Elaboration of selfish-mine strategy

As it was shown by Ittay Eyal and Emin Gün Sirer, the Bitcoin mining protocol is not incentive-compatible, because there exists an attack in which colluding miners obtain a revenue larger than their fair share. We describe an elaboration of Selfish-Mine Strategy and present an extended model of selfish mining based on independency hypothesis: both groups are made their work independently from each other. We describe a new state machine modelling selfish pool strategy. Let the selfish pool has mining power of $p$, $0<p<1/2$, and the others of $(1-p)$. We also consider the situation in which the others mine a block on the previously private branch (frequency $\gamma(1-p)$), and the others mine a block on the public branch (frequency $(1-\gamma)(1-p)$). Main result is an elaboration of an interval, in which selfish miners will earn more than their relative mining power: 1) for a given $p$, a pool of size $p$ obtains a revenue larger than its relative size for $p$ in the following range: $0 < p \le 0{.}429$ (the left bound coinsides with $\gamma =1$, and the right bound coinsides with $\gamma =0$); 2) for a given $p$, a pool of size $p$ obtains a revenue larger than a revenue of other group in the following range: $0{.}358 \le p \le 0{.}449$.