One of the suggestions to avoid the problem of fees going to zero is assurance contracts.  This lets users (perhaps large merchants or exchanges) pay to support the network.  If insufficient people pay for the contract, then it fails.

Mike Hearn suggests one way of achieving it, but it doesn't actually create an assurance contract.  Miners can exploit the system to convert the pledges into donations.

https://bitcointalk.org/index.php?topic=157141.msg1821770#msg1821770

Consider a situation in the future where the minting fee has dropped to almost zero.  A merchant wants to cause block number 1 million to effectively have a minting fee of 50BTC.

He creates a transaction with one input (0.1BTC) and one output (50BTC) and signs it using SIGHASH_ANYONE_CAN_PAY.  The output pays to OP_TRUE.  This means that anyone can spend it.  The miner who includes the transaction will send it to an address he controls (or pay to fee).  The transaction has a locktime of 1 million, so that it cannot be included before that point.

This transaction cannot be included in a block, since the inputs are lower than the outputs.  The SIGHASH_ANYONE_CAN_PAY field mean that others can pledge additional funds.  They add more input to add more money and the same sighash. 

There would need to be some kind of notice boeard system for these pledges, but if enough pledge, then a valid transaction can be created.  It is in miner's interests to maintain such a notice board.

The problem is that it counts as a pure donation.  Even if only 10BTC has been pledged, a miner can just add 40BTC of his own money and finish the transaction.  He nets the 10BTC of the pledges if he wins the block.  If he loses, nobody sees his 40BTC transaction.  The only risk is if his block is orphaned and somehow the miner who mines the winning block gets his 40BTC transaction into his block.

The assurance contract was supposed to mean "If the effective minting fee for block 1 million is 50 BTC, then I will pay 0.1BTC".  By adding his 40BTC to the transaction the miner converts it to a pure donation.

The key point is that other miners don't get 50BTC reward if they find the block, so it doesn't push up the total hashing power being committed to the blockchain, that a 50BTC minting fee would achieve.  This is the whole point of the assurance contract.

OP_CHECKLOCKTIMEVERIFY could be used to solve the problem.

Instead of paying to OP_TRUE, the transaction should pay 50 BTC to "<1 million> OP_CHECKLOCKTIMEVERIFY OP_TRUE" and 0.01BTC to "OP_TRUE".

This means that the transaction could be included into a block well in advance of the 1 million block point.  Once block 1 million arrives, any miner would be able to spend the 50 BTC.  The 0.01BTC is the fee for the block the transaction is included in.

If the contract hasn't been included in a block well in advance, pledgers would be recommended to spend their pledged input,

It can be used to pledge to many blocks at once.  The transaction could pay out to lots of 50BTC outputs but with the locktime increasing by for each output. 

For high value transactions, it isn't just the POW of the next block that matters but all the blocks that are built on top of it. 

A pledger might want to say "I will pay 1BTC if the next 100 blocks all have at least an effective minting fee of 50BTC"

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