Abstract
The nucleolus and the prenucleolus are solution concepts for TU games based on the excess vector that can be associated to any payoff vector. Here we explore some solution concepts resulting from a payoff vector selection based also on the excess vector but by means of an assessment of their relative fairness different from that given by the lexicographical order. We take the departure consisting of choosing the payoff vector which minimizes the variance of the resulting excesses of the coalitions. This procedure yields two interesting solution concepts, both a prenucleolus-like and a nucleolus-like notion, depending on which set is chosen to set up the minimizing problem: the set of efficient payoff vectors or the set of inputations. These solution concepts, which, paralleling the prenucleolus and the nucleolus, we call least square prenucleolus and least square nucleolus, are easy to calculate and exhibit nice properties. Different axiomatic characterizations of the former are established, some of them by means of consistency for a reasonable reduced game concept.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Banzhaf JF (1965) Weighted voting doesn't work: A mathematical analysis. Rutgers Law Review 19: 317–343
Charnes A, Kortanek K (1967) On classes of convex and preemptive nuclei for N-person games. S.R.M. No 176, The Technology Institute Northwestern University
Coleman JS (1971) Control of collectivities and the power of a collectivity to act. In: Lieberman B (Ed) Social Choice, London (Gordon and Breach)
Davis J, Maschler M (1965) The kernal of a cooperative game. Naval Research Logistics Quarterly 12: 223–259
Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Mathematics of Operation Research 4: 99–131
Hammer PL, Holzman R (1987) On Approximations of pseudo-Boolean functions. Rutcor Research Report ≠ 29–87, Dept. of Mathematics and Center for Operations Research, New Brunswick, Rutgers University, NJ
Hart S, Mas-Colell A (1989) Potential, Value and Consistency. Econometrica 57: 589–614
Maschler M (1992) The bargaining set, kernel and nucleolus. In: Aumann RJ, Hart S (Eds) Handbook of Game Theory Vol l, Elsevier Science Publishers BV
Maschler M, Peleg B, Shapley LS (1972) The kernel and bargaining set for convex games. International Journal of Game Theory 1: 73–93
Maschler M, Peleg B, Shapley LS (1979) Geometric properties of the kernel, nucleolus and related solution concepts. Mathematics of Operation Research 4: 303–338
Owen G (1982) Game Theory, 2nd edn. New York-London Academic Press
Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics 17: 1163–1170
Sobolev AI (1975) The characterization of optimality principles in cooperative games by functional equations (in Russian). In: Vorobjev NN (Ed) Mathematical Methods in the Social Sciences 6, Academy of Sciences of the Lithuanian SSR, Vilnius 94–151
Spinetto RD (1971) Solution concepts forn-person cooperative games as points in the game space. Technical report no 138, Department of Operations Research, College of Engineering, Cornell University, Ithaca, NY
Young HP (1985) Monotonic solutions of cooperative games. International Journal of Game Theory 14: 65–72.
Author information
Authors and Affiliations
Additional information
We want to thank M. Maschler for his helpful and encouraging comments at an early stage of this work. The research reported in this paper has received financial support from the Universidad del País Vasco (projects UPV 036.321-HA186/92 and UPV 036.321-HA127/93).
Rights and permissions
About this article
Cite this article
Ruiz, L.M., Valenciano, F. & Zarzuelo, J.M. The least square prenucleolus and the least square nucleolus. Two values for TU games based on the excess vector. Int J Game Theory 25, 113–134 (1996). https://doi.org/10.1007/BF01254388
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01254388