Fairness Considerations in Cooperative Games

This thesis motivates and introduces a way to model fairness considerations in cooperative game theory. Fairness can not be hoped to be modeled as a fixed concept, independent of personas or cultures. Even if one restricts attention to a rather narrow field, like the political debate in Germany, the word fairness carries a variety of meanings ranging from equal merits for equal achievement or meritocracy to equal distribution of chances or even, sometimes, distribution according to needs. Therefore the thesis does not introduce a definition of fairness, but rather shows how to model fairness concepts as predicates on the imputation space. For each game, each fairness concept will label imputations as either fair, or unfair. A range of traditional and new fairness concepts are developed within this setting, and studied. As a benchmark for the feasibility of a fairness concept, the concept of satisfiability within the core is introduced. The rationale behind this is that it would be very disadvantageous if all players had the same concept of fairness and this fairness culture would still prevent stable cooperation in some situations where the core is non-empty. The thesis consists of three parts. The first part starts out, of course, with an introduction of important concepts and examples of cooperative game theory. While the most relevant traditional examples are included, a number of examples are proposed for the first time. Then a model of fairness and a rationality-of-fairness argument based on satisfiability within the core are developed. The second part deals with more specific classes of games. A certain emphasis is put on the case of bankruptcy games, but some results are also developed for convex games, the positive cone of unanimity games and for 1-convex bankruptcy games. For convex games a conjecture is given as an open problem. The third part addresses the fact that dierent players might have dierent cultures of fairness – perceiving dierent imputations as fair, or unfair. An adapted form of the core inequalities gives the modified stability condition resulting in this setting. An example shows, how cooperation can become stable, interestingly enough, when certain players are culturally incompatible. A first strategy-proof-reporting result is also given on the reporting of individual fairness notions in a somewhat restricted setting. To a higher degree than in the other parts, the results of part three invite new questions and point towards open problems and opportunities for future work in this field