Matching Items (14)
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- All Subjects: Mathematics
- All Subjects: Game Theory
- Creators: Lanchier, Nicolas
Description
Parallel Monte Carlo applications require the pseudorandom numbers used on each processor to be independent in a probabilistic sense. The TestU01 software package is the standard testing suite for detecting stream dependence and other properties that make certain pseudorandom generators ineffective in parallel (as well as serial) settings. TestU01 employs two basic schemes for testing parallel generated streams. The first applies serial tests to the individual streams and then tests the resulting P-values for uniformity. The second turns all the parallel generated streams into one long vector and then applies serial tests to the resulting concatenated stream. Various forms of stream dependence can be missed by each approach because neither one fully addresses the multivariate nature of the accumulated data when generators are run in parallel. This dissertation identifies these potential faults in the parallel testing methodologies of TestU01 and investigates two different methods to better detect inter-stream dependencies: correlation motivated multivariate tests and vector time series based tests. These methods have been implemented in an extension to TestU01 built in C++ and the unique aspects of this extension are discussed. A variety of different generation scenarios are then examined using the TestU01 suite in concert with the extension. This enhanced software package is found to better detect certain forms of inter-stream dependencies than the original TestU01 suites of tests.
ContributorsIsmay, Chester (Author) / Eubank, Randall (Thesis advisor) / Young, Dennis (Committee member) / Kao, Ming-Hung (Committee member) / Lanchier, Nicolas (Committee member) / Reiser, Mark R. (Committee member) / Arizona State University (Publisher)
Created2013
Description
By the von Neumann min-max theorem, a two person zero sum game with finitely many pure strategies has a unique value for each player (summing to zero) and each player has a non-empty set of optimal mixed strategies. If the payoffs are independent, identically distributed (iid) uniform (0,1) random variables, then with probability one, both players have unique optimal mixed strategies utilizing the same number of pure strategies with positive probability (Jonasson 2004). The pure strategies with positive probability in the unique optimal mixed strategies are called saddle squares. In 1957, Goldman evaluated the probability of a saddle point (a 1 by 1 saddle square), which was rediscovered by many authors including Thorp (1979). Thorp gave two proofs of the probability of a saddle point, one using combinatorics and one using a beta integral. In 1965, Falk and Thrall investigated the integrals required for the probabilities of a 2 by 2 saddle square for 2 × n and m × 2 games with iid uniform (0,1) payoffs, but they were not able to evaluate the integrals. This dissertation generalizes Thorp's beta integral proof of Goldman's probability of a saddle point, establishing an integral formula for the probability that a m × n game with iid uniform (0,1) payoffs has a k by k saddle square (k ≤ m,n). Additionally, the probabilities of a 2 by 2 and a 3 by 3 saddle square for a 3 × 3 game with iid uniform(0,1) payoffs are found. For these, the 14 integrals observed by Falk and Thrall are dissected into 38 disjoint domains, and the integrals are evaluated using the basic properties of the dilogarithm function. The final results for the probabilities of a 2 by 2 and a 3 by 3 saddle square in a 3 × 3 game are linear combinations of 1, π2, and ln(2) with rational coefficients.
