#### Date of Submission

7-28-2015

#### Date of Award

7-28-2016

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Economics and Planning Unit (EPU-Delhi)

#### Supervisor

Sen, Arunava (EPU-Delhi; ISI)

#### Abstract (Summary of the Work)

In this thesis we consider situations where heterogenous objects are to be distributed among a set of claimants. The objects in question are indivisible, so they cannot be split or shared. There is no money in this economy, so objects cannot be simply bought and sold. Allocations to claimants must be based on their preferences over objects alone. We impose the additional restriction that each object has an exact capacity constraint, such that each object is assigned either to a pre-specified (and fixed) number of agents, or it is not assigned at all. Further, we assume that the capacity constraint is the same for all objects. We call such a situation an allocation problem. In Chapter 2 we consider the case where the exact capacity constraint is of size 2. Thus agents must be divided into pairs. Our quest is to find efficient rules in the framework. We propose a rule which we call the Partner Trading (PT) Rule, and show that it characterises the set of all rules in this model that satisfy the standard properties of strategy-proofness, limited influence, unanimity and neutrality. It is also group-strategy-proof and Pareto efficient. The PT rule can be thought of as a generalisation of the famous top trading cycles procedure to this particular environment. In Chapter 3 we consider fair rules in this framework. We extend the exact capacity constraints to be of any size. We demonstrate that the well-known incompatibility between fairness and Pareto efficiency persists in this model too. We propose a rule which we call the Deferred Acceptance with Improvements (DAI) rule, which is fair and constrained efficient. We also identify a Pareto improvement procedure that always leads us to a fair and constrained efficient allocation in one iteration. We show, however, that the DAI rule is not strategy-proof. In Chapter 4 we study group-strategy-proofness, which is the extension of strategy-proofness to groups of agents. This property comes in a standard form and a weak form. The distinction between the two forms is non-trivial as important rules in the literature fail the standard form but satisfy the weak form. It is well-known that in allocation models such as ours, a strategy-proof rule that is also non-bossy is (standard) group-strategy-proof. But the link between strategy-proofness and weak group-strategy-proofness is not as well established. We make steps towards this in this paper. We identify conditions (which we call ultra-weak Maskin monotonicity and weak nonbossiness) that are sufficient to ensure that a strategy-proof rule is weakly group-strategy-proof. These conditions are natural weaker forms of commonly used axioms in the literature. We also demonstrate that the conditions are â€˜weak enoughâ€™, in that a rule satisfying them may not be (standard) group-strategy-proof.

#### Control Number

ISILib-TH

#### 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

Raghavan, T. C. A. Madhav Dr., "Essays on Efficiency, Fairness and Strategy-Proofness in Allocation Problems with Exact Capacity Constraints." (2016). *Doctoral Theses*. 355.

https://digitalcommons.isical.ac.in/doctoral-theses/355

## 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:28843466