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- All Subjects: Interacting Particle Systems
- All Subjects: opinion dynamics
- Creators: Motsch, Sebastien
Description
This dissertation examines six different models in the field of econophysics using interacting particle systems as the basis of exploration. In each model examined, the underlying structure is a graph G = (V , E ), where each x ∈ V represents an individual who is characterized by the number of coins in her possession at time t. At each time step t, an edge (x, y) ∈ E is chosen at random, resulting in an exchange of coins between individuals x and y according to the rules of the model. Random variables ξt, and ξt(x) keep track of the current configuration and number of coins individual x has at time t respectively. Of particular interest is the distribution of coins in the long run. Considered first are the uniform reshuffling model, immediate exchange model and model with saving propensity. For each of these models, the number of coins an individual can have is nonnegative and the total number of coins in the system is conserved for all time. It is shown here that the distribution of coins converges to the exponential distribution, gamma distribution and a pseudo gamma distribution respectively. The next two models introduce debt, however, the total number of coins again remains fixed. It is shown here that when there is an individual debt limit, the number of coins per individual converges to a shifted exponential distribution. Alternatively, when a collective debt limit is imposed on the whole population, a heuristic argument is given supporting the conjecture that the distribution of coins converges to an asymmetric Laplace distribution. The final model considered focuses on the effect of cooperation on a population. Unlike the previous models discussed here, the total number of coins in the system at any given time is not bounded and the process evolves in continuous time rather than in discrete time. For this model, death of an individual will occur if they run out of coins. It is shown here that the survival probability for the population is impacted by the level of cooperation along with how productive the population is as whole.
ContributorsReed, Stephanie Jo (Author) / Lanchier, Nicolas (Thesis advisor) / Smith, Hal (Committee member) / Gumel, Abba (Committee member) / Motsch, Sebastien (Committee member) / Camacho, Erika (Committee member) / Arizona State University (Publisher)
Created2019
Description
This thesis explores and explains a stochastic model in Evolutionary Game Theory introduced by Dr. Nicolas Lanchier. The model is a continuous-time Markov chain that maps the two-dimensional lattice into the strategy space {1,2}. At every vertex in the grid there is exactly one player whose payoff is determined by its strategy and the strategies of its neighbors. Update times are exponential random variables with parameters equal to the absolute value of the respective cells' payoffs. The model is connected to an ordinary differential equation known as the replicator equation. This differential equation is analyzed to find its fixed points and stability. Then, by simulating the model using Java code and observing the change in dynamics which result from varying the parameters of the payoff matrix, the stochastic model's phase diagram is compared to the replicator equation's phase diagram to see what effect local interactions and stochastic update times have on the evolutionary stability of strategies. It is revealed that in the stochastic model altruistic strategies can be evolutionarily stable, and selfish strategies are only evolutionarily stable if they are more selfish than their opposing strategy. This contrasts with the replicator equation where selfishness is always evolutionarily stable and altruism never is.
ContributorsWehn, Austin Brent (Author) / Lanchier, Nicolas (Thesis director) / Kang, Yun (Committee member) / Motsch, Sebastien (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / School of International Letters and Cultures (Contributor)
Created2013-12
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
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
Description
This dissertation consists of three papers about opinion dynamics. The first paper is in collaboration with Prof. Lanchier while the other two papers are individual works.
Two models are introduced and studied analytically: the Deffuant model and the Hegselmann-Krause~(HK) model.
The main difference between the two models is that the Deffuant dynamics consists of pairwise interactions whereas the HK dynamics consists of group interactions.
Translated into graph, each vertex stands for an agent in both models.
In the Deffuant model, two graphs are combined: the social graph and the opinion graph.
The social graph is assumed to be a general finite connected graph where each edge is interpreted as a social link, such as a friendship relationship, between two agents.
At each time step, two social neighbors are randomly selected and interact if and only if their opinion distance does not exceed some confidence threshold, which results in the neighbors' opinions getting closer to each other.
The main result about the Deffuant model is the derivation of a positive lower bound for the probability of consensus that is independent of the size and topology of the social graph but depends on the confidence threshold, the choice of the opinion space and the initial distribution. For the HK model, agent~$i$ updates its opinion~$x_i$ by taking the average opinion of its neighbors, defined as the set of agents with opinion at most~$\epsilon$ apart from~$x_i$. Here,~$\epsilon > 0$ is a confidence threshold.
There are two types of HK models: the synchronous and the asynchronous HK models.
In the former, all the agents update their opinion simultaneously at each time step, whereas in the latter, only one agent is selected uniformly at random to update its opinion at each time step.
The mixed model is a variant of the HK model in which each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors.
The main results of this dissertation about HK models show conditions under which the asymptotic stability holds or a consensus can be achieved, and give a positive lower bound for the probability of consensus and, in the one-dimensional case, an upper bound for the probability of consensus.
I demonstrate the bounds for the probability of consensus on a unit cube and a unit interval.
ContributorsLi, Hsin-Lun (Author) / Lanchier, Nicolas (Thesis advisor) / Camacho, Erika (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Motsch, Sebastien (Committee member) / Arizona State University (Publisher)
Created2021