Welcome   Math 484   Math 485   Math 486   Math 555   Math 497G   Publications   Current and Past Students  

Current and Past Students

Current Students

Riley Mummah, Undergraduate Mathematics (ARL Honors Student)
Project: Swarm Control through Evolutionary Game Mechanisms
Abstract: Control of swarms is frequently complex because the swarm members must interact with each other, while a swarm controller prefers to shape the swarm structure to her requirements. In this work we investigate a mechanism design problem in which we assume members of a swarm evolve via an evolutionary game mechanism and a swarm shaper influences swarm evolution by global payoff matrix changes that are communicated infrequently.
Paper link coming soon...

Jim Fan, M.A. Mathematics (Ph.D. Supply Chain and Logistics)
Project: Quality Control Problem for Products in a Digital Distribution Ecosystem
Abstract: Digital distribution platforms (e.g., GooglePlay, iTunes, Steam) have revolutionized the way consumers purchase information-based products like music, software applications (apps), and video games. For example, Apple announced that their distribution platform, the App Store, recorded over $10 billion in sales during 2013 alone. Digital distribution platforms significantly decrease the time and complexity required to market software, connecting developers with consumers and making digital products readily available.
    With increased access to consumers and minimal barrier to entry, developers face a new set of strategic decisions. The software industry has traditional faced a speed-quality tradeoff in which firms that are first to market (even with quality impaired products) often gain a strategic advantage. On the other hand, even the most entrenched corporations can be unseated if quality suffers and alternatives are available (e.g., failures in Windows Vista coupled with Apple's move into the consumer electronics industry may have contributed to Microsoft's loss of dominance in the home desktop market).
    Our goal is to model the behavior of developers using a digital distribution platform. We begin with a single vendor looking to maximize profits by controlling the quality of its product. We take an optimal control approach in this situation and analyze control laws resulting under various assumptions.
Paper link coming soon...

Elisabeth Paulson, Undergraduate/Graduate Mathematics (Honors IUG Program)
Project: Better Timing of Cyber-Conflict
Abstract: We construct a model of cyber-weapon deployment and attempt to determine an optimal deployment time for cyberweapons using this model. We compare and contrast our approach to that in Axelrod and Iliev (R. Axelrod and R. Iliev. Timing of cyber conflict. Proceedings of the National Academy of Science, (1322638111), 2013.), showing that our model accurately captures four real-world scenarios and has fewer quantities that are difficult to measure than the aforementioned approach. Under simplifying assumptions, we prove rules of thumb for determining when and wether a cyber-weapon should be deployed.
ASE Cyber Security Conference Paper

Project: Bayesian Evolutionary Games with Lotka-Volterra Time Varying Populations
Abstract:We study the study the population and strategy evolution of two populations through a Bayesian game of both inter- and intra-species interactions. We refer to the two populations as predator and prey populations for consistency with other literature, and we call this game \textit{Bayesian} because we assume that the two species interact at random with one another. Thus, our dynamics account for three possible types of interactions: two types of intra-species interactions and an inter-species interaction. The strategy dynamics (i.e., the proportion of each population using each strategy) are modeled by a Bayesian version of the replicator dynamics, with which we argue that a variant of the Folk Theorem of Evolutionary Game Theory holds. We also merge the replicator dynamics with the Lotka-Volterra dynamics to study varying population proportions. With this coupled dynamical system, we find interesting and complex and results that we study numerically. Since there are three possible types of interactions, we can specify three different payoff matrices to govern these interactions. The game specifications are found to drastically alter the stability and convergence of both strategies and population proportions. We discuss a few results found from different classic game interactions: 1) The emergence of cooperation as a stable strategy when the predators play a Stag-Hunt game and the prey play a Prisonners' Dilemma game, 2) The existence of stable population proportions and limit cycles when both populations play a game of Chicken (Dove/Hawk), and 3) fast phase transitions and stable population proportions when the predators play an inverted Stag-Hunt game and the prey play a game of Chicken.
Paper link coming soon...

Seth Henry, Undergraduate Mathematics (Honors in Mathematics)
Project: Random Variable Toolbox
Abstract:The Random Variable Toolbox (RVT) is being developed to support modeling, analysis, and decision making in problem domains with multiple interacting random variables.

Past Students

Emily Battaglia, Undergraduate Mathematics/Economics (Honors in Mathematics, 2014)
Project: Two Mathematical Models of Inequality in Economics
Abstract: The issues surrounding income inequality are a topic that has garnered a lot of attention in recent years. Since the mid-1980s, the United States has become the most unequal of the advanced industrial countries, with income inequality growing at a pace that has not been seen since the Great Depression. With increasing income inequality, the potential benefits and harms become more widely debated. Robert Reich, former United States Secretary of Labor under President Bill Clinton, argues that income inequality impedes the buying power of the middle class while simultaneously allowing the upper class to store more of its wealth, rather than spend it. This thesis studies Reich's claim that income inequality is a fundamental detriment to growth. Income inequality is primarily studied in this thesis through two different lenses: a macro Solow model and a micro agent-based model. Once the theoretical foundation of each model is formulated, the models are used to run simulations to qualitatively examine income inequality. Our results from the Solow model show that in the presence of income inequality, the per capita capital of the wealthiest continues to rise, while the per capita capital of the middle class remains stagnant and the per capita capital of the lower class decreases. The agent-based model shows that the savings of the poor and middle classes are stagnant with inequality, while the savings of the rich continues to rise.
Thesis Link

