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This study aims to propose the Shapley value that originates from the game theory to quantify the relative risk of a security in an optimal portfolio.

Systematic risk as expressed by the relative covariance of stock returns to market returns is an essential measure in pricing risky securities. Although very much in use, the concept has become marginalized in recent years because of the difficulties that arise estimating beta. The idea is that portfolios can be viewed as cooperative games played by assets aiming at minimizing risk. With the Shapley value, investors can calculate the exact contribution of each risky asset to the joint payoff. For a portfolio of three stocks, this study exemplifies the Shapley value when risk is minimized regardless of portfolio return.

This study computes the Shapley value of stocks and indices for optimal mean-variance portfolios by using daily returns for the years 2016–2019. This results in the risk attributes allocated to securities in optimal portfolios. The Shapley values are analyzed and compared to the standard beta estimates to determine the ranking of assets with respect to pertinent risk and return.

An alternative approach to value risk and return in optimal portfolios is presented in this study. The logic and the mechanics of Shapley value theory in portfolio analysis have been explained, and its advantages relative to standard beta analysis are presented. Hence, financial analysts when adding or removing specific assets from present positions will have the true and exact impact of their actions by using the Shapley value instead of the beta.

When computing the Shapley value, portfolio risk is decomposed exactly among its assets because it considers all possible coalitions of portfolios. In that sense, financial analysts when adding or removing specific securities from present holdings will be able to predict the true and exact impact of their transactions by using the Shapley value instead of the beta. The main implication for investors is that risk is ultimately priced relative to their holdings. This prevents the subjective mispricing of securities, as standard beta is not used and might allow investors to gain from arbitrage conditions.

The logic and the methodology of Shapley value theory in portfolio analysis have been explained as an alternative to value risk and return in optimal portfolios by presenting its advantages relative to standard beta analysis. The conclusion is that the Shapley value theory contributes much more financial optimization than to standard systematic risk analysis because it enables looking at the contribution of each security to all possible coalitions of portfolios.

In the context of sharing economy and online shopping, establishing a stable urban joint distribution alliance (JDA) is extremely necessary for the entire logistics service market. The purpose of this paper is to rationally allocate the profits and determine the most stable allocation scheme for the urban JDA as well as provide a direction for cooperation between express enterprises and lead managers to pay more attention to the comprehensive performance.

Cooperative game-based methodologies including the proportion method, the core theory, nucleolus and Shapley value have been employed. Four criteria consisting of enterprise operation, customer satisfaction, environmental sustainability and information technology have been incorporated into Shapley value for improvement.

This paper reveals that express enterprises in logistics service market can achieve more benefit from JDA than those who operate separately. Among proposed profit allocation schemes, improved Shapley value scheme shows more rationality by considering partners’ asymmetric contribution. Besides, a stable alliance can be always ensured with partners’ lower propensity to disrupt and relatively balanced negotiation power under improved Shapley value scheme.

This paper makes a few attempts to contribute to the literature on the improvement of Shapley value and applies the concept of “propensity to disrupt” into the field of logistics. Besides, this paper provides various profit allocation schemes and incorporates influencing factors into Shapley value for an improvement thus helping policy-makers make better-informed decisions on urban distribution. Additionally, a case study based on urban express enterprises in Southwest China has been conducted to verify the proposed profit allocation schemes.

The study seeks to identify the impact of variables in a deep learning-based bankruptcy prediction model, which has achieved superior performance to other prediction models but cannot easily interpret hidden processes.

This study developed three LSTM-RNN–based models that predicted the probability of bankruptcy before 1, 2 and 3 years using financial, the construction market and macroeconomic variables as input variables. Then, the impacts of the input variables that affected prediction accuracy in each model were identified by using Shapley value and compared among the three models. This study also investigated the prediction accuracy using variants of input variables grouped sequentially by high-impact ranking.

The results showed that the prediction accuracies were largely impacted by “housing starts” in all models. As the prediction period increased, the effects of macroeconomic variables on prediction accuracy increased, whereas the impact of “return on assets” on prediction accuracy decreased. It also found that the “current ratio” and “debt ratio” significantly influenced the prediction accuracies in all models. Also, the results revealed that similar prediction accuracies could be achieved using only 8, 10, and 10 variables out of a total of 18 variables for the 1-, 2-, and 3-year prediction models, respectively.

This study provides a Shapley value-based approach to identify how each input variable in a deep-learning bankruptcy prediction model. The findings of this study can not only assist in obtaining better insights into the underlying concept of bankruptcy but also use to select variables by removing those identified as less significant.

This paper aims to extend the Shapley value (SV) into a discussion of Zakat, a Pillar of Islam. Lloyd Shapley was awarded the Nobel Prize in Economics in 2012. This study shows that their relationship is significant for all nations, that of levelling up. An important but neglected paper by Datta (1939) showed insights provided by the Power Law, or as it is sometimes called, the Pareto distribution, into the role of Zakat in raising the income of all above the subsistence level. The Pareto distribution describes the prevailing tendency. The SV illustrates the interdependence perspective of Zakat with the Pareto distribution, wealth, income and poverty. Payoffs apply equally to both givers and receivers. For this study’s purposes, payoffs are considered as transferable utilities. They are formed by individuals who willingly cooperate in society rather than atomistic individuals who act independently. Zakat represents the recognition that society needs to be cooperative rather than individualistic; people cooperate in groups or societies to create value. SV implications and axioms are evaluated with an illustration.

