The \(\tau\)-value, the core and semiconvex games. (English) Zbl 0582.90108
The notion of ”semiconvex game” (of n persons, in characteristic function form) is introduced and it is proved that any exact game (in Schmeidler’s sense) is semiconvex. Significant properties of Tijs’ ”\(\tau\)-value” \(\tau\) (v) associated with a semiconvex game v are pointed out. Among others: \(\tau\) (v) is contained into the core of v then \(n=4\) and \(\tau\) (v) is exactly the Shapley value when the gap function of v is a non- negative constant.
Reviewer: D.Butnariu
MSC:
| 91A12 | Cooperative games |
Keywords:
nucleolus; ”semiconvex game”; characteristic function form; exact game; \(\tau \) -value; core; Shapley value; gap functionReferences:
| [1] | Bennett, E., andM. Wooders: Income Distribution and Firm Formation. J. of Comparative Economics3, 1979, 304–317. · Zbl 0412.90023 · doi:10.1016/0147-5967(79)90032-5 |
| [2] | Chetty, V.K., D. Dasgupta andT.E.S. Raghavan: Power and Distribution of Profits. Discussion paper no. 139, Indian Statistical Institute, Delhi Centre 1976. |
| [3] | Driessen, T.S.H., andS.H. Tijs: The{\(\tau\)}-Value, the Nucleolus and the Core for a Subclass of Games. Methods of Operations Research46, 1983, 395–406. · Zbl 0514.90095 |
| [4] | –: Extensions and Modifications of the{\(\tau\)}-Value for Cooperative Games. In: Selected Topics in Operations Research and Mathematical Economics, ed. by G. Hammer and D. Pallaschke, Springer-Verlag, Berlin, 1984a, 252–261. |
| [5] | –: Game-Theoretic Solutions for Some Economic Situations. Cahiers du C.E.R.O. 26, 1984b, 51–58. · Zbl 0533.90099 |
| [6] | Kohlberg, E.: On the Nucleolus of a Characteristic Function Game. SIAM J. Appl. Math.20, 1971, 62–66. · Zbl 0228.90060 · doi:10.1137/0120009 |
| [7] | Rabie, M.A.: A Note on the Exact Games. Int. J. of Game Theory10, 1981, 131–132. · Zbl 0476.90093 · doi:10.1007/BF01755958 |
| [8] | Schmeidler, D.: The Nucleolus of a Characteristic Function Game. SIAM J. Appl. Math.17, 1969, 1163–1170. · Zbl 0191.49502 · doi:10.1137/0117107 |
| [9] | –: Cores of Exact Games. J. of Math. Analysis and Applications40, 1972, 214–225. · Zbl 0243.90071 · doi:10.1016/0022-247X(72)90045-5 |
| [10] | Shapley, L.S.: A Value forn-Person Games. Annals of Mathematics Studies28, 1953, 307–317. · Zbl 0050.14404 |
| [11] | –: Cores of Convex Games. Int. J. of Game Theory1, 1971, 11–26. · Zbl 0222.90054 · doi:10.1007/BF01753431 |
| [12] | Straffin, M., andJ.P. Heaney: Game Theory and the Tennessee Valley Authority. Int. J. of Game Theory10, 1981, 35–43. · Zbl 0452.90100 · doi:10.1007/BF01770069 |
| [13] | Tijs, S.H.: Bounds for the Core and the{\(\tau\)}-Value. In: Game Theory and Mathematical Economics, ed. by O. Moeschlin and D. Pallaschke, North-Holland Publ. Cie, 1981, 123–132. · Zbl 0467.90087 |
| [14] | Tijs, S.H., andF.A.S. Lipperts: The Hypercube and the Core Cover ofn-Person Cooperative Games. Cahiers du C.E.R.O.24, 1982, 27–37. · Zbl 0479.90093 |
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