  Mining for Bitcoins is a high-risk high-reward activity. Miners, seeking to
reduce their variance and earn steadier rewards, collaborate in pooling
strategies where they jointly mine for Bitcoins. Whenever some pool participant
is successful, the earned rewards are appropriately split among all pool
participants. Currently a dozen of different pooling strategies (i.e., methods
for distributing the rewards) are in use for Bitcoin mining.
  We here propose a formal model of utility and social welfare for Bitcoin
mining (and analogous mining systems) based on the theory of discounted
expected utility, and next study pooling strategies that maximize the social
welfare of miners. Our main result shows that one of the pooling strategies
actually employed in practice--the so-called geometric pay pool--achieves the
optimal steady-state utility for miners when its parameters are set
appropriately.
  Our results apply not only to Bitcoin mining pools, but any other form of
pooled mining or crowdsourcing computations where the participants engage in
repeated random trials towards a common goal, and where "partial" solutions can
be efficiently verified.