This study extends Datta’s approach by introducing distribution weights into the SV. The authors set out the concept of weighted Shapley values that retain the elements of randomness and marginal contribution to a coalition contained in pure/true SVs and weights that follow a ley-Pareto distribution. This paper is a viewpoint work that relies primarily on the author’s qualitative interpretation.

The findings indicate that individual members of a coalition make multiple contributions that are often unrewarded. The contribution of one member of a coalition is dependent upon the contribution of others. The measure of contributions is payoffs, which have both monetary and non-monetary aspects; transferable payoffs or utilities are usually assumed. Furthermore, the significant agents in society or an organisation are stakeholders rather than the usual categories: managers, staff, shareholders, etc.

Contextualising these concepts within the Islamic values and principles that guide Zakat administration is crucial to ensure that the distribution of Zakat funds is fair, equitable and meets the needs of all eligible recipients. By applying these concepts appropriately, Zakat administrators can ensure that the Zakat system functions effectively and fulfils its religious obligation.

The novelty of this paper is that it blends the SV and the idea behind Zakat by introducing the idea of alternatives of Shapley weights. The link between the institution of Zakat and SV in terms of equality, poverty elimination and wealth distribution should be at the top of the research agenda.

This paper aims to solve the problem of public resource allocation among vulnerable groups by proposing a new method called uncertain α-coordination value based on uncertain cooperative game.

First, explicit forms of uncertain Shapley value with Chouqet integral form and uncertain centre-of-gravity of imputation-set (CIS) value are defined separately on the basis of uncertainty theory and cooperative game. Then, a convex combination of the two values above called the uncertain α-coordination value is used as the best solution. This study proves that the proposed methods meet the basic properties of cooperative game.

The uncertain α-coordination value is used to solve a public medical resource allocation problem in fuzzy coalitions and uncertain payoffs. Compared with other methods, the α-coordination value can solve such problem effectively because it balances the worries of vulnerable group’s further development and group fairness.

In this paper, an extension of classical cooperative game called uncertain cooperative game is proposed, in which players choose any level of participation in a game and relate uncertainty with the value of the game. A new function called uncertain α-Coordination value is proposed to allocate public resources amongst vulnerable groups in an uncertain environment, a topic that has not been explored yet. The definitions of uncertain Shapley value with Choquet integral form and uncertain CIS value are proposed separately to establish uncertain α-Coordination value.

We examine the influence of ownership structure on a blockholder’s power in a firm. We first describe the presence and ownership stakes of blockholders in a comprehensive sample of US firms. We develop a measure of the influence of the ownership structure on a blockholder’s power and show that an average blockholder loses 12% of her potential power due to the presence and size of the ownership stakes of other blockholders. Further, the influence of ownership structure varies systematically with a blockholder’s rank and identity, with the second and nonfamily manager blockholders experiencing the largest loss of power.

The purpose of this paper is to propose a model for a target searching problem in a two-dimensional region with time constraints. The proposed model facilitates the search operation by minimizing the mission time and fuel usage, and the search operation is performed by a set of agents divided into a number of groups.

The authors have applied optimization techniques, Cartesian product, inclusion–exclusion principle, cooperative strategy, Shapley value, fuzzy Shapley function and Choquet integral to model the problem.

The proposed technique optimizes the placement of base stations that minimizes the sortie length of the agents. The results show that the cooperative strategy outperforms the non-cooperative strategy. The Shapley values quantify the rewards of each group based on their contributions to the search operation, whereas the fuzzy Shapley values determine the rewards of each group based on their contributions and level of cooperation in the search operation.

The proposed model can be applied to model many real-time problems such as patrolling in international borders, urban areas, forests and managing rescue operations after natural calamities, etc. Therefore, defence organizations, police departments and other operation management sectors will be benefitted by applying the proposed approach.

To the best of the authors’ knowledge, determining the optimal locations of base stations in a region is not explored in the existing works on target searching problems with fuel constraints. The proposed approach to cooperatively search the targets in a region is new. Introducing the Shapley function and fuzzy Shapley function is a novel idea to quantify the rewards of each group based on their contributions and level of cooperation in the search operation. This paper addresses these unexplored areas.

This paper examines the profit distribution of engineering projects in the integrated project delivery (IPD) mode. IPD is a new delivery method that can ameliorate many of the disadvantages of traditional delivery methods and improve project results. In the implementation of IPD, the profit distribution is key for ensuring the success of IPD projects.

This paper described a new method for characterizing profit distribution in the IPD mode. The payment function and Shapley value of the cooperative fuzzy game of fuzzy alliance were defined by considering the Choquet integral of the fuzzy measure. The participation of each player was considered, and the influence of participation on the profit distribution was discussed. Lastly, changes in the profit distribution of core participants under different alliance combinations were studied.

