Logics of preference when there is no best

Well-behaved preferences (e.g., total pre-orders) are a cornerstone of several areas in artificial intelligence, from knowledge representation, where preferences typically encode likelihood comparisons, to both game and decision theories, where preferences typically encode utility comparisons. Yet weaker (e.g., cyclical) structures of comparison have proven important in a number of areas, from argumentation theory to tournaments and social choice theory. In this paper we provide logical foundations for reasoning about this type of preference structures where no obvious best elements may exist. Concretely, we compare and axiomatize a number of ways in which the concepts of maximality and optimality can be lifted to this general class of preferences. In doing so we expand the scope of the long-standing tradition of the logical analysis of preference.