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How does Bitcoin transaction batching work exactly? How does this end up looking compared to a regular block of a bunch of transactions? Also, I have read people claim batching causes weaker privacy, how is this exactly?

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How does Bitcoin transaction batching work exactly?

Instead of paying multiple people using a new transaction for each person, you make one transaction that pays those multiple people.

For example, suppose you need to pay 5 people. Instead of making 5 different transactions with one output each, you make one transaction with 5 outputs to the 5 people you are paying.

How does this end up looking compared to a regular block of a bunch of transactions?

It is cheaper partially because each transaction has a fixed amount of overhead. By batching transactions, you only have to pay for the overhead once, not multiple times. Furthermore, usually each transaction will have a change output, and many times that change output must be spent again in the next transaction. This means that if you are sending to 5 people, you are creating 5 change outputs, and 4 inputs that have to spend from a change output. By batching, you only need to make one change output and you don't need to have those 4 additional inputs that spend the change. Those effectively cancel out and are thus removed from the transaction.

In a normal case with paying 3 people:

Input 1 -> Person 1
        -> Change 1 -> Spend Change 1 -> Person 2
                                      -> Change 2 -> Spend Change 2 -> Person 3
                                                                    -> Change 3

By batching:

Input 1 -> Person 1
        -> Person 2
        -> Person 3
        -> Change 3

So you have cut out Change 1 and 2, and Spend Change 1 and 2 which means you are paying less transaction fees.

Also, I have read people claim batching causes weaker privacy, how is this exactly?

All of the people being paid in a batched transaction can see who else you are paying and how much. However it is hard for observers to also know the true identities of those people.

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