We will discuss some famous fixed point theorems, such as Banach fixed point theorem (aka contraction mapping principle), Brouwer fixed point theorem, Schauder fixed point theorem, and Kakutani fixed point theorem. These theorems show up in unlikely areas in social sciences, such as in game theory, social choice theory, and economics. One example would be the proof of existence of Nash equilibrium in game theory, or existence of general equilibrium in market economies, using Kakutani fixed point theorem.
Our discussion will be based on the notions of convexity,...
Markov chains are certain kinds of random processes and are abundantly used as important tools to model many social or economic phenomena and predict their long-term behavior. They are used in statistical mechanics, queueing theory, information theory, speech recognition, game theory, social change models, marketing and finance, weather forecasts, biological modeling, bioinformatics, etc. Google uses them to define a PageRank for websites. It can be used to study migration patterns. The list goes on and on. We will look into the basics of Markov Chains and their...
Discrete dynamical systems can be used to model the evolution of some quantity/quantities over time, in discrete time steps. This technique is applied in engineering and physics, as well as population ecology.
Modeling a discrete dynamical system is like taking “a snapshot of the system at a sequence of times.” Seeing the state of a system at each of these times paints a picture of the rule, that given the initial conditions, governs the evolution of this system.
Jessica Yap will go over the predator-prey modeling system, and ...
We will discuss how to find the least squares line (regression line) and how it can help to make estimations for future data, using a couple of examples. We will also speak on the purpose of least squares and how they are used in social sciences. The method will also be presented through definition and picture, as well as examples and theorems. Finally, we will take up on a discuss of regression.
This workshop is an introduction to basic notions of metric spaces and topology, such as closed, open and convex sets, compactness, connectedness and their relation to continuity, as well as "Fixed Point Theorems", which would be the subject of a forthcoming workshop.
Metrics and Topology in their abstract form have many applications in Economics, Space Distribution, Game Theory (Nash's equilibrium theorem) and Social Choice Theory (Arrow's impossibility theorm).
We will discuss various classes and methods of fair division, such as divide-and-choose, filter-and-choose, and moving-knife procedures, as well as envy-freeness and equitability. The applications range from cake-cutting and rental harmony to negotiations and conflict of resolutions.
We wil cover basics of voting theory, starting from an overview of the essential mathematical concepts (transitivity and cardinal vs. ordinal preferences), as well as the notations used to represent individual preferences and their aggregation through voting procedures. Then, we will discuss some desired properties of voting procedures and Arrow's Impossibility Theorem.
The rest of the meeting will focus on concrete examples of voting procedures: majority, plurality (first-past-the-post), the Borda Count, and the sequential pairwise comparisons. In...
Factor analysis is a common data reduction technique used in the social sciences to uncover the latent structure or dimensions of a set of variables. This presentation will discuss the matrix algebra behind factor analysis. We will focus on the process of matrix diagonalization and the role of eigenvectors and eigenvalues and their relations to the underlying dimensions.
