Search Results (8)

With the rapid development of the prefabricated building industry, the green supply chain of prefabricated buildings has become a key driver of sustainable development and efficiency improvement in the industry. However, the issue of benefit distribution arising from cooperation has become the main challenge affecting the long-term stability of the supply chain. To address this, this study proposes an improved TFN-TOPSIS-Banzhaf value model, which optimizes the benefit distribution in the green supply chain of prefabricated buildings using cooperative game theory. This approach enhances both the fairness and accuracy of the distribution. The model integrates a combination of subjective and objective weighting methods based on triangular fuzzy numbers and the M-TOPSIS method for multi-factor evaluation, resulting in the corrected weight coefficients. By combining the weighting coefficients and least squares contributions, the improved Banzhaf value based on players’ weighted least squares contributions is constructed. The effectiveness and robustness of the model are verified through a case analysis, which significantly enhances the model’s ability to handle supply chain synergies and achieves a more fair and precise benefit distribution. This research provides an effective benefit distribution tool for the prefabricated building industry, promoting the continuous development of green building practices and supply chain cooperation. Full article

Intense market competition has driven small- and medium-sized enterprises (SMEs) in the manufacturing sector to collaborate and form supply chain coalitions, which can improve the information flow and resource sharing and significantly enhance supply chain management efficiency. However, the distribution of cooperative benefits poses a core challenge for the long-term stability of coalitions. This paper addresses the impact of dynamic changes in complex business environments by utilizing triangular fuzzy numbers to represent the value of coalition, effectively depicting the uncertainty and ambiguity in the cooperation process. Compared to traditional models (which do not use triangular fuzzy numbers), this model is better suited to dynamic changes, offering flexible response mechanisms that ensure adaptability and fairness in the decision-making process. In addition, considering the influence of each member’s weight in the coalition, the fuzzy comprehensive evaluation method is used to determine the weights. With the goal of minimizing the dissatisfaction of enterprises in benefit distribution, a least square contribution with triangular fuzzy numbers is constructed to replace the marginal contribution of the classical Banzhaf value, and an improved Banzhaf value based on the player’s triangular fuzzy number-weighted excess contribution is proposed to arrive at a fair and reasonable benefit allocation strategy in order to enhance the long-term stability and cooperative benefits of coalition. By analyzing an example of the supply chain coalition, the effectiveness of the proposed improved Banzhaf value is verified, which satisfies the uniqueness, the individual rationality, and the group rationality. It not only promotes the level of risk management and decision making under the uncertainty conditions of complex business, but also deepens the theoretical foundation of cooperative game theory and expands its possibilities in practical applications and future development. Full article

Computing Shapley values for large cooperative games is an NP-hard problem. For practical applications, stochastic approximation via permutation sampling is widely used. In the context of machine learning applications of the Shapley value, the concept of antithetic sampling has become popular. The idea is to employ the reverse permutation of a sample in order to reduce variance and accelerate convergence of the algorithm. We study this approach for the Shapley and Banzhaf values, as well as for the Owen value which is a solution concept for games with precoalitions. We combine antithetic samples with established stratified sampling algorithms. Finally, we evaluate the performance of these algorithms on four different types of cooperative games. Full article

We study the efficient computation of power indices for weighted voting games with precoalitions amongst subsets of players (reflecting, e.g., ideological proximity) using the paradigm of dynamic programming. Starting from the state-of-the-art algorithms for computing the Banzhaf and Shapley–Shubik indices for weighted voting games, we present a framework for fast algorithms for the three most common power indices with precoalitions, i.e., the Owen index, the Banzhaf–Owen index and the symmetric coalitional Banzhaf index, and point out why our new algorithms are applicable for large numbers of players. We discuss implementations of our algorithms for the three power indices with precoalitions in C++ and review computing times, as well as storage requirements. Full article

The aim of this paper is to extend the classical Banzhaf index of power to voting games in which players have weights representing different cooperation or bargaining abilities. The obtained value does not satisfy the classical total power property, which is justified by the imperfect cooperation. Nevertheless, it is monotonous in the weights. We also obtain three different characterizations of the value. Then we relate it to the Owen multilinear extension. Full article

A cooperative game represents a situation in which a set of agents form coalitions in order to achieve a common good. To allocate the benefits of the result of this cooperation there exist several values such as the Shapley value or the Banzhaf value. Sometimes it is considered that not all communications between players are feasible and a graph is introduced to represent them. Myerson (1977) introduced a Shapley-type value for these situations. Another model for cooperative games is the Owen model, Owen (1977), in which players that have similar interests form a priori unions that bargain as a block in order to get a fair payoff. The model of cooperation introduced in this paper combines these two models following Casajus (2007). The situation consists of a communication graph where a two-step value is defined. In the first step a negotiation among the connected components is made and in the second one players inside each connected component bargain. This model can be extended to fuzzy contexts such as proximity relations that consider leveled closeness between agents as we proposed in 2016. There are two extensions of the Banzhaf value to the Owen model, because the natural way loses the group symmetry property. In this paper we construct an appropriate value to extend the symmetric option for situations with a proximity relation and provide it with an axiomatization. Then we apply this value to a political situation. Full article

By using Moore’s subtraction operator and a total order on the set of closed intervals, we introduce a new variation of the Banzhaf value for cooperative interval games called the interval Banzhaf-like value which may accommodate the shortcomings of the interval Banzhaf value. We first reveal the relation between this introduced value and the interval Banzhaf value. Then, we present two sets of properties that may be used to determine whether an interval value is median-indifferent to the interval Banzhaf-like value. Finally, in order to overcome the disadvantages of the interval Banzhaf-like value, we propose the contracted interval Banzhaf-like value and give an axiomatization of this proposed value. Full article