Independence in the ordinal model and on closed set systems
Consensus theory has traditionally dealt with consensus functions acting on social choices. In Kenneth Arrow's original work, he assumed each voter in a population has a list of preferences, and a consensus function chooses a list of preferences that best reflects the "will" of the voters. He showed that some simple assumptions about the consensus function lead to a contradiction. One of these is that the consensus function decides the relative position of any subset of choices based on the voters' rankings of those choices, and not on the rankings of any other choices. Arrow called this assumption independence of irrelevant alternatives since the consensus function is supposed to ignore the irrelevant alternatives. Arrow proved that the only consensus function for weak orders that is independent of irrelevant alternatives and Pareto is a dictatorship.
Other authors realized that similar results hold for structures other than weak orders, such as classes of graphs and set systems. These other theorems all had notions of independence of irrelevant alternatives. Crown, Janowitz, and Powers (10) tried to provide a mathematical model that would integrate all Arrow type theorems into one common theory. Their characterization of neutral consensus functions was quite elegant and complete, but did not truly generalize Arrow's original result, nor the work done with other structures. Neutrality is a stronger condition than independence and leads to a stronger result.
A definition of independence that fits the ordinal model described in Crown et al. (10) will be presented. A generalized version of Arrow's theorem for the ordinal model will be proven. Subsequent chapters will show how this theorem applies to weak orders, n-trees, phylogenetic trees, and generalized weak hierarchies, with this latter application a new domain for Arrow's theorem.
Other possible definitions for independence on closed set systems will be presented and compared. These definitions were first presented in Barthelemy, McMorris, and Powers (6), but only in the context of n-trees. It will be shown that the relationships among the different independence conditions are true for much more general classes of closed set systems.
Finally, permutations on the closed set systems will be discussed. A relationship between different permutation independence conditions will be shown.