Essays On Decision Theory

Citation

Hamze Bajgiran, Hamed (2019) Essays On Decision Theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MVE7-HP81. https://resolver.caltech.edu/CaltechTHESIS:06072019-212943893

Abstract

This thesis introduces some general frameworks for studying problems in decision theory. The purpose of this dissertation is two-fold. First, I develop general mathematical frameworks and tools to explore different decision theoretic phenomena. Second, I apply my developed frameworks and tools in different topics of Microeconomics and Decision Theory.

Chapter 1 introduces a notion of the classifier, to represent the different classes of data revealed through some observations. I present a general model of classification, notion of complexity, and how a complicated classification procedure can be generated through some simpler classification procedures.

My goal is to show how an individual's complex behavior can be derived from some simple underlying heuristics. In this chapter, I model a classifier (as a general model for decision making) that based on observing some data points classifies them into different categories with a set of different labels. The only assumption for my model is that whenever a data point is in two categories, there should be an additional category representing the intersection of the two categories. First, I derive a duality result similar to the duality in convex geometry. Then, using my result, I find all representations of a complex classifier by aggregating simpler forms of classifiers. For example, I show how a complex classifier can be represented by simpler classifiers with only two categories (similar to a single linear classifier in a neural network). Finally, I show an application in the context of dynamic choice behaviors. Notably, I use my model to reinterpret the seminal works by Kreps (1979) and Dekel, Lipman, and Rustichini (2001) on representing preference ordering over menus with a subjective state space. I also show the connection between the notion of the minimal subjective state space in economics with my proposed notion of complexity of a classifier.

In Chapter 2, I provide a general characterization of recursive methods of aggregation and show that recursive aggregation lies behind many seemingly different results in economic theory. Recursivity means that the aggregate outcome of a model over two disjoint groups of features is a weighted average of the outcome of each group separately.

This chapter makes two contributions. The first contribution is to pin down any aggregation procedure that satisfies my definition of recursivity. The result unifies aggregation procedures across many different economic environments, showing that all of them rely on the same basic result. The second contribution is to show different extensions of the result in the context of belief formation, choice theory, and welfare economics.

In the context of belief formation, I model an agent who predicts the true state of nature, based on observing some signals in her information structure. I interpret each subset of signals as an event in her information structure. I show that, as long as the information structure has a finite cardinality, my weighted averaging axiom is the necessary and sufficient condition for the agent to behaves as a Bayesian updater. This result answers the question raised by Shmaya and Yariv (2007), regarding finding a necessary and sufficient condition for a belief formation process to act as a Bayesian updating rule.

In the context of choice theory, I consider the standard theory of discrete choice. An agent chooses randomly from a menu. The outcome of my model is the average choice (mean of the distribution of choices) rather than the entire distribution of choices. Average choice is easier to report and obtain than the entire distribution. However, an average choice does not uniquely reveal the underlying distribution of choices. In this context, I show that (1) it is possible to uniquely extract the underlying distribution of choices as long as the average choice satisfies weighted averaging axiom, and (2) there is a close connection between my weighted averaging axiom and the celebrated Luce (or Logit) model of discrete choice.

Chapter 3 is about the aggregation of the preference orderings of individuals over a set of alternatives. The role of an aggregation rule is to associate with each group of individuals another preference ordering of alternatives, representing the group's aggregated preference. I consider the class of aggregation rules satisfying the extended Pareto axiom. Extended Pareto means that whenever we partition a group of individuals into two subgroups, if both subgroups prefer one alternative over another (as indicated by their aggregated preferences), then the aggregated preference ordering of the union of the subgroups also prefers the first alternative over the second one.

I show that (1) the extended Pareto is equivalent to my weighted averaging axiom, and (2) I derive a generalization of Harsanyi's (1955) famous theorem on Utilitarianism. Harsanyi considers a single profile of individuals and a variant of Pareto to obtain Utilitarianism. However, in my approach, I partition a profile into smaller groups. Then, I aggregate the preference ordering of these smaller groups using the extended Pareto. Hence, I obtain Utilitarianism through this consistent form of aggregation. As a result, in my representation, the weight associated with each individual appears in all sub-profiles that contain her.

In another application, I find the class of extended Pareto social welfare functions. My result has a positive nature, compared to the claims by Kalai and Schmeidler (1977) and Hylland (1980) that the negative conclusion of Arrow's theorem holds even with vN-M preferences.

Finally, in Chapter 4, I derive a simple subjective conditional expectation theory of state-dependent preferences. In many applications such as models for buying health insurance, the standard assumption about the independence of the utility and the set of states is not a plausible one. Hence, I derive a model in which the main force behind the separation of beliefs and state-dependent utility comes from the extended Pareto condition. Moreover, I show that, as long as the model satisfies my strong minimal agreement condition, we can uniquely separate beliefs from the state-dependent utility.

Thesis Files