Essays on Matching Markets and Their Equilibria

Matching theory and matching markets are a core component of modern economic theory and market design. This dissertation presents three original contributions to this area. The first essay constructs a matching mechanism in an incomplete information matching market in which the positive assortative match is the unique efficient and unique stable match. The mechanism asks each agent in the matching market to reveal her privately known type. Through its novel payment rule, truthful revelation forms an ex post Nash equilibrium in this setting. This mechanism works in one-, two- and many-sided matching markets, thus offering the first mechanism to unify these matching markets under a single mechanism design framework. The second essay confronts a problem of matching in an environment in which no efficient and incentive compatible matching mechanism exists due to matching externalities. I develop a two-stage matching game in which a contracting stage facilitates subsequent conditionally efficient and incentive compatible Vickrey auction stage. Infinite repetition of this two-stage matching game enforces the contract in every period. This mechanism produces inequitably distributed social improvement: parties to the contract receive all of the gains and then some. The final essay demonstrates the existence of prices which stably and efficiently partition a single set of agents into firms and workers, and match those two sets to each other. This pricing system extends Kelso and Crawford's general equilibrium results in a labor market matching model and links one- and two-sided matching markets as well.

Leveraging Multiple Features for Image Retrieval and Matching

The goal of image retrieval and matching is to find and locate object instances in images from a large-scale image database. While visual features are abundant, how to combine them to improve performance by individual features remains a challenging task. In this work, we focus on leveraging multiple features for accurate and efficient image retrieval and matching. We first propose two graph-based approaches to rerank initially retrieved images for generic image retrieval. In the graph, vertices are images while edges are similarities between image pairs. Our first approach employs a mixture Markov model based on a random walk model on multiple graphs to fuse graphs. We introduce a probabilistic model to compute the importance of each feature for graph fusion under a naive Bayesian formulation, which requires statistics of similarities from a manually labeled dataset containing irrelevant images. To reduce human labeling, we further propose a fully unsupervised reranking algorithm based on a submodular objective function that can be efficiently optimized by greedy algorithm. By maximizing an information gain term over the graph, our submodular function favors a subset of database images that are similar to query images and resemble each other. The function also exploits the rank relationships of images from multiple ranked lists obtained by different features. We then study a more well-defined application, person re-identification, where the database contains labeled images of human bodies captured by multiple cameras. Re-identifications from multiple cameras are regarded as related tasks to exploit shared information. We apply a novel multi-task learning algorithm using both low level features and attributes. A low rank attribute embedding is joint learned within the multi-task learning formulation to embed original binary attributes to a continuous attribute space, where incorrect and incomplete attributes are rectified and recovered. To locate objects in images, we design an object detector based on object proposals and deep convolutional neural networks (CNN) in view of the emergence of deep networks. We improve a Fast RCNN framework and investigate two new strategies to detect objects accurately and efficiently: scale-dependent pooling (SDP) and cascaded rejection classifiers (CRC). The SDP improves detection accuracy by exploiting appropriate convolutional features depending on the scale of input object proposals. The CRC effectively utilizes convolutional features and greatly eliminates negative proposals in a cascaded manner, while maintaining a high recall for true objects. The two strategies together improve the detection accuracy and reduce the computational cost.

Top-K Query Processing in Edge-Labeled Graph Data

Edge-labeled graphs have proliferated rapidly over the last decade due to the increased popularity of social networks and the Semantic Web. In social networks, relationships between people are represented by edges and each edge is labeled with a semantic annotation. Hence, a huge single graph can express many different relationships between entities. The Semantic Web represents each single fragment of knowledge as a triple (subject, predicate, object), which is conceptually identical to an edge from subject to object labeled with predicates. A set of triples constitutes an edge-labeled graph on which knowledge inference is performed. Subgraph matching has been extensively used as a query language for patterns in the context of edge-labeled graphs. For example, in social networks, users can specify a subgraph matching query to find all people that have certain neighborhood relationships. Heavily used fragments of the SPARQL query language for the Semantic Web and graph queries of other graph DBMS can also be viewed as subgraph matching over large graphs. Though subgraph matching has been extensively studied as a query paradigm in the Semantic Web and in social networks, a user can get a large number of answers in response to a query. These answers can be shown to the user in accordance with an importance ranking. In this thesis proposal, we present four different scoring models along with scalable algorithms to find the top-k answers via a suite of intelligent pruning techniques. The suggested models consist of a practically important subset of the SPARQL query language augmented with some additional useful features. The first model called Substitution Importance Query (SIQ) identifies the top-k answers whose scores are calculated from matched vertices' properties in each answer in accordance with a user-specified notion of importance. The second model called Vertex Importance Query (VIQ) identifies important vertices in accordance with a user-defined scoring method that builds on top of various subgraphs articulated by the user. Approximate Importance Query (AIQ), our third model, allows partial and inexact matchings and returns top-k of them with a user-specified approximation terms and scoring functions. In the fourth model called Probabilistic Importance Query (PIQ), a query consists of several sub-blocks: one mandatory block that must be mapped and other blocks that can be opportunistically mapped. The probability is calculated from various aspects of answers such as the number of mapped blocks, vertices' properties in each block and so on and the most top-k probable answers are returned. An important distinguishing feature of our work is that we allow the user a huge amount of freedom in specifying: (i) what pattern and approximation he considers important, (ii) how to score answers - irrespective of whether they are vertices or substitution, and (iii) how to combine and aggregate scores generated by multiple patterns and/or multiple substitutions. Because so much power is given to the user, indexing is more challenging than in situations where additional restrictions are imposed on the queries the user can ask. The proposed algorithms for the first model can also be used for answering SPARQL queries with ORDER BY and LIMIT, and the method for the second model also works for SPARQL queries with GROUP BY, ORDER BY and LIMIT. We test our algorithms on multiple real-world graph databases, showing that our algorithms are far more efficient than popular triple stores.

Expedition in Data and Harmonic Analysis on Graphs

The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the i’th node is invertible if and only if all eigenvectors are nonzero on the i’th node. Because of this dependency on the support of eigenvectors we study the characteristic set of Laplacian eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on large neighborhoods and then explicitly construct a family of non-planar graphs that do exhibit this property. We then prove original results in modern analysis on graphs. We extend results on spectral graph wavelets to create vertex-dyanamic spectral graph wavelets whose support depends on both scale and translation parameters. We prove that Spielman’s Twice-Ramanujan graph sparsifying algorithm cannot outperform his conjectured optimal sparsification constant. Finally, we present numerical results on graph conditioning, in which edges of a graph are rescaled to best approximate the complete graph and reduce average commute time.