Brian Cai, Undergraduate Mathematics (Honors in Mathematics, 2014)
Project: An Optimal Control Problem Arising from an Evolutionary Game
Abstract: We execute an integrative study of evolutionary game theory and optimal control. We first study the basics of evolutionary game theory and introduce the model that we would like to study based on the game of Rock-Paper-Scissors. We then move on to an introduction of optimal control and identify the requirements that need to be fulfilled in order for a solution to become optimal. Finally, we explore different methods of modeling the Rock-Paper-Scissors game as an optimal control problem and try to find an optimal control that minimizes the cost of the game. Various linearization schemes are attempted and the results are discussed.
Thesis Link

Michael Thein, Graduate Industrial Engineering (M.S., 2014)
Project: An In Depth Analysis of Sudoku with Focus in Integer and Constraint Programming
Abstract: We analyze sudoku in detail. We study sudoku as it pertains to computational complexity, graph coloring, and various programming methods that can be used to solve sudoku. We construct integer and constraint programs to solve sudoku problems. We conduct empirical experiments using these programs. Our results exhibit constant solve times for our integer program and varied solve times for our constraint program depending on difficulty. For easier sudokus, constraint programming performs significantly faster,but as difficulty increases, our integer program exhibited faster solve times. Additionally, we applied a heuristic. When combined with a heuristic, the constraint program solve times were significantly improved. The solve times were drastically better than those of our integer program for easy, medium, and hard difficulty, and were only slightly worse for evil difficulty. We tested to see how solve times reduced when more information is provided to our constraint solver. We observe an exponential decay function in solve times as more cells were filled in. Finally, we present future research that we wish to conduct.
Thesis Link

Ryan Tatko, Undergraduate Economics (B.S., 2013)
Project: Game Theoretic Formation of a Centrality Based Network
Abstract: We model the formation of networks as a game where players aspire to maximize their own centrality by increasing the number of other players to which they are path-wise connected, while simultaneously incurring a cost for each added adjacent edge. We simulate the interactions between players using an algorithm that factors in rational strategic behavior based on a common objective function. The resulting networks exhibit pairwise stability, from which we derive necessary stable conditions for specific graph topologies. We then expand the model to simulate non-trivial games with large numbers of players. We show that using conditions necessary for the stability of star topologies we can induce the formation of hub players that positively impact the total welfare of the network.
Paper Link, Published Version

David Zach, Graduate Mathematics (M.A., 2013)
Project: Exploration of an m-player Voting Game on Discrete and Continuous Spaces
Abstract: We wish to model election politics as a game in which the candidates are the players and the objective of the game is to garner a plurality of the vote. The voters are distributed across some space, and the candidates must position themselves on this same space to appeal to as many voters as possible. The candidates are free to move about and revise their positions in their best interest to win the election. In this particular game, we are not taking into account the candidates' own beliefs, but rather only the optimal placement of their opinions based on an existing voter distribution.
Paper Link

Christopher Cluskey, Undergraduate Computer Science and Engineering (Honors in Mathematics, 2013)
Project: Exploration of an m-player Voting Game on Discrete and Continuous Spaces
Abstract: We provide an introduction to Hidden Markov Models (HMM) followed by a description of the Causal State Splitting and Recon- struction Algorithm, a variation on a HMM construction method. We then describe how to write the CSSR Algorithm as a linear integer program and demonstrate a simple example.
Thesis Link

Shaun Lichter, Graduate Mathematics (M.A., 2011)
Project: A Game Theoretic Perspective on Network Topologies
Abstract: We extend the results of Goyal and Joshi (S. Goyal and S. Joshi. Networks of collaboration in oligopoly. Games and Economic behavior, 43(1):57-85, 2003), who first considered the problem of collaboration networks of oligopolies and showed that under certain linear assumptions network collaboration produced a stable complete graph through selfish competition. We show with nonlinear cost functions and player payoff alteration that stable collaboration graphs with an arbitrary degree sequence can result. We also show a non-linear extension for the result in the aforementioned paper in which the complete graph is stable. As a by product, we prove a general result on the formation of graphs with arbitrary degree sequences as the result of selfish competition. Simple motivating examples are provided and we discuss a potential relation to Network Science in our conclusions.
Paper Link, Published Version, Thesis Link

Wilmarie Rios, Graduate Industrial Engineering (M.S., 2011)
Project: Multiple sourcing for department of defense supply chains
Abstract: Tactical operations and theater distribution in military supply chains play an important role in the success of a mission. Responsive and efficient delivery of supplies is essential to maintain equipment readiness, especially in combat operations. High uncertainty in demand and supply has a direct impact in readiness levels during combat military operations. Readiness levels are sensitive to sources of disruption, primarily from shortages, but also natural disasters, weather conditions, failure of communication and information systems, political instability, and terrorist attacks. This thesis measures uncertainty in a military supply chain from the brigade unit’s perspective using exponentially distributed lead times, and investigates multiple sourcing as a strategy to improve readiness by reducing the expected supply lead time while increasing the order yield or percentage of order successfully received by brigade units. Multiple sourcing can potentially increase readiness by 70% - 90% and increasing order yield by 15%-21%. This work also proposes a process which contains past data to model supply lead times and determine the number of depots that supply a brigade unit along with the quantity of supplies to order while keeping the net order cost low. The solutions are presented using a Value Path Approach.
Thesis Link