Essays in Decision Theory/ Bhavook Bhardwaj

Thesis (Ph.D.)- Indian statistical Institute, 2024

Includes references

Decisions over Sequences -- An Axiomatic Analysis of Rejection Behavior -- Copeland Choice Rules

Guided by Prof. Debasis Mishra

This thesis contains three chapters on individual decision-making and choice. The first chapter introduces a general model of decision-making where alternatives are sequentially examined by a decision maker. Our main object of study is a decision rule that maps infinite sequences of alternatives to a decision space. Within the class of decision rules, we focus on two natural subclasses: stopping and uniform stopping rules. Our main result establishes an equivalence between these two subclasses. Next, we introduce the notion of computability of decision rules using Turing machines and show that computable rules can be implemented using a simpler computational device: a finite automaton. We further show that computability of choice rules —a subclass of decision rules—is implied by their continuity with respect to a natural topology. Finally, we provide a revealed preference “toolkit” and characterize some natural choice procedures in our framework. The second chapter introduces a model of decision-making that formalizes the idea of rejection behavior using binary relations. We propose a procedure where a decision maker rejects the minimal alternatives from any decision problem. We provide an axiomatic foundation of this procedure and introduce a shortlisting model of choice where this procedure leads to a new type of a consideration set mapping: the rejection filter. We study the testable implications of this shortlisting model using observed reversals in choice. Next, we relate our findings to the existing literature and show that our model provides a novel explanation of some empirically observed behavior. Finally, we introduce and characterize a simple two-stage model of stochastic choice using rejection filters. The third chapter studies studies the Copeland set, a popular tournament solution, from a revealed preference perspective. Two choice procedures where a decision maker has a fixed underlying tournament are introduced and behaviorally characterized: (i) a deterministic choice rule that selects for every menu, the Copeland set of the tournament restricted to that menu; and (ii) a stochastic choice rule that assigns to every menu, a probability distribution over it in a “Luce” manner, where the Luce “weight” of each alternative is generated using its the Copeland score in that menu.