A case from a report of The American Institute of Architects (AIA) was used to verify the fuzzy alliance model. There was a significant correlation between the degree of participation of the owner, architect and builder and the profit distribution among these three participants.

The theoretical research in this paper has some limitations. Initially, this paper selects a case with only three key participants in order to simplify the research. When there are many core participants, how to establish the alliance in the IPD mode and how to establish the corresponding profit distribution model, further work is certainly required to disentangle these complexities in models. Second, in this case, BIM technology has little impact on the income of the whole project. Therefore, this paper does not consider the impact of BIM technology on the marginal effect of the IPD project. Third, the contract type in the case is a custom tri-party based on IFOA. There is no classified discussion of the impact of different contracts on the profit distribute in the paper.

Based on the in-depth study of cooperative game with alliance structure, this paper promotes the classic cooperative game with alliance structure. The authors define the payoff function of fuzzy cooperative games by Choquet integral of fuzzy measure, and introduce the idea into the field of IPD. It aims at extending the solution to a cooperative game without a core. It can be obtained through a simple calculation. In the IPD alliance, the fuzziness and uncertainty of the participation degree of each participant will affect the profit of the whole project. The authors find that the higher the participation rate of players, the more profit each participant has. The greater the influence weight of the designer on the alliance, the lower the influence weight of the contractor on the alliance, the lower the participation of the contractor and the designer, and the lower the income distribution value of the three core participants. It shows a monotonous decline status.

For any construction enterprise, it can make more profits if it joins the grand alliance. In the IPD alliance, each participant can maximize their own interests, which can also promote the enthusiasm of construction enterprises to participate in the alliance and increase the application of IPD mode in AEC industry. This research method provides a new fast, effective, and more realistic solution method for cooperative countermeasures. It can be further extended to other cooperative game fields and advance a new research perspective and solution for the distribution of cooperative interests.

The contribution of this paper is the development of a fuzzy alliance model that provides a tool for measuring the profit distribution in IPD. This is the first quantitative model to connect the degree of participation with the profit distribution in IPD using fuzzy alliance.

This paper aims to determine the conditions for the core of the Triple Helix game to exist. The Triple Helix of university-industry-government relationships is a three-person cooperative game with transferable utility. Then, the core, the Shapley value and the nucleolus were used as indicators of the synergy within an innovation system. Whereas the Shapley value and the nucleolus always exist, the core may not.

The core of a three-person cooperative game with transferable utility exists only if and only if the game is convex. The paper applies the convexity condition to the Triple Helix game.

The Triple Helix game is convex if and only if there is output within the system; it is strictly convex if and only if all the three bilateral and the trilateral relationships have an output.

Convex games are competitive situations in which there are strong incentives towards the formation of large coalitions; therefore, innovation actors must cooperate to maximise their interests. Furthermore, a Triple Helix game may be split into subgames for comprehensive analyses and several Triple Helix games may be combined for a global study.

This paper extends the meaning of the Shapley value and the nucleolus for Triple Helix innovation actors: the Shapley value indicates the quantity a player wins because of the coalitions he involves in and the nucleolus the return for solidarity of an innovation actor.

Public-private partnership (PPP) projects for construction waste recycling have become the main approach to construction waste treatment in China. Risk sharing and income distribution of PPP projects play a vital role in achieving project success. This paper is aimed at building a practical and effective risk sharing and income distribution model to achieve win–win situation among different stakeholders, thereby providing a systematic framework for governments to promote construction waste recycling.

Stakeholders of construction waste recycling PPP projects were reclassified according to the stakeholder theory. Best-worst multi–criteria decision-making method and comprehensive fuzzy evaluation method (BWM–FCE) risk assessment model was constructed to optimize the risk assessment of core stakeholders in construction waste recycling PPP projects. Based on the proposed risk evaluation model for construction waste recycling PPP projects, the Shapley value income distribution model was modified in combination with capital investment, contribution and project participation to obtain a more equitable and reasonable income distribution system.

The income distribution model showed that PPP Project Companies gained more transaction benefits, which proved that PPP Project Companies played an important role in the actual operation of PPP projects. The policy change risk, investment and financing risk and income risk were the most important risks and key factors for project success. Therefore, it is of great significance to strengthen the management of PPP Project Companies, and in the process of PPP implementation, the government should focus on preventing the risk of policy changes, investment and financing risks and income risks.

The findings from this study have advanced the application methods of risk sharing and income distribution for PPP projects and further improved PPP project-related theories. It helps to promote and rationalize fairness in construction waste recycling PPP projects and to achieve mutual benefits and win–win situation in risk sharing. It has also provided a reference for resource management of construction waste and laid a solid foundation for long-term development of construction waste resources.

PPP mode is an effective tool for construction waste recycling. How to allocate risks and distribute benefits has become the most important issue of waste recycling PPP projects, and also the key to project success. The originality of this study resides in its provision of a holistic approach of risk allocation and benefit distribution on construction waste PPP projects in China as a developing country. Accordingly, this study adds its value by promoting resource development of construction waste, extending an innovative risk allocation and benefit distribution method in PPP projects, and providing a valuable reference for policymakers and private investors who are planning to invest in PPP projects in China.