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1 Essays in Mechanism Design and Implementation Theory Dissertation Essays in Mechanism Design and Implementation Theory Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ritesh Jain Graduate Program in Economics The Ohio State University 2018 Dissertation Committee: Paul J Healy, Advisor Yaron Azrieli James Peck 1 Copyrighted by Ritesh Jain 2018 2 Abstract I broadly classify my research in micro-economic theory. This dissertation brings together my work in mechanism design and implementation theory. Mechanisms, (Jointly with Paul Healy) we study Nash implementation in public goods setting. Groves and Ledyard (1977) construct a mechanism for public goods procurement that can be viewed as a direct- revelation Groves mechanism in which agents announce a parameter of a quadratic approximation of their true preferences. The mechanism's Nash equilibrium outcomes are efficient. The budget is balanced because Groves mechanisms are balanced for the announced quadratic preferences. Tian (1996) subsequently discovered a richer set of budget-balancing preferences. We replicate the Groves-Ledyard construction using this expanded set of preferences, and uncover a new set of complex mechanisms that generalize the original Groves-Ledyard mechanism. In Chapter 2 titled Symmetric Mechanism Design, (Jointly with Yaron Azrieli) we study the extent to which regulators can guarantee fair outcomes by a policy requiring mechanisms to treat agents symmetrically. This is an exercise in mechanism design. Our main result is a characterization of the class of social choice functions that can be implemented under this constraint. In many environments, extremely discriminatory social choice functions can be implemented by symmetric mechanisms, but there are also cases ii in which symmetry is binding. Our characterization is based on a `revelation principle' type of result, where we show that a social choice function can be symmetrically implemented if and only if a particular kind of (indirect) symmetric mechanism implements it. We illustrate the result in environments of voting with private values, voting with a common value, and assignment of indivisible goods. In Chapter 3 titled Rationalizable Implementation of Social Choice Correspondences, I study the implementation of social choice correspondences (SCC), in a complete information setting, using rationalizability as the solution concept. I find a condition which I call r-monotonicity to be necessary for the rationalizable implementation of an SCC. r- monotonicity is strictly weaker than Maskin monotonicity, a condition introduced by Maskin (1999). If an SCC satisfies a no worst alternative condition and a condition which F- distinguishability, then it is shown that r-monotonicity is also sufficient for rationalizable implementation. We discuss the strength of these additional conditions. In particular, I find that, whenever there are more than 3 agents a social choice correspondence, which always selects at least two alternatives is rationalizably implementable if and only if it satisfies r-monotonicity. This paper, therefore, extends Bergemann et al. (2011) to the case of social choice correspondences. iii To my family iv Acknowledgments I am deeply indebted to Paul Healy and Yaron Azrieli for being inspirations for me in academia and beyond. They have been instrumental in helping conceptualize, write and rewrite the chapters in this dissertation. I thank James Peck for numerous conversations and comments which have helped shaped this dissertation. v Vita 2009................................................................B.A. in Economics, University of Delhi, India 2011................................................................M.A. in Economics, Jawaharlal Nehru University, New Delhi, India 2013................................................................M.A. in Economics, The Ohio State University, Columbus, OH 2013 to 2018 .................................................Graduate Associate, Department of Economics, The Ohio State University, Columbus, OH Fields of Study Major Field: Economics vi Table of Contents Abstract ............................................................................................................................... ii Dedication .......................................................................................................................... iv Acknowledgments............................................................................................................... v Vita ..................................................................................................................................... vi List of Tables ................................................................................................................... viii Chapter 1. Generalized Groves Ledyard Mechanisms ....................................................... 1 Chapter 2. Symmetric Mechanism Design ...................................................................... 20 Chapter 3. Rationalizable Implementation of Social Choice Correspondences .............. 50 References ......................................................................................................................... 78 Appendix for Chapter 1 .................................................................................................... 84 Appendix for Chapter 2 .................................................................................................... 91 Appendix for Chapter 3 .................................................................................................... 95 vii List of Tables Table 1 Preferences in Example 4 .................................................................................... 50 Table 2 Preferences in Example 5 .................................................................................... 58 Table 3 Preferences in Example 6 .................................................................................... 59 Table 4 Preferences in Example 10 .................................................................................. 75 Table 5 Preferences in Example 11 ................................................................................ 106 viii Chapter 1: Generalized Groves Ledyard Mechanism 1 Introduction Groves and Ledyard (1977) provided one of the first decentralized economic mechanisms to solve the free rider problem for general economies. Specifically, they devised a gov- ernment (or, mechanism) such that self-interested equilibrium behavior by all parties always leads to a Pareto optimal allocation. This work has been cited widely, with many researchers building off its original insights. What has not been appreciated, however, is the manner in which Groves and Ledyard (1977) actually construct their mechanism. The final mechanism looks simple: players announce a single number, and taxes are based on an proportional share of the cost and a quadratic penalty. In fact, the mechanism has a more complex foundation: Groves and Ledyard (1977) present it as being derived from a Groves mechanism (Groves, 1973) in which agents announce entire quasilinear utility functions and are taxed (or rewarded) based on the Marshallian surplus calculated from the announced preferences of all other agents in the economy. But agents may not actually have quasilinear preferences, so these announcements represents approximations of their true preferences. The space of allowable announcements is parameterized with a single parameter, which greatly simpli- fies agents' messages. In equilibrium, agents with non-quasilinear preferences announce quasilinear preferences that best approximate their true willingness to pay at the Pareto optimal allocation, and that allocation is selected by the mechanism.1 What is crucial to the Groves-cum-Groves-Ledyard construction is that the set of approximating preferences that the agents are allowed to announce always generate a budget-balanced allocation in the Groves mechanism. Budget balance does not always obtain with general quasilinear preferences; Groves and Loeb (1975) showed that balance is achieved if we further restrict agents to quadratic valuation functions. Following this 1Groves (1973) mechanisms are usually only discussed in settings where all agents actually have quasilinear preferences, and therefore have a dominant strategy of truthful revelation. Groves and Ledyard (1977) are novel in that they consider the Groves mechanism for more general preferences. 1 insight, Groves and Ledyard (1977) only allow agents to announce quadratic (approxi- mate) preferences, guaranteeing that the final allocation will balance the budget. Agents need only to announce the intercept of their (approximate) valuation function, and are charged according to the balanced Groves mechanism. Algebraic manipulation reveals that these outcome and tax functions reduce to the familiar Groves-Ledyard mechanism with its quadratic form. For two decades it was believed that quadratic preferences are the only ones for which a Groves mechanism could be budget balanced.2 This belief was overturned when Tian (1996) discovered a much richer class of preferences for which Groves mechanisms can be balanced. And, like the quadratic preferences, these are parameterized by a single parameter. In this paper we replicate the Groves-Ledyard procedure of turning a Groves mecha- nism into a one-dimensional, budget-balanced, Pareto efficient mechanism, but we do so on Tian's larger domain of preferences.
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