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Double serializing the exchange procedure, using both accounts' serializers,
protects the exchange operations from being interleaved with other account
operations from either account. This makes exchange atomic. So its behavior will be as expected.
With the original definition of exchange, the exchange operation will not be
atomic, but it will be composed of exactly four atomic operations, namely read the two account balances, make an atomic withdrawal from the larger account, and make an atomic deposit into the smaller account.
To see how the original exchange definition might not preserve the three
balances, consider the following three exchanges:
The original exchange allows us to interleave P3 between reads of P1 and P2:
However the sum of all three balances cannot change. This is because each
exchange adds and removes an equal amount between accounts. So total deposits will always equal total withdrawals.