Essays in Microeconomics

This dissertation has four chapters. The first chapter studies the testable implications of stable weighted (hedonic) coalitions. The second chapter explores an extension of Bayesian persuasion where the receiver can acquire additional information after receiving information from the sender. The third chapter studies observable implications when a decision-maker endogenously forms consideration sets. The last chapter examines weighted network formation where agents have social status concerns. Omitted proofs in each chapter are presented in the last section of each chapter. In the first chapter, we study the testable implications of a stable profile of weighted coalitions. We then apply our result to a model of weighted network formation, which subsumes aggregate matchings and the fractional stable roommates\u27 problem. In the second chapter, we study persuasion in a setting where a sender cares about a receiver\u27s action, and the receiver can acquire additional information after receiving information from the sender. Our main result indicates that the sender has considerable persuasion abilities. For binary actions, the sender always benefits from persuasion when there is a need for persuasion. For multiple actions, we give a sufficient condition for the sender to benefit from persuasion. We argue that this condition is frequently satisfied. In the third chapter, we model a decision-maker that is unable to consider all of the available alternatives due to costly attention. The decision-maker will optimally choose a subset of given alternatives by maximizing the expected utility of having that set minus the cost of attention required for considering that set. In particular, we provide a representation theorem for random choice rules where subsets of menus, which are interpreted as consideration sets, are formed by maximizing an objective function, and the probability of choosing alternatives outside this set is equal to zero. In the last chapter, we study environments where individuals allocate resources across relationships with others, creating a weighted, directed network. Value is achieved both through an exogenous factor and maintaining close connections to high-value individuals. We consider two cases corresponding to the direction benefits flow along links. In Model T (for ``taking\u27\u27) agents receive benefits through the links they create, whereas in Model G (for ``giving\u27\u27) the reverse is true: agents pass value along their links. Equilibrium and socially efficient networks are characterized. In Model G equilibrium networks do not necessarily maximize group welfare, but in Model T efficient networks and equilibrium networks coincide despite extensive network externalities

Learning Decentralized Linear Quadratic Regulator with T\sqrt{T} Regret

We study the problem of learning decentralized linear quadratic regulator when the system model is unknown a priori. We propose an online learning algorithm that adaptively designs a control policy as new data samples from a single system trajectory become available. Our algorithm design uses a disturbance-feedback representation of state-feedback controllers coupled with online convex optimization with memory and delayed feedback. We show that our controller enjoys an expected regret that scales as $\sqrt{T}$ with the time horizon $T$ for the case of partially nested information pattern. For more general information patterns, the optimal controller is unknown even if the system model is known. In this case, the regret of our controller is shown with respect to a linear sub-optimal controller. We validate our theoretical findings using numerical experiments

Online Mixed Discrete and Continuous Optimization: Algorithms, Regret Analysis and Applications

We study an online mixed discrete and continuous optimization problem where a decision maker interacts with an unknown environment for a number of $T$ rounds. At each round, the decision maker needs to first jointly choose a discrete and a continuous actions and then receives a reward associated with the chosen actions. The goal for the decision maker is to maximize the accumulative reward after $T$ rounds. We propose algorithms to solve the online mixed discrete and continuous optimization problem and prove that the algorithms yield sublinear regret in $T$. We show that a wide range of applications in practice fit into the framework of the online mixed discrete and continuous optimization problem, and apply the proposed algorithms to solve these applications with regret guarantees. We validate our theoretical results with numerical experiments

Large-scale single-photon imaging

Benefiting from its single-photon sensitivity, single-photon avalanche diode (SPAD) array has been widely applied in various fields such as fluorescence lifetime imaging and quantum computing. However, large-scale high-fidelity single-photon imaging remains a big challenge, due to the complex hardware manufacture craft and heavy noise disturbance of SPAD arrays. In this work, we introduce deep learning into SPAD, enabling super-resolution single-photon imaging over an order of magnitude, with significant enhancement of bit depth and imaging quality. We first studied the complex photon flow model of SPAD electronics to accurately characterize multiple physical noise sources, and collected a real SPAD image dataset (64 $\times$ 32 pixels, 90 scenes, 10 different bit depth, 3 different illumination flux, 2790 images in total) to calibrate noise model parameters. With this real-world physical noise model, we for the first time synthesized a large-scale realistic single-photon image dataset (image pairs of 5 different resolutions with maximum megapixels, 17250 scenes, 10 different bit depth, 3 different illumination flux, 2.6 million images in total) for subsequent network training. To tackle the severe super-resolution challenge of SPAD inputs with low bit depth, low resolution, and heavy noise, we further built a deep transformer network with a content-adaptive self-attention mechanism and gated fusion modules, which can dig global contextual features to remove multi-source noise and extract full-frequency details. We applied the technique on a series of experiments including macroscopic and microscopic imaging, microfluidic inspection, and Fourier ptychography. The experiments validate the technique's state-of-the-art super-resolution SPAD imaging performance, with more than 5 dB superiority on PSNR compared to the existing methods