Essays on information and networks - Tarbush, B (2013)
This thesis consists of three independent and self-contained chapters regarding information and networks. The abstract of each chapter is given below. CHAPTER 1: The seminal “agreeing to disagree” result of Aumann (1976) was generalized from a probabilistic setting to general decision functions over partitional information structures by Bacharach (1985). This was done by isolating the relevant properties of conditional probabilities that drive the original result – namely, the “Sure-Thing Principle” and “like-mindedness” – and imposing them as conditions on the decision functions of agents. Moses & Nachum (1990) identified conceptual flaws in the framework of Bacharach (1985), showing that his conditions require agents’ decision functions to be defined over events that are informationally meaningless for the agents. In this paper, we prove a new agreement theorem in information structures that contain “counterfactual” states, and where decision functions are defined, inter-alia, over the beliefs that agents hold at such states. We show that in this new framework, decisions are defined only over information that is meaningful for the agents. Furthermore, the version of the Sure-Thing Principle presented here, which accounts for beliefs at counterfactual states, sits well with the intuition of the original version proposed by Savage (1972). The paper also includes an additional self-contained appendix in which our framework is re-expressed syntactically, which allows us to provide further insights. CHAPTER 2: We develop a parsimonious and tractable dynamic social network formation model in which agents interact in overlapping social groups. The model allows us to analyze network properties and homophily patterns simultaneously. We derive closed-form analytical expressions for the distributions of degree and, importantly, of homophily indices, using mean-field approximations. We test the comparative static predictions of our model using a large dataset from Facebook covering student friendship networks in ten American colleges in 2005, and we calibrate the analytical solutions to these networks. We find good empirical support for our predictions. Furthermore, at the best-fitting parameters values, the homophily patterns, degree distribution, and individual clustering coefficients resulting from the simulations of our model fit well with the data. Our best-fitting parameter values indicate how American college students allocate their time across various activities when socializing. CHAPTER 3: We examine three models on graphs – an information transmission mechanism, a process of friendship formation, and a model of puzzle solving – in which the evolution of the process is conditioned on the multiple edge types of the graph. For example, in the model of information transmission, a node considers information to be reliable, and therefore transmits it to its neighbors, if and only if the same message was received on two distinct communication channels. For each model, we algorithmically characterize the set of all graphs that “solve” the model (in which, in finite time, all the nodes receive the message reliably, all potentially close friendships are realized, and the puzzle is completely solved). Furthermore, we establish results relating those sets of graphs to each other.