Essays in the theory of allocation and voting

Abstract

This thesis consists of two essays on allocation theory and one on voting theory. The first chapter analyses preference domains (called priority domains) where every strategy proof, non-bossy and neutral allocation rule is a priority rule. It considers two versions of neutrality: unanimous profile neutrality or UPN neutrality where the neutrality axiom applies only to preference profiles where all agents have a common preference ordering and full neutrality or FN neutrality, where the neutrality axiom applies generally. We show that a very simple condition characterises priority domains under the UPN axiom. If these domains satisfy a mild richness condition, they must be the universal domain. The class of priority domains under the FN axiom is larger than those satisfying only UPN. We identify an FN-priority domain that is of order 1 n relative to the universal domain. The second chapter analyses preference domains in voting environments where every strategy proof random social choice functions satisfying unanimity is a random dictatorships. We call these random-dictatorial domains. Pramanik (2015) identifies a class of domains called P-domains which are dictatorial i.e. every deterministic strategy proof social choice functions on these domains satisfying unanimity, is dictatorial. The main result of this chapter is that P-domain is random-dictatorial. A consequence of this result is that circular domains (Sato (2010)) are also random-dictatorial. The minimum size of a random-dictatorial domain satisfying minimal richness is shown to be twice the number of alternatives. This is the same as the corresponding lower bound for dictatorial domains. Our result stands in contrast to those in Chatterji et al. (2014) who showed that linked domains are not random-dictatorial. Linked domains were shown to be dictatorial in Aswal et al. (2003). The third chapter attempts to provide a justification of the non-bossiness axiom which is pervasive in the allocation literature. It has been criticised by Thomson (2016) on the grounds that it cannot be defended by appealing to various strategic and normative criteria. We show that in some special cases, non-bossiness is a simplifying assumption that can be imposed without loss of generality by an expected welfare maximising planner in a symmetric environment. We consider the case of three objects and three agents with a planner whose goal is to maximise the expected sum of welfare with respect to a uniform prior. We show that for every strategy proof, neutral and efficient allocation rule, there exists a strategy-proof, neutral and non-bossy allocation rule which yields the same expected welfare. We conjecture that this is true for an arbitrary number of agents. For the general case, we are able to show an equivalence in terms of expected welfare for a special class of bossy allocation rules