ContributorsManley, Michael (Author) / Kadell, Kevin W. J. (Thesis advisor) / Kao, Ming-Hung (Committee member) / Lanchier, Nicolas (Committee member) / Lohr, Sharon (Committee member) / Reiser, Mark R. (Committee member) / Arizona State University (Publisher)
Created2011
Description
One of the salient challenges of sustainability is the Tragedy of the Commons, where individuals acting independently and rationally deplete a common resource despite their understanding that it is not in the group's long term best interest to do so. Hardin presents this dilemma as nearly intractable and solvable only by drastic, government-mandated social reforms, while Ostrom's empirical work demonstrates that community-scale collaboration can circumvent tragedy without any elaborate outside intervention. Though more optimistic, Ostrom's work provides scant insight into larger-scale dilemmas such as climate change. Consequently, it remains unclear if the sustainable management of global resources is possible without significant government mediation. To investigate, we conducted two game theoretic experiments that challenged students in different countries to collaborate digitally and manage a hypothetical common resource. One experiment involved students attending Arizona State University and the Rochester Institute of Technology in the US and Mountains of the Moon University in Uganda, while the other included students at Arizona State and the Management Development Institute in India. In both experiments, students were randomly assigned to one of three production roles: Luxury, Intermediate, and Subsistence. Students then made individual decisions about how many units of goods they wished to produce up to a set maximum per production class. Luxury players gain the most profit (i.e. grade points) per unit produced, but they also emit the most externalities, or social costs, which directly subtract from the profit of everybody else in the game; Intermediate players produce a medium amount of profit and externalities per unit, and Subsistence players produce a low amount of profit and externalities per unit. Variables influencing and/or inhibiting collaboration were studied using pre- and post-game surveys. This research sought to answer three questions: 1) Are international groups capable of self-organizing in a way that promotes sustainable resource management?, 2) What are the key factors that inhibit or foster collective action among international groups?, and 3) How well do Hardin's theories and Ostrom's empirical models predict the observed behavior of students in the game? The results of gameplay suggest that international cooperation is possible, though likely sub-optimal. Statistical analysis of survey data revealed that heterogeneity and levels of trust significantly influenced game behavior. Specific traits of heterogeneity among students found to be significant were income, education, assigned production role, number of people in one's household, college class, college major, and military service. Additionally, it was found that Ostrom's collective action framework was a better predictor of game outcome than Hardin's theories. Overall, this research lends credence to the plausibility of international cooperation in tragedy of the commons scenarios such as climate change, though much work remains to be done.
ContributorsStanton, Albert Grayson (Author) / Clark, Susan Spierre (Thesis director) / Seager, Thomas (Committee member) / Civil, Environmental and Sustainable Engineering Programs (Contributor) / Barrett, The Honors College (Contributor)
Created2014-12
Description
Deconvolution of noisy data is an ill-posed problem, and requires some form of regularization to stabilize its solution. Tikhonov regularization is the most common method used, but it depends on the choice of a regularization parameter λ which must generally be estimated using one of several common methods. These methods can be computationally intensive, so I consider their behavior when only a portion of the sampled data is used. I show that the results of these methods converge as the sampling resolution increases, and use this to suggest a method of downsampling to estimate λ. I then present numerical results showing that this method can be feasible, and propose future avenues of inquiry.
ContributorsHansen, Jakob Kristian (Author) / Renaut, Rosemary (Thesis director) / Cochran, Douglas (Committee member) / Barrett, The Honors College (Contributor) / School of Music (Contributor) / Economics Program in CLAS (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2015-05
Description
This paper uses network theory to simulate Nash equilibria for selfish travel within a traffic network. Specifically, it examines the phenomenon of Braess's Paradox, the counterintuitive occurrence in which adding capacity to a traffic network increases the social costs paid by travelers in a new Nash equilibrium. It also employs the measure of the price of anarchy, a ratio between the social cost of the Nash equilibrium flow through a network and the socially optimal cost of travel. These concepts are the basis of the theory behind undesirable selfish routing to identify problematic links and roads in existing metropolitan traffic networks (Youn et al., 2008), suggesting applicative potential behind the theoretical questions this paper attempts to answer. New topologies of networks which generate Braess's Paradox are found. In addition, the relationship between the number of nodes in a network and the number of occurrences of Braess's Paradox, and the relationship between the number of nodes in a network and a network's price of anarchy distribution are studied.
ContributorsChotras, Peter Louis (Author) / Armbruster, Dieter (Thesis director) / Lanchier, Nicolas (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Economics Program in CLAS (Contributor)
Created2015-05
Description
There are multiple mathematical models for alignment of individuals moving within a group. In a first class of models, individuals tend to relax their velocity toward the average velocity of other nearby neighbors. These types of models are motivated by the flocking behavior exhibited by birds. Another class of models have been introduced to describe rapid changes of individual velocity, referred to as jump, which better describes behavior of smaller agents (e.g. locusts, ants). In the second class of model, individuals will randomly choose to align with another nearby individual, matching velocities. There are several open questions concerning these two type of behavior: which behavior is the most efficient to create a flock (i.e. to converge toward the same velocity)? Will flocking still emerge when the number of individuals approach infinity? Analysis of these models show that, in the homogeneous case where all individuals are capable of interacting with each other, the variance of the velocities in both the jump model and the relaxation model decays to 0 exponentially for any nonzero number of individuals. This implies the individuals in the system converge to an absorbing state where all individuals share the same velocity, therefore individuals converge to a flock even as the number of individuals approach infinity. Further analysis focused on the case where interactions between individuals were determined by an adjacency matrix. The second eigenvalues of the Laplacian of this adjacency matrix (denoted ƛ2) provided a lower bound on the rate of decay of the variance. When ƛ2 is nonzero, the system is said to converge to a flock almost surely. Furthermore, when the adjacency matrix is generated by a random graph, such that connections between individuals are formed with probability p (where 0
1/N. ƛ2 is a good estimator of the rate of convergence of the system, in comparison to the value of p used to generate the adjacency matrix..
