Ordinal Power Indices

The design of procedures aimed at ranking individuals according to how they behave in various groups is of great importance in many practical situations. The problem occurs in a variety of scenarios coming from social choice theory,cooperative game theory or multi-attribute decision theory, and examples include: comparing researchers in a scientificdepartment by taking into account their impact across different teams; finding the most influential political parties in aparliament based on past alliances within alternative majority coalitions; rating attributes according to their influence ina multi-attribute decision context, where independence of attributes is not verified because of mutual interactions. However, in many real world applications, a precise evaluation on the coalitions’ “power” may be hard for many reasons (e.g., uncertain data, complexity of the analysis, missing information or difficulties in the update, etc.). In this case, it may be interesting to consider only ordinal information concerning binary comparisons between coalitions. The main objectiveof this thesis is to study the problem of finding an ordinal ranking over the set N of individuals (called social ranking),given an ordinal ranking over its power set (called power relation). In order to do that, during the thesis we use notionsin classical voting theory and cooperative game theory. Mainly, we have defined solution concepts named ceteris paribusmajority rule, and ordinal Banzhad index, which are respectively inspired from classical voting theory and cooperativegame theory. Since the majority of our work in the thesis is to study solutions from property-driven approach, we axiomatically study the solutions by reformulating axioms in classical voting theory. Finally, exploring weighted extensionsof the ceteris paribus majority rule to rank more than two individuals result in an axiomatic study of families of weightedsolutions.