Essays on implementability and monotonicity
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In this thesis we study the implementation problem with regard to the relation between monotonicity and implementability. Recent work in the field has shown that the implementability of a social choice rule strongly depends upon the compatibility between the monotonicity structures of the social choice rule and of the solution concept according to which implementation takes place. Different degrees of monotonicity of the social choice rules and game theoretic solution concepts can be determined via a generalized monotonicity function, strongest of which is called self-monotonicity. In this study, we determine the unique self-monotonicity of the Nash equilibrium concept and show that the monotonicities of a social choice rule are inherited from the unique selfmonotonicity of the Nash equilibrium concept via the mechanisms that implement it. In particular, we show that the essential monotonicity is inherited via the Maskin-Vind type mechanism which is widely used in the characterization results. We also give a new characterization of strong Nash implementable social choice rules via critical profiles. We show that coalitional monotonicity when conjoined with three more conditions is both necessary and sufficient for implementability. Finally we determine a subset of subgame perfect Nash implementable social choice rules that satisfies conditions defined obtained by critical profiles. The results that are obtained in this thesis strongly support the view that implementation theory can be rewritten in terms of monotonicity and that this provides a better understanding of the theory.