ContributorsTrent, Austin L. (Author) / Motsch, Sebastien (Thesis director) / Lanchier, Nicolas (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
Division of Labor among social insects is frequently discussed in regards to the colony's worker population. However, before a colony achieves a worker population, a queen is required to perform all of the tasks necessary for her survival: foraging, building the colony, and brood care. A simple ODE model was developed through the use of a framework of replicator equations in dynamical environments to investigate how queen ants perform and distribute all of the tasks necessary for her and her colony's survival by incorporating individual internal thresholds and environmental stimulus. Modi�cations to the internal threshold, risk of performing the task, and the rate of increase of the environmental stimulus were also explored. Because of the simplicity of the model, it could also be used to measure the task performance of larger populations of social insects. However, the model has only been applied to the data collected from Pogonomyrmex barbatus single queen ants.
ContributorsKincade, Katherine Margaret (Author) / Kang, Yun (Thesis director) / Fewell, Jennifer (Committee member) / Lanchier, Nicolas (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Computer Science and Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2016-12
Description
This dissertation investigates the dynamics of evolutionary games based on the framework of interacting particle systems in which individuals are discrete, space is explicit, and dynamics are stochastic. Its focus is on 2-strategy games played on a d-dimensional integer lattice with a range of interaction M. An overview of related past work is given along with a summary of the dynamics in the mean-field model, which is described by the replicator equation. Then the dynamics of the interacting particle system is considered, first when individuals are updated according to the best-response update process and then the death-birth update process. Several interesting results are derived, and the differences between the interacting particle system model and the replicator dynamics are emphasized. The terms selfish and altruistic are defined according to a certain ordering of payoff parameters. In these terms, the replicator dynamics are simple: coexistence occurs if both strategies are altruistic; the selfish strategy wins if one strategy is selfish and the other is altruistic; and there is bistability if both strategies are selfish. Under the best-response update process, it is shown that there is no bistability region. Instead, in the presence of at least one selfish strategy, the most selfish strategy wins, while there is still coexistence if both strategies are altruistic. Under the death-birth update process, it is shown that regardless of the range of interactions and the dimension, regions of coexistence and bistability are both reduced. Additionally, coexistence occurs in some parameter region for large enough interaction ranges. Finally, in contrast with the replicator equation and the best-response update process, cooperators can win in the prisoner's dilemma for the death-birth process in one-dimensional nearest-neighbor interactions.
ContributorsEvilsizor, Stephen (Author) / Lanchier, Nicolas (Thesis advisor) / Kang, Yun (Committee member) / Motsch, Sebastien (Committee member) / Smith, Hal (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2016
Description
The decline of honeybee colonies around the world has been linked to the presence of the Varroa destructor, a mite acting as a virus vector for the Acute Bee Paralysis Virus. We developed a model of the infestation of the Apis melliifera honeybee colony by the Acute Bee Paralysis Virus, which is transmitted by the parasitic Varroa destructor. This is a four dimensional system of nonlinear ODE's for healthy and virus infected bees, total number of mites in the colony and number of mites that carry the virus. The Acute Bee Paralysis Virus can be transmitted between infected and uninfected bees, infected mite to adult bee, infected bee to phoretic mite, and reproductive mites to bee brood. This model is studied with analytical techniques deriving the conditions under which the bee colony can fight off an Acute Bee Paralysis Virus epidemic.
ContributorsDavis, Talia Lasandra (Author) / Kang, Yun (Thesis director) / Lanchier, Nicolas (Committee member) / Moore, Marianne (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2015-12
DescriptionUnderstanding the evolution of opinions is a delicate task as the dynamics of how one changes their opinion based on their interactions with others are unclear.
ContributorsWeber, Dylan (Author) / Motsch, Sebastien (Thesis advisor) / Lanchier, Nicolas (Committee member) / Platte, Rodrigo (Committee member) / Armbruster, Dieter (Committee member) / Fricks, John (Committee member) / Arizona State University (Publisher)
Created2021