## Doctoral Theses

### Essays on Random Social Choice Theory.

2-28-2020

2-28-2021

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Economic Research Unit (ERU-Kolkata)

#### Supervisor

Roy, Souvik (ERU-Kolkata; ISI)

#### Abstract (Summary of the Work)

This thesis comprises of six chapters related to random social choice theory. We provide a brief introduction of the chapters below.1.1 An Extreme Point Characterization of Strategy-proof and Unanimous Probabilistic Rules over Binary Restricted DomainsIn this chapter, we show 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.1.2 A Characterization of Random Min-max Domains and Its ApplicationsIn this chapter, we show 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.1.3 Formation of Committees through Random Voting RulesIn this chapter, we consider 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.1.4 A unified characterization of the randomized strategy-proof rulesIn this chapter, we show that 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.1.5 Restricted Probabilistic Fixed Ballot Rules and Hybrid DomainsIn this chapter, we study 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.

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:28843856

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