  Pool block withholding attack is performed among mining pools in digital
cryptocurrencies, such as Bitcoin. Instead of mining honestly, pools can be
incentivized to infiltrate their own miners into other pools. These
infiltrators report partial solutions but withhold full solutions, share block
rewards but make no contribution to block mining. The block withholding attack
among mining pools can be modeled as a non-cooperative game called "the miner's
dilemm", which reduces effective mining power in the system and leads to
potential systemic instability in the blockchain. However, existing literature
on the game-theoretic properties of this attack only gives a preliminary
analysis, e.g., an upper bound of 3 for the pure price of anarchy (PPoA) in
this game, with two pools involved and no miner betraying. Pure price of
anarchy is a measurement of how much mining power is wasted in the miner's
dilemma game. Further tightening its upper bound will bring us more insight
into the structure of this game, so as to design mechanisms to reduce the
systemic loss caused by mutual attacks. In this paper, we give a tight bound of
(1, 2] for the pure price of anarchy. Moreover, we show the tight bound holds
in a more general setting, in which infiltrators may betray.We also prove the
existence and uniqueness of pure Nash equilibrium in this setting. Inspired by
experiments on the game among three mining pools, we conjecture that similar
results hold in the N-player miner's dilemma game (N>=2).
