Essays on Game and Economic Theory

This dissertation studies a range of topics in game and economic theory. Chapter 1 proposes a new solution to the two-player bargaining problem of Nash (1950): The Consensus solution. The Consensus solution maximizes the total amount of options that both players agree are worse than the solution but better than no-cooperation. It can be characterized by a simple equality. It satisfies all the axioms of the Nash solution except Axiom IIA (Independence of Irrelevant Alternatives); the Nash solution satisfies all its axioms except one which says: when both players\u27 utilities of no-cooperation become lower creating additional room for players to cooperate, then as long as options within the additional room are worse than the current solution, the solution shall not change. At the same time, it is the same as the Nash solution in comprehensive bargaining problems, a class of bargaining problems where many good properties of the Nash solution are discovered. We discuss when bargaining problems are non-comprehensive. We conclude that the key difference between the two solutions is that the Consensus solution emphasize what players can achieve via cooperation whereas the Nash solution focus more on the anticipation of no-cooperation. Chapter 2, coauthored with with Treb Allen and Costas Arkolakis, studies a broad class of network models where a large number of heterogeneous agents simultaneously interact in many ways. We provide an iterative algorithm for calculating an equilibrium and offer sufficient and ``globally necessary\u27\u27 conditions under which the equilibrium is unique. The results arise from a multi-dimensional extension of the contraction mapping theorem which allows for the separate treatment of the different types of interactions. We illustrate that a wide variety of heterogeneous agent economies -- characterized by spatial, production, or social networks -- yield equilibrium representations amenable to our theorem\u27s characterization. Chapter 3, coauthored with with Treb Allen and Costas Arkolakis, develops a quantitative general equilibrium model that incorporates the many economic interactions that occur over the city, including commuting and spatial spillovers of productivities. Despite the many spatial linkages, the model allows for characterizing the existence and efficiency of the spatial equilibrium of the city when there are no spillovers. Additionally, we consider a city planner who can design zoning policy but leave the rest to the market. We show that even with the presence of spillovers, the city planner can still be efficient. We provide an explicit formula to evaluate welfare effects of zoning policies

CAF: Cluster Algorithm and A-Star with Fuzzy Approach for Lifetime Enhancement in Wireless Sensor Networks

Energy is a major factor in designing wireless sensor networks (WSNs). In particular, in the real world, battery energy is limited; thus the effective improvement of the energy becomes the key of the routing protocols. Besides, the sensor nodes are always deployed far away from the base station and the transmission energy consumption is index times increasing with the increase of distance as well. This paper proposes a new routing method for WSNs to extend the network lifetime using a combination of a clustering algorithm, a fuzzy approach, and an A-star method. The proposal is divided into two steps. Firstly, WSNs are separated into clusters using the Stable Election Protocol (SEP) method. Secondly, the combined methods of fuzzy inference and A-star algorithm are adopted, taking into account the factors such as the remaining power, the minimum hops, and the traffic numbers of nodes. Simulation results demonstrate that the proposed method has significant effectiveness in terms of balancing energy consumption as well as maximizing the network lifetime by comparing the performance of the A-star and fuzzy (AF) approach, cluster and fuzzy (CF)method, cluster and A-star (CA)method, A-star method, and SEP algorithm under the same routing criteria

PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization

In this paper, we propose a novel stochastic gradient estimator -- ProbAbilistic Gradient Estimator (PAGE) -- for nonconvex optimization. PAGE is easy to implement as it is designed via a small adjustment to vanilla SGD: in each iteration, PAGE uses the vanilla minibatch SGD update with probability $p_t$ or reuses the previous gradient with a small adjustment, at a much lower computational cost, with probability $1-p_t$. We give a simple formula for the optimal choice of $p_t$. Moreover, we prove the first tight lower bound $\Omega(n+\frac{\sqrt{n}}{\epsilon^2})$ for nonconvex finite-sum problems, which also leads to a tight lower bound $\Omega(b+\frac{\sqrt{b}}{\epsilon^2})$ for nonconvex online problems, where $b:= \min\{\frac{\sigma^2}{\epsilon^2}, n\}$. Then, we show that PAGE obtains the optimal convergence results $O(n+\frac{\sqrt{n}}{\epsilon^2})$ (finite-sum) and $O(b+\frac{\sqrt{b}}{\epsilon^2})$ (online) matching our lower bounds for both nonconvex finite-sum and online problems. Besides, we also show that for nonconvex functions satisfying the Polyak-\L{}ojasiewicz (PL) condition, PAGE can automatically switch to a faster linear convergence rate $O(\cdot\log \frac{1}{\epsilon})$. Finally, we conduct several deep learning experiments (e.g., LeNet, VGG, ResNet) on real datasets in PyTorch showing that PAGE not only converges much faster than SGD in training but also achieves the higher test accuracy, validating the optimal theoretical results and confirming the practical superiority of PAGE.Comment: 25 pages; accepted by ICML 2021 (long talk