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I'm trying to get a hold of the Casper FFG paper and I suddenly realized I have a very basic question about validators in PoS.

At any given time, the entire network has an active validator set. It is commonly said that every validator can "vote" with the weight depending on how much the validator "stakes".

But wealth is chain dependent: an account may have different balances in different chains.

So it seems to me that a validator can only vote in the branch of his "deposit/staking transaction" to the system.

But then, what does the "total deposit" mean, and how is "weight" calculated?

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You are absolutely right, in blockchains, wealth is chain dependant.

But one of the goals of blockchains is to reach consensus on what is the canonical chain. To reach this goal you will see a lot of PoS protocols that finalize blocks so that the finalized chain becomes the prefix of any canonical chain.

So to answer your question the "weight" is also chain dependant.

Another question can then arise: what if a validator has "stakes" on different chains, can he vote on each one? The answer is yes, but to prevent this behavior some protocols implement slashing conditions where if a validator is caught voting on two conflicting chains we can slash is stake.

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