ONE-POINT SOLUTIONS OBTAINED FROM BEST APPROXIMATION PROBLEMS FOR COOPERATIVE GAMES
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.
one-point solutions cooperative games additive games harsanyi dividends