Economics

Under the same assumptions made by Mas-Colell et al. (1995), I develop a short, simple, and complete proof of existence of equilibrium prices based on excess demand functions. The result is obtained by applying the Brouwer fixed point theorem to a trimmed simplex which does not contain prices equal to zero. The mathematical techniques are based on some results obtained in Neuefeind (1980) and Geanakoplos (2003).

We seek to take a different approach in deriving the optimal search policy for the repeated consumer search model found in Fishman and Rob (1995) with the main motivation of dropping the assumption of prior knowledge of the price distribution F(p) in each period. We will do this by incorporating the famous multi-armed bandit problem (MAB). We start by modifying the MAB framework to fit the setting of the repeated consumer search model and formulate the objective as a dynamic optimization problem. Then, given any sequence of exploration, we assign a value to each store in that sequence using Bellman equations. We then proceed to break down the problem into individual optimal stopping problems for each period which incidentally coincides with the framework of the famous secretary problem where we proceed to derive the optimal stopping policy. We will see that implementing the optimal stopping policy in each period solves the original dynamic optimization by `forward induction' reasoning.

In coordination games and speculative over-the-counter financial markets, solutions depend on higher-order average expectations: agents' expectations about what counterparties, on average, expect their counterparties to think, etc. We offer a unified analysis of these objects and their limits, for general information structures, priors, and networks of counterparty relationships. Our key device is an interaction structure combining the network and agents' beliefs, which we analyze using Markov methods. This device allows us to nest classical beauty contests and network games within one model and unify their results. Two applications illustrate the techniques: The first characterizes when slight optimism about counterparties' average expectations leads to contagion of optimism and extreme asset prices. The second describes the tyranny of the least-informed: agents coordinating on the prior expectations of the one with the worst private information, despite all having nearly common certainty, based on precise private signals, of the ex post optimal action.

A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result.

Voting is the aggregation of individual preferences in order to select a winning alternative. Selection of a winner is accomplished via a voting rule, e.g., rank-order voting, majority rule, plurality rule, approval voting. Which voting rule should be used? In social choice theory, desirable properties of voting rules are expressed as axioms to be satisfied. This thesis focuses on axioms concerning strategic manipulation by voters. Sometimes, voters may intentionally misstate their true preferences in order to alter the outcome for their own advantage. For example, in plurality rule, if a voter knows that their top-choice candidate will lose, then they might instead vote for their second-choice candidate just to avoid an even less desirable result. When no coalition of voters can strategically manipulate, then the voting rule is said to satisfy the axiom of Strategy-Proofness. A less restrictive axiom is Weak Strategy-Proofness (as defined by Dasgupta and Maskin (2019)), which allows for strategic manipulation by all but the smallest coalitions. Under certain intuitive conditions, Dasgupta and Maskin (2019) proved that the only voting rules satisfying Strategy-Proofness are rank-order voting and majority rule. In my thesis, I generalize their result, by proving that rank-order voting and majority rule are surprisingly still the only voting rules satisfying Weak Strategy-Proofness.

We study the incentive properties of envy-free mechanisms for the allocation of rooms and payments of rent among financially constrained roommates. Each agent reports her values for rooms, her housing earmark (soft budget), and an index that reflects the difficulty the agent experiences from having to pay over this amount. Then an envy-free allocation for these reports is recommended. The complete information non-cooperative outcomes of each of these mechanisms are exactly the envy-free allocations with respect to true preferences if and only if the admissible budget violation indices have a bound.

We propose an extended version of Gini index defined on the set of infinite utility streams, X= Y N where Y⊂R . For Y containing at most finitely many elements, the index satisfies the generalized Pigou-Dalton transfer principles in addition to the anonymity axiom.

We consider contest success functions (CSFs) that extract contestants' values of the prize. In the case in which the values are observable to the contest designer, in the more-than-two-contestant or common-value subcase, we present a CSF extractive in any equilibrium; in the other subcase, we present a CSF extractive in some equilibrium, but there exists no CSF extractive in any equilibrium. In the case in which the values are not observable, there exists no CSF extractive in some equilibrium. In the case in which the values are observable and common, we present extractive a CSF extractive in any equilibrium; we present a class of CSFs extractive in some equilibrium, and this class can control the number of active contestants.

COVID-19 has revealed limitations of existing mechanisms for rationing medical resources under emergency scenarios. Many argue that these mechanisms abandon various ethical values such as equity by discriminating against disadvantaged communities. Illustrating that these limitations are aggravated by a restrictive choice of mechanism, we formulate pandemic rationing of medical resources as a new application of market design and propose a reserve system as a resolution. We develop a general theory of reserve design, introduce new concepts such as cutoff equilibria and smart reserves, extend analysis of previously-known ones such as sequential reserve matching, relate these concepts to current debates, and present preliminary policy impact.

There is increasing regulatory interest in whether machine learning algorithms deployed in consequential domains (e.g. in criminal justice) treat different demographic groups "fairly." However, there are several proposed notions of fairness, typically mutually incompatible. Using criminal justice as an example, we study a model in which society chooses an incarceration rule. Agents of different demographic groups differ in their outside options (e.g. opportunity for legal employment) and decide whether to commit crimes. We show that equalizing type I and type II errors across groups is consistent with the goal of minimizing the overall crime rate; other popular notions of fairness are not.