A further study of some Markovian Bitcoin models from Göbel et al.

Abstract

We consider two different continuous-time Markov chain models recently studied in Göbel et al.[8 Göbel, J.; Keeler, H. P.; Krzesinski, A. E.; Taylor, P. G. Bitcoin blockchain dynamics: The selfish-mine strategy in the presence of propagation delay. Perform. Eval. 2016, 104, 23–41. DOI: 10.1016/j.peva.2016.07.001.[Crossref], [Web of Science ®] , [Google Scholar]], which were created to model the interactions between a small pool of miners, and a larger collection of miners, within the Bitcoin network. The first model we discuss represents the case where all miners behave honestly and follow the Bitcoin protocol, while the second model represents the case where the smaller pool of miners use the Selfish Mining strategy of Eyal and Sirer[3 Eyal, I.; Sirer, E. G. Majority is not enough: Bitcoin mining is vulnerable. In Financial Cryptography and Data Security; Springer, Berlin, 2013; 436–454. [Google Scholar]]. We give a new derivation of the stationary distribution of the process in the honest mining case and further build on the results of Göbel et al.[8 Göbel, J.; Keeler, H. P.; Krzesinski, A. E.; Taylor, P. G. Bitcoin blockchain dynamics: The selfish-mine strategy in the presence of propagation delay. Perform. Eval. 2016, 104, 23–41. DOI: 10.1016/j.peva.2016.07.001.[Crossref], [Web of Science ®] , [Google Scholar]] by showing that the normalizing constant can be expressed in closed-form. We also use similar techniques to derive expressions for the Laplace transforms of the transition functions. We then illustrate how these techniques yield similar expressions for the stationary distribution of the process when the smaller pool implements Selfish Mining: the Laplace transforms of the transition functions can be calculated as well. Lastly, we briefly explain how our methods can be extended to more general models of a similar type.