Cooperative games are very useful in considering profit allocation among multiple decision makers who cooperate with each other. In order to deal with cooperative games in practical situations, however, we have to deal with two additional factors. One is some restrictions on coalitions. This first factor has been taken into consideration through feasibility of coalitions. The other is partial cooperation of players. In order to describe this second factor, we consider fuzzy coalitions which permit partial participation in a coalition to a player. In this paper we take both of these factors into account in cooperative games. Namely, we analyze and discuss cooperative fuzzy games extended from ordinary cooperative games with restrictions on coalitions in two approaches. For the purpose of comparison of these two approaches, we define two special classes of extensions called U-extensions which satisfy linearity and W-extensions which satisfy U-extensions and two additional conditions, restriction invariance and monotonicity. Finally, we show sufficient conditions under which these obtained games in two approaches coincide.
In this paper, we extend the traditional linear regression methods to the (numerical input)-(interval output) data case assuming both the observation/measurement error and the indeterminacy of the input-output relationship. We propose three different models based on three different assumptions of interval output data. In each model, the errors are defined as intervals by solving the interval equation representing the relationship among the interval output, the interval function and the interval error. We formalize the estimation problem of parameters of the interval function so as to minimize the sum of square/absolute interval errors. Introducing suitable interpretation of minimization of an interval function, each estimation problem is well-formulated as a quadratic or linear programming problem. It is shown that the proposed methods have close relation to both traditional and interval linear regression methods which are formulated in different manners.
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.