I find this super-confusing, but there are two consensus algorithms under the name Casper:

What is the relation between these two designs? What is the rationale behind designing two similar protocols in parallel? Which one is going to be actually introduced in Ethereum?

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Here's a detailed post authored by an Ethereum researcher (with some bold added) https://medium.com/@jonchoi/ethereum-casper-101-7a851a4f1eb0

Casper FFG is what will be introduced in Ethereum first:

Casper the Friendly Finality Gadget (“FFG”) — aka “Vitalik’s Casper” — is a hybrid PoW/PoS consensus mechanism, which is the immediate candidate for Ethereum’s first bridge to proof of stake. More specifically, FFG implements a proof of stake mechanism as an overlay on top of a proof of work chain (such as Ethereum’s ethash PoW chain). Simply, the blockchain would grow every block with the familiar ethash PoW algorithm, but every 50 blocks is a PoS “checkpoint” where finality is assessed via a network of validators.

Here's a description of Casper CBC:

Casper the Friendly GHOST: Correct-by-Construction (“CBC”) — aka “Vlad’s Casper”—differs in approach from traditional protocol design: (1) the protocol is partially specified in the beginning and (2) and the rest of the protocol is derived in way that is proven to satisfy the desired/requisite properties (typically protocol is fully defined then tested to satisfy the said properties). In this case, one way to derive the full protocol is to implement an estimate safety oracle (“an ideal adversary”), which either raises exceptions of a fault of a justified estimate or enumerates the potential future faulty estimates. More specifically, Vlad’s work focuses on designing protocols where one can extend local views of a node’s estimate of safety to achieve consensus safety.

A short rationale:

Taking a step back, FFG focuses more on a multi-step transition to introducing PoS for the Ethereum network. This prepares for an iterative implementation that increases the role of PoS in the network over time. (PoS will claim a smaller portion of the rewards to start). In contrast, CBC focuses on formal methods that derive safety proofs “by construction” from first principles. Albeit confusing, the different approaches to solving this problem created two distinct work streams that complement each other well. The final form of Casper will likely draw from learnings from both FFG and CBC.

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