The complexity of the nucleolus in compact games

This is the author accepted manuscript. The final version is available from ACM via the DOI in this recordThe nucleolus is a well-known solution concept for coalitional games to fairly distribute the total available worth among the players. The nucleolus is known to be NP-hard to compute over compact coalitional games, that is, over games whose functions specifying the worth associated with each coalition are encoded in terms of polynomially computable functions over combinatorial structures. In particular, hardness results have been exhibited over minimum spanning tree games, threshold games, and flow games. However, due to its intricate definition involving reasoning over exponentially many coalitions, a nontrivial upper bound on its complexity was missing in the literature and looked for. This article faces this question and precisely characterizes the complexity of the nucleolus, by exhibiting an upper bound that holds on any class of compact games, and by showing that this bound is tight even on the (structurally simple) class of graph games. The upper bound is established by proposing a variant of the standard linear-programming based algorithm for nucleolus computation and by studying a framework for reasoning about succinctly specified linear programs, which are contributions of interest in their own. The hardness result is based on an elaborate combinatorial reduction, which is conceptually relevant for it provides a "measure" of the computational cost to be paid for guaranteeing voluntary participation to the distribution process. In fact, the pre-nucleolus is known to be efficiently computable over graph games, with this solution concept being defined as the nucleolus but without guaranteeing that each player is granted with it at least the worth she can get alone, that is, without collaborating with the other players. Finally, this article identifies relevant tractable classes of coalitional games, based on the notion of type of a player. Indeed, in most applications where many players are involved, it is often the case that such players do belong in fact to a limited number of classes, which is known in advance and may be exploited for computing the nucleolus in a fast way.Part of E. Malizia’s work was supported by the European Commission through the European Social Fund and by Calabria Regio

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

Complaint, compromise and solution concepts for cooperative games

This thesis mainly focuses on solution concepts for cooperative games. We investigate the solution concepts concerning the complaints of players. Motivated by the work the procedural values, we study the formation of the grand coalition and define a new kind of complaint for individual players. We then reveal that the solutions for both models coincide with the ENSC value either based on the lexicographic criterion or the least square criterion. We propose the so called alpha-ENSC value by considering the egoism of players. We implement the alpha-ENSC value by means of optimization and also the satisfier of a set of properties. Following the similar idea, we propose two kinds of complaints for coalitions and define the optimal compromise values based on the lexicographic criterion. It turns out that the optimal compromise values coincides with the ENSC value and the CIS value under corresponding complaint. We show an application of the previous mentioned method. We introduce and axiomatize a class of cost sharing methods for polluted river sharing systems that consists of the convex combinations of the known Local Responsibility Sharing (LR) method and the Upstream Equal Sharing (UES) method. We also deals with the solution concepts based on the compromise between the ideal and minimal payoffs for players, which is inspired by the definition of the tau value but in a more general way. We reveal the relations between the general compromise value with several well known solution concepts. Furthermore, we investigate the solution concepts for cooperative games with stochastic payoffs. We focus on a subset of all allocations and introduce the stochastic complaint for players. Under the least square criterion, the most stable solutions and the fairest solutions are proposed. Moreover, the optimal solution stays the same whether the optimization model depends on the coalitions or individual players