Abstract
Results of game theory are often the keys to decisions of economical and political executives. They are also used to create internal tools of many decision making software. For example, coordination games may be cooperative games, when the players choose the strategies by a consensus decision making process, and game trees are used to represent some key cooperative games. Our theory of cooperative games with transferable utilities makes it possible to deliver a formal certificate that contains statements and proofs with each result of any procedure in theory of cooperative TU-games. Such formal certificates can be archived and audited by independent experts to guarantee that the process that lead to the decision is sound and pertaining. As we use an automated proof checker, the review only has to guarantee that the statements of the certificate are correct. The proofs contained in the certificate are guaranteed automatically by the proof checker and our formal theory.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aarts, H., Driessen, T.: The irreducible core of a minimum cost spanning tree game. Mathematical Methods of Operations Research 38(2), 163–174 (1993), http://dx.doi.org/10.1007/BF01414212
Bird, C.G.: On cost allocation for a spanning tree: A game theoretic approach. Networks 6(4), 335–350 (1976), http://dx.doi.org/10.1002/net.3230060404
Bondareva, O.N.: Some applications of linear programming methods to the theory of cooperative games. Problemy kibernetiki 10, 119–139 (1963) (Russian)
Daumas, M., Lester, D., Martin-Dorel, É., Truffert, A.: Stochastic formal correctness of numerical algorithms. In: NASA Formal Methods Symposium, pp. 136–145 (2009a), http://ti.arc.nasa.gov/m/event/nfm09/NFM09Proceedings.pdf
Daumas, M., Lester, D., Muñoz, C.: Verified real number calculations: A library for interval arithmetic. IEEE Transactions on Computers 58(2), 226–237 (2009b), http://dx.doi.org/10.1109/TC.2008.213
Daumas, M., Martin-Dorel, É., Truffert, A.: Bornes quasi-certaines sur l’accumulation d’erreurs infimes dans les systèmes hybrides. In: MAnifestation des JEunes Chercheurs en Sciences et Technologies de l’Information et de la Communication, Avignon, France (2009c)
Davis, M., Maschler, M.: The kernel of a cooperative game. Naval Research Logistic Quarterly 12(3), 223–259 (1965), http://dx.doi.org/10.1002/nav.3800120303
Driessen, T.S.H.: A new characterization of the Shapley-value. Methods of Operations Research 50, 505–517 (1985)
Fougères, A., Truffert, A., Ventou, M.: Dualité probabiliste des jeux coopératifs : la stabilité comme principe d’équilibre, détermination algorithmique du nucleolus. Documents de travail 99A19, Groupement de Recherche en Economie Quantitative d’Aix Marseille (1999), http://www.vcharite.univ-mrs.fr/GREQAM/pdf/working_papers/1999/99a19s.pdf ; Travaux présentés aux journées du LEA (Jeux Coopératifs, Agrégation et Optimisation)
Fougères, A., Truffert, A., Ventou, M.: Minimal core generation. Technical Report 00A27, Groupement de Recherche en Economie Quantitative d’Aix Marseille (2000a), http://www.vcharite.univ-mrs.fr/GREQAM/pdf/working_papers/2000/00a27.pdf ; Work presented at the First World Congress of the Game Theory, Bilbao (2000)
Fougères, A., Truffert, A., Ventou, M.: The core revisited: conditional cores and leader coalitions. Technical Report 00A28, Groupement de Recherche en Economie Quantitative d’Aix Marseille (2000b), http://www.vcharite.univ-mrs.fr/GREQAM/pdf/working_papers/2000/00a28.pdf ; Work presented at the workshop on coalition formation, Barcelona (2000)
Gillies, D.B.: Some theorems on n-person games. PhD thesis, Princeton University (1953)
Gonthier, G., Mahboubi, A.: A small scale reflection extension for the Coq system. Research Report 6455, Institut National de Recherche en Informatique et en Automatique, Le Chesnay, France (2008), http://hal.inria.fr/inria-00258384/
Granot, D., Huberman, G.: On minimum cost spanning tree games. Mathematical Programming 21(1), 1–18 (1981), http://dx.doi.org/10.1007/BF01584227
Huet, G., Kahn, G., Paulin-Mohring, C.: The Coq proof assistant: a tutorial: version 8.0 (2004), ftp://ftp.inria.fr/INRIA/coq/current/doc/Tutorial.pdf.gz
Kern, W., Paulusna, D.: Matching game: the least core and the nucleolus. Mathematics of Operations Research 28(2), 294–308 (2003), http://dx.doi.org/10.1287/moor.28.2.294.14477
Maschler, M., Peleg, B., Shapley, L.S.: The kernel and bargaining set for convex games. International Journal of Game Theory 1(1), 73–93 (1971), http://dx.doi.org/10.1007/BF01753435
Maschler, M., Peleg, B., Shapley, L.S.: Geometric properties of the kernel, nucleolus, and related solution concepts. Mathematics of Operations Research 4(4), 303–338 (1979), http://dx.doi.org/10.1287/10.1287/moor.4.4.303
Owre, S., Rushby, J.M., Shankar, N.: PVS: a prototype verification system. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 748–752. Springer, Heidelberg (1992), http://pvs.csl.sri.com/papers/cade92-pvs/cade92-pvs.ps
Owre, S., Shankar, N., Rushby, J.M., Stringer-Calvert, D.W.J.: PVS Language Reference. In: SRI International, Version 2.4 (2001), http://pvs.csl.sri.com/doc/pvs-language-reference.pdf
Potters, J.A.M.: An axiomatization of the nucleolus. International Journal of Game Theory 19(4), 365–373 (1991), http://dx.doi.org/10.1007/BF01766427
Schmeidler, D.: The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics 17(6), 1163–1170 (1969), http://dx.doi.org/10.1137/0117107
Shapley, L.S.: A value for n-person games. In: Contributions to the Theory of Games II, pp. 307–317. Princeton University Press, Princeton (1953)
Shapley, L.S.: On balanced sets and cores. Naval Research Logistic Quarterly 14(4), 453–460 (1967), http://dx.doi.org/10.1002/nav.3800140404
Shapley, L.S., Shubik, M.: The assignment game: the core. International Journal of Game Theory 1(1), 111–130 (1971), http://dx.doi.org/10.1007/BF01753437
Snijders, C.: Axiomatization of the nucleolus. Mathematics of Operation Research 20(1), 189–196 (1995), http://dx.doi.org/10.1287/moor.20.1.189
Sobolev, A.I.: The characterization of optimality principles in cooperative games by functional equations. Mathematical Methods in Social Sciences 6, 94–151 (1975) (Russian)
Sudhölter, P., Peleg, B.: A note on an axiomatization of the core of market games. Mathematics of Operations Research 27(2), 441–444 (2002), http://dx.doi.org/10.1287/moor.27.2.441.326
Sudhölter, P., Peleg, B.: Nucleoli as maximizers of collective satisfaction functions. Social Choice and Welfare 15(3), 383–411 (1998), http://dx.doi.org/10.1007/s003550050113
Sudhölter, P., Potters, J.A.M.: The semireactive bargaining set of a cooperative game. International Journal of Game Theory 30(1), 117–139 (2001), http://dx.doi.org/10.1007/s001820100068
von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Daumas, M., Martin-Dorel, É., Truffert, A., Ventou, M. (2009). A Formal Theory of Cooperative TU-Games. In: Torra, V., Narukawa, Y., Inuiguchi, M. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2009. Lecture Notes in Computer Science(), vol 5861. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04820-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-04820-3_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04819-7
Online ISBN: 978-3-642-04820-3
eBook Packages: Computer ScienceComputer Science (R0)