Bargaining is a process to decide how to divide shared resources between two or more players. And axiomatic bargaining specifies desirable and simple properties the outcome of the bargaining should satisfy and identifies the solution that produces this outcome. This approach was first developed by John Nash in his seminal work(Nash 1950). Since then, numerous studies have been done on bargaining problems with convex feasible set or with non-convex but comprehensive feasible set. There is, however, little work on finite bargaining problems. In this dissertation, we study finite bargaining problems systematically by extending the standard bargaining model to the one consisting of all finite bargaining problems. For our bargaining problems, we first propose the Nash, Maximin, Leximin, Maxiproportionalmin, Lexiproportianlmin solutions, which are the counterparts of those that have been studied extensively in both convex and non-convex but comprehensive problems. We then axiomatically characterize these solutions in our context. We next introduce two new solutions, the maximin-utilitarian solution and the utilitarian-maximin solution, each of which combines the maximin solution and utilitarian solution in different ways. The maximin-utilitarian solution selects the alternatives from the maximin solution that have the greatest sum of individuals’ utilities, and the utilitarian-maximin solution selects the maximin alternatives from the utilitarian solution. These two solutions attempt to combine two important but very different ethical principles to produce compromised solutions to bargaining problems. Finally, we discuss several variants of the egalitarian solution. The egalitarian solution in finite bargaining problems is more complicated than its counterpart in either convex or non-convex but comprehensive bargaining problems. Given its complexity in our context, we start our inquiry by investigating two-person, finite bargaining problems, and then extend some of the analysis to n-person, finite bargaining problems. Our analysis of finite bargaining problems and axiomatic characterizations of the extensions of various standard solutions of convex/non-convex but comprehensive bargaining problems to finite bargaining problems will shed new light on the behavior of these solutions. Our new solutions will expand our understanding of the bargaining theory and distributive justice from a different perspective.

Cette thèse traite de l’analyse théorique et l’application d’une nouvelle famille de jeux coopératifs, où la valeur de chaque coalition peut être calculée à partir des contributions des joueurs par un opérateur additif qui décrit comme les capacités individuelles interagissent au sein de groupes. Précisément, on introduit une grande classe de jeux, les Generalized Additive Games, qui embrasse plusieurs classes de jeux coopératifs dans la littérature, et en particulier de graph games, où un réseau décrit les restrictions des possibilités d’interaction entre les joueurs. Des propriétés et solutions pour cette classe de jeux sont étudiées, avec l’objectif de fournir des outils pour l’analyse de classes de jeux connues, ainsi que pour la construction de nouvelles classes de jeux avec des propriétés intéressantes d’un point de vue théorique. De plus, on introduit une classe de solutions pour les communication situations, où la formation d’un réseau est décrite par un mécanisme additif, et dans la dernière partie de cette thèse on présente des approches avec notre modèle à des problèmes réels modélisés par des graph games, dans les domaines de la théorie de l’argumentation et de la biomédecine. / This thesis deals with the theoretical analysis and the application of a new family of cooperative games, where the worth of each coalition can be computed from the contributions of single players via an additive operator describing how the individual abilities interact within groups. Specifically, we introduce a large class of games, namely the Generalized Additive Games, which encompasses several classes of cooperative games from the literature, and in particular of graph games, where a network describes the restriction of the interaction possibilities among players. Some properties and solutions of such class of games are studied, with the objective of providing useful tools for the analysis of known classes of games, as well as for the construction of new classes of games with interesting properties from a theoretic point of view. Moreover, we introduce a class of solution concepts for communication situations, where the formation of a network is described by means of an additive pattern, and in the last part of the thesis we present two approaches using our model to real-world problems described by graph games, in the fields of Argumentation Theory and Biomedicine.

Researchers conceptually recommend inquiry-based learning as a necessary means to alleviate the problems of learning but this study has embarked on practical implementation of inquiry-based facilitation and learning in Euclidean Geometry. Inquiry-based learning is student-centred. Therefore, the teaching or monitoring of inquiry-based learning in this study is referred to as inquiry-based facilitation. The null hypothesis discarded in this study explains that there is no difference between inquiry-based facilitation and traditional axiomatic approach in teaching Euclidean Geometry, that is, H0: μinquiry-based facilitation = μtraditional axiomatic approach. This study emphasises a pragmatist view that constructivism is fundamental to realism, that is, inductive inquiry supplements deductive inquiry in teaching and learning. Participants in this study comprise schools in Tshwane North district that served as experimental group and Tshwane West district schools classified as comparison group. The two districts are in the Gauteng Province of South Africa. The total number of students who participated is 166, that is, 97 students in the experimental group and 69 students in the comparison group. Convenient sampling applied and three experimental and three comparison group schools were sampled. Embedded mixed-method methodology was employed. Quantitative and qualitative methodologies are integrated in collecting data; analysis and interpretation of data. Inquiry-based-facilitation occurred in experimental group when the facilitator probed asking students to research, weigh evidence, explore, share discoveries, allow students to display authentic knowledge and skills and guiding students to apply knowledge and skills to solve problems for the classroom and for the world out of the classroom. In response to inquiry-based facilitation, students engaged in cooperative learning, exploration, self-centred and self-regulated learning in order to acquire knowledge and skills. In the comparison group, teaching progressed as usual. Quantitative data revealed that on average, participant that received intervention through inquiry-based facilitation acquired inquiry-based learning skills and improved (M= -7.773, SE= 0.7146) than those who did not receive intervention (M= -0.221, SE = 0.4429). This difference (-7.547), 95% CI (-8.08, 5.69), was significant at t (10.88), p = 0.0001, p<0.05 and represented a large effect size of 0.55. The large effect size emphasises that inquiry-based facilitation contributed significantly towards improvement in inquiry-based learning and that the framework contributed by this study can be considered as a framework of inquiry-based facilitation in Euclidean Geometry. This study has shown that the traditional axiomatic approach promotes rote learning; passive, deductive and algorithmic learning that obstructs application of knowledge in problem-solving. Therefore, this study asserts that the application of Inquiry-based facilitation to implement inquiry-based learning promotes deeper, authentic, non-algorithmic, self-regulated learning that enhances problem-solving skills in Euclidean Geometry. / Mathematics Education / Ph. D. (Mathematics, Science and Technology Education)