#### Date of Submission

Winter 2-28-2020

#### Date of Award

Winter 2-28-2021

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Quantitative Economics

#### Department

Economic Research Unit (ERU-Kolkata)

#### Supervisor

Roy, Souvik (ERU-Kolkata; ISI)

#### Abstract (Summary of the Work)

The thesis covers an extreme point characterization of strategy-proof and unanimous probabilistic rules over binary restricted domains. It shows that every strategy-proof and unanimous probabilistic rule on a binary restricted domain has binary support, and is a probabilistic mixture of strategy-proof and unanimous deterministic rules. Examples of binary restricted domains are single-dipped domains, which are of interest when considering the location of public bads. We also provide an extension to infinitely many alternatives. It also discusses that a random rule on a top-connected single-peaked domain is unanimous and strategy-proof if and only if it is a random min-max rule. As a by-product of this result, it follows that a top-connected single-peaked domain is tops-only for random rules. We further provide a characterization of the random min-max domains.This thesis considered the problem of choosing a committee from a set of finite candidates based on the preferences of the agents in a society. The preference of an agent over a candidate is binary in the sense that either she wants the candidate to be included in a(ny) committee or she does not - she is never indifferent. A collection of preferences of an agent, one for each candidate, is extended to a preference over all subsets of candidates (i.e., potential committees) in a separable manner. Separability means if an agents wants a particular candidate to be in some committee, then she wants her to be in every committee. Further attention has been made on a large class of restricted domains such as single-peaked, single-crossing, single-dipped, tree-single-peaked with top-set along a path, Euclidean, multi-peaked, intermediate ([58]), etc., can be characterized by using betweenness property, and we present a unified characterization of unanimous and strategy-proof random rules on these domains. We do separate analysis for both the cases where the number of alternatives is finite or infinite. As corollaries of our result, we show that the domains we consider in this paper satisfy tops-onlyness and deterministic extreme point property.It also studied Random Social Choice Functions (or RSCFs) in a standard ordinal mechanism design model. We introduce a new preference domain called a hybrid domain which includes as special cases as the complete domain and the single-peaked domain. We characterize the class of unanimous and strategy-proof RSCFs on these domains and refer to them as Restricted Probabilistic Fixed Ballot Rules (or RPFBRs). These RSCFs are not necessarily decomposable, i.e., cannot be written as a convex combination of their deterministic counterparts. We identify a necessary and sufficient condition under which decomposability holds for anonymous RPFBRs. Finally, we provide an axiomatic justification of hybrid domains and show that every connected domain satisfying some mild conditions is a hybrid domain where the RPFBR characterization still prevails. Finally, it highlighted the problem where finitely many agents have preferences on a finite set of alternatives, single-peaked with respect to a connected graph with these alternatives as vertices. A 2 probabilistic rule assigns to each preference profile a probability distribution over the alternatives. First, all unanimous and strategy-proof probabilistic rules are characterized when the graph is a tree. These rules are uniquely determined by their outcomes at those preference profiles where all peaks are on leafs of the tree, and thus extend the known case of a line graph. Second, it is shown that every unanimous and strategy-proof probabilistic rule is random dictatorial if and only if the graph has no leafs. Finally, the two results are combined to obtain a general characterization for every connected graph by using its block tree representation.

#### Control Number

ISI-Lib-TH502

#### Creative Commons License

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

#### DOI

http://hdl.handle.net/10263/7111

#### Recommended Citation

Sadhukhan, Soumyarup Dr., "Essays on Random Social Choice Theory" (2021). *Doctoral Theses*. 1.

https://digitalcommons.isical.ac.in/doctoral-theses/1

## Comments

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Total pages: 197