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An open distributed system can be secured by requiring participants to present proof of work and rewarding them for participation. The Bit coin digital currency introduced this mechanism, which is adopted by almost all contemporary digital currencies and related services. A natural process leads participants of such systems to form pools, where members aggregate their power and share the rewards. Experience with Bit coin shows that the largest pools are often open, allowing anyone to join. It has long been known that a member can sabotage an open pool by seemingly joining it but never sharing its proofs of work. The pool shares its revenue with the attacker, and so each of its participants earns less. We define and analyze a game where pools use some of their participants to infiltrate other pools and perform such an attack. With any number of pools, no-pool-attacks is not a Nash equilibrium. We study the …