Mathematical strategame theory

Abstract

This study focuses on a paradoxical effect generally known as Parrondo’s paradox. Parrondo’s paradox describes the counterintuitive phenomenon whereby two losing dynamics combined to produce winning dynamics. Parrondo’s paradox is originally demonstrated using game setting in the form of capital dependent game. In the first part of this study, Parrondo’s paradox is expanded to 3rd order path dependent game. The 3rd order game is then generalized to higher order path dependent game, which include longer term memory into the dynamics. A sequential game is then developed by using multiple path dependent game. The sequential game demonstrates the possibility of multiple losing games combined to result in a winning game. The sequential game is also shown to be sensitive to initial condition. Brief discussion on the implication of asymmetry of genetic code along with Parrondo’s paradox is done. Experimental study on DNA-mediated charge transport and the effect of magnetic field has been performed. The result demonstrates paradoxical effect similar to a form of Parrondo’s paradox. In the second part of this study, the paradoxical effect is extended to independent processes instead of dependent processes. Using the example of population dynamics, two sink habitats are shown to be able to sustain an increased population though rebalance. This is counter to the general understanding that at least a source habitat would be needed to sustain a healthy population. A rebalance model is developed to study such phenomenon. Empirical studies and data are included in this report to support the model. To better reflect reality, the model is extended to include cost associated with rebalance. Strategies such as localization, globalization and strategy switching are discussed. Depending on the surrounding condition, different strategies have their own pros and cons and have varied performance.