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Proof-of-Importance, proof-of-work, and proof-of-stake all have one thing in common. They are all algorithms, which when applied to cryptocurrency are designed to prevent service abuses such as double spending. This is where money is spent more than once (fraudulently). For example, some currencies use verification of each transaction in the blockchain to prevent this.
This is a companion discussion topic for the original entry at http://blog.nem.io/what-are-poi-and-vesting/
