# zbMATH — the first resource for mathematics

Minimum norm solutions for cooperative games. (English) Zbl 1131.91008
Summary: A solution $$f$$ for cooperative games is a minimum norm solution, if the space of games has a norm such that $$f (v)$$ minimizes the distance (induced by the norm) between the game $$v$$ and the set of additive games. We show that each linear solution having the inessential game property is a minimum norm solution. Conversely, if the space of games has a norm, then the minimum norm solution w.r.t. this norm is linear and has the inessential game property. Both claims remain valid also if solutions are required to be efficient. A minimum norm solution, the least square solution, is given an axiomatic characterization.

##### MSC:
 91A12 Cooperative games
##### Keywords:
cooperative games; solutions; minimum norm; Banzhaf value
Full Text:
##### References:
 [1] Charnes A, Rousseau J, Seiford L (1978) Complements, mollifiers and the propensity to disrupt. Int J Game Theory 7:37–50 · Zbl 0373.90094 · doi:10.1007/BF01763119 [2] Dubey P, Neyman A, Weber J (1981) Value theory without efficiency. Math Oper Res 4:99–131 · Zbl 0409.90008 · doi:10.1287/moor.4.2.99 [3] Kalai E, Samet D (1987) On weighted Shapley values. Int J Game Theory 16:205–222 · Zbl 0633.90100 · doi:10.1007/BF01756292 [4] Keane M (1969) Some topics in n-person game theory. Ph.D. dissertation, Northwestern University, Evanston [5] Kleinberg NL, Weiss JH (1985) A new formula for the Shapley value. Econ Lett 17:311–315 · Zbl 1273.91035 · doi:10.1016/0165-1765(85)90249-6 [6] Kleinberg NL, Weiss JH (1986) The orthogonal decomposition of games and an averaging formula for the Shapley value. Math Oper Res 11:117–124 · Zbl 0592.90102 · doi:10.1287/moor.11.1.117 [7] Kleinberg NL, Weiss JH (1988) The orthogonal decomposition of games and an averaging formula for the Shapley value: a correction. Math Oper Res 13:190 · Zbl 0709.90536 · doi:10.1287/moor.13.1.190 [8] Luenberger D (1969) Optimization by vector space methods. Wiley, New York · Zbl 0176.12701 [9] Maschler M, Peleg B (1996) A characterization, existence proof and dimension bounds for the kernel of a game. Pac J Math 18:289–328 · Zbl 0144.43403 [10] Rockafellar RT, Wets RJ-B (1998) Variational analysis. Springer, Berlin [11] Rothblum UG (1985) A simple proof for the Kleinberg-Weiss representation of the Shapley value. Econ Lett 19:137–139 · Zbl 1273.91041 · doi:10.1016/0165-1765(85)90009-6 [12] Ruiz LM, Valenciano F, Zarzuelo FM (1996) 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 · Zbl 0855.90147 · doi:10.1007/BF01254388 [13] Ruiz LM, Valenciano F, Zarzuelo FM (1998a) The family of least square values for transferable utility games. Games Econ Behav 24:109–130 · Zbl 0910.90276 · doi:10.1006/game.1997.0622 [14] Ruiz LM, Valenciano F, Zarzuelo FM (1998b) Some new results on least square values for TU games. TOP 6:139–158 · Zbl 0907.90285 · doi:10.1007/BF02564802 [15] Shapley LS (1953) A value for n-person games. Ann Math Stud 28:307–317 [16] Shapley LS (1971) Cores of convex games. Int J Game Theory 1:11–26 · Zbl 0222.90054 · doi:10.1007/BF01753431 [17] Winter E (2002) The Shapley value. Chap 53 In: Aumann R, Hart S (eds) Handbook of game theory with economic applications, vol 3. Elsevier, Amsterdam
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.