## Date of Submission

5-22-2018

## Date of Award

5-22-2019

## 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)

In Chapter 2, we consider domains of admissible preferences with a natural property called top-circularity. Several domains with practical applications such as multidimensional single-peaked domain in [9], union of a single-peaked and a single dipped domain, etc. satisfy top-circularity. We show that if such a domain satisfies either the maximal conflict property or the weak conflict property, then it is dictatorial. We show that this result can be applied to the problem of locating a public facility where the planner does not know whether agents derive positive or negative externality from the facility. The union of a single-peaked and a single-dipped domain captures such situations and such domains are top-circular satisfying the maximal conflict property. It follows from our results that such domains are dictatorial. Further, we obtain the result in [74] as a corollary. In Chapter 3, we consider social choice problems where the set of alternatives can be ordered over a real line and the admissible set of preferences of each agent is single-peaked. A preference is called single-peaked if the preference falls as one moves away from its top-ranked alternative. First, we show that if all the agents have the same admissible set of single-peaked preferences, then every unanimous and strategy-proof social choice function is tops-only. A social choice function is called tops-only if it is insensitive to changes in agentsâ€™ preferences below the top-ranked alternative. Next, we consider situations where different agents have different admissible sets of single-peaked preferences. We show by means of an example that unanimous and strategy-proof social choice functions need not be tops-only int his situation, and consequently provide a sufficient condition on the admissible sets of preferences of the agents so that unanimity and strategy-proofness guarantee tops-onlyness. Finally, we characterize all domains on which (i) every unanimous and strategy-proof social choice function is a min-max rule ([54]), and (ii) every min-max rule is strategy-proof. As an application of our result, we obtain a characterization of the unanimous and strategy-proof social choice functions on maximal single-peaked domains ([54], [86]), minimally rich single-peaked domains ([61]), maximal regular single-crossing domains ([72], [73]), and distance based single-peaked domains. In Chapter 4, we consider domains that exhibit single-peakedness only over a subset of alternatives. We call such domains partially single-peaked domains and provide a characterization of the unanimous and strategy-proof social choice functions on these domains. As an application of this result, we obtain a characterization of the unanimous and strategy-proof social choice functions on multi-peaked domains ([80], [78], [37]), single-peaked domains with respect to a partial order ([18]), multiple single-peaked domains ([67]) and single-peaked domains on graphs ([76]). As a by-product of our results, it follows that strategy-proofness implies tops-onlyness on these domains. Moreover, we show that strategy-proofness and group strategy-proofness are equivalent on these domains. In Chapter 5, we consider a social choice setting where the set of alternatives can be ordered over a real line. In Chapter 3, we have characterized domains where the set of unanimous and strategy-proof rules coincide with the set of min-max rules. Min-max rules satisfy an interesting property called uncompromisingness ([17]). A social choice function is uncompromising if no agent can influence the outcome by taking extreme positions. It follows from our result in Chapter 3 that a domain is not top-connected single-peaked then unanimous and strategy-proof rules may violate uncompromisingness. In this chapter, we consider arbitrary single-peaked domains (not necessarily top-connected) and provide a general characterization of the unanimous and strategy-proof social choice functions on those domains. We show that every unanimous and strategy-proof social choice function defined on such domains satisfy a property called weak uncompromisingness. Weak uncompromisingness implies that whenever an agentâ€™s top-ranked alternative moves closer to the outcome, the outcome does not change. Moreover, if an agent moves his top-ranked alternative away from the outcome, the outcome can change only in a restricted way.

## Control Number

ISILib-TH457

## 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

Achuthankutty, Gopakumar Dr., "Domain Restrictions in Strategy-Proof Social Choice." (2019). *Doctoral Theses*. 427.

https://digitalcommons.isical.ac.in/doctoral-theses/427

## 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:28843847