The Nucleolus, the Kernel, and the Bargaining Set: An Update

One of David Schmeidler’s many important contributions in his distinguished career was the introduction of the nucleolus, one of the central single-valued solution concepts in cooperative game theory. This paper is an updated survey on the nucleolus and its two related supersolutions, i.e., the kernel and the bargaining set. As a first approach to these concepts, we refer the reader to the great survey by Maschler (1992); see also the relevant chapters in Peleg and Sudholter (2003). Building on the notes of four lectures on the nucleolus and the kernel delivered by one of the authors at the Hebrew University of Jerusalem in 1999, we have updated Maschler’s survey by adding more recent contributions to the literature. Following a similar structure, we have also added a new section that covers the bargaining set. The nucleolus has a number of desirable properties, including nonemptiness, uniqueness, core selection, and consistency. The first way to understand it is based on an egalitarian principle among coalitions. However, by going over the axioms that characterize it, what comes across as important is its connection with coalitional stability, as formalized in the notion of the core. Indeed, if one likes a single-valued version of core stability that always yields a prediction, one should consider the nucleolus as a recommendation. The kernel, which contains the nucleolus, is based on the idea of “bilateral equilibrium” for every pair of players. And the bargaining set, which contains the kernel, checks for the credibility of objections coming from coalitions. In this paper, section 2 presents preliminaries, section 3 is devoted to the nucleolus, section 4 to the kernel, and section 5 to the bargaining set.Iñarra acknowledges research support from the Spanish Government grant ECO2015-67519-P, and Shimomura from Grant-in-Aid for Scientific Research (A)18H03641 and (C)19K01558

Contributions to Game Theory and Management. Vol. III. Collected papers presented on the Third International Conference Game Theory and Management.

The collection contains papers accepted for the Third International Conference Game Theory and Management (June 24-26, 2009, St. Petersburg University, St. Petersburg, Russia). The presented papers belong to the field of game theory and its applications to management. The volume may be recommended for researches and post-graduate students of management, economic and applied mathematics departments.

Pattern Clustering using Cooperative Game Theory

In this paper, we approach the classical problem of clustering using solution concepts from cooperative game theory such as Nucleolus and Shapley value. We formulate the problem of clustering as a characteristic form game and develop a novel algorithm DRAC (Density-Restricted Agglomerative Clustering) for clustering. With extensive experimentation on standard data sets, we compare the performance of DRAC with that of well known algorithms. We show an interesting result that four prominent solution concepts, Nucleolus, Shapley value, Gately point and \tau-value coincide for the defined characteristic form game. This vindicates the choice of the characteristic function of the clustering game and also provides strong intuitive foundation for our approach.Comment: 6 pages, 6 figures, published in Proceedings of Centenary Conference - Department of Electrical Engineering, Indian Institute of Science : 653-658, 201

Modelling human fairness in cooperative games : a goal programming approach

The issues of rationality in human behavior and fairness in cooperation have gained interest in various economic studies. In many prescriptive models of games, rationality of human decision makers implicitly assumes exchange-ability. This means that real people are assumed to adopt the beliefs of a player as expressed in the game when placed in the shoes of that particular player. However, it is a well debated topic in the literature that this modeling assumption is not in accordance to what behavioral economists have observed in some games played with real human subjects. Even when assuming the role of the same player in the game, different people think differently about the fairness of a particular outcome. People also view fairness as an essential ingredient of their decision making processes in games on cooperation. The aim of this research is to develop a new modeling approach to decision making in games on cooperation in which fairness is an important consideration. The satisficing and egilitarian philosophies on which weighted and Chebyshev Goal Programming (GP) rely, seem to offer an adequate and natural way for modeling human decision processes in at least the single-shot games of coordination that are investigated in this work. The solutions returned by the proposed GP approach aim to strike the right balance on several dimensions of con icting goals that are set by players themselves and that arise in the mental models these players have of other relevant players.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Optimization and Mechanism Design for Ridesharing Services

Ridesharing services, whose aim is to gather travelers with similar itineraries and compatible schedules, are able to provide substantial environmental and social benefits through reducing the use of private vehicles. When the operations of a ridesharing system is optimized, it can also save travelers a significant amount of transportation cost. The economic benefits associated with ridesharing in turn attract more travelers to participate in ridesharing services and thereby improve the utilization of transportation infrastructure capacity. This study addresses two of the most challenging issues in designing an efficient and sustainable ridesharing service: ridesharing optimization and ridesharing market design. The first part of the dissertation formally defines the large-scale ridesharing optimization problem, characterizes its complexity and discusses its relation to classic relevant problems like the traveling salesman problem (TSP) and the vehicle routing problem (VRP). A mixed-integer program (MIP) model is developed to solve the ridesharing optimization problem. Since the ridesharing optimization problem is NP-hard, the MIP model is not able to solve larger instances within a reasonable time. An insertion-based heuristic is developed to get approximate solutions to the ridesharing optimization problem. Experiments showed that ridesharing can significantly reduce the system-wide travel cost and vehicle trips. Evaluation of the heuristic solution method showed that the heuristic approach can solve the problem very fast and provide nearly-optimal (98%) solutions, thus, confirming its efficiency and accuracy. From a societal perspective, the ridesharing optimization model proposed in this dissertation provided substantial system-wide travel cost saving (25%+) and vehicle-trip saving (50%) compared to non-ridesharing situation. However, the system-level optimal solution might not completely align with individual participant interest. The second part of this dissertation formulates this issue as a fair cost allocation problem through the lens of the cooperative game theory. A special property of the cooperative ridesharing game is that, its characteristic function values are calculated by solving an optimization problem. We characterize the game to be monotone and subadditive, but non-convex. Several concepts of fairness are investigated and special attention is paid to a solution concept named nucleolus, which aims to minimize the maximum dissatisfaction in the system. However, finding the nucleolus is very challenging because it requires solving the ridesharing optimization problem for every possible coalition, whose number grows exponentially as the number of participants increases in the system. We break the cost allocation (nucleolus finding) problem into a master-subproblem structure and two subproblems are developed to generate constraints for the master problem. We propose a coalition generation procedure to find the nucleolus and approximate nucleolus of the game. When the game has a non-empty core, in the approximate nucleolus scheme the coalitions are computed only when it is necessary, and the approximate nucleolus scheme produces the actual nucleolus. Experimental results showed that, when the game has an empty core, the approximate nucleolus is close to the actual nucleolus. Results also showed that, regardless of the emptiness of the game, by using our algorithm, only a small fraction (1:6%) of the total coalition constraints were generated to compute the approximate nucleolus, and the approximate nucleolus is close to the actual nucleolus