complaint, compromise and solution concepts for cooperative games
Panfei Sun is a PhD student in the research group Discrete Mathematics and Mathematical Programming. His supervisors are prof.dr. M.J. Uetz from the Faculty of Electrical Engineering, Mathematics and Computer Science and prof.dr. H. Sun from the Northwestern Polytechnical University, Xian, China.
This thesis mainly focuses on solution concepts for cooperative games. We try to investigate the solution concepts concerning the complaints of players. Various principles are taken into account in order to define the complaints for either individual players or coalitions. Then based on different optimization criteria, values are defined as the solutions to corresponding optimization models. Alternatively, we deal with the complaints in the view of compromise motivated by the definition of the tau value. In such a way, we define a more general compromise value, which has a very close relation to many well-known compromise values.
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, which is in some sense a compromise among all players when considering all possible permutations. Followed by the definition of the (pre-)nucleolus and the family of least square values, we then reveal that the optimal solutions for both models coincide with the ENSC value either based on the lexicographic criterion or the least square criterion.
Furthermore, we propose the so called alpha-ENSC value by considering the egoism of players. First of all, the coefficient alpha is introduced to measure the degree of egoism of players. Then similar to the procedural method, we implement the alpha-ENSC value by means of optimization and also the satisfier of a set of properties. Notice that, we can also implement the corresponding (alpha-)CIS value, due to the fact that the ENSC value and the CIS value are dual to each other. Following the similar idea, we propose another 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. For instance, we give the necessary and sufficient condition ensuring that the new value belongs to the core. We also provide sufficient conditions when the general compromise value coincides with the tau value and the chi value respectively.
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. For further research, there is still a long way to go in this area. The first concern is how to axiomatize these solutions, where the main difficulty is the stochastic property of the payoffs. Another question is that whether the results of this chapter are valid for a more general class of allocations. We could also investigate the relationship between the proposed solution concepts and existing solution concepts, for instance, various core concepts for stochastic cooperative games.
