Coding and Quantization in Communications and Microeconomics.
详细信息
文摘
Since information theory was developed by Claude E. Shannon, in addition to its primary role in communications and networking, it has broadened to find applications in many other areas of science and technology, such as microeconomics, statistics, and neuroscience. This thesis investigates the application of information theoretic viewpoints to two interesting and significant topics. One is the fundamental topic in the field of communications and networking. The other is a long-term open issue in the area of microeconomics. We study the capacity of multiuser networks, which has been a long-standing problem in the information theory. Recently, Avestimehr, Diggavi, and Tse have proposed a deterministic network model to approximate multiuser wireless networks. For wireless multicast relay networks, they have shown that the capacity for the deterministic model is equal to the minimal rank of the incidence matrix of a certain cut between the source and any of the sinks. Their proposed code construction, however, is not guaranteed to be efficient and may potentially involve an infinite block length. We propose an efficient linear code construction for the deterministic wireless relay network model. Unlike several previous coding schemes, we do not attempt to find flows in the network. Instead, for a layered network, we maintain an invariant where it is required that at each stage of the code construction, certain sets of codewords are linearly independent. We then apply ideas from information theory to solve a canonical problem in microeconomics with information constraints. Specifically, we analyze the problem of nonlinear pricing with limited information. Due to information constraints, the seller is limited to offering a menu with a finite number of choices to a continuum of buyers with a continuum of possible valuations. By revealing an underlying connection to the quantization in the information theory, we introduce the conditions which the optimal finite menu for the socially efficient and that for the revenue-maximizing mechanism must satisfy. In both cases, we provide an estimate of the loss resulting from using the finite menu. We then extend our nonlinear pricing model to multi-product environment where each buyer can purchase a variety of multiple goods at a time. His preference over these goods is represented by a multi-dimensional vector in some compact subset with a continuum of possible valuations. We generalize our results to multi-product environment via vector quantization. We discuss the benefit of bundling the consumers preferences over multiple goods, and designing the finite menus jointly in higher dimensions. This benefit arises by using vector quantization methods that take advantage of the dimensionality, the shape of the marginal density function, and the correlation among the consumers types over multiple products.