Kybernetika 49 no. 3, 395-403, 2013

One-point solutions obtained from best approximation problems for cooperative games

Tetsuzo Tanino


In this paper we focus on one-point (point-valued) solutions for transferable utility games (TU-games). Since each allocated profit vector is identified with an additive game, a solution can be regarded as a mapping which associates an additive game with each TU-game. Recently Kultti and Salonen proposed a minimum norm problem to find the best approximation in the set of efficient additive games for a given TU-game. They proved some interesting properties of the obtained solution. However, they did not show how to choose the inner product defining the norm to obtain a special class of solutions such as the Shapley value and more general random order values. In this paper, noting that there is a one-to-one correspondence between a game and a Harsanyi dividend vector, we propose a minimum norm problem in the dividend space, not in the game space. Since the dividends for any set with more than one elements are all zero for an additive game, our approach enables us to deal with simpler problems. We will make clear how to choose an inner product, i. e., a positive definite symmetric matrix, to obtain a Harsanyi payoff vector, a random order value and the Shapley value.


cooperative games, one-point solutions, additive games, Harsanyi dividends




  1. A. Charnes, J. Rousseau and L. Seiford: Complements, molifiers and the propensity to disrupt. Internat. J. Game Theory 7 (1978), 37-50.   CrossRef
  2. J. Derks, G. van der Laan and V. Vasil'ev: Characterizations of the random order values by Harsanyi payoff vectors. Math. Methods Oper. Res. 64 (2006), 155-163.   CrossRef
  3. M. Grabisch: $k$-order additive discrete fuzzy measures and their representations. Fuzzy Sets and Systems 92 (1997), 167-189.   CrossRef
  4. J. C. Harsanyi: A simplified bargaining model for the $n$-person cooperative game. Internat. Econom. Rev. 4 (1963), 194-220.   CrossRef
  5. K. Kultti and H. Salonen: Minimum norm solutions for cooperative games. Internat. J. Game Theory 35 (2007), 591-602.   CrossRef
  6. D. Monderer and D. Samet: Variations on the Shapley value. In: Handbook of Game Theory with Economic Applications, Vol. 3 (R. Aumann and S. Hart, eds.), Elsevier Science Publishers, Amsterdam 2002, pp. 2055-2076.   CrossRef
  7. L. M. Ruiz, F. Valenciano and F. M. Zarzuelo: The family of least square values for transferable utility games. Games and Econom. Behavior 24 (1998), 109-130.   CrossRef
  8. L. M. Ruiz, F. Valenciano and F. M. Zarzuelo: Some new results on least square values for TU games. TOP 6 (1998), 139-158.   CrossRef