  The Bitcoin protocol prescribes certain behavior by the miners who are
responsible for maintaining and extending the underlying blockchain; in
particular, miners who successfully solve a puzzle, and hence can extend the
chain by a block, are supposed to release that block immediately. Eyal and
Sirer showed, however, that a selfish miner is incentivized to deviate from the
protocol and withhold its blocks under certain conditions.
  The analysis by Eyal and Sirer, as well as in followup work, considers a
\emph{single} deviating miner (who may control a large fraction of the hashing
power in the network) interacting with a remaining pool of honest miners. Here,
we extend this analysis to the case where there are \emph{multiple}
(non-colluding) selfish miners. We find that with multiple strategic miners,
specific deviations from honest mining by multiple strategic agents can
outperform honest mining, even if individually miners would not be incentivised
to be dishonest. This previous point effectively renders the Bitcoin protocol
to be less secure than previously thought.
