games (Vol. 34). Springer Science & Business Media.
Peters, H. (2008). The Shapley Value (Vol. 28). Springer.
Pirilina, E. (2022). A survey on cooperative stochastic games with finite and infinite duration. Contributions to Game Theory and Management, 11, 129–195.
Roth, A. G. (1977). The Shapley value as a von Neumann-Morgenstern utility. Econometrica, 45, 657–664.
Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics, 17, 1163–1170.
Shapley, L. S. (1953). A value for n-person game. Annals of Mathematical Studies, 28, 307–317.
Soltan, V. (2015). Lectures on Convex Sets. World Scientific.
Tijs, S. H. (1981). Bounds for the core of a game and the \(\tau \)-value. Game Theory and Mathematical Economics, 123–132.
Tijs, S. H. (1987). An axiomatization of the \(\tau \)-value. Mathematical Social Sciences, 13, 177–181.
Tijs, S. H., & Lipperts, F. A. S. (1982). The hypercube and the core cover of n-person cooperative games. Cahiers du Centre d’Études de Researche Opérationelle, 24, 27–37.
van den Brink, J. (2002). An axiomatization of the Shapley value using a fairness property. International Journal of Game Theory, 30(3), 309–319.
van der Brink, J. R. (1995). An axiomatization of the Shapley value using component efficiency and fairness. Tinbergen Discussin Paper.
Willson, S. J. (1993). A value for partially defined cooperative games. International Journal of Game Theory, 21(4), 371–384.
Xiaohui, Y. (2021). Extension of Owen value for the game with a coalition structure under the limited feasible coalition. Soft Computing, 25(8), 6139–6156.
Young, C. (1989). A new axiomatization of the Shapley value. Games and Economic Behaviour, 1, 119–130.
Acknowledgements
The authors would like to thank Milan Hladík and David Hartman for initial discussions regarding the paper. We also thank anonymous referees for their feedback leading to improvements of the manuscript.
Funding
Both authors were supported by SVV–2020–260578, by the Charles University Grant Agency (GAUK 341721), and by the Czech Science Foundation Grant P403-22-11117 S.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bok, J., Černý, M. 1-convex extensions of incomplete cooperative games and the average value. Theory Decis 96, 239–268 (2024). https://doi.org/10.1007/s11238-023-09946-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11238-023-09946-8