Is using digital signatures to authorize off-chain transactions a good idea?

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In the future, as for micro-transactions, transaction fees will probably be relatively more expensive than they are today. For on-chain transactions, digital signatures have been used to authorize transactions. And this method guarantees security and accountability.

What about off-chain transactions? Can and should digital signatures be used to authorize off-chain transactions? Well, not sure. But, why not?

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Here's a simple scheme for employing digital signatures for off-line transactions:

Mr. Goodman: trusted 3rd party
Andy: an identity (not necessarily a person)
Bree: an identity (not necessarily a person)

[Scene 1: Opening up an account with Mr. Goodman]

Andy: Hi, Mr. Goodman, I want to open up an account with you.
Mr. Goodman: Choose a wallet address you like, and sign this <Random-String>
Andy: Here you are: {<Wallet-Address>, <Signature-for-the-Random-String>}
Mr. Goodman: Here is your deposit address <Deposit-Address> and you are all set!

[Scene 2: Depositing]

Andy: Hi, Mr. Goodman, I want to make a deposit.
Mr. Goodman: Sure, just send fund to your deposit address.
(Andy sends funds to his deposit address)
(Mr. Goodman consults the Blockchain and confirms Andy's deposit)
Mr. Goodman: Your account has been credited.

[Scene 3: Withdrawing]

Andy: Hi, Mr. Goodman, I want to make a withdrawal of 0.33 BTC.
Mr. Goodman: Sure thing. You just need to sign this: {withdraw, {from, Andy}, {to, <External-Wallet-Address>}, {amount, 0.33 BTC}} and pay a miner fee of 0.0001 BTC.
Andy: Done, and this is the signature <...>
Mr. Goodman: (After verifying the signature) The fund is on its way.

[Scene 4: Off-chain transferring (Finally!)]

Andy: Hi, Mr. Goodman, I want to transfer 0.00007 BTC to Bree.
Mr. Goodman: Sure thing. You just need to sign this: {transfer, {from, Andy}, {to, Bree}, {amount, 0.00007 BTC}} and pay a fee of 0.00000007 BTC.
Andy: Done, and this is the signature <...>
Mr. Goodman: (After verifying the signature) Done.

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[Scene 5: Closing an account]

Andy: Hi, Mr. Goodman, I want to close my account registered with you.
Mr. Goodman: Sure, please sign this {close_account, <Andy's-Wallet-Address>}
Andy: Done, and this is the signature <...>
Mr. Goodman: (After verifying the signature) You are all set.

Bree: Hi, Mr. Goodman, I want to transfer 0.0003 BTC to Andy, and this is the signature <sig...> for {transfer, {from, Bree}, {to, Andy}, {amount, 0.0003 BTC}}.
Mr. Goodman: Oh, I'm sorry, but Andy is no longer a customer with us, so this transaction cannot go through. You'd better make a on-chain transaction.
Bree: OK, I'll call him later, then.