#### Date of Submission

6-28-2014

#### Date of Award

6-28-2015

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Quantitative Economics

#### Department

Economics and Planning Unit (EPU-Delhi)

#### Supervisor

Sen, Arunava (EPU-Delhi; ISI)

#### Abstract (Summary of the Work)

This thesis comprises three chapters that investigate the structure of dominant strategy incentive compatible (strategy-proof) social choice functions on restricted domains. The first chapter deals with standard mechanism design problem in quasi-linear environments with multidimensional non-convex type spaces. It identifies a class of type spaces called ordinal type spaces and shows that the simple and familiar 2-cycle monotonicity condition is necessary and sufficient for implementation. This result covers the single-peaked domain. The second chapter deals with the standard social choice problem (no monetary transfers) but one where agents have indifference in their preference orderings generated via an exogenous partition of the set of alternatives. A domain is dictatorial if a strategy-proof and unanimous social choice function defined on this domain is dictatorial. The chapter explores the relationship between agent partitions and dictatorial domains. The final chapter considers the standard social choice model with linear preferences. It provides new results on dictatorial domains in this model.We provide a brief description of each chapter below. 1.1 Multidimensional Mechanism Design with Ordinal RestrictionsIn this chapter, we study multidimensional mechanism design in private values and quasilinear utility environments when the set of alternatives is finite and the allocation rule is deterministic. A standard goal in mechanism design is to investigate conditions that are necessary and sufficient for implementing an allocation rule. An allocation rule in such an environment is implementable if there exists a payment rule such that truth-telling is a dominant strategy for the agents in the resulting mechanism. Our main result is that in a large class of multidimensional type spaces that satisfy some ordinal restrictions, implementabil1 ity is equivalent to a simple condition called 2-cycle monotonicity. By virtue of revenue equivalence, which holds in these type spaces, we are able to characterize the entire class of dominant strategy incentive compatible mechanisms. The 2-cycle monotonicity condition requires the following: given the types of other agents, if the alternative chosen by the allocation rule is a when agent i reports its type to be t and the alternative chosen by the allocation rule is b when agent i reports its type to be s, then it must be thatt(a) âˆ’ t(b) â‰¥ s(a) âˆ’ s(b),where for any alternative x, t(x) and s(x) denote the values of alternative x in types t and s respectively showed that a significantly stronger condition called cycle monotonicity is necessary and sufficient for implementability in any type space - see also Rockafellar (1970). Myerson (1981) formally establishes that in the single object auction set up, where the type is single dimensional, 2-cycle monotonicity is necessary and sufficient for implementation. When the type space is multidimensional, if the set of alternatives is finite and the type space is convex, 2-cycle monotonicity implies cycle monotonicity (Bikhchandani et al., 2006; Saks and Yu, 2005; Ashlagi et al., 2010). Though convexity is a natural geometric property satisfied in many economic environments, it excludes many interesting type spaces. A primary objective of this chapter is to formulate restrictions on type spaces without the convexity assumption made in the literature and answer the question of implementability in such multidimensional type spaces. Indeed, our restrictions allow many interesting multidimensional non-convex type spaces. Prominent type spaces covered by our formulation are the single peaked type spaces and its generalizations. We show that 2-cycle monotonicity is necessary and sufficient for implementability in all these type spaces.

#### Control Number

ISILib-TH389

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Pramanik, Anup Dr., "Essays on Incentive Compatibility on Restricted Domains." (2015). *Doctoral Theses*. 267.

https://digitalcommons.isical.ac.in/doctoral-theses/267

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843047